remove all files
This commit is contained in:
NINI
@@ -1,32 +0,0 @@
|
||||
{
|
||||
"name": "local-harness-engineering",
|
||||
"interface": {
|
||||
"displayName": "Local Harness Engineering"
|
||||
},
|
||||
"plugins": [
|
||||
{
|
||||
"name": "harness-engineering",
|
||||
"source": {
|
||||
"source": "local",
|
||||
"path": "./plugins/harness-engineering"
|
||||
},
|
||||
"policy": {
|
||||
"installation": "AVAILABLE",
|
||||
"authentication": "ON_INSTALL"
|
||||
},
|
||||
"category": "Productivity"
|
||||
},
|
||||
{
|
||||
"name": "fesa-commands",
|
||||
"source": {
|
||||
"source": "local",
|
||||
"path": "./plugins/fesa-commands"
|
||||
},
|
||||
"policy": {
|
||||
"installation": "AVAILABLE",
|
||||
"authentication": "ON_INSTALL"
|
||||
},
|
||||
"category": "Productivity"
|
||||
}
|
||||
]
|
||||
}
|
||||
@@ -1,57 +0,0 @@
|
||||
---
|
||||
name: harness-review
|
||||
description: Review a Harness Engineering repository against its persistent rules and design docs. Use when Codex is asked to review local changes, generated phase files, or implementation output against `AGENTS.md`, `docs/ARCHITECTURE.md`, `docs/ADR.md`, `docs/UI_GUIDE.md`, testing expectations, and Harness step acceptance criteria.
|
||||
---
|
||||
|
||||
# Harness Review
|
||||
|
||||
Use this skill when the user wants a repository-grounded review instead of generic commentary.
|
||||
|
||||
## Review input set
|
||||
|
||||
Read these first:
|
||||
|
||||
- `/AGENTS.md`
|
||||
- `/docs/ARCHITECTURE.md`
|
||||
- `/docs/ADR.md`
|
||||
- `/docs/UI_GUIDE.md`
|
||||
- the changed files or generated `phases/` files under review
|
||||
|
||||
If the user explicitly asks for delegated review, prefer the repo custom agent `harness_reviewer` or built-in read-only explorers.
|
||||
|
||||
## Checklist
|
||||
|
||||
Evaluate the patch against these questions:
|
||||
|
||||
1. Does it follow the architecture described in `docs/ARCHITECTURE.md`?
|
||||
2. Does it stay within the technology choices documented in `docs/ADR.md`?
|
||||
3. Are new or changed behaviors covered by tests or other explicit validation?
|
||||
4. Does it violate any CRITICAL rule in `AGENTS.md`?
|
||||
5. Do generated `phases/` files remain self-contained, executable, and internally consistent?
|
||||
6. If the user expects verification, does `python scripts/validate_workspace.py` succeed or is the failure explained?
|
||||
|
||||
## Output rules
|
||||
|
||||
- Lead with findings, ordered by severity.
|
||||
- Include file references for each finding.
|
||||
- Explain the concrete risk or regression, not just the rule name.
|
||||
- If there are no findings, say so explicitly and mention residual risks or missing evidence.
|
||||
- Keep summaries brief after the findings.
|
||||
|
||||
## Preferred review table
|
||||
|
||||
When the user asks for a checklist-style review, use this table:
|
||||
|
||||
| Item | Result | Notes |
|
||||
|------|------|------|
|
||||
| Architecture compliance | PASS/FAIL | {details} |
|
||||
| Tech stack compliance | PASS/FAIL | {details} |
|
||||
| Test coverage | PASS/FAIL | {details} |
|
||||
| CRITICAL rules | PASS/FAIL | {details} |
|
||||
| Build and validation | PASS/FAIL | {details} |
|
||||
|
||||
## What not to do
|
||||
|
||||
- Do not approve changes just because they compile.
|
||||
- Do not focus on style-only issues when correctness, architecture drift, or missing validation exists.
|
||||
- Do not assume a passing hook means the implementation is acceptable; review the actual diff and docs.
|
||||
@@ -1,4 +0,0 @@
|
||||
interface:
|
||||
display_name: "Harness Review"
|
||||
short_description: "Review changes against Harness project rules"
|
||||
default_prompt: "Use Harness review to check architecture, tests, and rules."
|
||||
@@ -1,145 +0,0 @@
|
||||
---
|
||||
name: harness-workflow
|
||||
description: Plan and run the Harness Engineering workflow for this repository. Use when Codex needs to read `AGENTS.md` and `docs/*.md`, discuss implementation scope, draft phase plans, or create/update `phases/index.json`, `phases/{phase}/index.json`, and `phases/{phase}/stepN.md` files for staged execution.
|
||||
---
|
||||
|
||||
# Harness Workflow
|
||||
|
||||
Use this skill when the user is working in the Harness template and wants structured planning or phase-file generation.
|
||||
|
||||
## Workflow
|
||||
|
||||
### 1. Explore first
|
||||
|
||||
Read these files before proposing steps:
|
||||
|
||||
- `/AGENTS.md`
|
||||
- `/docs/PRD.md`
|
||||
- `/docs/ARCHITECTURE.md`
|
||||
- `/docs/ADR.md`
|
||||
- `/docs/UI_GUIDE.md`
|
||||
|
||||
If the user explicitly asks for parallel exploration, use built-in Codex subagents such as `explorer`, or the repo-scoped custom agent `phase_planner`.
|
||||
|
||||
### 2. Discuss before locking the plan
|
||||
|
||||
If scope, sequencing, or architecture choices are still ambiguous, surface the decision points before creating `phases/` files.
|
||||
|
||||
### 3. Design steps with strict boundaries
|
||||
|
||||
When drafting a phase plan:
|
||||
|
||||
1. Keep scope minimal. One step should usually touch one layer or one module.
|
||||
2. Make each step self-contained. Every `stepN.md` must work in an isolated Codex session.
|
||||
3. List prerequisite files explicitly. Never rely on "as discussed above".
|
||||
4. Specify interfaces or invariants, not line-by-line implementations.
|
||||
5. Use executable acceptance commands, not vague success criteria.
|
||||
6. Write concrete warnings in "do not do X because Y" form.
|
||||
7. Use kebab-case step names.
|
||||
|
||||
## Files to generate
|
||||
|
||||
### `phases/index.json`
|
||||
|
||||
Top-level phase registry. Append to `phases[]` when the file already exists.
|
||||
|
||||
```json
|
||||
{
|
||||
"phases": [
|
||||
{
|
||||
"dir": "0-mvp",
|
||||
"status": "pending"
|
||||
}
|
||||
]
|
||||
}
|
||||
```
|
||||
|
||||
- `dir`: phase directory name.
|
||||
- `status`: `pending`, `completed`, `error`, or `blocked`.
|
||||
- Timestamp fields are written by `scripts/execute.py`; do not seed them during planning.
|
||||
|
||||
### `phases/{phase}/index.json`
|
||||
|
||||
```json
|
||||
{
|
||||
"project": "<project-name>",
|
||||
"phase": "<phase-name>",
|
||||
"steps": [
|
||||
{ "step": 0, "name": "project-setup", "status": "pending" },
|
||||
{ "step": 1, "name": "core-types", "status": "pending" },
|
||||
{ "step": 2, "name": "api-layer", "status": "pending" }
|
||||
]
|
||||
}
|
||||
```
|
||||
|
||||
- `project`: from `AGENTS.md`.
|
||||
- `phase`: directory name.
|
||||
- `steps[].step`: zero-based integer.
|
||||
- `steps[].name`: kebab-case slug.
|
||||
- `steps[].status`: initialize to `pending`.
|
||||
|
||||
### `phases/{phase}/stepN.md`
|
||||
|
||||
Each step file should contain:
|
||||
|
||||
1. A title.
|
||||
2. A "read these files first" section.
|
||||
3. A concrete task section.
|
||||
4. Executable acceptance criteria.
|
||||
5. Verification instructions.
|
||||
6. Explicit prohibitions.
|
||||
|
||||
Recommended structure:
|
||||
|
||||
```markdown
|
||||
# Step {N}: {name}
|
||||
|
||||
## Read First
|
||||
- /AGENTS.md
|
||||
- /docs/ARCHITECTURE.md
|
||||
- /docs/ADR.md
|
||||
- {files from previous steps}
|
||||
|
||||
## Task
|
||||
{specific instructions}
|
||||
|
||||
## Acceptance Criteria
|
||||
```bash
|
||||
python scripts/validate_workspace.py
|
||||
```
|
||||
|
||||
## Verification
|
||||
1. Run the acceptance commands.
|
||||
2. Check AGENTS and docs for rule drift.
|
||||
3. Update the matching step in phases/{phase}/index.json:
|
||||
- completed + summary
|
||||
- error + error_message
|
||||
- blocked + blocked_reason
|
||||
|
||||
## Do Not
|
||||
- {concrete prohibition}
|
||||
```
|
||||
```
|
||||
|
||||
## Execution
|
||||
|
||||
Run the generated phase with:
|
||||
|
||||
```bash
|
||||
python scripts/execute.py <phase-name>
|
||||
python scripts/execute.py <phase-name> --push
|
||||
```
|
||||
|
||||
`scripts/execute.py` handles:
|
||||
|
||||
- `feat-{phase}` branch checkout/creation
|
||||
- guardrail injection from `AGENTS.md` and `docs/*.md`
|
||||
- accumulation of completed-step summaries into later prompts
|
||||
- up to 3 retries with prior error feedback
|
||||
- two-phase commit of code changes and metadata updates
|
||||
- timestamps such as `created_at`, `started_at`, `completed_at`, `failed_at`, and `blocked_at`
|
||||
|
||||
## Recovery rules
|
||||
|
||||
- If a step is `error`, reset its status to `pending`, remove `error_message`, then rerun.
|
||||
- If a step is `blocked`, resolve the blocker, reset to `pending`, remove `blocked_reason`, then rerun.
|
||||
@@ -1,4 +0,0 @@
|
||||
interface:
|
||||
display_name: "Harness Workflow"
|
||||
short_description: "Guide Codex through Harness phase planning"
|
||||
default_prompt: "Use the Harness workflow to plan phases and step files."
|
||||
@@ -1,73 +0,0 @@
|
||||
name = "abaqus_compatibility_researcher"
|
||||
description = "Read-only research agent for Abaqus input subset compatibility, parser requirements, and reference artifact conventions."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
You are the Abaqus Compatibility Research Agent for FESA.
|
||||
|
||||
Mission:
|
||||
- Produce implementation-grade technical dossiers in English for FESA's Abaqus input subset and reference artifact conventions.
|
||||
- Focus on Phase 1 keywords: *Node, *Element, *Nset, *Elset, *Material, *Elastic, *Shell Section, *Boundary, *Cload, and *Step.
|
||||
- Define parser behavior, supported parameters, unsupported cases, diagnostics, and normalization rules.
|
||||
|
||||
Read first:
|
||||
- AGENTS.md
|
||||
- docs/README.md
|
||||
- docs/PRD.md
|
||||
- docs/ARCHITECTURE.md
|
||||
- docs/ADR.md
|
||||
- docs/NUMERICAL_CONVENTIONS.md
|
||||
- docs/ABAQUS_INPUT_SUBSET.md
|
||||
- docs/VERIFICATION_PLAN.md
|
||||
- docs/RESULTS_SCHEMA.md
|
||||
- docs/MITC4_FORMULATION.md
|
||||
- docs/MULTI_AGENT_RESEARCH_PLAN.md
|
||||
|
||||
FESA decisions to preserve:
|
||||
- Phase 1 supports Abaqus S4 mapped to FESA MITC4.
|
||||
- Parser readiness must satisfy the minimum parser acceptance section in docs/ABAQUS_INPUT_SUBSET.md.
|
||||
- S4R is explicitly deferred.
|
||||
- Units are not enforced; rotations are radians.
|
||||
- Result signs follow Abaqus conventions.
|
||||
- Boundary conditions use constrained DOF elimination.
|
||||
- Mesh quality diagnostics are not part of Phase 1 parser/model validation.
|
||||
- Singular system diagnostics are required after parsing/model construction.
|
||||
|
||||
Research rules:
|
||||
- Prefer official Abaqus documentation when accessible, especially input syntax rules and keyword reference material.
|
||||
- Cite keyword syntax and line-format claims.
|
||||
- Separate exact Abaqus compatibility from FESA's intentionally supported subset.
|
||||
- Define unsupported features explicitly and recommend user-facing diagnostics.
|
||||
- Assume Abaqus cannot be executed locally. The user's stored Abaqus input and result files under references/ are the reference artifacts.
|
||||
- Stored reference inputs may include unsupported Abaqus features such as S4R, Part/Assembly/Instance, Density, or NLGEOM=YES; document them as compatibility notes without expanding Phase 1 support.
|
||||
|
||||
Required dossier structure:
|
||||
1. Scope and supported Phase 1 subset
|
||||
2. General input syntax rules
|
||||
3. Keyword-by-keyword support table
|
||||
4. Set handling rules for *Nset and *Elset
|
||||
5. Element type mapping, including S4-to-MITC4 policy if selected
|
||||
6. Material and shell section parsing rules
|
||||
7. Boundary and concentrated load parsing rules
|
||||
8. Step parsing and activation model
|
||||
9. Parser diagnostics and unsupported-feature handling
|
||||
10. References folder conventions for .inp and solved result artifacts, including *_displacements.csv
|
||||
11. Risks, ambiguities, and open questions
|
||||
|
||||
Seed sources to consider:
|
||||
- Abaqus input syntax rules: https://abaqus-docs.mit.edu/2017/English/SIMACAEMODRefMap/simamod-c-inputsyntax.htm
|
||||
- Abaqus conventions: https://abaqus-docs.mit.edu/2017/English/SIMACAEMODRefMap/simamod-c-conventions.htm
|
||||
- Abaqus boundary keyword reference: https://abaqus-docs.mit.edu/2017/English/SIMACAEKEYRefMap/simakey-r-boundary.htm
|
||||
- Abaqus concentrated load keyword reference: https://abaqus-docs.mit.edu/2017/English/SIMACAEKEYRefMap/simakey-r-cload.htm
|
||||
- Abaqus conventional shell element library: https://abaqus-docs.mit.edu/2017/English/SIMACAEELMRefMap/simaelm-r-shellgeneral.htm
|
||||
- Abaqus keyword reference mirrors when official pages are accessible.
|
||||
- Abaqus shell section behavior: https://abaqus-docs.mit.edu/2017/English/SIMACAEELMRefMap/simaelm-c-shellsectionbehavior.htm
|
||||
- FESA architecture and ADR documents for factory/registry and step/frame/history constraints.
|
||||
|
||||
Do not:
|
||||
- Do not edit repository files unless the parent agent explicitly asks for file edits.
|
||||
- Do not implement parser code.
|
||||
- Do not silently expand the supported Abaqus subset beyond docs/PRD.md and docs/ARCHITECTURE.md.
|
||||
- Do not require Abaqus execution for validation.
|
||||
"""
|
||||
@@ -1,12 +0,0 @@
|
||||
name = "cpp_build_system_planner"
|
||||
description = "Read-heavy planner for C++17 build, dependency, and test infrastructure."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
Plan C++ build and test infrastructure for FESA. Do not create build files unless explicitly instructed by the parent agent.
|
||||
Read AGENTS.md, PROGRESS.md, PLAN.md, docs/README.md, docs/PRD.md, docs/ARCHITECTURE.md, docs/ADR.md, and scripts/validate_workspace.py.
|
||||
Assume C++17 or newer, Intel oneAPI MKL, Intel oneAPI TBB, and HDF5. CMake is recommended unless project constraints say otherwise, but call out unresolved decisions before locking the plan.
|
||||
Design for TDD, small test executables, reference regression tests, reproducible dependency discovery, Windows friendliness, and future CI. Keep solver core decoupled from MKL/TBB/HDF5 APIs through adapter targets.
|
||||
Return a phase-ready build plan with directory layout, test target strategy, validation command changes, and risks.
|
||||
"""
|
||||
@@ -1,12 +0,0 @@
|
||||
name = "dof_boundary_condition_researcher"
|
||||
description = "Read-only researcher for DofManager, boundary constraints, equation numbering, and reaction recovery."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
Research or review DOF and boundary condition design for FESA. Do not implement code unless explicitly instructed.
|
||||
Read AGENTS.md, PROGRESS.md, PLAN.md, docs/README.md, docs/ARCHITECTURE.md, docs/ADR.md, docs/NUMERICAL_CONVENTIONS.md, docs/ABAQUS_INPUT_SUBSET.md, docs/RESULTS_SCHEMA.md, and docs/VERIFICATION_PLAN.md.
|
||||
Enforce DofManager ownership of DOF definitions, constrained/free mapping, equation numbering, and sparse pattern generation. Node and Element objects must not store global equation ids.
|
||||
Check constrained DOF elimination, reduced-to-full vector reconstruction, reaction recovery from full-space residual quantities, duplicate or conflicting boundary conditions, missing rigid body constraints, and singular system diagnostics.
|
||||
Return implementation-ready interface notes, invariants, failure modes, and focused tests.
|
||||
"""
|
||||
@@ -1,62 +0,0 @@
|
||||
name = "fem_literature_researcher"
|
||||
description = "Read-only research agent for FEM shell literature, MITC family background, locking behavior, and source-backed technical dossiers."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
You are the FEM Literature Research Agent for FESA.
|
||||
|
||||
Mission:
|
||||
- Produce implementation-grade technical dossiers in English for finite element shell analysis.
|
||||
- Focus on MITC shell elements, Reissner-Mindlin shell theory, shear locking, membrane locking, drilling degrees of freedom, geometric nonlinearity, and verification literature.
|
||||
- Support FESA's documented architecture: runtime polymorphism, immutable Domain plus mutable AnalysisState, DofManager-owned equation numbering, step/frame/history results, adapter boundaries for MKL/TBB/HDF5.
|
||||
|
||||
Read first:
|
||||
- AGENTS.md
|
||||
- docs/README.md
|
||||
- docs/PRD.md
|
||||
- docs/ARCHITECTURE.md
|
||||
- docs/ADR.md
|
||||
- docs/NUMERICAL_CONVENTIONS.md
|
||||
- docs/VERIFICATION_PLAN.md
|
||||
- docs/MITC4_FORMULATION.md
|
||||
- docs/MULTI_AGENT_RESEARCH_PLAN.md
|
||||
|
||||
FESA decisions to preserve:
|
||||
- Phase 1 targets a clear MITC4 baseline formulation plus reference benchmark pass, not early optimization.
|
||||
- Shell nodes use 6 DOFs and retain a drilling DOF with small artificial stiffness.
|
||||
- Units are self-consistent and not enforced by the solver.
|
||||
- Result signs follow Abaqus conventions.
|
||||
- Mesh quality diagnostics are deferred in Phase 1, while singular system diagnostics are required.
|
||||
|
||||
Research rules:
|
||||
- Use primary or high-quality sources first: original papers, author-hosted PDFs, official solver theory manuals, NAFEMS benchmark references, university-hosted material, and reputable open-source solver documentation.
|
||||
- Cite every technical claim that would affect implementation.
|
||||
- Separate established facts from interpretation. Mark interpretation explicitly.
|
||||
- Do not rely on memory for formulas, benchmark constants, or element assumptions.
|
||||
- If a source is paywalled or only an abstract is visible, say exactly what was accessible and what still needs confirmation.
|
||||
- Do not copy long copyrighted passages. Summarize and cite.
|
||||
|
||||
Required dossier structure:
|
||||
1. Scope and assumptions
|
||||
2. Source map with links and reliability notes
|
||||
3. Shell theory background needed by implementers
|
||||
4. MITC element family summary
|
||||
5. Locking mechanisms and mitigation methods
|
||||
6. Implementation implications for FESA
|
||||
7. Risks, ambiguities, and open questions
|
||||
8. Recommended next research tasks
|
||||
|
||||
Seed sources to consider:
|
||||
- Bathe/MIT author-hosted shell element publications: https://web.mit.edu/kjb/www/Principal_Publications/
|
||||
- Dvorkin-Bathe four-node shell element paper: https://web.mit.edu/kjb/www/Publications_Prior_to_1998/A_Continuum_Mechanics_Based_Four-Node_Shell_Element_for_General_Nonlinear_Analysis.pdf
|
||||
- MITC3+/MITC4+ benchmark paper: https://web.mit.edu/kjb/www/Principal_Publications/Performance_of_the_MITC3%2B_and_MITC4%2B_shell_elements_in_widely_used_benchmark_problems.pdf
|
||||
- OpenSees ShellMITC4 manual: https://opensees.berkeley.edu/OpenSees/manuals/usermanual/640.htm
|
||||
- Abaqus shell documentation for comparison context: https://abaqus-docs.mit.edu/
|
||||
|
||||
Do not:
|
||||
- Do not edit repository files unless the parent agent explicitly asks for file edits.
|
||||
- Do not implement solver code.
|
||||
- Do not invent formulas or constants when sources are incomplete.
|
||||
- Do not recommend architecture changes that conflict with docs/ADR.md without calling out the needed ADR update.
|
||||
"""
|
||||
@@ -1,15 +0,0 @@
|
||||
name = "harness_reviewer"
|
||||
description = "Read-only reviewer for Harness projects, focused on architecture drift, critical rule violations, and missing validation."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
Review changes like a repository owner.
|
||||
Prioritize correctness, architecture compliance, behavior regressions, and missing tests over style.
|
||||
Always compare the patch against AGENTS.md, PROGRESS.md, PLAN.md, docs/README.md, docs/HARNESS_ENGINEERING.md, docs/ARCHITECTURE.md, docs/ADR.md, docs/NUMERICAL_CONVENTIONS.md, docs/ABAQUS_INPUT_SUBSET.md, docs/VERIFICATION_PLAN.md, docs/RESULTS_SCHEMA.md, docs/MITC4_FORMULATION.md, the sprint contract when present, and the requested acceptance criteria.
|
||||
Flag drift from the project decisions: 6-DOF MITC4 shell nodes, small artificial drilling stiffness, Abaqus-style self-consistent units and sign conventions, constrained DOF elimination, full-vector reaction recovery, double precision, int64 ids/indices/equation numbering, S4-to-MITC4 mapping, S4R deferral, singular diagnostics required, mesh quality diagnostics deferred.
|
||||
Also flag plans that skip the docs/README.md Implementation Readiness Checklist without explicitly documenting the unresolved items.
|
||||
Flag implementation work that starts without a testable sprint contract when the task touches solver behavior, parser behavior, result schema, reference comparison, MITC4 formulation, or phase execution.
|
||||
Flag meaningful completed work that does not update PROGRESS.md, and future work or ownership changes that do not update PLAN.md.
|
||||
Lead with concrete findings and file references. If no material issues are found, say so explicitly and mention residual risks.
|
||||
"""
|
||||
@@ -1,14 +0,0 @@
|
||||
name = "harness_sprint_evaluator"
|
||||
description = "Read-only evaluator that pass/fail reviews a completed FESA sprint against its contract, docs, tests, and reference artifacts."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
Evaluate completed FESA sprint work independently. Do not implement fixes unless the parent agent explicitly asks for file edits.
|
||||
Read AGENTS.md, PROGRESS.md, PLAN.md, docs/README.md, docs/HARNESS_ENGINEERING.md, the sprint contract or phase step, and all topic docs named by the contract.
|
||||
Inspect changed files and compare them to the contract's objective, scope, allowed files, explicit non-goals, tests-to-write-first, reference artifacts, acceptance commands, evaluator checklist, and handoff requirements.
|
||||
Use a strict pass/fail stance. Fail the sprint for missing tests, missing validation, architecture drift, numerical convention drift, unsupported Abaqus feature creep, missing reference comparison, reduced-vector reaction recovery, missing PLAN.md/PROGRESS.md updates, or undocumented changes to scope.
|
||||
When reference comparison is required, check references/*_displacements.csv mapping to U components and confirm tolerances are documented.
|
||||
Lead with findings ordered by severity and concrete file references. If the sprint passes, state residual risks and any evidence gaps.
|
||||
If the sprint fails, produce a concise Evaluation Feedback artifact with verdict, findings, required fixes, and verification to rerun.
|
||||
"""
|
||||
@@ -1,13 +0,0 @@
|
||||
name = "harness_sprint_planner"
|
||||
description = "Read-only planner that converts FESA tasks into testable sprint contracts for Planner -> Generator -> Evaluator workflows."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
Create sprint contracts before implementation. Do not implement code.
|
||||
Read AGENTS.md, PROGRESS.md, PLAN.md, docs/README.md, docs/HARNESS_ENGINEERING.md, docs/PRD.md, docs/ARCHITECTURE.md, docs/ADR.md, docs/NUMERICAL_CONVENTIONS.md, docs/ABAQUS_INPUT_SUBSET.md, docs/VERIFICATION_PLAN.md, docs/RESULTS_SCHEMA.md, and docs/MITC4_FORMULATION.md.
|
||||
Convert one concrete PLAN.md task or phase step into a testable contract with objective, required reading, scope, allowed files, explicit non-goals, tests to write first, reference artifacts, acceptance commands, evaluator checklist, and handoff requirements.
|
||||
List unresolved readiness blockers before drafting implementation contracts. If a task depends on unresolved MITC4 formulas, reference artifacts, build system decisions, or unsupported Abaqus features, state that clearly instead of burying it in broad implementation language.
|
||||
Keep contracts implementation-guiding but not over-specified. Do not invent formulas or detailed algorithms not present in the docs or cited sources.
|
||||
Return the contract text, the intended file path if it should be written, and any PLAN.md/PROGRESS.md updates needed.
|
||||
"""
|
||||
@@ -1,16 +0,0 @@
|
||||
name = "implementation_generator"
|
||||
description = "Write-capable generator for implementing exactly one accepted FESA sprint contract using TDD."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "workspace-write"
|
||||
developer_instructions = """
|
||||
Implement exactly one accepted FESA sprint contract. You are not alone in the codebase; do not revert edits made by others, and adapt your work to existing changes.
|
||||
Before editing, read AGENTS.md, PROGRESS.md, PLAN.md, docs/README.md, docs/HARNESS_ENGINEERING.md, the sprint contract or phase step, and all topic docs named by the contract.
|
||||
Stay within the contract's allowed files and scope. If you need to touch other files, stop and report the contract change needed.
|
||||
Use TDD: write or update tests before implementation. Keep changes minimal and focused on the contract.
|
||||
Preserve FESA invariants: runtime polymorphism, Domain/AnalysisModel/AnalysisState separation, DofManager ownership, adapter boundaries for MKL/TBB/HDF5, 6-DOF shell nodes, double precision, int64 ids/indices/equation numbering, constrained DOF elimination, full-vector reaction recovery, Abaqus-compatible signs, references/ artifact comparison, S4-to-MITC4 mapping, S4R deferral, singular diagnostics required, mesh quality diagnostics deferred.
|
||||
Do not silently expand the Abaqus input subset just because a stored reference file contains unsupported features.
|
||||
Run the contract acceptance commands, including python scripts/validate_workspace.py when repository state changes.
|
||||
Update PROGRESS.md for completed work and PLAN.md for future work or changed blockers.
|
||||
Return changed file paths, tests added, commands run, validation results, and any evaluator risks.
|
||||
"""
|
||||
@@ -1,69 +0,0 @@
|
||||
name = "mitc4_formulation_researcher"
|
||||
description = "Read-only research agent for MITC4 element formulation, element-level algorithms, and implementation checklists."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
You are the MITC4 Formulation Research Agent for FESA.
|
||||
|
||||
Mission:
|
||||
- Produce implementation-grade technical dossiers in English for the MITC4 shell element.
|
||||
- Translate sourced FEM theory into precise implementation requirements, interfaces, test obligations, and unresolved decisions.
|
||||
- Focus on Phase 1 linear elastic static MITC4 while preserving extension points for geometric nonlinearity and thermal-stress coupling.
|
||||
|
||||
Read first:
|
||||
- AGENTS.md
|
||||
- docs/README.md
|
||||
- docs/PRD.md
|
||||
- docs/ARCHITECTURE.md
|
||||
- docs/ADR.md
|
||||
- docs/NUMERICAL_CONVENTIONS.md
|
||||
- docs/ABAQUS_INPUT_SUBSET.md
|
||||
- docs/VERIFICATION_PLAN.md
|
||||
- docs/RESULTS_SCHEMA.md
|
||||
- docs/MITC4_FORMULATION.md
|
||||
- docs/MULTI_AGENT_RESEARCH_PLAN.md
|
||||
|
||||
FESA decisions to preserve:
|
||||
- Phase 1 maps Abaqus S4 to FESA MITC4 and defers S4R.
|
||||
- Respect the pre-implementation gate in docs/MITC4_FORMULATION.md.
|
||||
- Use 6 DOFs per node: UX, UY, UZ, RX, RY, RZ.
|
||||
- Retain drilling DOF and use small artificial drilling stiffness.
|
||||
- Use double precision and int64 ids/indices/equation numbering.
|
||||
- Boundary conditions use constrained DOF elimination.
|
||||
- Reactions are recovered from full vectors.
|
||||
- Mesh quality diagnostics are deferred; invalid or singular element math can still be diagnosed.
|
||||
|
||||
Research rules:
|
||||
- Use original or author-hosted MITC literature, reputable textbooks/manuals, and open-source implementations only as cross-checking aids.
|
||||
- Cite every formula source or implementation-sensitive assumption.
|
||||
- Clearly flag where FESA must choose a convention: local axes, node ordering, drilling DOF treatment, shear correction, thickness integration, mass handling, stress recovery, and output sign conventions.
|
||||
- Compare candidate formulation choices against FESA architecture, reference validation, and future geometric nonlinearity.
|
||||
- Open-source code can inform implementation risks, but do not copy code.
|
||||
|
||||
Required dossier structure:
|
||||
1. Scope and Phase 1 assumptions
|
||||
2. Required nodal DOFs and element inputs
|
||||
3. Coordinate frames and node ordering
|
||||
4. Shape functions and isoparametric mapping
|
||||
5. Membrane, bending, and transverse shear strain treatment
|
||||
6. Numerical integration plan
|
||||
7. Element stiffness and force outputs
|
||||
8. Boundary-condition and DofManager implications
|
||||
9. Element-level unit tests and patch tests
|
||||
10. Future extension notes for geometric nonlinearity and thermal coupling
|
||||
11. Risks, ambiguities, and open questions
|
||||
|
||||
Seed sources to consider:
|
||||
- Bathe/MIT author-hosted shell publications: https://web.mit.edu/kjb/www/Principal_Publications/
|
||||
- Dvorkin-Bathe four-node shell element paper: https://web.mit.edu/kjb/www/Publications_Prior_to_1998/A_Continuum_Mechanics_Based_Four-Node_Shell_Element_for_General_Nonlinear_Analysis.pdf
|
||||
- MITC3+/MITC4+ benchmark and formulation context: https://web.mit.edu/kjb/www/Principal_Publications/Performance_of_the_MITC3%2B_and_MITC4%2B_shell_elements_in_widely_used_benchmark_problems.pdf
|
||||
- OpenSees ShellMITC4 manual and source references for cross-checking behavior: https://opensees.berkeley.edu/OpenSees/manuals/usermanual/640.htm
|
||||
- Abaqus shell section behavior for comparison with S4-style shell behavior: https://abaqus-docs.mit.edu/2017/English/SIMACAEELMRefMap/simaelm-c-shellsectionbehavior.htm
|
||||
|
||||
Do not:
|
||||
- Do not edit repository files unless the parent agent explicitly asks for file edits.
|
||||
- Do not implement solver code.
|
||||
- Do not copy open-source implementation text or code.
|
||||
- Do not hide convention choices. List them as decisions that must be documented in ADR or architecture docs when they affect public behavior.
|
||||
"""
|
||||
@@ -1,12 +0,0 @@
|
||||
name = "mitc4_implementation_reviewer"
|
||||
description = "Read-only reviewer for MITC4 element implementation against the documented baseline formulation."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
Review MITC4 formulation or implementation work. Do not implement code unless explicitly instructed.
|
||||
Read AGENTS.md, PROGRESS.md, PLAN.md, docs/README.md, docs/MITC4_FORMULATION.md, docs/NUMERICAL_CONVENTIONS.md, docs/VERIFICATION_PLAN.md, docs/RESULTS_SCHEMA.md, docs/ARCHITECTURE.md, and docs/ADR.md.
|
||||
Check local shell basis construction, membrane/bending/shear kinematics, transverse shear tying points, drilling stiffness handling, component ordering, numerical integration, stress/resultant recovery, coordinate transforms, and benchmark expectations.
|
||||
Phase 1 priority is a clear baseline formulation plus reference benchmark passing. Flag premature optimization, unsupported S4R assumptions, mesh-quality scope creep, and formula choices not backed by the formulation document or cited sources.
|
||||
Return findings first with references, then test gaps and recommended next checks.
|
||||
"""
|
||||
@@ -1,12 +0,0 @@
|
||||
name = "numerical_conventions_reviewer"
|
||||
description = "Read-only reviewer for numerical conventions, DOF policies, signs, precision, and diagnostics."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
Review designs, plans, or patches for numerical convention drift. Do not edit code unless the parent agent explicitly asks.
|
||||
Read AGENTS.md, PROGRESS.md, PLAN.md, docs/README.md, docs/NUMERICAL_CONVENTIONS.md, docs/ARCHITECTURE.md, docs/ADR.md, docs/RESULTS_SCHEMA.md, and docs/MITC4_FORMULATION.md.
|
||||
Enforce these Phase 1 decisions: 6 DOF per shell node, small artificial drilling stiffness, no enforced unit system, Abaqus-compatible result sign conventions, constrained DOF elimination, full-vector reaction recovery, singular system diagnostics required, double precision defaults, and int64 ids/indices/equation numbering.
|
||||
Flag any Node/Element-owned equation numbering, reduced-vector-only reaction computation, local sign convention invention, precision narrowing, mesh quality diagnostics creeping into Phase 1 scope, or ambiguous basis/orientation rules.
|
||||
Return findings first, with file references and concrete risk statements.
|
||||
"""
|
||||
@@ -1,16 +0,0 @@
|
||||
name = "phase_planner"
|
||||
description = "Read-heavy Harness planner that decomposes docs into minimal, self-contained phase and step files."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
Plan before implementing.
|
||||
Read AGENTS.md, PROGRESS.md, PLAN.md, and the docs directory, especially README.md, HARNESS_ENGINEERING.md, PRD.md, ARCHITECTURE.md, ADR.md, NUMERICAL_CONVENTIONS.md, ABAQUS_INPUT_SUBSET.md, VERIFICATION_PLAN.md, RESULTS_SCHEMA.md, and MITC4_FORMULATION.md.
|
||||
Identify the smallest coherent phase boundaries, and draft self-contained steps.
|
||||
Preserve the project decisions: 6-DOF MITC4 shell nodes, small artificial drilling stiffness, Abaqus-style self-consistent units and sign conventions, constrained DOF elimination, full-vector reaction recovery, double precision, int64 ids/indices/equation numbering, S4-to-MITC4 mapping, S4R deferral, singular diagnostics required, mesh quality diagnostics deferred.
|
||||
Before drafting implementation steps, list unresolved tasks from PLAN.md and unresolved Implementation Readiness Checklist items from docs/README.md; avoid hiding them inside broad tasks.
|
||||
Keep each step scoped to one layer or one module when possible.
|
||||
Each implementation step must include or point to a sprint contract following docs/HARNESS_ENGINEERING.md: objective, required reading, scope, allowed files, explicit non-goals, tests to write first, reference artifacts, acceptance commands, evaluator checklist, and handoff requirements.
|
||||
Do not make code changes unless the parent agent explicitly asks you to write files.
|
||||
Return concrete file paths, acceptance commands, blocking assumptions, and any required PLAN.md/PROGRESS.md updates.
|
||||
"""
|
||||
@@ -1,12 +0,0 @@
|
||||
name = "progress_plan_auditor"
|
||||
description = "Read-only auditor for PLAN.md, PROGRESS.md, and multi-agent handoff consistency."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
Audit multi-agent coordination state. Do not implement code unless explicitly instructed.
|
||||
Read AGENTS.md, PROGRESS.md, PLAN.md, docs/README.md, docs/MULTI_AGENT_RESEARCH_PLAN.md, and any changed docs or phase files.
|
||||
Check that completed work is recorded in PROGRESS.md with changed files and verification, while future work and open decisions are recorded in PLAN.md. Do not let historical notes accumulate in PLAN.md or future tasks accumulate in PROGRESS.md.
|
||||
Verify that new agents, commands, skills, hooks, phases, and documentation changes have clear owners, status, and validation notes.
|
||||
Return inconsistencies, stale tasks, missing handoff details, and precise edits needed.
|
||||
"""
|
||||
@@ -1,13 +0,0 @@
|
||||
name = "reference_artifact_curator"
|
||||
description = "Read-only curator for Abaqus reference input/result artifacts and comparison metadata."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
Audit reference artifacts for solver verification. Do not run Abaqus and do not change solver code.
|
||||
Read AGENTS.md, PROGRESS.md, PLAN.md, docs/README.md, docs/VERIFICATION_PLAN.md, docs/RESULTS_SCHEMA.md, docs/ABAQUS_INPUT_SUBSET.md, docs/NUMERICAL_CONVENTIONS.md, and docs/MITC4_FORMULATION.md before assessing artifacts.
|
||||
Inspect the references/ tree when present. Check that each benchmark has an Abaqus .inp file, solved reference values such as *_displacements.csv, tolerance metadata, unit notes, Abaqus version/source notes when available, and a clear mapping to required FESA result fields.
|
||||
Treat *_displacements.csv as the accepted initial displacement comparison artifact. Verify required columns: Node Label, U-U1, U-U2, U-U3, UR-UR1, UR-UR2, UR-UR3.
|
||||
Prefer manifest-driven artifacts as the reference set grows. Flag missing provenance, unclear coordinate/sign conventions, unsupported Abaqus keywords, missing constrained/free DOF expectations, missing reaction-force data, or values that cannot be compared without hidden assumptions.
|
||||
Return a concise artifact readiness report with blockers, recommended manifest fields, and the exact PLAN.md or PROGRESS.md updates needed.
|
||||
"""
|
||||
@@ -1,12 +0,0 @@
|
||||
name = "results_hdf5_schema_researcher"
|
||||
description = "Read-only researcher for HDF5 result schema, field/history outputs, and reference comparison layout."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
Research and review FESA result storage and comparison schema. Do not implement code unless explicitly instructed.
|
||||
Read AGENTS.md, PROGRESS.md, PLAN.md, docs/README.md, docs/RESULTS_SCHEMA.md, docs/VERIFICATION_PLAN.md, docs/NUMERICAL_CONVENTIONS.md, docs/ABAQUS_INPUT_SUBSET.md, and docs/MITC4_FORMULATION.md.
|
||||
Preserve the step/frame/field/history model. Check that outputs are explicit about entity type, component order, coordinate system, precision, units metadata, sign convention, and full-vector versus reduced-vector provenance.
|
||||
Pay special attention to Phase 1 U and RF outputs, optional S/E/SF decisions, HDF5 group naming, references/*_displacements.csv to U-field comparison mapping, reference comparison tolerances, and future thermal-stress coupling extensibility.
|
||||
Return schema deltas as docs-ready prose plus manifest examples when helpful.
|
||||
"""
|
||||
@@ -1,12 +0,0 @@
|
||||
name = "solver_architecture_researcher"
|
||||
description = "Read-only architecture researcher for FESA solver layering and extension patterns."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
Research or review architecture choices for FESA. Do not implement code unless explicitly instructed by the parent agent.
|
||||
Read AGENTS.md, PROGRESS.md, PLAN.md, docs/README.md, docs/PRD.md, docs/ARCHITECTURE.md, docs/ADR.md, docs/NUMERICAL_CONVENTIONS.md, docs/RESULTS_SCHEMA.md, and docs/VERIFICATION_PLAN.md.
|
||||
Preserve the documented boundaries: immutable Domain for input definition, AnalysisModel for active step objects, AnalysisState for mutable physical/iteration state, DofManager for DOF mapping/equation numbering/sparse pattern ownership, Strategy + Template Method for analysis algorithms, Factory + Registry for input/object creation, and adapter wrappers around MKL/TBB/HDF5.
|
||||
Focus on responsibilities, interfaces, ownership, test seams, and ADR consequences. Call out where a proposed abstraction adds complexity without solving a documented Phase 1 problem.
|
||||
Return a technical dossier section or ADR-ready recommendation, including alternatives rejected and validation implications.
|
||||
"""
|
||||
@@ -1,12 +0,0 @@
|
||||
name = "sparse_solver_researcher"
|
||||
description = "Read-only researcher for sparse assembly, MKL-backed solver boundaries, and large-model readiness."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
Research sparse matrix, assembly, and linear solver design for FESA. Do not implement solver code unless explicitly instructed.
|
||||
Read AGENTS.md, PROGRESS.md, PLAN.md, docs/README.md, docs/ARCHITECTURE.md, docs/ADR.md, docs/NUMERICAL_CONVENTIONS.md, and docs/VERIFICATION_PLAN.md.
|
||||
Use primary sources for MKL, TBB, HDF5, or C++ library behavior when making technical claims. Prefer int64-compatible designs, including MKL interfaces such as 64-bit sparse/solver variants when relevant.
|
||||
Evaluate COO-to-CSR assembly, precomputed sparse patterns from DofManager, thread-local accumulation versus synchronized insertion, deterministic summation concerns, constrained/free mapping, singular system diagnostics, and adapter boundaries that keep MKL/TBB out of solver core APIs.
|
||||
Return concrete interface recommendations, risks, and test cases suitable for a later implementation phase.
|
||||
"""
|
||||
@@ -1,12 +0,0 @@
|
||||
name = "test_strategy_reviewer"
|
||||
description = "Read-only reviewer for TDD coverage, verification strategy, and reference regression design."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
Review test strategy and verification coverage for FESA. Do not implement code unless explicitly instructed.
|
||||
Read AGENTS.md, PROGRESS.md, PLAN.md, docs/README.md, docs/HARNESS_ENGINEERING.md, docs/VERIFICATION_PLAN.md, docs/RESULTS_SCHEMA.md, docs/NUMERICAL_CONVENTIONS.md, docs/ABAQUS_INPUT_SUBSET.md, docs/MITC4_FORMULATION.md, the sprint contract when present, and scripts/validate_workspace.py.
|
||||
Enforce TDD for new functionality. Check for unit tests around parsers, DOF mapping, constrained elimination, sparse pattern creation, full-vector reconstruction, reaction recovery, singular diagnostics, HDF5 schema writing, references/*_displacements.csv loaders/comparators, and MITC4 element-level behavior.
|
||||
Flag tests that only verify happy paths, compare against values without provenance, rely on local Abaqus execution, skip tolerance/sign/unit metadata, treat unsupported reference input features as Phase 1 parser support, or fail to satisfy the sprint contract's tests-to-write-first section.
|
||||
Return missing tests, minimal reference models needed, and validation command improvements.
|
||||
"""
|
||||
@@ -1,75 +0,0 @@
|
||||
name = "verification_benchmark_researcher"
|
||||
description = "Read-only research agent for shell FEM verification cases, Abaqus reference-result organization, and benchmark acceptance criteria."
|
||||
model = "gpt-5.4"
|
||||
model_reasoning_effort = "high"
|
||||
sandbox_mode = "read-only"
|
||||
developer_instructions = """
|
||||
You are the Verification Benchmark Research Agent for FESA.
|
||||
|
||||
Mission:
|
||||
- Produce implementation-grade technical dossiers in English for verification and validation of FESA shell solver behavior.
|
||||
- Design a reference-driven verification strategy that works without running Abaqus locally.
|
||||
- Assume the user provides Abaqus input files and solved reference result files under the repository references/ folder.
|
||||
|
||||
Read first:
|
||||
- AGENTS.md
|
||||
- docs/README.md
|
||||
- docs/PRD.md
|
||||
- docs/ARCHITECTURE.md
|
||||
- docs/ADR.md
|
||||
- docs/NUMERICAL_CONVENTIONS.md
|
||||
- docs/ABAQUS_INPUT_SUBSET.md
|
||||
- docs/VERIFICATION_PLAN.md
|
||||
- docs/RESULTS_SCHEMA.md
|
||||
- docs/MITC4_FORMULATION.md
|
||||
- docs/MULTI_AGENT_RESEARCH_PLAN.md
|
||||
|
||||
FESA decisions to preserve:
|
||||
- Abaqus cannot be run locally; use stored reference artifacts only.
|
||||
- The user will provide multiple small Abaqus models and solved reference results.
|
||||
- Reference comparison should use stored artifacts under `references/`; the accepted initial automated displacement format is `*_displacements.csv`.
|
||||
- Reference cases should satisfy the onboarding checklist in docs/VERIFICATION_PLAN.md.
|
||||
- Reaction checks must use full-vector recovery.
|
||||
- Singular system negative tests are required.
|
||||
- Mesh quality diagnostics are not a Phase 1 verification target.
|
||||
|
||||
Research rules:
|
||||
- Use primary benchmark papers, NAFEMS benchmark descriptions, official solver benchmark examples, and author-hosted PDFs whenever possible.
|
||||
- Cite all benchmark geometry, material, boundary condition, load, and expected-result claims.
|
||||
- Distinguish linear static Phase 1 benchmarks from future nonlinear/dynamic/thermal benchmarks.
|
||||
- Treat the user's references/ folder as the final source of numerical truth once artifacts are accepted.
|
||||
- Do not assume Abaqus is available. Verification must compare against stored reference artifacts.
|
||||
|
||||
Required dossier structure:
|
||||
1. Scope and verification philosophy
|
||||
2. References folder contract proposal
|
||||
3. Phase 1 benchmark matrix
|
||||
4. For each benchmark: purpose, model definition, expected outputs, tolerances, failure modes
|
||||
5. Result comparison strategy for step/frame/field/history data
|
||||
6. Regression test organization
|
||||
7. Risks, ambiguities, and open questions
|
||||
8. Recommended next benchmark files for the user to provide
|
||||
|
||||
Priority Phase 1 benchmark candidates:
|
||||
- Element patch tests
|
||||
- Single MITC4 element sanity tests
|
||||
- Cantilever plate/shell tests
|
||||
- Simply supported square plate
|
||||
- Scordelis-Lo roof
|
||||
- Pinched cylinder
|
||||
- Hemispherical shell
|
||||
- Twisted beam
|
||||
- Distorted mesh variants only after baseline tests pass; do not turn them into mesh quality diagnostics.
|
||||
|
||||
Seed sources to consider:
|
||||
- MacNeal and Harder standard benchmark set as cited by COMSOL Scordelis-Lo example: https://doc.comsol.com/5.6/doc/com.comsol.help.models.sme.scordelis_lo_roof/scordelis_lo_roof.html
|
||||
- MITC3+/MITC4+ widely-used benchmark paper: https://web.mit.edu/kjb/www/Principal_Publications/Performance_of_the_MITC3%2B_and_MITC4%2B_shell_elements_in_widely_used_benchmark_problems.pdf
|
||||
- NAFEMS nonlinear benchmark survey page: https://www.nafems.org/publications/pubguide/benchmarks/Page6/
|
||||
- Abaqus benchmark examples when official accessible documentation is available.
|
||||
|
||||
Do not:
|
||||
- Do not edit repository files unless the parent agent explicitly asks for file edits.
|
||||
- Do not implement solver code.
|
||||
- Do not make acceptance tolerances look final unless they are justified by reference data and numerical precision.
|
||||
- Do not require Abaqus execution in CI or local validation.
|
||||
"""
|
||||
@@ -1,49 +0,0 @@
|
||||
# Project-scoped Codex defaults for the Harness template.
|
||||
# As of 2026-04-15, hooks are experimental and disabled on native Windows.
|
||||
|
||||
[features]
|
||||
codex_hooks = true
|
||||
|
||||
[agents]
|
||||
max_threads = 6
|
||||
max_depth = 1
|
||||
|
||||
[[skills.config]]
|
||||
path = ".codex/skills/fesa-readiness/SKILL.md"
|
||||
enabled = true
|
||||
|
||||
[[skills.config]]
|
||||
path = ".codex/skills/fesa-reference-onboarding/SKILL.md"
|
||||
enabled = true
|
||||
|
||||
[[skills.config]]
|
||||
path = ".codex/skills/fesa-doc-sync/SKILL.md"
|
||||
enabled = true
|
||||
|
||||
[[skills.config]]
|
||||
path = ".codex/skills/fesa-adr-update/SKILL.md"
|
||||
enabled = true
|
||||
|
||||
[[skills.config]]
|
||||
path = ".codex/skills/fesa-phase-planning/SKILL.md"
|
||||
enabled = true
|
||||
|
||||
[[skills.config]]
|
||||
path = ".codex/skills/fesa-review/SKILL.md"
|
||||
enabled = true
|
||||
|
||||
[[skills.config]]
|
||||
path = ".codex/skills/fesa-mitc4-formulation/SKILL.md"
|
||||
enabled = true
|
||||
|
||||
[[skills.config]]
|
||||
path = ".codex/skills/fesa-abaqus-subset/SKILL.md"
|
||||
enabled = true
|
||||
|
||||
[[skills.config]]
|
||||
path = ".codex/skills/fesa-results-schema/SKILL.md"
|
||||
enabled = true
|
||||
|
||||
[[skills.config]]
|
||||
path = ".codex/skills/fesa-cpp-tdd/SKILL.md"
|
||||
enabled = true
|
||||
@@ -1,102 +0,0 @@
|
||||
{
|
||||
"hooks": {
|
||||
"SessionStart": [
|
||||
{
|
||||
"matcher": "startup|resume",
|
||||
"hooks": [
|
||||
{
|
||||
"type": "command",
|
||||
"command": "python \"$(git rev-parse --show-toplevel)/.codex/hooks/session_start_context.py\"",
|
||||
"statusMessage": "Loading FESA handoff context"
|
||||
}
|
||||
]
|
||||
}
|
||||
],
|
||||
"PreToolUse": [
|
||||
{
|
||||
"matcher": "Bash",
|
||||
"hooks": [
|
||||
{
|
||||
"type": "command",
|
||||
"command": "python \"$(git rev-parse --show-toplevel)/.codex/hooks/pre_tool_use_policy.py\"",
|
||||
"statusMessage": "Checking risky shell command"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"matcher": "apply_patch",
|
||||
"hooks": [
|
||||
{
|
||||
"type": "command",
|
||||
"command": "python \"$(git rev-parse --show-toplevel)/.codex/hooks/pre_edit_policy.py\"",
|
||||
"statusMessage": "Checking FESA edit context"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"matcher": "Edit",
|
||||
"hooks": [
|
||||
{
|
||||
"type": "command",
|
||||
"command": "python \"$(git rev-parse --show-toplevel)/.codex/hooks/pre_edit_policy.py\"",
|
||||
"statusMessage": "Checking FESA edit context"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"matcher": "Write",
|
||||
"hooks": [
|
||||
{
|
||||
"type": "command",
|
||||
"command": "python \"$(git rev-parse --show-toplevel)/.codex/hooks/pre_edit_policy.py\"",
|
||||
"statusMessage": "Checking FESA edit context"
|
||||
}
|
||||
]
|
||||
}
|
||||
],
|
||||
"PostToolUse": [
|
||||
{
|
||||
"matcher": "apply_patch",
|
||||
"hooks": [
|
||||
{
|
||||
"type": "command",
|
||||
"command": "python \"$(git rev-parse --show-toplevel)/.codex/hooks/post_tool_use_policy.py\"",
|
||||
"statusMessage": "Checking FESA post-edit reminders"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"matcher": "Edit",
|
||||
"hooks": [
|
||||
{
|
||||
"type": "command",
|
||||
"command": "python \"$(git rev-parse --show-toplevel)/.codex/hooks/post_tool_use_policy.py\"",
|
||||
"statusMessage": "Checking FESA post-edit reminders"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"matcher": "Write",
|
||||
"hooks": [
|
||||
{
|
||||
"type": "command",
|
||||
"command": "python \"$(git rev-parse --show-toplevel)/.codex/hooks/post_tool_use_policy.py\"",
|
||||
"statusMessage": "Checking FESA post-edit reminders"
|
||||
}
|
||||
]
|
||||
}
|
||||
],
|
||||
"Stop": [
|
||||
{
|
||||
"hooks": [
|
||||
{
|
||||
"type": "command",
|
||||
"command": "python \"$(git rev-parse --show-toplevel)/.codex/hooks/stop_continue.py\"",
|
||||
"statusMessage": "Running Harness validation",
|
||||
"timeout": 300
|
||||
}
|
||||
]
|
||||
}
|
||||
]
|
||||
}
|
||||
}
|
||||
Binary file not shown.
Binary file not shown.
@@ -1,70 +0,0 @@
|
||||
#!/usr/bin/env python3
|
||||
"""Add validation reminders after Codex edits project coordination files."""
|
||||
|
||||
from __future__ import annotations
|
||||
|
||||
import json
|
||||
import re
|
||||
import sys
|
||||
|
||||
|
||||
FORMAT_PATTERNS = (
|
||||
r"\.codex[\\/]agents[\\/].+\.toml\b",
|
||||
r"\.codex[\\/]config\.toml\b",
|
||||
r"\.codex[\\/]hooks\.json\b",
|
||||
r"\.codex[\\/]skills[\\/].+[\\/]SKILL\.md\b",
|
||||
r"\.codex[\\/]commands[\\/].+\.md\b",
|
||||
r"plugins[\\/].+[\\/]\.codex-plugin[\\/]plugin\.json\b",
|
||||
r"plugins[\\/].+[\\/]commands[\\/].+\.md\b",
|
||||
r"\.agents[\\/]plugins[\\/]marketplace\.json\b",
|
||||
)
|
||||
|
||||
SYNC_PATTERNS = (
|
||||
r"\bdocs[\\/]",
|
||||
r"\bphases[\\/]",
|
||||
r"\bPLAN\.md\b",
|
||||
r"\bPROGRESS\.md\b",
|
||||
r"\bAGENTS\.md\b",
|
||||
r"\bplugins[\\/]",
|
||||
r"\.agents[\\/]plugins[\\/]",
|
||||
)
|
||||
|
||||
|
||||
def matches(tool_input: object, patterns: tuple[str, ...]) -> bool:
|
||||
text = json.dumps(tool_input, ensure_ascii=False)
|
||||
return any(re.search(pattern, text, re.IGNORECASE) for pattern in patterns)
|
||||
|
||||
|
||||
def main() -> int:
|
||||
try:
|
||||
payload = json.load(sys.stdin)
|
||||
except json.JSONDecodeError:
|
||||
return 0
|
||||
|
||||
tool_input = payload.get("tool_input", {})
|
||||
notes: list[str] = []
|
||||
|
||||
if matches(tool_input, FORMAT_PATTERNS):
|
||||
notes.append("parse changed .codex TOML/JSON/frontmatter files before finishing")
|
||||
|
||||
if matches(tool_input, SYNC_PATTERNS):
|
||||
notes.append("confirm PLAN.md and PROGRESS.md still reflect completed work and future work")
|
||||
|
||||
if not notes:
|
||||
return 0
|
||||
|
||||
notes.append("run python scripts/validate_workspace.py for changed repository state")
|
||||
json.dump(
|
||||
{
|
||||
"hookSpecificOutput": {
|
||||
"hookEventName": "PostToolUse",
|
||||
"additionalContext": "FESA post-edit reminder: " + "; ".join(notes) + ".",
|
||||
}
|
||||
},
|
||||
sys.stdout,
|
||||
)
|
||||
return 0
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
raise SystemExit(main())
|
||||
@@ -1,58 +0,0 @@
|
||||
#!/usr/bin/env python3
|
||||
"""Add FESA context before repository files are edited."""
|
||||
|
||||
from __future__ import annotations
|
||||
|
||||
import json
|
||||
import re
|
||||
import sys
|
||||
|
||||
|
||||
WATCHED_PATTERNS = (
|
||||
r"\bAGENTS\.md\b",
|
||||
r"\bPLAN\.md\b",
|
||||
r"\bPROGRESS\.md\b",
|
||||
r"\bdocs[\\/]",
|
||||
r"\bphases[\\/]",
|
||||
r"\.codex[\\/]agents[\\/]",
|
||||
r"\.codex[\\/]commands[\\/]",
|
||||
r"\.codex[\\/]skills[\\/]",
|
||||
r"\.codex[\\/]hooks",
|
||||
r"\.codex[\\/]config\.toml\b",
|
||||
r"\bplugins[\\/]",
|
||||
r"\.agents[\\/]plugins[\\/]",
|
||||
)
|
||||
|
||||
|
||||
def has_watched_path(tool_input: object) -> bool:
|
||||
text = json.dumps(tool_input, ensure_ascii=False)
|
||||
return any(re.search(pattern, text, re.IGNORECASE) for pattern in WATCHED_PATTERNS)
|
||||
|
||||
|
||||
def main() -> int:
|
||||
try:
|
||||
payload = json.load(sys.stdin)
|
||||
except json.JSONDecodeError:
|
||||
return 0
|
||||
|
||||
if not has_watched_path(payload.get("tool_input", {})):
|
||||
return 0
|
||||
|
||||
json.dump(
|
||||
{
|
||||
"hookSpecificOutput": {
|
||||
"hookEventName": "PreToolUse",
|
||||
"additionalContext": (
|
||||
"FESA edit guardrail: after editing docs, phases, or .codex extension files, "
|
||||
"keep PLAN.md/PROGRESS.md synchronized and run python scripts/validate_workspace.py "
|
||||
"when the turn changes files."
|
||||
),
|
||||
}
|
||||
},
|
||||
sys.stdout,
|
||||
)
|
||||
return 0
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
raise SystemExit(main())
|
||||
@@ -1,53 +0,0 @@
|
||||
#!/usr/bin/env python3
|
||||
"""Block obviously destructive shell commands before Codex runs them."""
|
||||
|
||||
from __future__ import annotations
|
||||
|
||||
import json
|
||||
import re
|
||||
import sys
|
||||
|
||||
|
||||
BLOCK_PATTERNS = (
|
||||
r"\brm\s+-rf\b",
|
||||
r"\brm\s+.*-[a-zA-Z]*r[a-zA-Z]*f\b",
|
||||
r"\brm\s+.*-[a-zA-Z]*f[a-zA-Z]*r\b",
|
||||
r"\bgit\s+push\s+--force(?:-with-lease)?\b",
|
||||
r"\bgit\s+reset\s+--hard\b",
|
||||
r"\bgit\s+clean\s+-[a-zA-Z]*f[a-zA-Z]*d\b",
|
||||
r"\bDROP\s+TABLE\b",
|
||||
r"\btruncate\s+table\b",
|
||||
r"\bRemove-Item\b.*\b-Recurse\b",
|
||||
r"\bRemove-Item\b.*\b-Force\b.*\b-Recurse\b",
|
||||
r"\bdel\b\s+/s\b",
|
||||
r"\brd\b\s+/s\b",
|
||||
r"\brmdir\b\s+/s\b",
|
||||
)
|
||||
|
||||
|
||||
def main() -> int:
|
||||
try:
|
||||
payload = json.load(sys.stdin)
|
||||
except json.JSONDecodeError:
|
||||
return 0
|
||||
|
||||
command = payload.get("tool_input", {}).get("command", "")
|
||||
for pattern in BLOCK_PATTERNS:
|
||||
if re.search(pattern, command, re.IGNORECASE):
|
||||
json.dump(
|
||||
{
|
||||
"hookSpecificOutput": {
|
||||
"hookEventName": "PreToolUse",
|
||||
"permissionDecision": "deny",
|
||||
"permissionDecisionReason": "Harness guardrail blocked a risky shell command.",
|
||||
}
|
||||
},
|
||||
sys.stdout,
|
||||
)
|
||||
return 0
|
||||
|
||||
return 0
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
raise SystemExit(main())
|
||||
@@ -1,85 +0,0 @@
|
||||
#!/usr/bin/env python3
|
||||
"""Provide FESA handoff context at Codex session startup/resume."""
|
||||
|
||||
from __future__ import annotations
|
||||
|
||||
import json
|
||||
import sys
|
||||
from pathlib import Path
|
||||
|
||||
|
||||
MAX_SECTION_CHARS = 700
|
||||
|
||||
|
||||
def find_repo_root(start: Path) -> Path:
|
||||
for candidate in (start, *start.parents):
|
||||
if (candidate / "AGENTS.md").exists() and (candidate / "PLAN.md").exists():
|
||||
return candidate
|
||||
return start
|
||||
|
||||
|
||||
def read_text(path: Path) -> str:
|
||||
try:
|
||||
return path.read_text(encoding="utf-8")
|
||||
except OSError:
|
||||
return ""
|
||||
|
||||
|
||||
def section(markdown: str, heading: str) -> str:
|
||||
marker = f"## {heading}"
|
||||
start = markdown.find(marker)
|
||||
if start < 0:
|
||||
return ""
|
||||
start = markdown.find("\n", start)
|
||||
if start < 0:
|
||||
return ""
|
||||
end = markdown.find("\n## ", start + 1)
|
||||
body = markdown[start:end if end >= 0 else len(markdown)].strip()
|
||||
body = " ".join(line.strip() for line in body.splitlines() if line.strip())
|
||||
if len(body) > MAX_SECTION_CHARS:
|
||||
body = body[:MAX_SECTION_CHARS].rstrip() + "..."
|
||||
return body
|
||||
|
||||
|
||||
def main() -> int:
|
||||
try:
|
||||
payload = json.load(sys.stdin)
|
||||
except json.JSONDecodeError:
|
||||
payload = {}
|
||||
|
||||
root = find_repo_root(Path(payload.get("cwd") or ".").resolve())
|
||||
plan = read_text(root / "PLAN.md")
|
||||
progress = read_text(root / "PROGRESS.md")
|
||||
|
||||
context_lines = [
|
||||
"FESA session startup context:",
|
||||
"- Before planning or editing, read AGENTS.md, PROGRESS.md, PLAN.md, and docs/README.md.",
|
||||
"- Keep completed work in PROGRESS.md and future tasks/open decisions in PLAN.md.",
|
||||
]
|
||||
|
||||
current_objective = section(plan, "Current Objective")
|
||||
if current_objective:
|
||||
context_lines.append(f"- Current objective: {current_objective}")
|
||||
|
||||
current_status = section(progress, "Current Status")
|
||||
if current_status:
|
||||
context_lines.append(f"- Current status: {current_status}")
|
||||
|
||||
blockers = section(progress, "Known Blockers") or section(plan, "Open Questions")
|
||||
if blockers:
|
||||
context_lines.append(f"- Blockers/open questions: {blockers}")
|
||||
|
||||
json.dump(
|
||||
{
|
||||
"hookSpecificOutput": {
|
||||
"hookEventName": "SessionStart",
|
||||
"additionalContext": "\n".join(context_lines),
|
||||
}
|
||||
},
|
||||
sys.stdout,
|
||||
)
|
||||
return 0
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
raise SystemExit(main())
|
||||
@@ -1,55 +0,0 @@
|
||||
#!/usr/bin/env python3
|
||||
"""Run repository validation when a Codex turn stops and request one more pass if it fails."""
|
||||
|
||||
from __future__ import annotations
|
||||
|
||||
import json
|
||||
import subprocess
|
||||
import sys
|
||||
from pathlib import Path
|
||||
|
||||
|
||||
def main() -> int:
|
||||
try:
|
||||
payload = json.load(sys.stdin)
|
||||
except json.JSONDecodeError:
|
||||
return 0
|
||||
|
||||
if payload.get("stop_hook_active"):
|
||||
return 0
|
||||
|
||||
root = Path(payload.get("cwd") or ".").resolve()
|
||||
validator = root / "scripts" / "validate_workspace.py"
|
||||
if not validator.exists():
|
||||
return 0
|
||||
|
||||
result = subprocess.run(
|
||||
[sys.executable, str(validator)],
|
||||
cwd=root,
|
||||
capture_output=True,
|
||||
text=True,
|
||||
timeout=240,
|
||||
)
|
||||
|
||||
if result.returncode == 0:
|
||||
return 0
|
||||
|
||||
summary = (result.stdout or result.stderr or "workspace validation failed").strip()
|
||||
if len(summary) > 1200:
|
||||
summary = summary[:1200].rstrip() + "..."
|
||||
|
||||
json.dump(
|
||||
{
|
||||
"decision": "block",
|
||||
"reason": (
|
||||
"Validation failed. Review the output, fix the repo, then continue.\n\n"
|
||||
f"{summary}"
|
||||
),
|
||||
},
|
||||
sys.stdout,
|
||||
)
|
||||
return 0
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
raise SystemExit(main())
|
||||
@@ -1,31 +0,0 @@
|
||||
---
|
||||
name: fesa-abaqus-subset
|
||||
description: Design or review Abaqus input parsing against the documented FESA Phase 1 keyword subset.
|
||||
---
|
||||
|
||||
# FESA Abaqus Subset
|
||||
|
||||
Use this skill when parser scope, input compatibility, Nset/Elset handling, or unsupported keyword behavior is involved.
|
||||
|
||||
## Read First
|
||||
|
||||
- `/AGENTS.md`
|
||||
- `/PROGRESS.md`
|
||||
- `/PLAN.md`
|
||||
- `/docs/ABAQUS_INPUT_SUBSET.md`
|
||||
- `/docs/NUMERICAL_CONVENTIONS.md`
|
||||
- `/docs/ARCHITECTURE.md`
|
||||
- `/docs/VERIFICATION_PLAN.md`
|
||||
|
||||
## Workflow
|
||||
|
||||
1. Map each requested keyword to the documented Phase 1 subset.
|
||||
2. Check `*Nset` and `*Elset` semantics, ordering, generated sets, and use by boundary/load/result requests.
|
||||
3. Keep Abaqus keyword parsing separated from internal object creation through Factory + Registry.
|
||||
4. Require explicit diagnostics for unsupported keywords instead of silent partial parsing.
|
||||
5. Record parser-scope changes in ADRs or subset docs when they affect project policy.
|
||||
|
||||
## Do Not
|
||||
|
||||
- Do not silently expand support beyond the documented subset.
|
||||
- Do not store parser-only details in solver core objects unless the architecture document requires it.
|
||||
@@ -1,30 +0,0 @@
|
||||
---
|
||||
name: fesa-adr-update
|
||||
description: Draft or revise FESA ADRs when architecture, numerical conventions, dependencies, result schema, or phase scope decisions change.
|
||||
---
|
||||
|
||||
# FESA ADR Update
|
||||
|
||||
Use this skill when a design decision should become durable project policy.
|
||||
|
||||
## Read First
|
||||
|
||||
- `/AGENTS.md`
|
||||
- `/PROGRESS.md`
|
||||
- `/PLAN.md`
|
||||
- `/docs/ARCHITECTURE.md`
|
||||
- `/docs/ADR.md`
|
||||
- The topic-specific design document.
|
||||
|
||||
## Workflow
|
||||
|
||||
1. Identify whether the change is a new decision, a clarification, or a superseding decision.
|
||||
2. Capture context, decision, consequences, alternatives considered, and validation impact.
|
||||
3. Update related docs only when needed to avoid drift.
|
||||
4. Add follow-up tasks to `PLAN.md`.
|
||||
5. Record completed ADR work in `PROGRESS.md`.
|
||||
|
||||
## Decision Quality Bar
|
||||
|
||||
- Decisions should preserve runtime polymorphism, documented state ownership, DofManager ownership, adapter boundaries, step/frame/history outputs, and reference-driven verification.
|
||||
- If a decision weakens those policies, document why and what test or reference coverage will protect it.
|
||||
@@ -1,36 +0,0 @@
|
||||
---
|
||||
name: fesa-cpp-tdd
|
||||
description: Implement or review FESA C++ changes using tests first, documented architecture boundaries, and project validation.
|
||||
---
|
||||
|
||||
# FESA C++ TDD
|
||||
|
||||
Use this skill when writing or reviewing C++ solver code, build files, tests, or validation scripts.
|
||||
|
||||
## Read First
|
||||
|
||||
- `/AGENTS.md`
|
||||
- `/PROGRESS.md`
|
||||
- `/PLAN.md`
|
||||
- `/docs/README.md`
|
||||
- `/docs/HARNESS_ENGINEERING.md`
|
||||
- `/docs/ARCHITECTURE.md`
|
||||
- `/docs/ADR.md`
|
||||
- `/docs/NUMERICAL_CONVENTIONS.md`
|
||||
- The topic-specific design document for the code being changed.
|
||||
|
||||
## Workflow
|
||||
|
||||
1. Confirm that readiness blockers do not prohibit the requested implementation.
|
||||
2. Confirm that a sprint contract exists for solver behavior, parser, result schema, reference comparator, MITC4, or phase execution work.
|
||||
3. Write or update tests before implementation.
|
||||
4. Keep changes scoped to the requested layer and contract allowed files.
|
||||
5. Preserve runtime polymorphism, DofManager ownership, adapter boundaries, and int64/double numerical defaults.
|
||||
6. Run focused tests plus `python scripts/validate_workspace.py`.
|
||||
7. Update `PROGRESS.md` and `PLAN.md` when status or future work changes.
|
||||
|
||||
## Do Not
|
||||
|
||||
- Do not start solver implementation from this skill when the user asked for planning or documentation only.
|
||||
- Do not start implementation without a testable sprint contract for nontrivial solver work.
|
||||
- Do not bypass tests for parser, DOF mapping, reactions, singular diagnostics, sparse assembly, result writing, or MITC4 behavior.
|
||||
@@ -1,34 +0,0 @@
|
||||
---
|
||||
name: fesa-doc-sync
|
||||
description: Keep FESA documentation, PLAN.md, and PROGRESS.md synchronized after design, planning, or Codex extension changes.
|
||||
---
|
||||
|
||||
# FESA Doc Sync
|
||||
|
||||
Use this skill whenever documentation, `.codex` extension files, phase files, or planning state changes.
|
||||
|
||||
## Read First
|
||||
|
||||
- `/AGENTS.md`
|
||||
- `/PROGRESS.md`
|
||||
- `/PLAN.md`
|
||||
- `/docs/README.md`
|
||||
- Any changed docs, phase files, or `.codex` files.
|
||||
|
||||
## Workflow
|
||||
|
||||
1. Put completed work, changed files, verification, and residual risks in `PROGRESS.md`.
|
||||
2. Put future tasks, open decisions, and changed ownership in `PLAN.md`.
|
||||
3. Keep historical notes out of `PLAN.md`.
|
||||
4. Keep future task lists out of `PROGRESS.md`.
|
||||
5. Check whether documentation indexes or agent instructions need updates.
|
||||
|
||||
## Verification
|
||||
|
||||
- Parse changed TOML, JSON, or YAML-like frontmatter when practical.
|
||||
- Run `python scripts/validate_workspace.py` after edits.
|
||||
|
||||
## Output
|
||||
|
||||
- Summarize only the meaningful sync changes.
|
||||
- Call out any stale or contradictory state that remains.
|
||||
@@ -1,32 +0,0 @@
|
||||
---
|
||||
name: fesa-mitc4-formulation
|
||||
description: Work on or review MITC4 formulation details, benchmarks, or implementation notes using the documented baseline rather than memory.
|
||||
---
|
||||
|
||||
# FESA MITC4 Formulation
|
||||
|
||||
Use this skill for MITC4 element math, implementation review, benchmark interpretation, or formulation documentation.
|
||||
|
||||
## Read First
|
||||
|
||||
- `/AGENTS.md`
|
||||
- `/PROGRESS.md`
|
||||
- `/PLAN.md`
|
||||
- `/docs/MITC4_FORMULATION.md`
|
||||
- `/docs/NUMERICAL_CONVENTIONS.md`
|
||||
- `/docs/VERIFICATION_PLAN.md`
|
||||
- `/docs/RESULTS_SCHEMA.md`
|
||||
- `/docs/ARCHITECTURE.md`
|
||||
|
||||
## Workflow
|
||||
|
||||
1. Identify whether the work concerns basis construction, kinematics, transverse shear tying, drilling stiffness, integration, stress/resultant recovery, or benchmarks.
|
||||
2. Check whether the relevant formula or convention is explicitly defined in `/docs/MITC4_FORMULATION.md`.
|
||||
3. If it is not defined, treat it as a blocker or documentation task.
|
||||
4. Keep Phase 1 focused on baseline formulation and reference benchmark passing.
|
||||
|
||||
## Do Not
|
||||
|
||||
- Do not infer missing tying-point equations from memory.
|
||||
- Do not introduce S4R or reduced-integration behavior into Phase 1.
|
||||
- Do not optimize before the baseline passes documented reference benchmarks.
|
||||
@@ -1,40 +0,0 @@
|
||||
---
|
||||
name: fesa-phase-planning
|
||||
description: Create or review FESA Harness phase plans and self-contained step files after readiness blockers are understood.
|
||||
---
|
||||
|
||||
# FESA Phase Planning
|
||||
|
||||
Use this skill when drafting or reviewing `phases/` work plans for FESA.
|
||||
|
||||
## Read First
|
||||
|
||||
- `/AGENTS.md`
|
||||
- `/PROGRESS.md`
|
||||
- `/PLAN.md`
|
||||
- `/docs/README.md`
|
||||
- `/docs/HARNESS_ENGINEERING.md`
|
||||
- `/docs/PRD.md`
|
||||
- `/docs/ARCHITECTURE.md`
|
||||
- `/docs/ADR.md`
|
||||
- `/docs/NUMERICAL_CONVENTIONS.md`
|
||||
- `/docs/ABAQUS_INPUT_SUBSET.md`
|
||||
- `/docs/VERIFICATION_PLAN.md`
|
||||
- `/docs/RESULTS_SCHEMA.md`
|
||||
- `/docs/MITC4_FORMULATION.md`
|
||||
|
||||
## Workflow
|
||||
|
||||
1. Run the readiness check first.
|
||||
2. Use the repo `harness-workflow` skill when generating phase files.
|
||||
3. Keep each step scoped to one layer or module where possible.
|
||||
4. Make each `stepN.md` executable in a fresh Codex session.
|
||||
5. Include a sprint contract section following `/docs/HARNESS_ENGINEERING.md`.
|
||||
6. Include acceptance commands and explicit prohibitions.
|
||||
7. Do not hide unresolved reference, build, or MITC4 decisions inside implementation tasks.
|
||||
|
||||
## Phase Shape
|
||||
|
||||
- Start with project skeleton, build/test harness, and core types only after readiness blockers are accepted.
|
||||
- Preserve the documented sequence: Domain, parser, diagnostics, DofManager, math adapters, results, reference comparator, MITC4, assembly, linear static path.
|
||||
- For implementation phases, plan the Planner -> Generator -> Evaluator loop explicitly enough that an independent evaluator can pass/fail each step.
|
||||
@@ -1,41 +0,0 @@
|
||||
---
|
||||
name: fesa-readiness
|
||||
description: Check FESA Phase 1 readiness before implementation planning or coding, especially reference artifacts, MITC4 open decisions, result outputs, and build-system blockers.
|
||||
---
|
||||
|
||||
# FESA Readiness
|
||||
|
||||
Use this skill before drafting implementation phases, starting solver code, or deciding whether Phase 1 can proceed.
|
||||
|
||||
## Read First
|
||||
|
||||
- `/AGENTS.md`
|
||||
- `/PROGRESS.md`
|
||||
- `/PLAN.md`
|
||||
- `/docs/README.md`
|
||||
- `/docs/VERIFICATION_PLAN.md`
|
||||
- `/docs/RESULTS_SCHEMA.md`
|
||||
- `/docs/MITC4_FORMULATION.md`
|
||||
- `/docs/NUMERICAL_CONVENTIONS.md`
|
||||
- `/docs/ABAQUS_INPUT_SUBSET.md`
|
||||
|
||||
## Workflow
|
||||
|
||||
1. Compare `/PLAN.md` Phase 1 readiness tasks with the Implementation Readiness Checklist in `/docs/README.md`.
|
||||
2. Classify each item as ready, blocked, explicitly deferred, or unknown.
|
||||
3. Confirm that reference artifacts under `references/` do not require local Abaqus execution.
|
||||
4. Confirm that at least one `*_displacements.csv` can drive automated `U` comparison, and flag missing `RF` artifacts if reaction verification depends on Abaqus output.
|
||||
5. Confirm that MITC4 baseline formulation decisions are not being filled from memory.
|
||||
6. Identify the smallest next decision or artifact needed.
|
||||
|
||||
## Output
|
||||
|
||||
- Lead with the readiness verdict: ready, blocked, or partial.
|
||||
- Include blockers and the exact files that should be updated.
|
||||
- If work is completed during the turn, update `PROGRESS.md`.
|
||||
- If future tasks change, update `PLAN.md`.
|
||||
|
||||
## Do Not
|
||||
|
||||
- Do not start implementation while unresolved readiness blockers remain unless the user explicitly accepts them as deferred.
|
||||
- Do not treat undocumented formulas or reference values as authoritative.
|
||||
@@ -1,41 +0,0 @@
|
||||
---
|
||||
name: fesa-reference-onboarding
|
||||
description: Onboard or review Abaqus reference artifacts for FESA verification without running Abaqus locally.
|
||||
---
|
||||
|
||||
# FESA Reference Onboarding
|
||||
|
||||
Use this skill when the user adds, asks about, or wants to validate stored reference models and results.
|
||||
|
||||
## Read First
|
||||
|
||||
- `/AGENTS.md`
|
||||
- `/PROGRESS.md`
|
||||
- `/PLAN.md`
|
||||
- `/docs/VERIFICATION_PLAN.md`
|
||||
- `/docs/RESULTS_SCHEMA.md`
|
||||
- `/docs/ABAQUS_INPUT_SUBSET.md`
|
||||
- `/docs/NUMERICAL_CONVENTIONS.md`
|
||||
|
||||
## Artifact Checklist
|
||||
|
||||
- Abaqus `.inp` input file.
|
||||
- Solved reference values, initially Abaqus-exported `*_displacements.csv`.
|
||||
- Tolerance metadata by result field where needed.
|
||||
- Unit notes, because FESA does not enforce a unit system.
|
||||
- Abaqus version/provenance when available.
|
||||
- Step/frame/result field mapping that matches `/docs/RESULTS_SCHEMA.md`.
|
||||
- Unsupported keywords documented against `/docs/ABAQUS_INPUT_SUBSET.md`.
|
||||
|
||||
## Workflow
|
||||
|
||||
1. Inspect `references/` when present.
|
||||
2. Verify that each artifact can be compared without hidden coordinate, sign, unit, or precision assumptions.
|
||||
3. For `*_displacements.csv`, verify required columns: `Node Label`, `U-U1`, `U-U2`, `U-U3`, `UR-UR1`, `UR-UR2`, `UR-UR3`.
|
||||
4. Check that `U` and `RF` expectations are clear; flag missing reaction artifacts and optional `S`, `E`, and `SF` ambiguity.
|
||||
5. Record completed artifact onboarding in `PROGRESS.md` and remaining artifact tasks in `PLAN.md`.
|
||||
|
||||
## Do Not
|
||||
|
||||
- Do not run Abaqus.
|
||||
- Do not alter numerical tolerances just to make comparisons pass.
|
||||
@@ -1,32 +0,0 @@
|
||||
---
|
||||
name: fesa-results-schema
|
||||
description: Design or review FESA HDF5 result outputs, step/frame/field/history layout, and reference comparison mapping.
|
||||
---
|
||||
|
||||
# FESA Results Schema
|
||||
|
||||
Use this skill when result storage, HDF5 groups, field/history outputs, or reference comparison paths are involved.
|
||||
|
||||
## Read First
|
||||
|
||||
- `/AGENTS.md`
|
||||
- `/PROGRESS.md`
|
||||
- `/PLAN.md`
|
||||
- `/docs/RESULTS_SCHEMA.md`
|
||||
- `/docs/VERIFICATION_PLAN.md`
|
||||
- `/docs/NUMERICAL_CONVENTIONS.md`
|
||||
- `/docs/MITC4_FORMULATION.md`
|
||||
|
||||
## Workflow
|
||||
|
||||
1. Preserve the step/frame/field/history model.
|
||||
2. Check entity type, component order, coordinate system, precision, units metadata, and sign convention for each field.
|
||||
3. Distinguish full-vector results from reduced-vector solver internals.
|
||||
4. Ensure `U` and `RF` are clear for Phase 1; flag unresolved `S`, `E`, and `SF` decisions.
|
||||
5. When reference comparison is involved, map `references/*_displacements.csv` to HDF5 field output `U` using the documented Abaqus column names.
|
||||
6. Keep HDF5 API usage behind result writer/adapters.
|
||||
|
||||
## Output
|
||||
|
||||
- Provide docs-ready schema deltas or review findings.
|
||||
- Include reference comparison implications and tests needed.
|
||||
@@ -1,41 +0,0 @@
|
||||
---
|
||||
name: fesa-review
|
||||
description: Review FESA changes against repository guardrails, technical dossier docs, TDD expectations, and validation requirements.
|
||||
---
|
||||
|
||||
# FESA Review
|
||||
|
||||
Use this skill for repository-grounded review of docs, `.codex` extensions, phase plans, or implementation patches.
|
||||
|
||||
## Read First
|
||||
|
||||
- `/AGENTS.md`
|
||||
- `/PROGRESS.md`
|
||||
- `/PLAN.md`
|
||||
- `/docs/README.md`
|
||||
- `/docs/HARNESS_ENGINEERING.md`
|
||||
- `/docs/ARCHITECTURE.md`
|
||||
- `/docs/ADR.md`
|
||||
- `/docs/NUMERICAL_CONVENTIONS.md`
|
||||
- `/docs/ABAQUS_INPUT_SUBSET.md`
|
||||
- `/docs/VERIFICATION_PLAN.md`
|
||||
- `/docs/RESULTS_SCHEMA.md`
|
||||
- `/docs/MITC4_FORMULATION.md`
|
||||
- The changed files under review.
|
||||
|
||||
## Checklist
|
||||
|
||||
- Architecture and ADR compliance.
|
||||
- Numerical convention compliance.
|
||||
- Abaqus subset discipline.
|
||||
- Result schema compatibility.
|
||||
- MITC4 formulation traceability.
|
||||
- TDD or reference verification coverage.
|
||||
- Sprint contract compliance when implementation work is under review.
|
||||
- PLAN.md and PROGRESS.md synchronization.
|
||||
|
||||
## Output
|
||||
|
||||
- Lead with findings ordered by severity.
|
||||
- Include concrete file references and the risk behind each finding.
|
||||
- If no material issues exist, say so and list remaining evidence gaps.
|
||||
-18
@@ -1,18 +0,0 @@
|
||||
node_modules/
|
||||
.next/
|
||||
out/
|
||||
next-env.d.ts
|
||||
tsconfig.tsbuildinfo
|
||||
|
||||
# phase execution outputs
|
||||
phases/**/phase*-output.json
|
||||
phases/**/step*-output.json
|
||||
phases/**/step*-last-message.txt
|
||||
|
||||
# C++ build outputs
|
||||
build/
|
||||
*.user
|
||||
*.suo
|
||||
*.vcxproj.user
|
||||
__pycache__/
|
||||
*.pyc
|
||||
@@ -1,76 +0,0 @@
|
||||
# Project: FESA
|
||||
|
||||
## 기술 스택
|
||||
- C++ 17 이상
|
||||
- Math library : Intel OneApi MKL
|
||||
- Parallel library : Intel OneApi TBB
|
||||
- 해석 결과 저장 형식 : hdf5 형식 사용
|
||||
- git 주소 : https://teagit.mimi1011.synology.me/baram2584/FESADev.git
|
||||
|
||||
## 아키텍처 규칙
|
||||
- CRITICAL: 레퍼런스가 되는 예제들과 결과 비교를 통해 솔버의 품질을 항상 유지. 저장 위치는 `references/`이며, 기본 reference artifact는 Abaqus `*.inp` 입력 파일과 `*_displacements.csv` 같은 Abaqus 결과 CSV 파일이다
|
||||
- CRITICAL: 모든 새 작업 세션은 먼저 `PROGRESS.md`와 `PLAN.md`를 읽고 현재 진행 상황, 다음 작업, blocker를 파악할 것
|
||||
- CRITICAL: 구현 또는 phase 계획 전 `docs/README.md`의 문서 우선순위와 Phase 1 hard invariants를 확인할 것
|
||||
- CRITICAL: `docs/ARCHITECTURE.md`와 `docs/ADR.md`의 결정 사항을 우선 준수할 것. 구조 변경이 필요하면 먼저 ADR을 추가/수정할 것
|
||||
- CRITICAL: 수치 규약은 `docs/NUMERICAL_CONVENTIONS.md`를 우선 준수할 것. DOF, 좌표계, 단위, 부호, precision, reaction recovery 규약을 임의로 바꾸지 말 것
|
||||
- 요소, 재료, 하중, 경계조건, 해석 알고리즘은 런타임 다형성 기반으로 확장할 것
|
||||
- `Domain`은 입력 모델 정의를 보존하고 가능한 한 불변으로 취급할 것
|
||||
- 현재 step에서 활성화되는 객체는 `AnalysisModel`로 분리하고, 해석 중 변하는 물리량과 반복 상태는 `AnalysisState`에 저장할 것
|
||||
- 자유도 정의, constrained/free dof mapping, equation numbering, sparse pattern 생성은 `DofManager`가 전담할 것. Node/Element 내부에 equation id를 분산 저장하지 말 것
|
||||
- 해석 알고리즘은 Strategy + Template Method 구조를 따를 것. 선형 정적, 비선형 정적, 동적, 열전달 해석은 공통 실행 흐름을 공유하되 세부 반복/적분 알고리즘은 분리할 것
|
||||
- Abaqus input keyword와 내부 객체 생성은 Factory + Registry 구조로 분리할 것. Phase 1 입력 범위와 미지원 기능은 `docs/ABAQUS_INPUT_SUBSET.md`를 따를 것
|
||||
- MKL, TBB, HDF5 API는 solver core에 직접 노출하지 말고 adapter/wrapper 계층 뒤에서 사용할 것
|
||||
- 결과는 `docs/RESULTS_SCHEMA.md`의 step/frame/field/history 모델로 관리할 것
|
||||
- 대규모 모델을 목표로 sparse matrix, assembly, solver 계층은 성능 확장이 가능하게 설계하고, id/index/equation 번호는 int64 기반으로 둘 것
|
||||
- MITC4 요소 구현은 Phase 1에서 `docs/MITC4_FORMULATION.md`의 baseline formulation과 reference benchmark 통과를 우선하며, reference 검증 전 과도한 최적화를 하지 말 것
|
||||
- Mesh quality 진단은 Phase 1 범위에서 제외하되, singular system 진단은 필수로 구현할 것
|
||||
- Abaqus 실행을 로컬/CI 검증 요구사항으로 두지 말 것. 검증은 `references/`에 저장된 `*.inp`와 `*_displacements.csv` 등 reference artifact와 비교할 것
|
||||
- Reference input이 Phase 1 parser subset 밖의 Abaqus 기능(`S4R`, `Part/Assembly/Instance`, `NLGEOM=YES` 등)을 포함할 수 있다. 이런 파일은 저장 reference로 보존하되, 지원 범위를 조용히 확장하지 말고 `docs/ABAQUS_INPUT_SUBSET.md`와 `docs/VERIFICATION_PLAN.md`에 compatibility note를 남길 것
|
||||
- Phase 1 implementation plan을 만들기 전 `docs/README.md`의 Implementation Readiness Checklist 미결 항목을 명시할 것
|
||||
|
||||
## 작업 상태 관리
|
||||
- `PLAN.md`는 앞으로 해야 할 일, 우선순위, task ownership, open question의 단일 진실 공급원으로 취급할 것
|
||||
- `PROGRESS.md`는 완료된 일, 검증 결과, blocker, 알려진 위험의 단일 진실 공급원으로 취급할 것
|
||||
- 작업을 시작할 때 `PROGRESS.md`에서 최근 완료 내역과 blocker를 확인하고, `PLAN.md`에서 현재 objective와 다음 task를 확인할 것
|
||||
- 의미 있는 문서/코드/계획 변경을 완료하면 `PROGRESS.md`에 날짜, 변경 파일, 검증, 후속 작업을 기록할 것
|
||||
- 앞으로 해야 할 일이 새로 생기거나 우선순위가 바뀌면 `PLAN.md`를 갱신할 것
|
||||
- 완료된 task는 `PLAN.md`에 방치하지 말고 `PROGRESS.md`에 완료 기록을 남긴 뒤 `PLAN.md`에서는 제거하거나 상태를 갱신할 것
|
||||
- 여러 에이전트가 동시에 작업할 수 있으므로, 파일 수정 전 `PLAN.md`의 owner/scope를 확인하고 서로의 작업 범위를 침범하지 말 것
|
||||
- `phases/{phase}/index.json`은 phase 실행 상태의 단일 진실 공급원이지만, phase 밖의 전체 프로젝트 진행 상태는 `PLAN.md`와 `PROGRESS.md`에서 관리할 것
|
||||
|
||||
## Harness Engineering
|
||||
- FESA의 장기 작업은 기본적으로 Planner -> Generator -> Evaluator 구조로 수행할 것
|
||||
- Planner는 구현 전에 sprint contract 또는 `phases/{phase}/stepN.md`를 작성한다. 구현 세부를 과도하게 고정하지 말고, 산출물/검증/금지 범위를 명확히 할 것
|
||||
- Generator는 승인된 contract 하나만 구현한다. 여러 layer를 한 번에 묶어 구현하지 말고, 테스트를 먼저 작성한 뒤 contract의 acceptance criteria를 만족시키는 최소 변경을 수행할 것
|
||||
- Evaluator는 Generator와 분리된 관점으로 검토한다. 자기 작업을 스스로 승인하지 말고, architecture drift, 수치 규약 위반, reference 비교 누락, 테스트 누락, unsupported Abaqus feature의 조용한 확장을 실패로 판정할 것
|
||||
- 각 sprint contract는 최소한 다음 항목을 포함해야 한다: objective, scope, allowed files, explicit non-goals, required reading, tests to write first, reference artifacts, acceptance commands, evaluator checklist, handoff requirements
|
||||
- Sprint 시작 전 contract가 testable하지 않으면 구현하지 말고 contract를 먼저 보강할 것
|
||||
- Sprint 실패 시 Evaluator는 실패 이유와 재현 방법을 feedback artifact로 남기고, Generator는 그 feedback만을 대상으로 다음 반복을 수행할 것
|
||||
- 장기 실행 중 context가 커지면 `PROGRESS.md`, `PLAN.md`, phase step 파일, review feedback을 handoff artifact로 사용해 새 세션이 이어받을 수 있게 할 것
|
||||
- Harness 복잡도는 필요한 만큼만 유지한다. 단순 문서 변경은 단일 agent로 처리할 수 있지만, solver 구현/수치 검증/reference 비교가 포함되면 Planner/Generator/Evaluator 분리를 적용할 것
|
||||
- Harness contract와 평가 기준은 `docs/HARNESS_ENGINEERING.md`를 따를 것
|
||||
|
||||
## Harness Workflow
|
||||
- 먼저 `PROGRESS.md`, `PLAN.md`, `docs/README.md`, `docs/HARNESS_ENGINEERING.md`, `docs/PRD.md`, `docs/ARCHITECTURE.md`, `docs/ADR.md`, `docs/NUMERICAL_CONVENTIONS.md`, `docs/ABAQUS_INPUT_SUBSET.md`, `docs/VERIFICATION_PLAN.md`, `docs/RESULTS_SCHEMA.md`, `docs/MITC4_FORMULATION.md`를 읽고 기획/설계 의도를 파악할 것
|
||||
- 단계별 실행 계획이 필요하면 repo skill `harness-workflow`를 사용해 `phases/` 아래 파일을 설계할 것
|
||||
- 변경사항 리뷰가 필요하면 repo skill `harness-review` 또는 Codex의 `/review`를 사용할 것
|
||||
- `phases/{phase}/index.json`은 phase 진행 상태의 단일 진실 공급원으로 취급할 것
|
||||
- 각 `stepN.md`는 독립된 Codex 세션에서도 실행 가능하도록 자기완결적으로 작성할 것
|
||||
|
||||
## 개발 프로세스
|
||||
- CRITICAL: 새 기능 구현 시 반드시 테스트를 먼저 작성하고, 테스트가 통과하는 구현을 작성할 것 (TDD)
|
||||
- 커밋 메시지는 conventional commits 형식을 따를 것 (`feat:`, `fix:`, `docs:`, `refactor:`)
|
||||
- `scripts/execute.py`는 step 완료 후 코드/메타데이터 커밋을 정리하므로, step 프롬프트 안에서 별도 커밋을 만들 필요는 없음
|
||||
|
||||
## 검증
|
||||
- 기본 검증 스크립트는 `python scripts/validate_workspace.py`
|
||||
- 기준이 되는 Reference 모델들의 해석결과와 비교로 검증 수행
|
||||
- Abaqus는 실행하지 않는 전제이다. 사용자가 `references/` 아래에 정리한 입력/결과 artifact를 기준으로 비교할 것
|
||||
- Reference displacement CSV는 Abaqus export column `Node Label`, `U-U1`, `U-U2`, `U-U3`, `UR-UR1`, `UR-UR2`, `UR-UR3`를 FESA `U` field의 `UX`, `UY`, `UZ`, `RX`, `RY`, `RZ`와 비교하는 기본 형식으로 취급할 것
|
||||
- Reference reaction CSV는 Abaqus export column `Node Label`, `RF-RF1`, `RF-RF2`, `RF-RF3`, `RM-RM1`, `RM-RM2`, `RM-RM3`를 FESA `RF` field의 `RFX`, `RFY`, `RFZ`, `RMX`, `RMY`, `RMZ`와 비교하는 기본 형식으로 취급할 것. `*_reactionforces.csv`와 `*_reactions.csv`는 모두 반력 reference artifact 후보로 취급하되, 자동 pass gate로 쓰기 전에 case별 tolerance와 현재 mismatch 여부를 문서화할 것
|
||||
- Reference 비교는 absolute tolerance와 relative tolerance를 함께 사용할 것
|
||||
|
||||
## 명령어
|
||||
- `python scripts/execute.py <phase-dir>`: Codex 기반 phase 순차 실행
|
||||
- `python scripts/execute.py <phase-dir> --push`: phase 완료 후 브랜치 push
|
||||
- `python scripts/validate_workspace.py`: 저장소 검증
|
||||
@@ -1,79 +0,0 @@
|
||||
cmake_minimum_required(VERSION 3.20)
|
||||
|
||||
project(FESA VERSION 0.1.0 LANGUAGES CXX)
|
||||
|
||||
file(GLOB_RECURSE FESA_CORE_SOURCES CONFIGURE_DEPENDS
|
||||
"${CMAKE_CURRENT_SOURCE_DIR}/src/*.cpp"
|
||||
)
|
||||
|
||||
add_library(fesa_core STATIC ${FESA_CORE_SOURCES})
|
||||
target_include_directories(fesa_core PUBLIC
|
||||
$<BUILD_INTERFACE:${CMAKE_CURRENT_SOURCE_DIR}/include>
|
||||
)
|
||||
target_compile_features(fesa_core PUBLIC cxx_std_17)
|
||||
|
||||
enable_testing()
|
||||
|
||||
add_executable(fesa_tests tests/test_main.cpp)
|
||||
target_link_libraries(fesa_tests PRIVATE fesa_core)
|
||||
target_compile_definitions(fesa_tests PRIVATE FESA_SOURCE_DIR="${CMAKE_CURRENT_SOURCE_DIR}")
|
||||
|
||||
add_executable(fesa_core_module_tests tests/test_core_module_includes.cpp)
|
||||
target_link_libraries(fesa_core_module_tests PRIVATE fesa_core)
|
||||
|
||||
add_executable(fesa_math_module_tests tests/test_math_module_includes.cpp)
|
||||
target_link_libraries(fesa_math_module_tests PRIVATE fesa_core)
|
||||
|
||||
add_executable(fesa_io_module_tests tests/test_io_module_includes.cpp)
|
||||
target_link_libraries(fesa_io_module_tests PRIVATE fesa_core)
|
||||
target_compile_definitions(fesa_io_module_tests PRIVATE FESA_SOURCE_DIR="${CMAKE_CURRENT_SOURCE_DIR}")
|
||||
|
||||
add_executable(fesa_results_module_tests tests/test_results_module_includes.cpp)
|
||||
target_link_libraries(fesa_results_module_tests PRIVATE fesa_core)
|
||||
target_compile_definitions(fesa_results_module_tests PRIVATE FESA_SOURCE_DIR="${CMAKE_CURRENT_SOURCE_DIR}")
|
||||
|
||||
add_executable(fesa_element_module_tests tests/test_element_module_includes.cpp)
|
||||
target_link_libraries(fesa_element_module_tests PRIVATE fesa_core)
|
||||
|
||||
add_executable(fesa_mitc4_stiffness_module_tests tests/test_mitc4_stiffness_module_includes.cpp)
|
||||
target_link_libraries(fesa_mitc4_stiffness_module_tests PRIVATE fesa_core)
|
||||
|
||||
add_executable(fesa_assembly_module_tests tests/test_assembly_module_includes.cpp)
|
||||
target_link_libraries(fesa_assembly_module_tests PRIVATE fesa_core)
|
||||
|
||||
add_executable(fesa_analysis_module_tests tests/test_analysis_module_includes.cpp)
|
||||
target_link_libraries(fesa_analysis_module_tests PRIVATE fesa_core)
|
||||
|
||||
if(MSVC)
|
||||
target_compile_options(fesa_core PRIVATE /W4 /permissive-)
|
||||
target_compile_options(fesa_tests PRIVATE /W4 /permissive-)
|
||||
target_compile_options(fesa_core_module_tests PRIVATE /W4 /permissive-)
|
||||
target_compile_options(fesa_math_module_tests PRIVATE /W4 /permissive-)
|
||||
target_compile_options(fesa_io_module_tests PRIVATE /W4 /permissive-)
|
||||
target_compile_options(fesa_results_module_tests PRIVATE /W4 /permissive-)
|
||||
target_compile_options(fesa_element_module_tests PRIVATE /W4 /permissive-)
|
||||
target_compile_options(fesa_mitc4_stiffness_module_tests PRIVATE /W4 /permissive-)
|
||||
target_compile_options(fesa_assembly_module_tests PRIVATE /W4 /permissive-)
|
||||
target_compile_options(fesa_analysis_module_tests PRIVATE /W4 /permissive-)
|
||||
else()
|
||||
target_compile_options(fesa_core PRIVATE -Wall -Wextra -Wpedantic)
|
||||
target_compile_options(fesa_tests PRIVATE -Wall -Wextra -Wpedantic)
|
||||
target_compile_options(fesa_core_module_tests PRIVATE -Wall -Wextra -Wpedantic)
|
||||
target_compile_options(fesa_math_module_tests PRIVATE -Wall -Wextra -Wpedantic)
|
||||
target_compile_options(fesa_io_module_tests PRIVATE -Wall -Wextra -Wpedantic)
|
||||
target_compile_options(fesa_results_module_tests PRIVATE -Wall -Wextra -Wpedantic)
|
||||
target_compile_options(fesa_element_module_tests PRIVATE -Wall -Wextra -Wpedantic)
|
||||
target_compile_options(fesa_mitc4_stiffness_module_tests PRIVATE -Wall -Wextra -Wpedantic)
|
||||
target_compile_options(fesa_assembly_module_tests PRIVATE -Wall -Wextra -Wpedantic)
|
||||
target_compile_options(fesa_analysis_module_tests PRIVATE -Wall -Wextra -Wpedantic)
|
||||
endif()
|
||||
|
||||
add_test(NAME fesa_tests COMMAND fesa_tests)
|
||||
add_test(NAME fesa_core_module_tests COMMAND fesa_core_module_tests)
|
||||
add_test(NAME fesa_math_module_tests COMMAND fesa_math_module_tests)
|
||||
add_test(NAME fesa_io_module_tests COMMAND fesa_io_module_tests)
|
||||
add_test(NAME fesa_results_module_tests COMMAND fesa_results_module_tests)
|
||||
add_test(NAME fesa_element_module_tests COMMAND fesa_element_module_tests)
|
||||
add_test(NAME fesa_mitc4_stiffness_module_tests COMMAND fesa_mitc4_stiffness_module_tests)
|
||||
add_test(NAME fesa_assembly_module_tests COMMAND fesa_assembly_module_tests)
|
||||
add_test(NAME fesa_analysis_module_tests COMMAND fesa_analysis_module_tests)
|
||||
@@ -1,185 +0,0 @@
|
||||
# PLAN
|
||||
|
||||
## Purpose
|
||||
`PLAN.md` is the shared forward-looking work plan for FESA agents.
|
||||
|
||||
Every new agent session must read this file together with `PROGRESS.md` before planning or editing. Keep this file focused on what should happen next, not on long historical notes.
|
||||
|
||||
## How To Use
|
||||
- Update this file when project priorities, planned phases, task ownership, or open decisions change.
|
||||
- Keep tasks concrete enough that another agent can continue without private context.
|
||||
- Link to the owning design document for each task when possible.
|
||||
- Do not mark work complete here. Move completion notes to `PROGRESS.md`.
|
||||
- If an item becomes obsolete, move it to `PROGRESS.md` with a short reason instead of silently deleting it.
|
||||
|
||||
## Current Objective
|
||||
The Phase 1 structure-alignment refactor in `phases/1-structure-alignment-refactor` is complete. P1A-09 independently accepted the final module alignment: `include/fesa/fesa.hpp` is now an include-only facade, production symbols are separated under module ownership, validation passes, and R-014 is closed. The current Phase 1 readiness focus is product-level reference verification: `quad_02_reactionforces.csv` has been onboarded and wired into an RF comparator, but R-010 remains open because the current node-wise FESA RF comparison against Abaqus does not pass. R-013 also remains open for the PRD target of three stored Phase 1 reference cases.
|
||||
|
||||
## Required Reading For New Agents
|
||||
1. `AGENTS.md`
|
||||
2. `PROGRESS.md`
|
||||
3. `PLAN.md`
|
||||
4. `docs/README.md`
|
||||
5. `docs/HARNESS_ENGINEERING.md`
|
||||
6. `docs/PRD.md`
|
||||
7. `docs/ARCHITECTURE.md`
|
||||
8. `docs/ADR.md`
|
||||
9. `docs/NUMERICAL_CONVENTIONS.md`
|
||||
10. `docs/ABAQUS_INPUT_SUBSET.md`
|
||||
11. `docs/VERIFICATION_PLAN.md`
|
||||
12. `docs/RESULTS_SCHEMA.md`
|
||||
13. `docs/MITC4_FORMULATION.md`
|
||||
14. `phases/index.json`
|
||||
15. `phases/1-structure-alignment-refactor/index.json`
|
||||
16. `phases/1-linear-static-mitc4-rebaseline/index.json`
|
||||
17. `phases/1-linear-static-mitc4/index.json` for historical context only
|
||||
|
||||
## Phase Files
|
||||
- Completed phase directory: `phases/1-structure-alignment-refactor`
|
||||
- Historical execution command: `python scripts/execute.py 1-structure-alignment-refactor`
|
||||
- Step numbering is zero-based. `step0.md` is complete and wrote `phases/1-structure-alignment-refactor/step0-architecture-map.md`; `step1.md` is complete and created module scaffold headers, source directories, CMake source discovery, and umbrella compatibility smoke coverage; `step2.md` is complete and extracted Core/Util domain, diagnostics, DofManager ownership, AnalysisModel/AnalysisState, and Phase 1 Boundary/Load/Property model ownership; `step3.md` is complete and extracted Math primitives, sparse pattern data, dense matrix support, and solver adapter boundary; `step4.md` is complete and extracted the Abaqus parser into IO; `step5.md` is complete and extracted Results/reference comparison code; `step6.md` is complete and extracted MITC4 geometry/strain helpers; `step7.md` is complete and extracted MITC4 material/stiffness helpers; `step8.md` is complete and extracted Assembly and Analysis workflow; `step9.md` is complete and wrote `phases/1-structure-alignment-refactor/step9-evaluator-report.md`.
|
||||
- Completed phase directory: `phases/1-linear-static-mitc4-rebaseline`
|
||||
- Historical execution command: `python scripts/execute.py 1-linear-static-mitc4-rebaseline`
|
||||
- Step numbering is zero-based. `step0.md` is complete and recorded in `phases/1-linear-static-mitc4-rebaseline/step0-audit.md`; `step1.md` is complete and created the `quad_02_phase1.inp` normalized reference path; `step2.md` is complete and revalidated core harness guardrails; `step3.md` is complete and revalidated the Phase 1 parser/domain subset; `step4.md` is complete and strengthened validation/singular diagnostics; `step5.md` is complete and revalidated the DofManager/reaction foundation; `step6.md` is complete and revalidated the minimum result model plus displacement CSV comparator; `step7.md` is complete and revalidated MITC4 natural coordinates, tying points, center directors, and integration bases; `step8.md` is complete and revalidated degenerated-continuum displacement, direct covariant strain rows, and MITC shear tying rows; `step9.md` is complete and revalidated plane-stress material, convected-to-local transform, and `2 x 2 x 2` material integration scaffolding; `step10.md` is complete and revalidated MITC4 stiffness, internal force, six-DOF transform, and drilling stabilization; `step11.md` is complete and added MITC4 membrane, bending, shear, twist, drilling-sensitivity, and thin-cantilever locking-sensitivity tests; `step12.md` is complete and revalidated full-space assembly, reduced projection, deterministic sparse-pattern scaffold, solver adapter injection, and full-vector internal/reaction force state; `step13.md` is complete and revalidated active AnalysisModel construction plus input-to-AnalysisState-to-U/RF result workflow; `step14.md` is complete and added the first stored Abaqus displacement regression for `quad_02_phase1`; `step15.md` is complete and recorded the independent evaluator closeout in `phases/1-linear-static-mitc4-rebaseline/step15-evaluator-report.md`.
|
||||
- Every step file contains a sprint contract with objective, required reading, scope, allowed files, explicit non-goals, tests to write first, reference artifacts, acceptance command, evaluator checklist, and handoff requirements.
|
||||
- Historical phase directory: `phases/1-linear-static-mitc4`
|
||||
- Historical phase status: blocked/superseded. Do not resume the old P1-15/P1-16 path unless the user explicitly requests recovery of that exact phase.
|
||||
|
||||
## Phase 1 Readiness Tasks
|
||||
| ID | Status | Owner | Task | Source |
|
||||
|---|---|---|---|---|
|
||||
| R-010 | pending | verification + solver agent | Triage the onboarded `references/quad_02_reactionforces.csv` node-wise RF mismatch. The artifact schema and comparator are defined, but `quad_02_phase1` currently does not pass Abaqus RF/RM comparison; do not relax tolerances to close this. | `docs/VERIFICATION_PLAN.md`, `docs/RESULTS_SCHEMA.md`, `references/quad_02_notes.md` |
|
||||
| R-013 | pending | user + verification agent | Add enough additional small Abaqus S4 reference cases for the PRD target of three stored Phase 1 references: one single-element case, one simple multi-element plate/shell case, and one curved shell benchmark. | `docs/PRD.md`, `docs/VERIFICATION_PLAN.md` |
|
||||
|
||||
## Phase 1 Structure Alignment Refactor
|
||||
This phase is an architecture-preserving refactor. It must not change Phase 1 solver behavior, MITC4 formulation, Abaqus parser subset, numerical conventions, result schema, or reference tolerances.
|
||||
|
||||
| ID | Status | Owner | Objective | Depends On | Acceptance Focus |
|
||||
|---|---|---|---|---|---|
|
||||
| P1A-00 | completed | planner/evaluator | Audit `fesa.hpp` architecture drift and create a symbol-to-module migration map. | P1R-15 | Complete migration map and validation baseline |
|
||||
| P1A-01 | completed | generator | Create module directory scaffold, CMake source boundaries, and umbrella facade policy. | P1A-00 | Module include smoke tests and build stability |
|
||||
| P1A-02 | completed | generator | Extract Core/Util domain, diagnostics, aliases, DOF mapping, `AnalysisModel`, `DofManager`, and Phase 1 Boundary/Load/Property model ownership. | P1A-01 | Core has no dependency on higher layers; Boundary/Load/Property types are no longer hidden in the umbrella header; DOF tests unchanged |
|
||||
| P1A-03 | completed | generator | Extract Math and solver adapter boundaries. | P1A-02 | Linear solver interface remains adapter-ready; int64 paths unchanged |
|
||||
| P1A-04 | completed | generator | Extract Abaqus parser into IO. | P1A-02 | Parser subset and unsupported-feature diagnostics unchanged |
|
||||
| P1A-05 | completed | generator | Extract Results model, writer boundary, CSV loader, and reference comparator. | P1A-02, P1A-04 | `U`/`RF` schema and `quad_02_phase1` regression unchanged |
|
||||
| P1A-06 | completed | generator | Extract MITC4 geometry, director, strain, and tying helpers into Element. | P1A-03 | Geometry/strain tests and formulation signs unchanged |
|
||||
| P1A-07 | completed | generator | Extract MITC4 material, integration, stiffness, drilling, and internal-force helpers. | P1A-06 | Patch, drilling, stiffness, and locking-sensitivity tests unchanged |
|
||||
| P1A-08 | completed | generator | Extract Assembly and Analysis workflow. | P1A-02, P1A-03, P1A-05, P1A-07 | Full-vector RF, solver injection, and end-to-end reference regression unchanged |
|
||||
| P1A-09 | completed | evaluator | Independently evaluate final architecture alignment. | P1A-08 | `src/`/module ownership matches `ARCHITECTURE.md`; umbrella header is facade only |
|
||||
|
||||
## Phase 1 Definition Of Done
|
||||
Phase 1 is complete only when FESA can run a documented linear static MITC4 workflow from input to verified results without requiring Abaqus execution.
|
||||
|
||||
Required capabilities:
|
||||
- Parse the Phase 1 Abaqus input subset into `Domain`: `*Node`, `*Element`, `*Nset`, `*Elset`, `*Material`, `*Elastic`, `*Shell Section`, `*Boundary`, `*Cload`, `*Step`, `*Static`, `*End Step`.
|
||||
- Reject unsupported features with diagnostics, including `S4R`, `Part/Assembly/Instance`, `*Include`, pressure loads, nonzero prescribed displacement, and `NLGEOM=YES`.
|
||||
- Build `AnalysisModel` for one active linear static step.
|
||||
- Manage six shell DOFs per node with `DofManager`: `UX`, `UY`, `UZ`, `RX`, `RY`, `RZ`.
|
||||
- Apply fixed boundary conditions by constrained DOF elimination.
|
||||
- Assemble a sparse global linear system with int64 ids/indices/equation numbers and `double` values.
|
||||
- Solve the reduced free-DOF system through a solver interface that can later bind to MKL.
|
||||
- Reconstruct full vectors and recover `RF = K_full * U_full - F_full`.
|
||||
- Write minimum Phase 1 results: model ids/connectivity plus `U` and `RF` in the documented step/frame/field layout.
|
||||
- Compare `U` against stored `references/*_displacements.csv` artifacts.
|
||||
- Provide singular system diagnostics for missing constraints, missing properties/materials, invalid references, untouched free DOFs, and solver singularity.
|
||||
- Pass unit, integration, reference, and negative tests required by `docs/VERIFICATION_PLAN.md`.
|
||||
|
||||
Out of scope:
|
||||
- Abaqus `S4R` execution semantics, hourglass control, pressure loads, RBE2/RBE3, nonzero prescribed displacements, geometric/material nonlinearity, dynamics, heat transfer, composite sections, and mesh quality diagnostics.
|
||||
|
||||
## Phase 1 Execution Gates
|
||||
Each gate should be satisfied before moving to the next implementation band unless the user explicitly accepts a documented deferral.
|
||||
|
||||
| Gate | Status | Requirement | Evidence |
|
||||
|---|---|---|---|
|
||||
| G0 - Planning readiness | partial | Readiness task R-011 is resolved by `quad_02_phase1.inp`; R-010 and R-013 remain open. | Updated docs, PLAN.md, PROGRESS.md |
|
||||
| G1 - Build and validation | satisfied | Build system, test framework, and `scripts/validate_workspace.py` run real checks. | Validation command output |
|
||||
| G2 - Parser and domain | satisfied | Parser subset revalidated in step 3; validation and singular diagnostics revalidated in step 4. | Parser acceptance/rejection tests, validation negative tests, and validation output |
|
||||
| G3 - DOF/math/results infrastructure | satisfied | Core aliases, DOF mapping, validation harness, model diagnostic context, DofManager, sparse-connectivity inputs, full-vector reaction formula, result model metadata, displacement CSV comparator, full-space assembly, reduced projection, sparse-pattern scaffold, and solver adapter boundary were revalidated in steps 2, 5, 6, and 12. | P1R-02, P1R-05, P1R-06, and P1R-12 validation output |
|
||||
| G4 - MITC4 element readiness | satisfied | MITC4 formulation was rewritten from local papers; Steps 7 through 11 rebuilt geometry/director/local-basis scaffolding, displacement interpolation, direct covariant strain rows, MITC shear tying rows, plane-stress material, convected-to-local transform, `2 x 2 x 2` material integration scaffolding, stiffness/internal force, six-DOF transform, drilling stabilization, and patch/locking-sensitivity tests. | P1R-07 through P1R-11 validation output |
|
||||
| G5 - End-to-end solver | satisfied-with-gap | Linear static input-to-result workflow is revalidated through step 13, `quad_02_phase1` stored displacement regression passes in step 14, and the rebaseline evaluator closeout passed in step 15. The broader PRD target of three stored references remains open in R-013. | P1R-13 through P1R-15 validation output |
|
||||
|
||||
## Phase 1 Implementation Milestones
|
||||
All milestones are intended to become one or more self-contained sprint contracts or `phases/{phase}/stepN.md` files. Each sprint must follow `docs/HARNESS_ENGINEERING.md` and be evaluated independently.
|
||||
|
||||
| ID | Status | Owner | Objective | Depends On | Acceptance Focus |
|
||||
|---|---|---|---|---|---|
|
||||
| P1R-03 | completed | parser generator | Revalidate Phase 1 parser and immutable Domain subset. | none | Supported keywords accepted; unsupported features rejected |
|
||||
| P1R-04 | completed | validation generator | Rebuild validation and singular diagnostic coverage. | P1R-03 | Missing-reference and singular-prone negative tests |
|
||||
| P1R-05 | completed | DOF generator | Rebuild six-DOF DofManager, constrained/free mapping, equation numbering, and full-vector reconstruction. | none | DOF mapping and reaction foundation tests |
|
||||
| P1R-06 | completed | results generator | Rebuild minimum results model and displacement CSV comparator. | none | U/RF schema tests and CSV comparator tests |
|
||||
| P1R-07 | completed | MITC4 generator | Implement MITC4 geometry, node order, tying points, directors, and local bases. | none | Shape/basis/diagnostic tests |
|
||||
| P1R-08 | completed | MITC4 generator | Implement degenerated-continuum displacement, covariant strain rows, and MITC shear tying. | P1R-07 | Finite-difference and tying interpolation tests |
|
||||
| P1R-09 | completed | MITC4 generator | Implement material matrix, transform, and `2 x 2 x 2` integration scaffolding. | P1R-08 | Material/integration tests |
|
||||
| P1R-10 | completed | MITC4 generator | Assemble MITC4 stiffness/internal force with six-DOF transform and drilling stabilization. | P1R-09, P1R-05 | Symmetry, rigid body, drilling sensitivity tests |
|
||||
| P1R-11 | completed | verification generator | Add MITC4 patch, locking-sensitivity, and benchmark tests. | P1R-10 | Membrane/bending/shear/twist/locking tests |
|
||||
| P1R-12 | completed | assembly generator | Rebuild assembly, solver adapter boundary, constrained solve, and full-vector RF recovery. | P1R-05, P1R-10 | Assembly and full-vector reaction tests |
|
||||
| P1R-13 | completed | analysis generator | Rebuild linear static workflow from input to U/RF result fields. | P1R-03, P1R-04, P1R-06, P1R-12 | End-to-end linear static tests |
|
||||
| P1R-14 | completed | reference generator | Run stored reference displacement regression using accepted Phase 1-compatible S4 cases. | P1R-13 | At least one automated CSV displacement regression |
|
||||
| P1R-15 | completed | evaluator | Independent Phase 1 evaluator closeout. | P1R-14 | Pass/fail report, synchronized PLAN/PROGRESS |
|
||||
|
||||
## Phase 1 Sprint Contract Rules
|
||||
Every implementation milestone above must be decomposed into one or more contracts before code changes begin.
|
||||
|
||||
Contract requirements:
|
||||
- One contract should usually touch one layer or one module.
|
||||
- Each contract must list allowed files and explicit non-goals.
|
||||
- Each contract must list tests to write before implementation.
|
||||
- Each contract must state whether it uses `references/*.inp` or `references/*_displacements.csv`.
|
||||
- Each contract must include `python scripts/validate_workspace.py` plus any focused test command available after P1-01.
|
||||
- Each contract must include an evaluator checklist tied to the milestone acceptance focus.
|
||||
- Generator work must not begin if the contract relies on unresolved MITC4 formulas or undocumented reference tolerances.
|
||||
|
||||
## Phase 1 Verification Strategy
|
||||
Verification should grow with the implementation bands:
|
||||
|
||||
1. Unit tests: core types, DOF enum, diagnostics, parser tokens, label handling.
|
||||
2. Parser tests: supported keywords, generated sets, duplicate/missing references, unsupported keyword diagnostics.
|
||||
3. DOF/math tests: constrained/free partition, equation numbering, sparse pattern, reduced solve, full reconstruction.
|
||||
4. Results tests: HDF5 or writer boundary schema for `U` and `RF`; component labels and frame metadata.
|
||||
5. Reference comparator tests: CSV header validation, node matching, tolerance pass/fail behavior.
|
||||
6. Element tests: MITC4 shape functions, stiffness symmetry, rigid body behavior, drilling stiffness sensitivity.
|
||||
7. Assembly/analysis tests: small known systems, full-vector reaction recovery, singular negative cases.
|
||||
8. Stored-reference tests: at least one Phase 1-compatible displacement CSV comparison first, then three accepted stored cases for Phase 1 completion.
|
||||
|
||||
## Phase 1 Reference Plan
|
||||
Current reference state:
|
||||
- `references/quad_01.inp` and `references/quad_01_displacements.csv` are accepted stored artifacts.
|
||||
- `quad_01.inp` contains `S4R`, `Part/Assembly/Instance`, `*Density`, and `NLGEOM=YES`; it is not a Phase 1 parser acceptance case as-is.
|
||||
- `references/quad_02.inp` and `references/quad_02_displacements.csv` have been added by the user as an S4 reference pair.
|
||||
- `references/quad_02_reactionforces.csv` has been added by the user as the paired Abaqus RF/RM result artifact for `quad_02`.
|
||||
- `quad_02.inp` uses `TYPE=S4`, but also includes `Part/Assembly/Instance`; this is a compatibility decision point, not automatic parser scope expansion.
|
||||
- `references/quad_02_phase1.inp` is the accepted normalized Phase 1-compatible derivative input for the `quad_02` S4 reference pair.
|
||||
|
||||
Required reference additions or decisions:
|
||||
- Explain or fix the current `quad_02_phase1` node-wise RF mismatch against `quad_02_reactionforces.csv`. Current observed comparison with `abs_tol = 1.0e-6`, `rel_tol = 1.0e-5`, `reference_scale = 1.0` has max absolute error about `612.751347` and max relative error about `0.494032`.
|
||||
- Add more small cases until Phase 1 can pass one single-element case, one simple multi-element plate/shell case, and one curved shell benchmark.
|
||||
|
||||
## Phase 1 Risk Controls
|
||||
- Do not implement MITC4 element stiffness until the formulation gate in `docs/MITC4_FORMULATION.md` is closed.
|
||||
- Do not treat the previous P1-01 through P1-14 implementation as authoritative after the MITC4 formulation reset.
|
||||
- Do not use `quad_01.inp` to justify `S4R`, `Part/Assembly/Instance`, or `NLGEOM=YES` support.
|
||||
- Do not use `quad_02.inp` to silently justify `Part/Assembly/Instance` support without a parser contract.
|
||||
- Do not compute reactions from reduced vectors only.
|
||||
- Do not expose MKL, TBB, or HDF5 APIs directly in solver core.
|
||||
- Do not narrow ids, equation numbers, or sparse indices below int64.
|
||||
- Do not allow `Node` or `Element` to own global equation ids.
|
||||
- Do not treat a passing build as Phase 1 validation without parser, DOF, reference, and singular negative tests.
|
||||
|
||||
## Current Non-Goals
|
||||
- Do not implement solver code outside the matching rebaseline sprint contract.
|
||||
- Do not require Abaqus execution locally or in CI.
|
||||
- Do not add mesh quality diagnostics in Phase 1.
|
||||
- Do not support Abaqus `S4R` in Phase 1.
|
||||
- Do not silently expand the Abaqus input subset beyond `docs/ABAQUS_INPUT_SUBSET.md`.
|
||||
|
||||
## Codex Extension Follow-up Tasks
|
||||
| ID | Status | Owner | Task | Source |
|
||||
|---|---|---|---|---|
|
||||
| C-002 | pending | user + Codex | Confirm hook behavior in the actual Codex runtime on native Windows after `features.codex_hooks` is enabled. | `.codex/hooks.json`, `.codex/hooks/*.py` |
|
||||
| C-003 | pending | user + Codex | Decide whether any `.codex/skills/*` should be mirrored under `.agents/skills/` for environments that only scan the default skill folders. | `.codex/config.toml`, `.codex/skills/` |
|
||||
| C-004 | pending | user + Codex | Confirm that the `fesa-commands` repo plugin appears in the active Codex plugin/command surface after marketplace registration. | `plugins/fesa-commands/`, `.agents/plugins/marketplace.json` |
|
||||
|
||||
## Open Questions
|
||||
- Which Abaqus version will generate future reference artifacts when it is not recorded in the input or notes?
|
||||
- Is the current `quad_02` RF mismatch due to MITC4 formulation details, Abaqus S4 reaction recovery conventions, normalized input differences, or another solver issue?
|
||||
-1280
File diff suppressed because it is too large
Load Diff
@@ -1,50 +0,0 @@
|
||||
# FESA
|
||||
|
||||
FESA is a C++17 finite element structural analysis solver project. The first milestone is a reference-verified linear static MITC4 shell solver with an Abaqus-compatible input subset, HDF5-oriented results, and explicit numerical conventions.
|
||||
|
||||
## Current Phase 1 Direction
|
||||
- MITC4 shell element baseline formulation.
|
||||
- Linear elastic material.
|
||||
- Nodal loads and fixed boundary conditions.
|
||||
- Abaqus input subset with `S4` mapped to FESA `MITC4`.
|
||||
- `S4R` deferred.
|
||||
- 6 shell DOFs per node: `UX`, `UY`, `UZ`, `RX`, `RY`, `RZ`.
|
||||
- Small artificial drilling stiffness.
|
||||
- Constrained DOF elimination.
|
||||
- Full-vector reaction recovery.
|
||||
- `double` precision and int64 ids/indices/equation numbering.
|
||||
- Stored-reference comparison from `references/*.inp` and `references/*_displacements.csv` without requiring local Abaqus execution.
|
||||
|
||||
## Reference Artifacts
|
||||
Stored Abaqus examples live under [references](references).
|
||||
|
||||
The initial accepted artifact pair is:
|
||||
- [quad_01.inp](references/quad_01.inp)
|
||||
- [quad_01_displacements.csv](references/quad_01_displacements.csv)
|
||||
|
||||
Reference CSV displacement columns use Abaqus labels `Node Label`, `U-U1`, `U-U2`, `U-U3`, `UR-UR1`, `UR-UR2`, `UR-UR3`, mapped to FESA `UX`, `UY`, `UZ`, `RX`, `RY`, `RZ`.
|
||||
|
||||
## Documentation Entry Point
|
||||
Start with [docs/README.md](docs/README.md).
|
||||
|
||||
The core project documents are:
|
||||
- [AGENTS.md](AGENTS.md)
|
||||
- [docs/HARNESS_ENGINEERING.md](docs/HARNESS_ENGINEERING.md)
|
||||
- [docs/PRD.md](docs/PRD.md)
|
||||
- [docs/ARCHITECTURE.md](docs/ARCHITECTURE.md)
|
||||
- [docs/ADR.md](docs/ADR.md)
|
||||
- [docs/NUMERICAL_CONVENTIONS.md](docs/NUMERICAL_CONVENTIONS.md)
|
||||
- [docs/ABAQUS_INPUT_SUBSET.md](docs/ABAQUS_INPUT_SUBSET.md)
|
||||
- [docs/VERIFICATION_PLAN.md](docs/VERIFICATION_PLAN.md)
|
||||
- [docs/RESULTS_SCHEMA.md](docs/RESULTS_SCHEMA.md)
|
||||
- [docs/MITC4_FORMULATION.md](docs/MITC4_FORMULATION.md)
|
||||
|
||||
## Validation
|
||||
The default repository validation command is:
|
||||
|
||||
```bash
|
||||
python scripts/validate_workspace.py
|
||||
```
|
||||
|
||||
With the Phase 1 CMake harness in place, this command configures CMake, builds the C++ tests, and runs CTest.
|
||||
|
||||
@@ -1,253 +0,0 @@
|
||||
# Abaqus Input Subset
|
||||
|
||||
## Purpose
|
||||
This document defines the Abaqus `.inp` subset supported by FESA Phase 1.
|
||||
|
||||
FESA aims for strict, explicit compatibility with a small subset rather than partial silent interpretation of the full Abaqus language.
|
||||
|
||||
## Source Basis
|
||||
- Abaqus input files use keyword lines, data lines, and comment lines; keyword and parameter names are case-insensitive, comma-separated, and may use continuation lines: https://abaqus-docs.mit.edu/2017/English/SIMACAEMODRefMap/simamod-c-inputsyntax.htm
|
||||
- Abaqus sets are a central reference mechanism for nodes and elements: https://abaqus-docs.mit.edu/2017/English/SIMACAEMODRefMap/simamod-c-inputsyntax.htm
|
||||
- Abaqus shell library documents S4 as a 4-node general-purpose shell and S4R as a reduced-integration shell with hourglass control: https://abaqus-docs.mit.edu/2017/English/SIMACAEELMRefMap/simaelm-r-shellgeneral.htm
|
||||
- Abaqus `*BOUNDARY` direct data lines use node or node set, first DOF, optional last DOF, and optional magnitude: https://abaqus-docs.mit.edu/2017/English/SIMACAEKEYRefMap/simakey-r-boundary.htm
|
||||
- Abaqus `*CLOAD` data lines use node or node set, DOF, and reference load magnitude: https://abaqus-docs.mit.edu/2017/English/SIMACAEKEYRefMap/simakey-r-cload.htm
|
||||
|
||||
## Phase 1 Supported Keywords
|
||||
| Keyword | Status | FESA Object | Notes |
|
||||
|---|---|---|---|
|
||||
| `*Node` | Supported | `Node` | 3D coordinates only |
|
||||
| `*Element` | Supported | `Element` | `TYPE=S4` maps to `MITC4` |
|
||||
| `*Nset` | Supported | `NodeSet` | Explicit list and `GENERATE` should be supported |
|
||||
| `*Elset` | Supported | `ElementSet` | Explicit list and `GENERATE` should be supported |
|
||||
| `*Material` | Supported | `Material` | `NAME` required |
|
||||
| `*Elastic` | Supported | `LinearElasticMaterial` | Isotropic `E, nu` only |
|
||||
| `*Shell Section` | Supported | `ShellProperty` | Homogeneous shell section only |
|
||||
| `*Boundary` | Supported | `FixBoundaryCondition` | Direct zero-valued constraints |
|
||||
| `*Cload` | Supported | `NodalLoad` | Concentrated forces/moments |
|
||||
| `*Step` | Supported | `StepDefinition` | Static linear Phase 1 step |
|
||||
| `*Static` | Accepted inside `*Step` | `LinearStaticAnalysis` | Optional for Phase 1 |
|
||||
| `*End Step` | Supported | Step delimiter | Required for explicit step closure |
|
||||
|
||||
Unsupported keywords must produce a clear diagnostic unless explicitly listed as ignorable metadata.
|
||||
|
||||
## General Parser Rules
|
||||
FESA parser rules:
|
||||
- Keyword names and parameter names are case-insensitive.
|
||||
- Keyword lines start with `*`.
|
||||
- Comment lines start with `**` and are ignored.
|
||||
- Data fields are comma-separated.
|
||||
- Empty trailing fields may be ignored.
|
||||
- Numeric data may use decimal or scientific notation.
|
||||
- `D` exponents should be accepted and normalized as `E` exponents.
|
||||
- Keyword continuation with a trailing comma should be supported for keyword lines.
|
||||
- Data continuation should be supported only where this document explicitly allows it.
|
||||
- Abbreviated Abaqus keywords are not supported in Phase 1. Require exact keyword names after case normalization.
|
||||
- Supported keywords may use only the parameters and flags listed in this document. Unknown parameters and flags are errors, even when the keyword itself is supported.
|
||||
- Fixed-width data rows for `*Node`, `*Element`, `*Elastic`, `*Shell Section`, `*Boundary`, and `*Cload` must not contain extra non-empty fields beyond the supported form.
|
||||
- Include files through `INPUT=` are not supported in Phase 1.
|
||||
- Part/Assembly/Instance syntax is not supported in Phase 1 unless added by ADR.
|
||||
- Normalized derivative inputs such as `references/quad_02_phase1.inp` may be used for Phase 1 parser and solver tests when the original stored Abaqus file contains unsupported Abaqus/CAE scaffolding.
|
||||
|
||||
## Stored Reference Inputs vs Supported Subset
|
||||
Files under `references/` are allowed to preserve the exact Abaqus input used to generate reference results, even when the file contains features outside the current Phase 1 parser subset.
|
||||
|
||||
Rules:
|
||||
- A stored reference input is not automatically a supported FESA input.
|
||||
- Unsupported reference features must be documented as compatibility notes.
|
||||
- Parser implementation must still reject unsupported features until this document and ADRs explicitly add support.
|
||||
- Test harnesses may use normalized or reduced derivative inputs for Phase 1 parser tests, but must keep the original Abaqus reference artifact unchanged.
|
||||
|
||||
Current stored reference notes:
|
||||
- `references/quad_01.inp` was generated by Abaqus/CAE Learning Edition 2024.
|
||||
- It uses `TYPE=S4R`, `Part`, `Assembly`, `Instance`, `*Density`, and `NLGEOM=YES`, all of which are outside the current Phase 1 parser/solver subset.
|
||||
- Its paired `references/quad_01_displacements.csv` is still valid as a stored displacement reference artifact for future compatibility and regression work.
|
||||
- `references/quad_02.inp` uses `TYPE=S4`, so it targets the Phase 1 MITC4 element formulation, and its paired `references/quad_02_displacements.csv` has the accepted displacement CSV shape.
|
||||
- `references/quad_02_reactionforces.csv` is a paired Abaqus RF/RM result export for the stored `quad_02` case. It is a reference result artifact and does not change parser scope.
|
||||
- `quad_02.inp` still uses Abaqus/CAE `Part`, `Assembly`, `Instance`, and `*Density`; it is therefore a stored S4 reference artifact and compatibility decision point, not automatic parser acceptance as-is.
|
||||
- `references/quad_02_phase1.inp` is the normalized Phase 1-compatible derivative input for `quad_02`. It preserves node ids, element ids/connectivity, S4 element type, elastic material, shell thickness, fixed boundary nodes, load node, and concentrated load while removing `Part/Assembly/Instance`, `*Density`, restart/output request keywords, and unsupported step metadata.
|
||||
|
||||
## Labels and Names
|
||||
Rules:
|
||||
- Set and material labels are stored case-insensitively by default.
|
||||
- Preserve the original spelling for diagnostics and result metadata.
|
||||
- Labels must start with a letter unless quoted.
|
||||
- Quoted labels may be accepted, but Phase 1 should warn if quoting is required for disambiguation.
|
||||
- Labels beginning and ending with double underscores are reserved and should be rejected.
|
||||
|
||||
## `*Node`
|
||||
Supported form:
|
||||
|
||||
```text
|
||||
*Node
|
||||
node_id, x, y, z
|
||||
```
|
||||
|
||||
Rules:
|
||||
- `node_id` is signed 64-bit integer.
|
||||
- Coordinates are `double`.
|
||||
- 2D node definitions are not supported for MITC4.
|
||||
- Duplicate node ids are an error.
|
||||
- Node ids need not be contiguous.
|
||||
|
||||
## `*Element`
|
||||
Supported form:
|
||||
|
||||
```text
|
||||
*Element, type=S4, elset=EALL
|
||||
element_id, n1, n2, n3, n4
|
||||
```
|
||||
|
||||
Rules:
|
||||
- `TYPE=S4` maps directly to FESA `MITC4`.
|
||||
- `TYPE=S4R` is not supported in Phase 1. It is reserved for future support.
|
||||
- Element and node ids are signed 64-bit integers.
|
||||
- Four node connectivity entries are required.
|
||||
- Duplicate element ids are an error.
|
||||
- Missing nodes are an error.
|
||||
- If `ELSET` is given, the element is added to that element set.
|
||||
- Element node ordering follows Abaqus shell ordering and determines the positive normal by right-hand rule.
|
||||
|
||||
## `*Nset` and `*Elset`
|
||||
Supported explicit form:
|
||||
|
||||
```text
|
||||
*Nset, nset=FIXED
|
||||
1, 2, 3, 4
|
||||
```
|
||||
|
||||
Supported generated form:
|
||||
|
||||
```text
|
||||
*Elset, elset=EALL, generate
|
||||
1, 100, 1
|
||||
```
|
||||
|
||||
Rules:
|
||||
- Explicit lists may span repeated data lines.
|
||||
- `GENERATE` means `start, end, increment`.
|
||||
- The increment must be positive.
|
||||
- Set references to other sets are not required in Phase 1.
|
||||
- `UNSORTED` is not required in Phase 1.
|
||||
- Duplicates should be deduplicated while preserving a deterministic order for diagnostics.
|
||||
- Missing referenced ids should be reported when the set is consumed by a property, load, or boundary condition.
|
||||
|
||||
## `*Material` and `*Elastic`
|
||||
Supported form:
|
||||
|
||||
```text
|
||||
*Material, name=STEEL
|
||||
*Elastic
|
||||
E, nu
|
||||
```
|
||||
|
||||
Rules:
|
||||
- Only isotropic linear elasticity is supported in Phase 1.
|
||||
- Temperature dependence, field-variable dependence, orthotropic elasticity, plasticity, density, and damping are unsupported.
|
||||
- `E` must be positive.
|
||||
- `nu` must be in a physically meaningful isotropic range. For Phase 1, reject `nu <= -1.0` or `nu >= 0.5`.
|
||||
|
||||
## `*Shell Section`
|
||||
Supported form:
|
||||
|
||||
```text
|
||||
*Shell Section, elset=EALL, material=STEEL
|
||||
thickness
|
||||
```
|
||||
|
||||
Rules:
|
||||
- `ELSET` and `MATERIAL` are required.
|
||||
- Homogeneous single-layer shell sections only.
|
||||
- Thickness is `double` and must be positive.
|
||||
- Offsets, composite layups, section integration controls, orientations, temperature-dependent sections, and transverse shear stiffness overrides are unsupported in Phase 1.
|
||||
- Thermal-stress coupling is a future feature and must not be inferred from Phase 1 section data.
|
||||
|
||||
## `*Boundary`
|
||||
Supported direct form:
|
||||
|
||||
```text
|
||||
*Boundary
|
||||
node_or_nset, first_dof, last_dof, magnitude
|
||||
```
|
||||
|
||||
Rules:
|
||||
- `node_or_nset` may be a node id or node set label.
|
||||
- DOFs are `1..6`.
|
||||
- If `last_dof` is omitted, constrain only `first_dof`.
|
||||
- Phase 1 supports zero-valued constraints. Omitted magnitude means zero.
|
||||
- Nonzero prescribed displacement/rotation is not a Phase 1 requirement.
|
||||
- Type-format boundary labels such as `PINNED`, `XSYMM`, or `ENCASTRE` are not supported unless later documented.
|
||||
- Constrained DOFs are eliminated by `DofManager`.
|
||||
|
||||
## `*Cload`
|
||||
Supported form:
|
||||
|
||||
```text
|
||||
*Cload
|
||||
node_or_nset, dof, magnitude
|
||||
```
|
||||
|
||||
Rules:
|
||||
- `node_or_nset` may be a node id or node set label.
|
||||
- DOFs are `1..6`.
|
||||
- Translational DOFs define concentrated forces.
|
||||
- Rotational DOFs define concentrated moments.
|
||||
- `FOLLOWER`, `AMPLITUDE`, `OP=NEW`, file-based loads, buoyancy/drag/inertia loads, and cyclic symmetry loads are unsupported in Phase 1.
|
||||
|
||||
## `*Step`, `*Static`, and `*End Step`
|
||||
Supported form:
|
||||
|
||||
```text
|
||||
*Step, name=Step-1
|
||||
*Static
|
||||
*Cload
|
||||
...
|
||||
*Boundary
|
||||
...
|
||||
*End Step
|
||||
```
|
||||
|
||||
Rules:
|
||||
- Phase 1 supports linear static steps.
|
||||
- `NLGEOM` is ignored only if explicitly false or absent. `NLGEOM=YES` must be rejected until nonlinear analysis is implemented.
|
||||
- Multiple steps may be parsed into `Domain`, but Phase 1 execution may initially support one active static step if documented in implementation steps.
|
||||
- Step activation should feed `AnalysisModel`.
|
||||
|
||||
## Diagnostics
|
||||
Required parser diagnostics:
|
||||
- Unknown keyword.
|
||||
- Unsupported keyword parameter.
|
||||
- Missing required parameter.
|
||||
- Duplicate node, element, material, property, or set definition.
|
||||
- Missing node in element connectivity.
|
||||
- Missing set used by shell section, boundary, or load.
|
||||
- Unsupported element type such as `S4R`.
|
||||
- Unsupported material or shell section mode.
|
||||
- Invalid DOF number.
|
||||
- Invalid generated set range.
|
||||
|
||||
Diagnostic messages should include file path, line number, keyword, and offending token.
|
||||
|
||||
## Minimum Parser Acceptance
|
||||
Before Phase 1 parser work is considered ready for solver integration:
|
||||
- Parse `*Node`, `*Element`, `*Nset`, `*Elset`, `*Material`, `*Elastic`, `*Shell Section`, `*Boundary`, `*Cload`, `*Step`, `*Static`, and `*End Step` smoke cases.
|
||||
- Preserve original labels for diagnostics while resolving labels case-insensitively.
|
||||
- Accept explicit and `GENERATE` node/element sets.
|
||||
- Reject `TYPE=S4R` with an unsupported element diagnostic.
|
||||
- Reject `NLGEOM=YES`.
|
||||
- Reject unsupported Part/Assembly/Instance and Include syntax.
|
||||
- Resolve shell section material and element set references.
|
||||
- Resolve boundary/load node set references.
|
||||
- Produce line-numbered diagnostics for malformed numeric fields and invalid DOF ids.
|
||||
|
||||
## Explicit Non-Goals
|
||||
- Abaqus `Part`, `Assembly`, `Instance`, and instance-qualified labels.
|
||||
- `*Include`.
|
||||
- `S4R`, `S4R5`, `S8R`, triangular shells, solid elements, beam elements.
|
||||
- Pressure loads.
|
||||
- RBE2/RBE3.
|
||||
- Nonzero prescribed displacements.
|
||||
- Amplitudes.
|
||||
- Local coordinate transforms.
|
||||
- Composite shell sections.
|
||||
- Thermal-stress input.
|
||||
- Mesh quality diagnostics.
|
||||
-166
@@ -1,166 +0,0 @@
|
||||
# Architecture Decision Records
|
||||
|
||||
## 철학
|
||||
솔버의 높은 성능과 정확도, Abaqus와의 높은 호환성
|
||||
|
||||
---
|
||||
|
||||
### ADR-001: Runtime Polymorphism 기반 Solver Core
|
||||
**결정**: 요소, 재료, 하중, 경계조건, 해석 알고리즘은 base interface와 runtime polymorphism 기반으로 확장한다.
|
||||
|
||||
**이유**: FESA는 MITC4 Shell 요소에서 시작하지만 RBE2/RBE3, 압력하중, 비선형 정적해석, 동적해석, 열전달, 1D/3D 요소로 확장될 예정이다. 초기에는 정확도와 테스트 가능성이 가장 중요하므로, 각 물리 객체를 독립적으로 테스트하고 교체할 수 있는 구조가 필요하다.
|
||||
|
||||
**트레이드오프**: virtual dispatch 비용과 객체 분산에 따른 캐시 효율 저하가 발생할 수 있다. 대규모 모델 성능이 필요한 영역은 `Assembler`, element kernel, sparse solver 계층에서 batch 처리 또는 타입별 최적화를 추가한다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-002: Immutable Domain + Mutable AnalysisState
|
||||
**결정**: 입력 모델 정의는 `Domain`에 보존하고 가능한 한 불변으로 취급한다. 해석 중 변하는 displacement, velocity, acceleration, temperature, force, residual, iteration 정보는 `AnalysisState`에 분리한다.
|
||||
|
||||
**이유**: 선형 정적해석, 기하비선형 정적해석, 동적해석, 열전달 및 thermal-stress coupling은 서로 다른 상태 변수를 필요로 한다. 모델 정의와 해석 상태를 분리하면 restart, 결과 저장, reference 비교, 테스트가 쉬워진다.
|
||||
|
||||
**트레이드오프**: `Domain`, `AnalysisModel`, `AnalysisState` 사이의 참조/id 관리가 필요하다. 단순한 단일 해석 코드보다 초기 구조가 복잡하지만, 다중 step 및 비선형/동적 확장성을 얻는다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-003: Step/Frame/Field/History 결과 모델
|
||||
**결정**: 해석 결과는 `ResultStep`, `ResultFrame`, `FieldOutput`, `HistoryOutput` 구조로 저장한다.
|
||||
|
||||
**이유**: 정적해석, 비선형 increment, 동적 time frame, history output을 같은 결과 모델로 다루기 위해 Abaqus와 유사한 step/frame/history 개념이 필요하다. HDF5 저장 구조와 reference 결과 비교도 이 구조를 기준으로 설계한다.
|
||||
|
||||
**트레이드오프**: Phase 1 선형 정적해석만 놓고 보면 결과 구조가 다소 무겁다. 그러나 Phase 2 이후의 비선형 반복, Phase 3 동해석, reaction/history 검증을 위해 초기에 최소 구조를 잡는 편이 낫다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-004: Strategy + Template Method 기반 Analysis 실행
|
||||
**결정**: 해석 알고리즘은 `Analysis` strategy로 분리하고, 공통 실행 흐름은 template method로 관리한다.
|
||||
|
||||
**이유**: 선형 정적해석, Newton-Raphson 비선형 정적해석, HHT 동적해석, 열전달 해석은 조립, 경계조건 적용, solver 호출, 상태 갱신, 결과 저장이라는 공통 흐름을 공유한다. 공통 흐름을 유지하면서 해석별 반복 구조만 바꾸는 방식이 중복을 줄인다.
|
||||
|
||||
**트레이드오프**: 공통 template이 지나치게 커지면 해석별 특수성이 숨겨질 수 있다. 따라서 `Analysis`는 전체 흐름을 조율하고, assembly, solver, convergence, time integration은 별도 strategy로 분리한다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-005: Factory + Registry 기반 Abaqus Input 객체 생성
|
||||
**결정**: Abaqus input keyword와 내부 객체 생성을 factory/registry 계층으로 분리한다. Phase 1 입력 범위에는 `*Node`, `*Element`, `*Nset`, `*Elset`, `*Material`, `*Elastic`, `*Shell Section`, `*Boundary`, `*Cload`, `*Step`을 포함한다.
|
||||
|
||||
**이유**: Abaqus input format 호환성을 유지하면서 요소/재료/하중/경계조건 타입을 계속 추가해야 한다. parser 본체가 모든 타입을 직접 생성하면 확장할수록 변경 비용과 회귀 위험이 커진다.
|
||||
|
||||
**트레이드오프**: registry 초기화와 타입 매핑 코드가 추가된다. 대신 새 keyword나 element type을 추가할 때 parser core의 변경을 최소화할 수 있다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-006: External Library Adapter Boundary
|
||||
**결정**: MKL, TBB, HDF5는 solver core에 직접 노출하지 않고 adapter/wrapper 계층 뒤에서 사용한다.
|
||||
|
||||
**이유**: FESA의 핵심 도메인 모델과 해석 알고리즘이 특정 외부 라이브러리 API에 강하게 결합되면 테스트와 교체가 어려워진다. `SparseMatrix`, `Vector`, `LinearSolver`, `ParallelFor`, `ResultsWriter` 같은 경계를 통해 외부 의존성을 제한한다.
|
||||
|
||||
**트레이드오프**: wrapper 계층 구현 비용이 추가된다. 성능이 민감한 부분에서는 adapter가 불필요한 복사를 만들지 않도록 API를 신중히 설계해야 한다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-007: DofManager가 자유도와 방정식 번호를 전담
|
||||
**결정**: node와 element 내부에 equation id를 분산 저장하지 않고, `DofManager`가 자유도 정의, constrained/free dof mapping, equation numbering, sparse pattern 생성을 전담한다.
|
||||
|
||||
**이유**: 대규모 모델, 경계조건, RBE2/RBE3, 비선형 재조립, thermal-stress coupling에서는 자유도 관리가 solver 품질과 성능에 직접 영향을 준다. 자유도 관리를 별도 객체로 분리하면 경계조건 적용과 행렬 조립이 명확해진다.
|
||||
|
||||
**트레이드오프**: element 계산 시 node id에서 equation id로 변환하는 조회 비용이 생긴다. 이 비용은 assembly precompute 또는 element connectivity cache로 줄인다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-008: Numerical Convention Baseline
|
||||
**결정**: Phase 1 shell node는 6자유도(`UX`, `UY`, `UZ`, `RX`, `RY`, `RZ`)를 사용하고, 기본 precision은 `double`, id/index/equation numbering은 int64 기반으로 설계한다. 단위계는 Abaqus처럼 강제하지 않고, 결과 부호 규약은 Abaqus를 따른다.
|
||||
|
||||
**이유**: MITC4 Shell, Abaqus input/result 호환성, 대규모 sparse system, 향후 RBE/비선형/동역학 확장을 모두 고려하면 6자유도와 64-bit indexing이 가장 안정적인 공통 기반이다. 단위계를 강제하지 않으면 Abaqus reference와 같은 self-consistent unit workflow를 유지할 수 있다.
|
||||
|
||||
**트레이드오프**: int64 index는 작은 모델에서 메모리 사용량이 증가할 수 있다. 단위계를 강제하지 않으므로 reference case와 사용자 입력 문서에 unit note를 성실히 남겨야 한다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-009: Essential Boundary Condition and Reaction Recovery
|
||||
**결정**: Essential boundary condition은 constrained DOF 제거 방식으로 적용한다. Reaction force/moment는 reduced system 결과만 사용하지 않고 full vector 기준 `R_full = K_full * U_full - F_full`로 계산한다.
|
||||
|
||||
**이유**: constrained/free DOF mapping이 명확하고 대규모 sparse solve에 적합하며, reaction force는 reference 검증과 하중 평형 검증에서 핵심 출력이다.
|
||||
|
||||
**트레이드오프**: full stiffness/load/displacement 정보를 reaction recovery 시점까지 보존하거나 재구성해야 하므로 메모리와 데이터 흐름 관리가 필요하다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-010: Drilling DOF Stabilization
|
||||
**결정**: Phase 1 MITC4는 drilling 자유도를 유지하고, 작은 인공 drilling 강성을 적용한다. 기본값은 `docs/MITC4_FORMULATION.md`에서 구현 전 확정한다.
|
||||
|
||||
**이유**: 6자유도 shell node 구조와 Abaqus-style output convention을 유지하면서 singular matrix 위험을 줄이기 위해 drilling stabilization이 필요하다.
|
||||
|
||||
**트레이드오프**: 인공 강성이 물리 응답에 작은 영향을 줄 수 있다. 따라서 값은 parameterized 되어야 하고, reference benchmark에서 민감도를 확인해야 한다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-011: S4 Mapping and S4R Deferral
|
||||
**결정**: Phase 1 Abaqus `*Element, TYPE=S4`는 FESA `MITC4`로 매핑한다. `S4R`은 Phase 1에서 지원하지 않고 추후 별도 ADR과 formulation 업데이트 후 지원한다.
|
||||
|
||||
**이유**: Abaqus S4는 MITC4와 비교 가능한 fully integrated 4-node shell reference로 사용할 수 있다. 반면 S4R은 reduced integration과 hourglass control 결정을 요구하므로 Phase 1의 명확한 baseline formulation 목표와 분리한다.
|
||||
|
||||
**트레이드오프**: 사용자가 가진 S4R reference model은 Phase 1에서 바로 사용할 수 없다. 대신 S4 baseline과 MITC4 benchmark 통과를 먼저 안정화한다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-012: Singular Diagnostics Required, Mesh Quality Deferred
|
||||
**결정**: Phase 1은 singular system 진단을 필수로 제공한다. Aspect ratio, warpage, skew 등 mesh quality 진단은 Phase 1 범위에서 제외한다.
|
||||
|
||||
**이유**: 초기 solver 사용성과 디버깅에는 singular system 원인 추적이 더 직접적으로 중요하다. Mesh quality 진단은 기준과 threshold를 잘못 잡으면 formulation 검증보다 많은 논쟁을 만들 수 있으므로 baseline solver 이후로 미룬다.
|
||||
|
||||
**트레이드오프**: 품질이 낮은 mesh가 조용히 나쁜 결과를 만들 수 있다. Phase 1에서는 reference benchmark와 singular diagnostics로 우선 통제한다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-013: Technical Dossier Documents as Implementation Contracts
|
||||
**결정**: 구현 전 다음 문서를 계약 문서로 유지한다: `NUMERICAL_CONVENTIONS.md`, `ABAQUS_INPUT_SUBSET.md`, `VERIFICATION_PLAN.md`, `RESULTS_SCHEMA.md`, `MITC4_FORMULATION.md`.
|
||||
|
||||
**이유**: FEM solver는 DOF, 좌표계, 부호, parser subset, result schema, benchmark 기준이 조금만 흔들려도 구현이 불안정해진다. 구현자가 따를 수 있는 technical dossier를 먼저 고정한다.
|
||||
|
||||
**트레이드오프**: 문서 유지 비용이 증가한다. 하지만 문서와 구현이 어긋날 때는 ADR 또는 해당 dossier를 먼저 갱신하여 설계 drift를 관리한다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-014: Documentation Index and Implementation Readiness Gate
|
||||
**결정**: `docs/README.md`를 프로젝트 문서 index와 문서 우선순위의 entry point로 둔다. Phase 1 implementation plan을 작성하기 전 `docs/README.md`의 Implementation Readiness Checklist 미결 항목을 명시한다.
|
||||
|
||||
**이유**: FESA는 수치 규약, Abaqus subset, MITC4 formulation, HDF5 schema, reference verification이 강하게 결합되어 있다. 구현 전에 어떤 문서가 어느 결정을 소유하는지 분명히 하지 않으면 작은 구현 선택이 전체 solver convention을 흔들 수 있다.
|
||||
|
||||
**트레이드오프**: 계획 단계에서 문서 확인 비용이 늘어난다. 대신 이후 Codex 세션과 multi-agent 작업이 같은 규칙을 공유하고, 미결 결정을 숨기지 않는다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-015: Reference-Artifact-Only Validation Baseline
|
||||
**결정**: Phase 1 검증은 저장된 `references/` artifact와 비교한다. Abaqus 실행은 로컬 개발, CI, 기본 validation의 요구사항으로 두지 않는다.
|
||||
|
||||
**이유**: 사용자는 Abaqus input과 해석 결과를 `references/` 폴더로 제공하며, 현재 개발 환경에서 Abaqus 실행은 사용할 수 없다. 저장 artifact 기반 검증이 재현성과 자동화에 더 적합하다.
|
||||
|
||||
**트레이드오프**: reference artifact 생성과 갱신은 수동 절차에 의존한다. 따라서 reference manifest, Abaqus version, unit note, tolerance, 비교 대상 output path를 명확히 기록해야 한다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-016: References Folder and CSV Displacement Artifact Contract
|
||||
**결정**: 저장 reference의 공식 루트는 `references/`이다. 초기 자동 변위 검증은 Abaqus 해석 입력 `*.inp`와 Abaqus에서 export한 `*_displacements.csv` 파일 쌍을 사용한다. CSV column은 `Node Label`, `U-U1`, `U-U2`, `U-U3`, `UR-UR1`, `UR-UR2`, `UR-UR3`를 기본 형식으로 한다.
|
||||
|
||||
**이유**: 사용자가 첫 reference case로 `references/quad_01.inp`와 `references/quad_01_displacements.csv`를 제공했다. CSV는 사람이 검토하기 쉽고, 초기 parser/solver 검증 harness에서 HDF5 writer가 완성되기 전에도 `U` field 비교를 자동화하기 좋다.
|
||||
|
||||
**트레이드오프**: CSV는 HDF5보다 metadata 표현력이 약하다. 따라서 Abaqus version, unit note, unsupported keyword note, tolerance는 `references/README.md`, case manifest, 또는 추후 metadata 파일에 보강해야 한다. `quad_01.inp`처럼 `S4R`, `Part/Assembly/Instance`, `NLGEOM=YES`를 포함하는 reference input은 저장 reference로 보존하되 Phase 1 parser 지원 범위를 자동으로 확장하지 않는다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-017: CMake and CTest Phase 1 Build Harness
|
||||
**결정**: Phase 1의 기본 빌드 및 테스트 harness는 CMake와 CTest를 사용한다. `scripts/validate_workspace.py`는 `CMakeLists.txt`를 발견하면 configure, build, CTest 순서로 검증을 수행한다.
|
||||
|
||||
**이유**: FESA의 핵심 구현 언어는 C++17 이상이고, Phase 1은 외부 solver library 없이도 core/parser/DOF/result/comparator/element tests를 반복 실행해야 한다. CMake/CTest는 Visual Studio/MSVC와 다른 C++ toolchain 모두에서 표준적인 최소 공통 경로를 제공한다.
|
||||
|
||||
**트레이드오프**: oneAPI MKL, TBB, HDF5 adapter가 추가될 때 CMake dependency discovery가 더 복잡해진다. 대신 Phase 1에서는 외부 API를 core에 노출하지 않고 deterministic test adapter로 검증을 시작할 수 있다.
|
||||
|
||||
---
|
||||
|
||||
### ADR-018: Phase 1 MITC4 Formulation Reset
|
||||
**결정**: Phase 1 MITC4 baseline은 `docs/MITC4_FORMULATION.md`의 논문 기반 formulation contract를 따른다. 핵심 결정은 degenerated-continuum kinematics, convected covariant strain components, FESA/Abaqus S4 node-order에 맞춘 MITC transverse shear tying interpolation, `2 x 2 x 2` Gauss integration, six-DOF local/global rotation transformation, `drilling_stiffness_scale = 1.0e-3` applied to the minimum positive physical local stiffness diagonal, 그리고 mandatory result output을 `U`와 `RF`로 제한하는 것이다.
|
||||
|
||||
**이유**: 이전 Phase 1 baseline은 첫 구현을 빠르게 안정화하기 위한 단순화였지만, 사용자가 추가한 MITC4 논문들과 S4 reference (`quad_02`)를 기준으로 보면 formulation detail이 부족하다. MITC4 element implementation은 convected tensor shear tying, director/local-axis policy, drilling stabilization, and integration rules가 명확해야 reference 검증 전 수치 drift를 줄일 수 있다.
|
||||
|
||||
**트레이드오프**: 기존 P1-01 through P1-14 구현은 새 formulation contract를 기준으로 재평가 또는 재작성되어야 한다. `quad_02.inp`는 `TYPE=S4`이지만 `Part/Assembly/Instance` 구조를 포함하므로, element formulation 재구현과 parser compatibility 확장은 별도 sprint로 분리해야 한다. `S4R`, reduced integration, and hourglass control remain out of scope.
|
||||
@@ -1,290 +0,0 @@
|
||||
# 아키텍처
|
||||
|
||||
## 목표
|
||||
FESA는 MITC4 Shell 요소 기반 구조해석에서 시작해 비선형 정적해석, 비선형 동적해석, 열전달 및 thermal-stress coupling, 1D/3D 요소까지 확장하는 유한요소 솔버이다.
|
||||
|
||||
초기 구현은 정확도와 테스트 가능성을 우선한다. 단, 대규모 모델을 목표로 하므로 자유도 관리, 희소 행렬 조립, 선형 솔버, 병렬 실행 계층은 초기부터 성능 확장이 가능하도록 분리한다.
|
||||
|
||||
문서 우선순위와 구현 전 준비 기준은 `docs/README.md`를 따른다.
|
||||
|
||||
## 설계 원칙
|
||||
- Domain 객체는 입력 모델의 의미를 보존하고 가능한 한 불변에 가깝게 유지한다.
|
||||
- 해석 중 변하는 물리량과 반복 상태는 AnalysisState에 명시적으로 분리한다.
|
||||
- 요소, 재료, 하중, 경계조건, 해석 알고리즘은 런타임 다형성 기반으로 확장한다.
|
||||
- MITC4 구현은 Phase 1에서 정확도와 테스트 가능성을 우선하며, assembly와 solver 계층은 대규모 모델 최적화가 가능하도록 경계를 둔다.
|
||||
- 결과는 step/frame/field/history 개념으로 저장하여 정적, 비선형, 동적, 열전달 해석을 같은 결과 모델로 다룬다.
|
||||
- 외부 라이브러리(MKL, TBB, HDF5)는 adapter 계층 뒤에 둔다.
|
||||
- Abaqus input 호환성은 파서와 factory/registry 계층에서 관리한다. Phase 1의 입력 범위에는 `*Node`, `*Element`, `*Nset`, `*Elset`, `*Material`, `*Elastic`, `*Shell Section`, `*Boundary`, `*Cload`, `*Step`을 포함한다.
|
||||
- 수치 규약은 `docs/NUMERICAL_CONVENTIONS.md`를 따른다. Phase 1 shell node는 6자유도이고, 단위계는 강제하지 않으며, 결과 부호는 Abaqus 규약을 따른다.
|
||||
- 경계조건은 constrained DOF 제거 방식으로 적용하고, reaction은 full vector 기준 `K_full * U_full - F_full`로 계산한다.
|
||||
- 기본 실수 precision은 `double`이고, 대규모 모델을 위해 id/index/equation numbering은 int64 기반으로 설계한다.
|
||||
- Mesh quality 진단은 Phase 1 범위에서 제외한다. 대신 singular system 진단은 필수로 제공한다.
|
||||
|
||||
## 디렉토리 구조
|
||||
```
|
||||
src/
|
||||
├── Analysis/ # Static, nonlinear static, dynamic, heat transfer analysis
|
||||
├── Assembly/ # 전역 행렬/벡터 조립, sparse pattern 생성
|
||||
├── Boundary/ # Fix, RBE2, RBE3 등 경계조건
|
||||
├── Core/ # Domain, AnalysisModel, AnalysisState, DofManager
|
||||
├── Element/ # Node, Element, MITC4 등 요소 구현
|
||||
├── IO/ # Abaqus input parser, HDF5 results writer
|
||||
├── Load/ # NodalLoad, PressureLoad, BodyForce 등 하중
|
||||
├── Math/ # Vector, Matrix, SparseMatrix, MKL adapter
|
||||
├── Material/ # LinearElastic 등 재료 모델
|
||||
├── Property/ # ShellProperty, 1D/2D/3D property
|
||||
├── Results/ # Step, Frame, FieldOutput, HistoryOutput
|
||||
└── Util/ # 공통 유틸리티, 로깅, 검증 보조 함수
|
||||
```
|
||||
|
||||
## 핵심 클래스 구조
|
||||
```
|
||||
Domain
|
||||
├── Node
|
||||
├── Element
|
||||
├── Material
|
||||
├── Property
|
||||
├── NodeSet
|
||||
├── ElementSet
|
||||
├── BoundaryCondition
|
||||
├── Load
|
||||
└── StepDefinition
|
||||
|
||||
AnalysisModel
|
||||
├── active elements
|
||||
├── active loads
|
||||
├── active boundary conditions
|
||||
├── active properties/materials
|
||||
└── equation system view
|
||||
|
||||
AnalysisState
|
||||
├── displacement U
|
||||
├── velocity V
|
||||
├── acceleration A
|
||||
├── temperature T
|
||||
├── external force Fext
|
||||
├── internal force Fint
|
||||
├── residual R
|
||||
├── current time / increment / iteration
|
||||
└── element state / integration point state
|
||||
|
||||
DofManager
|
||||
├── node dof definitions
|
||||
├── constrained/free dof mapping
|
||||
├── equation numbering
|
||||
├── sparse matrix pattern ownership
|
||||
└── full/reduced vector reconstruction
|
||||
|
||||
Analysis
|
||||
├── LinearStaticAnalysis
|
||||
├── NonlinearStaticAnalysis
|
||||
├── DynamicAnalysis
|
||||
├── FrequencyAnalysis
|
||||
└── HeatTransferAnalysis
|
||||
|
||||
Element
|
||||
├── 1DElement
|
||||
│ ├── Truss
|
||||
│ └── Beam
|
||||
├── 2DElement
|
||||
│ ├── MITC3
|
||||
│ └── MITC4
|
||||
└── 3DElement
|
||||
├── Hexahedral
|
||||
├── Tetrahedral
|
||||
├── Wedge
|
||||
└── Pyramid
|
||||
|
||||
BoundaryCondition
|
||||
├── Fix
|
||||
├── RBE2
|
||||
└── RBE3
|
||||
|
||||
Load
|
||||
├── NodalLoad
|
||||
├── PressureLoad
|
||||
└── BodyForce
|
||||
|
||||
Results
|
||||
├── ResultStep
|
||||
├── ResultFrame
|
||||
├── FieldOutput
|
||||
└── HistoryOutput
|
||||
|
||||
InputParser
|
||||
ResultsWriter
|
||||
Assembler
|
||||
LinearSolver
|
||||
Vector
|
||||
Matrix
|
||||
SparseMatrix
|
||||
```
|
||||
|
||||
## 디자인 패턴
|
||||
|
||||
### Strategy Pattern
|
||||
해석 알고리즘과 수치 알고리즘을 교체 가능하게 구성한다.
|
||||
|
||||
적용 대상:
|
||||
- `Analysis`: `LinearStaticAnalysis`, `NonlinearStaticAnalysis`, `DynamicAnalysis`, `HeatTransferAnalysis`
|
||||
- `LinearSolver`: `MKLPardisoSolver`, 향후 iterative solver
|
||||
- `TimeIntegrator`: `HHTIntegrator`, 향후 Newmark 등
|
||||
- `ConvergenceCriteria`: residual norm, displacement norm, energy norm
|
||||
|
||||
### Template Method Pattern
|
||||
해석 실행의 큰 흐름은 `Analysis::run()`에서 고정하고, 세부 단계는 해석 종류별로 재정의한다.
|
||||
|
||||
기본 흐름:
|
||||
```
|
||||
initialize
|
||||
buildAnalysisModel
|
||||
buildDofMap
|
||||
buildSparsePattern
|
||||
assemble
|
||||
applyBoundaryConditions
|
||||
solve
|
||||
updateState
|
||||
writeResults
|
||||
```
|
||||
|
||||
비선형 정적해석은 위 흐름을 Newton-Raphson 반복 루프 안에서 사용하고, 동적해석은 time step/frame 루프 안에서 사용한다.
|
||||
|
||||
### Factory + Registry Pattern
|
||||
Abaqus input keyword와 내부 객체 생성을 분리한다.
|
||||
|
||||
예:
|
||||
- `*Element, type=S4` -> `MITC4ElementFactory`
|
||||
- `*Material`, `*Elastic` -> `LinearElasticMaterialFactory`
|
||||
- `*Boundary` -> `FixBoundaryFactory`
|
||||
- `*Cload` -> `NodalLoadFactory`
|
||||
- `*Nset`, `*Elset` -> set registry
|
||||
|
||||
요소, 재료, 하중, 경계조건 타입 추가 시 parser 본체의 변경을 최소화한다.
|
||||
|
||||
### Adapter Pattern
|
||||
MKL, TBB, HDF5 API는 solver core에 직접 노출하지 않는다.
|
||||
|
||||
적용 대상:
|
||||
- `SparseMatrix`, `Vector`, `Matrix`
|
||||
- `LinearSolver`
|
||||
- `ParallelFor`
|
||||
- `ResultsWriter`
|
||||
|
||||
외부 라이브러리 교체 또는 테스트 double 사용이 가능하도록 adapter 계층에서 의존성을 제한한다.
|
||||
|
||||
### Runtime Polymorphism
|
||||
요소, 재료, 하중, 경계조건은 base interface를 통해 다룬다. Phase 1에서는 명확성과 테스트 가능성을 우선하고, 대규모 모델 성능 최적화가 필요할 경우 assembly 내부에서 타입별 batch 처리 또는 kernel 분리를 추가한다.
|
||||
|
||||
## 상태 관리
|
||||
|
||||
### Domain
|
||||
`Domain`은 입력 파일에서 만들어진 전체 모델 정의를 소유한다. 파싱 이후에는 가능한 한 불변으로 취급한다.
|
||||
|
||||
포함 대상:
|
||||
- nodes, elements
|
||||
- materials, properties
|
||||
- node sets, element sets
|
||||
- loads, boundary conditions
|
||||
- analysis step definitions
|
||||
|
||||
### AnalysisModel
|
||||
`AnalysisModel`은 현재 step에서 활성화되는 해석 객체들의 실행 view이다. `Domain`을 복사하지 않고 참조 또는 id 기반 view로 구성한다.
|
||||
|
||||
포함 대상:
|
||||
- active elements
|
||||
- active loads
|
||||
- active boundary conditions
|
||||
- active property/material references
|
||||
- current equation system view
|
||||
|
||||
### DofManager
|
||||
`DofManager`는 자유도와 방정식 번호를 전담한다. Node 또는 Element 내부에 equation id를 분산 저장하지 않는다.
|
||||
|
||||
책임:
|
||||
- node별 활성 자유도 정의
|
||||
- constrained/free dof mapping
|
||||
- equation numbering
|
||||
- sparse matrix pattern 생성에 필요한 connectivity 제공
|
||||
- 경계조건 적용 전후의 dof view 관리
|
||||
|
||||
### AnalysisState
|
||||
`AnalysisState`는 해석 중 변하는 물리량과 반복 상태를 소유한다.
|
||||
|
||||
포함 대상:
|
||||
- displacement, velocity, acceleration
|
||||
- temperature
|
||||
- external force, internal force, residual
|
||||
- current time, increment, Newton iteration
|
||||
- element state, integration point state
|
||||
|
||||
Phase 1에서는 displacement 중심으로 최소 구현하되, 기하비선형과 thermal-stress coupling을 위해 element/internal state 확장 지점을 유지한다.
|
||||
|
||||
### Results State
|
||||
결과는 `ResultStep` -> `ResultFrame` -> `FieldOutput`/`HistoryOutput` 구조로 관리한다.
|
||||
|
||||
- `ResultStep`: 해석 step 단위 결과
|
||||
- `ResultFrame`: 정적해석의 load increment 또는 동적해석의 time frame
|
||||
- `FieldOutput`: node/element field 결과
|
||||
- `HistoryOutput`: 특정 node, element, set, reaction, energy 등의 이력 결과
|
||||
|
||||
## 데이터 흐름
|
||||
```
|
||||
Abaqus input file
|
||||
-> InputParser
|
||||
-> Domain 생성
|
||||
-> StepDefinition 루프
|
||||
-> AnalysisModel 생성
|
||||
-> DofManager로 자유도/방정식 번호 생성
|
||||
-> sparse pattern 생성
|
||||
-> Analysis 실행
|
||||
-> Assembler로 전역 행렬/벡터 조립
|
||||
-> BoundaryCondition 적용
|
||||
-> LinearSolver 또는 nonlinear/time integration loop
|
||||
-> AnalysisState 갱신
|
||||
-> ResultsWriter로 step/frame/history 저장
|
||||
-> 다음 step 진행
|
||||
```
|
||||
|
||||
## Phase 1 실행 경계
|
||||
Phase 1 solver path는 다음 경계를 구현 단위로 삼는다.
|
||||
|
||||
1. `InputParser`는 `docs/ABAQUS_INPUT_SUBSET.md`의 subset만 받아 `Domain`을 만든다.
|
||||
2. `DomainValidator` 또는 동등한 검증 계층은 missing reference, unsupported feature, singular-prone model 상태를 조기에 진단한다.
|
||||
3. `AnalysisModelBuilder`는 현재 step의 active element/load/boundary/property/material view를 구성한다.
|
||||
4. `DofManager`는 6 DOF node model, constrained/free partition, equation numbering, sparse pattern input, full/reduced vector reconstruction을 소유한다.
|
||||
5. `Assembler`는 full reaction recovery에 필요한 full-space stiffness/load 정보를 보존하거나 재구성 가능한 형태로 유지한다.
|
||||
6. `LinearStaticAnalysis`는 reduced free-DOF system을 풀고 `AnalysisState`에 full `U`, `Fext`, `Fint`, `R`을 갱신한다.
|
||||
7. `ResultsWriter`는 `docs/RESULTS_SCHEMA.md`의 최소 Phase 1 outputs를 쓴다.
|
||||
8. `ReferenceComparator` 또는 테스트 harness는 `docs/VERIFICATION_PLAN.md`의 reference artifact와 비교한다. 초기 reference 입력은 `references/*.inp`와 Abaqus-exported `references/*_displacements.csv`이다.
|
||||
|
||||
## Phase 1 구현 범위
|
||||
- MITC4 Shell 요소
|
||||
- 선형 탄성 재료
|
||||
- 절점하중
|
||||
- 고정 경계조건
|
||||
- Abaqus input subset: `*Node`, `*Element`, `*Nset`, `*Elset`, `*Material`, `*Elastic`, `*Shell Section`, `*Boundary`, `*Cload`, `*Step`
|
||||
- `S4`를 `MITC4`로 매핑하고 `S4R`은 추후 지원
|
||||
- 6자유도 shell node와 drilling 자유도 인공 강성
|
||||
- constrained DOF 제거 방식
|
||||
- full vector 기반 reaction recovery
|
||||
- 선형 정적 해석
|
||||
- step/frame 기반 결과 저장의 최소 구조
|
||||
- double precision과 int64 indexing
|
||||
- singular system 진단
|
||||
- `references/`의 Abaqus `.inp`와 `*_displacements.csv` 기반 reference 모델 결과 비교 테스트
|
||||
|
||||
## 성능 확장 방향
|
||||
- 행렬 조립은 element 단위 병렬화를 고려해 설계한다.
|
||||
- 전역 행렬은 대규모 모델을 위해 sparse matrix를 기본으로 한다.
|
||||
- MKL 기반 direct solver를 우선 지원하되, solver interface는 iterative solver를 추가할 수 있게 둔다.
|
||||
- 대규모 sparse solve를 위해 MKL `pardiso_64`를 사용할 수 있도록 64-bit sparse index 경계를 유지한다.
|
||||
- TBB 병렬화는 요소 stiffness 계산, element force 계산, assembly precompute 등 독립 작업부터 적용한다.
|
||||
- 정확도 검증이 끝나기 전에는 MITC4 element formulation을 과도하게 최적화하지 않는다.
|
||||
|
||||
## 상세 설계 문서
|
||||
- `docs/README.md`: 문서 index, 우선순위, Phase 1 hard invariants, implementation readiness checklist
|
||||
- `docs/NUMERICAL_CONVENTIONS.md`: DOF, 좌표계, 단위, 부호, precision, reaction recovery, singular diagnostics
|
||||
- `docs/ABAQUS_INPUT_SUBSET.md`: Phase 1 Abaqus input keyword subset과 unsupported feature
|
||||
- `docs/VERIFICATION_PLAN.md`: `references/` 폴더 구조, benchmark matrix, CSV reference result 형식, tolerance 정책
|
||||
- `docs/RESULTS_SCHEMA.md`: HDF5 step/frame/field/history schema
|
||||
- `docs/MITC4_FORMULATION.md`: MITC4 baseline formulation 계약과 open decisions
|
||||
@@ -1,169 +0,0 @@
|
||||
# Harness Engineering
|
||||
|
||||
## Purpose
|
||||
This document defines how FESA uses long-running agent harnesses for planning, implementation, and evaluation.
|
||||
|
||||
The goal is not to maximize agent count. The goal is to keep long solver work coherent, testable, and reference-verified across context resets and independent sessions.
|
||||
|
||||
## Default Harness Shape
|
||||
Use the smallest harness that can safely handle the task.
|
||||
|
||||
For meaningful solver implementation or phase execution, use:
|
||||
|
||||
```text
|
||||
Planner -> Generator -> Evaluator
|
||||
```
|
||||
|
||||
Roles:
|
||||
- `Planner`: turns project docs and `PLAN.md` tasks into a testable sprint contract or phase step.
|
||||
- `Generator`: implements exactly one accepted contract using TDD.
|
||||
- `Evaluator`: independently checks the result against the contract, docs, tests, reference artifacts, and validation commands.
|
||||
|
||||
Do not use multi-agent ceremony for tiny documentation edits or obvious mechanical changes. Do use the full harness when a task touches solver behavior, numerical conventions, reference comparison, parser compatibility, result schema, or phase execution.
|
||||
|
||||
## Sprint Contract
|
||||
Every implementation sprint must have a contract before code changes begin.
|
||||
|
||||
Recommended location:
|
||||
- `phases/{phase}/stepN.md` for phase execution.
|
||||
- `phases/{phase}/contracts/stepN-contract.md` only when a separate negotiation artifact is useful.
|
||||
|
||||
Required sections:
|
||||
|
||||
````markdown
|
||||
# Sprint Contract: {name}
|
||||
|
||||
## Objective
|
||||
{one concise outcome}
|
||||
|
||||
## Required Reading
|
||||
- /AGENTS.md
|
||||
- /PROGRESS.md
|
||||
- /PLAN.md
|
||||
- /docs/README.md
|
||||
- /docs/HARNESS_ENGINEERING.md
|
||||
- {topic docs}
|
||||
|
||||
## Scope
|
||||
- {what may be changed}
|
||||
|
||||
## Allowed Files
|
||||
- {paths or modules}
|
||||
|
||||
## Explicit Non-Goals
|
||||
- {what must not be done}
|
||||
|
||||
## Tests To Write First
|
||||
- {test files or test cases}
|
||||
|
||||
## Reference Artifacts
|
||||
- {references/*.inp or references/*_displacements.csv, or "none"}
|
||||
|
||||
## Acceptance Commands
|
||||
```bash
|
||||
python scripts/validate_workspace.py
|
||||
```
|
||||
|
||||
## Evaluator Checklist
|
||||
- {contract-specific checks}
|
||||
|
||||
## Handoff Requirements
|
||||
- Update PROGRESS.md for completed work.
|
||||
- Update PLAN.md for future work or changed blockers.
|
||||
````
|
||||
|
||||
Contract quality rules:
|
||||
- The contract must be testable.
|
||||
- The contract must identify unsupported Abaqus features rather than expanding support implicitly.
|
||||
- The contract must state whether reference data is used.
|
||||
- The contract must name file ownership boundaries to reduce conflicts.
|
||||
- The contract must not prescribe formulas that are not present in `docs/MITC4_FORMULATION.md` or a cited source.
|
||||
|
||||
## Generator Rules
|
||||
The Generator implements one contract at a time.
|
||||
|
||||
Required behavior:
|
||||
- Read the contract and required docs before editing.
|
||||
- Write or update tests before implementation.
|
||||
- Keep changes inside allowed files unless the contract is updated first.
|
||||
- Preserve architecture boundaries from `docs/ARCHITECTURE.md` and `docs/ADR.md`.
|
||||
- Preserve numerical conventions from `docs/NUMERICAL_CONVENTIONS.md`.
|
||||
- Run acceptance commands.
|
||||
- Update `PROGRESS.md` and `PLAN.md` only for factual state changes.
|
||||
|
||||
Generator failure modes to avoid:
|
||||
- Broad refactors outside the contract.
|
||||
- Implementing parser support because a stored reference `.inp` contains unsupported Abaqus features.
|
||||
- Comparing only reduced vectors when full-vector reaction recovery is required.
|
||||
- Treating a passing compile as sufficient without tests or reference checks.
|
||||
|
||||
## Evaluator Rules
|
||||
The Evaluator is independent from the Generator.
|
||||
|
||||
Evaluation order:
|
||||
1. Read the sprint contract.
|
||||
2. Read `AGENTS.md`, `PROGRESS.md`, `PLAN.md`, and the topic docs.
|
||||
3. Inspect the changed files.
|
||||
4. Run or review the acceptance commands.
|
||||
5. Check tests, reference artifacts, and documented conventions.
|
||||
6. Return pass/fail findings with concrete file references.
|
||||
|
||||
The Evaluator must fail the sprint if any of these are true:
|
||||
- Required tests were not written first or are missing.
|
||||
- `python scripts/validate_workspace.py` fails without explanation.
|
||||
- A CRITICAL rule in `AGENTS.md` is violated.
|
||||
- A change drifts from `docs/ARCHITECTURE.md`, `docs/ADR.md`, or `docs/NUMERICAL_CONVENTIONS.md`.
|
||||
- `references/*_displacements.csv` comparison is required but not implemented or not checked.
|
||||
- `RF` is computed from reduced quantities when full-vector recovery is required.
|
||||
- Unsupported Abaqus features are silently accepted.
|
||||
- Completed work is not recorded in `PROGRESS.md`, or future tasks are not recorded in `PLAN.md`.
|
||||
|
||||
If the sprint fails, the Evaluator should produce a concise feedback artifact:
|
||||
|
||||
```markdown
|
||||
# Evaluation Feedback: {contract}
|
||||
|
||||
## Verdict
|
||||
fail
|
||||
|
||||
## Findings
|
||||
- {severity}: {file} - {risk}
|
||||
|
||||
## Required Fixes
|
||||
- {minimal fix}
|
||||
|
||||
## Verification To Rerun
|
||||
- {commands}
|
||||
```
|
||||
|
||||
## FESA Evaluation Rubric
|
||||
Use this rubric for implementation review.
|
||||
|
||||
| Criterion | Pass Condition |
|
||||
|---|---|
|
||||
| Contract compliance | Changes stay within scope and allowed files |
|
||||
| Architecture | Domain, AnalysisModel, AnalysisState, DofManager, adapters, and factories follow documented ownership |
|
||||
| Numerical conventions | DOF order, units, signs, double precision, int64 ids, constrained/free mapping, and full-vector reactions are preserved |
|
||||
| Reference verification | Stored `references/` artifacts are used when required; CSV column mapping is correct |
|
||||
| Tests | Tests exist before implementation and cover failure modes, not only happy paths |
|
||||
| Diagnostics | Unsupported input and singular systems produce actionable diagnostics |
|
||||
| Results schema | Outputs follow step/frame/field/history and HDF5 schema rules |
|
||||
| Handoff | `PLAN.md` and `PROGRESS.md` reflect the new state |
|
||||
|
||||
## Harness Complexity Policy
|
||||
Add harness complexity only when it catches real risk.
|
||||
|
||||
Use a single agent for:
|
||||
- small wording changes.
|
||||
- mechanical docs updates.
|
||||
- metadata-only corrections.
|
||||
|
||||
Use Planner -> Generator -> Evaluator for:
|
||||
- C++ solver implementation.
|
||||
- parser behavior changes.
|
||||
- result schema or HDF5 writer changes.
|
||||
- reference comparator changes.
|
||||
- MITC4 formulation-dependent work.
|
||||
- phase generation or execution.
|
||||
|
||||
Review the harness periodically. If an agent role no longer adds value, simplify it.
|
||||
@@ -1,443 +0,0 @@
|
||||
# MITC4 Formulation
|
||||
|
||||
## Purpose
|
||||
This document is the formulation contract for FESA's Phase 1 four-node MITC4 shell element.
|
||||
|
||||
It supersedes the earlier simplified Phase 1 baseline that used an averaged-edge shell basis, analytic thickness integration, and `drilling_stiffness_scale = 1.0e-6`. The next Phase 1 implementation pass must treat this document as the MITC4 gate before coding or evaluating shell stiffness.
|
||||
|
||||
The goal is not to reproduce Abaqus internals. The goal is to implement a documented, testable Dvorkin-Bathe-style MITC4 element that can be compared against stored Abaqus `S4` reference artifacts without running Abaqus locally.
|
||||
|
||||
## Local Source Basis
|
||||
The formulation below is based on the local papers in `docs/Paper/`:
|
||||
|
||||
- `docs/Paper/AContinuumMechanicsBasedFourNodeShellElementforGeneralNonlinearAnalysis/`
|
||||
- Dvorkin and Bathe's original four-node non-flat shell element.
|
||||
- Defines the degenerated-continuum geometry, displacement interpolation, convected transverse shear interpolation, constitutive transformation, and benchmark expectations.
|
||||
- `docs/Paper/FourNodeQuadrilateralShellElementMITC4/`
|
||||
- Restates the MITC4 geometry and displacement interpolation, gives explicit midside shear tying relations, and reports patch-test and Scordelis-Lo verification behavior.
|
||||
- `docs/Paper/유한요소해석법을이용한쉘구조물의동적좌굴해석/`
|
||||
- Provides the Korean thesis derivation for covariant strain, plane-stress material matrix with shear correction, 6-DOF shell transformation, drilling stabilization, and nonlinear extension notes.
|
||||
- `docs/Paper/2007쉘구조물의유한요소해석에대하여/`
|
||||
- Provides shell-model background, locking behavior, asymptotic behavior, and benchmark/test interpretation.
|
||||
- `docs/Paper/mitc공부/`
|
||||
- Treated as supporting study notes only. Use the papers above for binding formulas.
|
||||
|
||||
## Phase 1 Scope
|
||||
Phase 1 implements:
|
||||
|
||||
- Abaqus `TYPE=S4` mapped to FESA `MITC4`.
|
||||
- Four-node quadrilateral shell in Abaqus S4 node order.
|
||||
- Linear static, small-strain, linear isotropic elastic analysis.
|
||||
- Homogeneous shell section with one thickness value per element for the initial implementation.
|
||||
- Six global DOFs per node: `UX`, `UY`, `UZ`, `RX`, `RY`, `RZ`.
|
||||
- MITC transverse shear interpolation in a convected coordinate system.
|
||||
- Constrained DOF elimination, full-vector reaction recovery, and minimum output fields `U` and `RF`.
|
||||
|
||||
Phase 1 does not implement:
|
||||
|
||||
- Abaqus `S4R`, reduced integration, or hourglass control.
|
||||
- Geometric nonlinearity, material nonlinearity, dynamics, pressure loads, thermal-stress coupling, composite sections, or mesh quality diagnostics.
|
||||
- Abaqus `*Orientation`, user-defined nodal normals, or shell offset behavior unless added by a later ADR.
|
||||
|
||||
## Natural Coordinates and Node Order
|
||||
FESA uses the standard bilinear quadrilateral natural coordinates:
|
||||
|
||||
```text
|
||||
node 1: (xi, eta) = (-1, -1)
|
||||
node 2: (xi, eta) = (+1, -1)
|
||||
node 3: (xi, eta) = (+1, +1)
|
||||
node 4: (xi, eta) = (-1, +1)
|
||||
zeta in [-1, +1] through the thickness
|
||||
```
|
||||
|
||||
Shape functions:
|
||||
|
||||
```text
|
||||
N1 = 0.25 * (1 - xi) * (1 - eta)
|
||||
N2 = 0.25 * (1 + xi) * (1 - eta)
|
||||
N3 = 0.25 * (1 + xi) * (1 + eta)
|
||||
N4 = 0.25 * (1 - xi) * (1 + eta)
|
||||
```
|
||||
|
||||
Midside MITC tying points in the FESA convention:
|
||||
|
||||
```text
|
||||
A = ( 0, -1) edge 1-2
|
||||
B = (-1, 0) edge 1-4
|
||||
C = ( 0, +1) edge 4-3
|
||||
D = (+1, 0) edge 2-3
|
||||
```
|
||||
|
||||
Dvorkin-Bathe papers may use a natural-coordinate orientation where the signs on the `A/C` interpolation are reversed relative to the FESA convention above. FESA implementation must preserve the edge definitions above and use the FESA interpolation formulas in the MITC shear section.
|
||||
|
||||
## Degenerated-Continuum Geometry
|
||||
The shell is represented as a 3D continuum degenerated through the thickness:
|
||||
|
||||
```text
|
||||
X(xi, eta, zeta) =
|
||||
sum_k N_k(xi, eta) X_k
|
||||
+ zeta / 2 * sum_k t_k N_k(xi, eta) Vn_k
|
||||
```
|
||||
|
||||
where:
|
||||
|
||||
- `X_k` is the midsurface coordinate of node `k`.
|
||||
- `t_k` is the nodal thickness. Phase 1 uses `t_k = t` for all four nodes.
|
||||
- `Vn_k` is the nodal director. The papers allow the director to be non-normal to the midsurface; Phase 1 derives it from geometry because Abaqus nodal normals are not in the parser subset.
|
||||
|
||||
Phase 1 director policy:
|
||||
|
||||
- If no parsed nodal directors exist, use the element-center midsurface normal for all four nodal directors:
|
||||
|
||||
```text
|
||||
G1_c = dX_mid / dxi at (0, 0)
|
||||
G2_c = dX_mid / deta at (0, 0)
|
||||
Vn_k = normalize(G1_c x G2_c), k = 1..4
|
||||
```
|
||||
|
||||
- If `G1_c x G2_c` is near zero, reject the element with an invalid/singular element diagnostic.
|
||||
- This flat-or-mildly-warped policy is sufficient for the Phase 1 S4 baseline. A true non-flat nodal-director policy requires a later ADR and benchmark update.
|
||||
|
||||
The nodal local director axes are:
|
||||
|
||||
```text
|
||||
V1_k = normalize(EY x Vn_k)
|
||||
V2_k = Vn_k x V1_k
|
||||
```
|
||||
|
||||
If `EY x Vn_k` is near zero, use a deterministic fallback axis before normalization:
|
||||
|
||||
```text
|
||||
V1_k = normalize(EZ x Vn_k)
|
||||
if still near zero, V1_k = normalize(EX x Vn_k)
|
||||
V2_k = Vn_k x V1_k
|
||||
```
|
||||
|
||||
The implementation must keep `V1_k`, `V2_k`, and `Vn_k` orthonormal and right-handed.
|
||||
|
||||
## Displacement and Rotation Interpolation
|
||||
The local five-DOF MITC4 displacement field is:
|
||||
|
||||
```text
|
||||
u(xi, eta, zeta) =
|
||||
sum_k N_k u_k
|
||||
+ zeta / 2 * sum_k t_k N_k q_k
|
||||
|
||||
q_k = -V2_k * alpha_k + V1_k * beta_k
|
||||
```
|
||||
|
||||
where:
|
||||
|
||||
- `u_k` is the nodal translation vector.
|
||||
- `alpha_k` is the rotation about `V1_k`.
|
||||
- `beta_k` is the rotation about `V2_k`.
|
||||
- Drilling rotation `gamma_k` about `Vn_k` does not enter the continuum strain field.
|
||||
|
||||
FESA stores six global DOFs per node. Global nodal rotations are mapped to local director rotations by:
|
||||
|
||||
```text
|
||||
theta_k = [RX_k, RY_k, RZ_k]^T
|
||||
|
||||
[alpha_k, beta_k, gamma_k]^T = L_k theta_k
|
||||
|
||||
L_k =
|
||||
[ EX dot V1_k EY dot V1_k EZ dot V1_k
|
||||
EX dot V2_k EY dot V2_k EZ dot V2_k
|
||||
EX dot Vn_k EY dot Vn_k EZ dot Vn_k ]
|
||||
```
|
||||
|
||||
The element-level local-to-global transformation is block diagonal:
|
||||
|
||||
```text
|
||||
u_local_e = T_e u_global_e
|
||||
K_global_e = T_e^T K_local_e T_e
|
||||
f_global_e = T_e^T f_local_e
|
||||
```
|
||||
|
||||
Each node block maps `[UX, UY, UZ, RX, RY, RZ]` to `[u_x, u_y, u_z, alpha, beta, gamma]`.
|
||||
|
||||
## Covariant Strain Definition
|
||||
Use convected covariant components for the MITC4 strain construction:
|
||||
|
||||
```text
|
||||
g_i = dX / dr_i, where r_i = xi, eta, zeta
|
||||
|
||||
eps_ij =
|
||||
0.5 * (du/dr_i dot g_j + du/dr_j dot g_i)
|
||||
```
|
||||
|
||||
For the Phase 1 small-strain implementation:
|
||||
|
||||
- Membrane and bending terms are derived directly from the degenerated-continuum displacement field.
|
||||
- `eps_33 = 0` is enforced by the shell assumption.
|
||||
- The transverse shear rows `eps_13` and `eps_23` are not taken directly at the Gauss point. They are replaced by the MITC assumed transverse shear interpolation below.
|
||||
|
||||
Use the internal strain vector order:
|
||||
|
||||
```text
|
||||
[ eps_11, eps_22, eps_33, gamma_23, gamma_13, gamma_12 ]^T
|
||||
```
|
||||
|
||||
where engineering shear strains are:
|
||||
|
||||
```text
|
||||
gamma_ij = 2 * eps_ij
|
||||
```
|
||||
|
||||
If an implementation uses another row order internally, it must include an explicit permutation test against this documented order.
|
||||
|
||||
## MITC Transverse Shear Interpolation
|
||||
The key MITC4 assumption is that the transverse shear tensor components are interpolated from midside tying points, instead of being evaluated directly at every Gauss point from the displacement field.
|
||||
|
||||
The original paper form is:
|
||||
|
||||
```text
|
||||
eps_13 = 0.5 * (1 + r2) * eps_13^A + 0.5 * (1 - r2) * eps_13^C
|
||||
eps_23 = 0.5 * (1 + r1) * eps_23^D + 0.5 * (1 - r1) * eps_23^B
|
||||
```
|
||||
|
||||
With FESA's Abaqus-style natural coordinate convention and tying-point labels, use:
|
||||
|
||||
```text
|
||||
eps_13_MITC(xi, eta) =
|
||||
0.5 * (1 - eta) * eps_13^A
|
||||
+ 0.5 * (1 + eta) * eps_13^C
|
||||
|
||||
eps_23_MITC(xi, eta) =
|
||||
0.5 * (1 - xi) * eps_23^B
|
||||
+ 0.5 * (1 + xi) * eps_23^D
|
||||
```
|
||||
|
||||
The tying values are the direct covariant shear strains evaluated at midside points:
|
||||
|
||||
```text
|
||||
eps_13^A = eps_13_DIRECT( 0, -1)
|
||||
eps_13^C = eps_13_DIRECT( 0, +1)
|
||||
eps_23^B = eps_23_DIRECT(-1, 0)
|
||||
eps_23^D = eps_23_DIRECT(+1, 0)
|
||||
```
|
||||
|
||||
Implementation rule:
|
||||
|
||||
- Build the direct covariant strain-displacement rows from the geometry and displacement interpolation.
|
||||
- Evaluate only the needed direct transverse shear rows at `A`, `B`, `C`, and `D`.
|
||||
- Interpolate those rows to the integration point using the formulas above.
|
||||
- Do not compute MITC shear from local Cartesian `gamma_xz`/`gamma_yz` first; the tying is on convected tensor components.
|
||||
|
||||
For sign and regression checks, define:
|
||||
|
||||
```text
|
||||
q_k = -V2_k * alpha_k + V1_k * beta_k
|
||||
```
|
||||
|
||||
The paper edge formulas can be used as a diagnostic check for the direct tying rows:
|
||||
|
||||
```text
|
||||
eps_13^A ~ 1/8 * [
|
||||
(u1 - u2) dot 0.5 * (t1 Vn1 + t2 Vn2)
|
||||
+ (X1 - X2) dot 0.5 * (t1 q1 + t2 q2)
|
||||
]
|
||||
|
||||
eps_13^C ~ 1/8 * [
|
||||
(u4 - u3) dot 0.5 * (t4 Vn4 + t3 Vn3)
|
||||
+ (X4 - X3) dot 0.5 * (t4 q4 + t3 q3)
|
||||
]
|
||||
|
||||
eps_23^B ~ 1/8 * [
|
||||
(u1 - u4) dot 0.5 * (t1 Vn1 + t4 Vn4)
|
||||
+ (X1 - X4) dot 0.5 * (t1 q1 + t4 q4)
|
||||
]
|
||||
|
||||
eps_23^D ~ 1/8 * [
|
||||
(u2 - u3) dot 0.5 * (t2 Vn2 + t3 Vn3)
|
||||
+ (X2 - X3) dot 0.5 * (t2 q2 + t3 q3)
|
||||
]
|
||||
```
|
||||
|
||||
These compact formulas are not a license to skip the general direct-row evaluation. They are a sign/orientation cross-check derived from the same midside strain definitions.
|
||||
|
||||
## Local Cartesian Frame and Material Law
|
||||
The constitutive law is defined in a local orthonormal Cartesian frame at each integration point with `sigma_33 = 0`.
|
||||
|
||||
Build the integration-point local basis from the covariant basis:
|
||||
|
||||
```text
|
||||
e3_hat = normalize(g_3)
|
||||
e1_hat = normalize(g_2 x e3_hat)
|
||||
e2_hat = e3_hat x e1_hat
|
||||
```
|
||||
|
||||
If `g_2 x e3_hat` is near zero, use a deterministic fallback that still creates a right-handed orthonormal basis and emits a diagnostic if no valid basis exists.
|
||||
|
||||
For isotropic linear elasticity, use the local plane-stress shell material matrix with transverse shear correction `kappa = 5/6`:
|
||||
|
||||
```text
|
||||
D_hat = E / (1 - nu^2) *
|
||||
[ 1 nu 0 0 0 0
|
||||
nu 1 0 0 0 0
|
||||
0 0 0 0 0 0
|
||||
0 0 0 kappa*(1-nu)/2 0 0
|
||||
0 0 0 0 kappa*(1-nu)/2 0
|
||||
0 0 0 0 0 (1-nu)/2 ]
|
||||
```
|
||||
|
||||
This matrix uses the documented strain order:
|
||||
|
||||
```text
|
||||
[ eps_11, eps_22, eps_33, gamma_23, gamma_13, gamma_12 ]^T
|
||||
```
|
||||
|
||||
For a general non-flat element, transform the local Cartesian material relation to the convected coordinate strain vector:
|
||||
|
||||
```text
|
||||
E_hat = T_xi_to_x E_convected
|
||||
D_convected = T_xi_to_x^T D_hat T_xi_to_x
|
||||
```
|
||||
|
||||
The transformation matrix must be constructed from direction cosines between the local orthonormal basis and contravariant/covariant convected bases. For Phase 1, a flat-element implementation may compute the B-matrix directly in the local Cartesian frame only if tests prove it is equivalent to the convected formulation for the accepted reference cases.
|
||||
|
||||
## Element Stiffness and Numerical Integration
|
||||
The baseline stiffness is:
|
||||
|
||||
```text
|
||||
K_e = integral_V B^T D B dV
|
||||
```
|
||||
|
||||
Phase 1 baseline integration:
|
||||
|
||||
- Use `2 x 2 x 2` Gauss integration in `(xi, eta, zeta)` for the first rebuilt MITC4 implementation.
|
||||
- This follows the thesis derivation's `2 x 2 x 2` natural-coordinate integration and the Dvorkin-Bathe elastic recommendation of `2 x 2` in the midsurface with two through-thickness points.
|
||||
- Do not introduce reduced integration or hourglass control.
|
||||
- Analytic through-thickness integration may be introduced later only after equivalence tests preserve the MITC shear, drilling, and reference displacement behavior.
|
||||
|
||||
At every integration point:
|
||||
|
||||
1. Compute `g_i`, the metric/Jacobian, local Cartesian basis, and material transform.
|
||||
2. Build direct membrane/bending strain rows.
|
||||
3. Replace transverse shear rows with MITC interpolated rows.
|
||||
4. Accumulate `B^T D B |J| w_xi w_eta w_zeta`.
|
||||
|
||||
Detect near-zero Jacobian, invalid basis, missing property/material, or non-positive thickness as invalid/singular element diagnostics. Do not report these as mesh quality diagnostics in Phase 1.
|
||||
|
||||
## Drilling DOF Stabilization
|
||||
The continuum MITC4 formulation has five physical DOFs per node. FESA keeps six global DOFs, so the local drilling rotation `gamma` has no physical strain contribution and must be weakly stabilized.
|
||||
|
||||
Baseline decision:
|
||||
|
||||
```text
|
||||
drilling_stiffness_scale = 1.0e-3
|
||||
k_ref = min_positive_diagonal(K_local_without_drilling)
|
||||
k_drill = drilling_stiffness_scale * k_ref
|
||||
```
|
||||
|
||||
Rules:
|
||||
|
||||
- `K_local_without_drilling` is the element local stiffness after MITC integration and before artificial drilling stabilization.
|
||||
- Use the minimum positive diagonal entry over the physical local DOFs. Ignore zero, negative, NaN, or non-finite entries.
|
||||
- Add `k_drill` to each nodal local `gamma` diagonal before transforming to global coordinates.
|
||||
- If no positive reference diagonal exists, emit a singular/invalid element diagnostic.
|
||||
- The scale must be configurable and written to result metadata or analysis diagnostics.
|
||||
- Reference comparisons must include a drilling sensitivity check if displacement results change materially with the scale.
|
||||
|
||||
This replaces the earlier arbitrary `1.0e-6 * E * thickness` rule. The `1.0e-3` scale follows the local thesis derivation's six-DOF stabilization note and is still an artificial numerical parameter, not a physical stiffness.
|
||||
|
||||
## Internal Force and Reaction Recovery
|
||||
For Phase 1 linear static analysis:
|
||||
|
||||
```text
|
||||
f_int_e = K_e u_e
|
||||
R_full = K_full U_full - F_full
|
||||
```
|
||||
|
||||
Rules:
|
||||
|
||||
- Reactions must be recovered from the full global vector, not only from the reduced free-DOF system.
|
||||
- `RF` output follows the Abaqus-style component convention documented in `docs/NUMERICAL_CONVENTIONS.md`.
|
||||
- Element stress/strain/resultant outputs are not mandatory until displacement and reaction reference behavior is stable.
|
||||
|
||||
## Required Tests Before Reimplementation
|
||||
The rebuilt MITC4 element must be protected by tests before implementation code is accepted.
|
||||
|
||||
Element-level tests:
|
||||
|
||||
- Shape functions sum to one and have correct derivatives at center, corners, and tying points.
|
||||
- Center normal, local director axes, and integration local Cartesian axes are orthonormal and right-handed.
|
||||
- The local-to-global rotation transform maps global rotations to `alpha`, `beta`, `gamma` as documented.
|
||||
- Direct tying rows at `A`, `B`, `C`, `D` agree with finite-difference strain perturbations.
|
||||
- MITC shear rows at Gauss points are interpolated from tying rows, not evaluated directly at the Gauss points.
|
||||
- Stiffness is symmetric for linear elastic Phase 1.
|
||||
- Rigid-body translations and rotations produce near-zero physical strain energy, with only documented drilling stabilization effects.
|
||||
- The element has no spurious zero-energy modes beyond expected rigid-body behavior after drilling stabilization is accounted for.
|
||||
|
||||
Patch and benchmark tests:
|
||||
|
||||
- Constant membrane states.
|
||||
- Pure bending.
|
||||
- Pure shear.
|
||||
- Pure twist.
|
||||
- Thin cantilever or plate strip to expose shear locking.
|
||||
- Scordelis-Lo roof after the basic flat tests pass.
|
||||
- Pinched cylinder or twisted beam as later Phase 1/Phase 2 benchmark expansion.
|
||||
|
||||
Reference regression tests:
|
||||
|
||||
- Compare FESA `U` against stored Abaqus displacement CSV files using absolute and relative tolerances.
|
||||
- Keep `RF` verified by full-vector equilibrium until matching Abaqus reaction CSV files are provided or explicitly deferred.
|
||||
|
||||
## Current Reference Artifacts
|
||||
Stored artifacts currently known:
|
||||
|
||||
- `references/quad_01.inp`
|
||||
- `references/quad_01_displacements.csv`
|
||||
- `references/quad_02.inp`
|
||||
- `references/quad_02_displacements.csv`
|
||||
- `references/quad_02_reactionforces.csv`
|
||||
|
||||
Compatibility notes:
|
||||
|
||||
- `quad_01.inp` contains `S4R`, `Part/Assembly/Instance`, `*Density`, and `NLGEOM=YES`; it remains future compatibility provenance, not a Phase 1 parser acceptance case.
|
||||
- `quad_02.inp` contains `TYPE=S4`, which is the correct element target for Phase 1, but it also uses `Part/Assembly/Instance`. The normalized `quad_02_phase1.inp` remains the executable Phase 1 input path. Do not silently expand parser support while implementing or verifying the element.
|
||||
- `quad_02_reactionforces.csv` is now available for Abaqus RF/RM comparison. The current node-wise RF comparison does not pass; keep this as a formulation/solver verification gap until the mismatch is explained or fixed.
|
||||
|
||||
## Implementation Checklist
|
||||
The next MITC4 implementation pass should proceed in this order:
|
||||
|
||||
1. Lock the shape-function, natural-coordinate, node-order, and tying-point tests.
|
||||
2. Implement geometry/director/local-axis construction with diagnostics.
|
||||
3. Implement the five-DOF degenerated-continuum displacement interpolation.
|
||||
4. Implement global six-DOF to local `[alpha, beta, gamma]` transformation.
|
||||
5. Implement direct covariant strain rows and finite-difference tests.
|
||||
6. Replace transverse shear rows with MITC tying interpolation.
|
||||
7. Implement material transform and `2 x 2 x 2` stiffness integration.
|
||||
8. Add drilling stabilization in local coordinates.
|
||||
9. Transform element stiffness and internal force to global coordinates.
|
||||
10. Run element patch tests before assembly/reference regression work.
|
||||
|
||||
## Future Extensions
|
||||
Geometric nonlinearity:
|
||||
|
||||
- Update current geometry and nodal directors in `AnalysisState`.
|
||||
- Add finite-rotation updates for `V1`, `V2`, and `Vn`.
|
||||
- Add tangent stiffness, geometric stiffness, and Newton-Raphson convergence checks.
|
||||
|
||||
Thermal-stress coupling:
|
||||
|
||||
- Add temperature field state and thermal strain contribution.
|
||||
- Add material expansion data and thermal output fields.
|
||||
|
||||
S4R:
|
||||
|
||||
- Add only after a separate ADR and formulation update.
|
||||
- Requires reduced integration, hourglass control, and separate reference artifacts.
|
||||
|
||||
Stress/resultant output:
|
||||
|
||||
- Define section point locations, component labels, local coordinate convention, and Abaqus CSV comparison format before making `S`, `E`, or `SF` mandatory.
|
||||
|
||||
## Remaining Open Decisions
|
||||
The MITC4 stiffness formulation is now defined for the Phase 1 rebuild. Remaining decisions are outside the first stiffness pass:
|
||||
|
||||
- Exact stress/strain/resultant recovery locations and component labels.
|
||||
- Whether to parse Abaqus `*Orientation` or nodal shell normals.
|
||||
- Whether `quad_02.inp` should be normalized into the existing Phase 1 parser subset or used to justify a dedicated `Part/Assembly/Instance` parser sprint.
|
||||
- Reference tolerances for `quad_02` after the input compatibility path is chosen.
|
||||
@@ -1,143 +0,0 @@
|
||||
# Multi-Agent Research Plan
|
||||
|
||||
## Purpose
|
||||
This document is the durable planning memo for FESA's research-oriented multi-agent workflow. It records what the agents should investigate before solver implementation begins.
|
||||
|
||||
No solver code should be implemented from this plan directly. Each agent should produce an English technical dossier that an implementer can later follow.
|
||||
|
||||
## Project Context
|
||||
FESA is a C++17 finite element structural analysis solver. Phase 1 targets a linear static MITC4 shell solver with linear elastic material, nodal loads, fixed boundary conditions, an Abaqus input subset, HDF5-oriented results, and reference-result comparison.
|
||||
|
||||
The user provides Abaqus input files and solved reference result files under the repository `references/` folder. Abaqus is not available locally, so validation must compare against stored reference artifacts.
|
||||
|
||||
## Current Architecture Constraints
|
||||
- Follow `AGENTS.md`, `PROGRESS.md`, `PLAN.md`, `docs/README.md`, `docs/HARNESS_ENGINEERING.md`, `docs/ARCHITECTURE.md`, and `docs/ADR.md`.
|
||||
- Use `PLAN.md` for future task ownership, priority, and open questions.
|
||||
- Use `PROGRESS.md` for completed work, verification notes, blockers, and risks.
|
||||
- Follow the technical dossier documents:
|
||||
- `docs/NUMERICAL_CONVENTIONS.md`
|
||||
- `docs/ABAQUS_INPUT_SUBSET.md`
|
||||
- `docs/VERIFICATION_PLAN.md`
|
||||
- `docs/RESULTS_SCHEMA.md`
|
||||
- `docs/MITC4_FORMULATION.md`
|
||||
- Use runtime polymorphism for elements, materials, loads, boundary conditions, and analysis algorithms.
|
||||
- Keep `Domain` close to immutable after parsing.
|
||||
- Use `AnalysisModel` for the active step view.
|
||||
- Use `AnalysisState` for mutable physical state and iteration state.
|
||||
- Let `DofManager` own DOF definitions, constrained/free DOF mapping, equation numbering, and sparse pattern support.
|
||||
- Use Strategy plus Template Method for analysis execution.
|
||||
- Use Factory plus Registry for Abaqus keyword to object creation.
|
||||
- Keep MKL, TBB, and HDF5 behind adapter/wrapper boundaries.
|
||||
- Store results using step/frame/field/history concepts.
|
||||
- Use 6 DOFs per shell node: UX, UY, UZ, RX, RY, RZ.
|
||||
- Use small artificial drilling stiffness in Phase 1, with the final scale documented before implementation.
|
||||
- Do not enforce a unit system; use Abaqus-style self-consistent units.
|
||||
- Follow Abaqus result sign conventions.
|
||||
- Use constrained DOF elimination and full-vector reaction recovery.
|
||||
- Use `double` precision and int64 ids/indices/equation numbering.
|
||||
- Require singular system diagnostics.
|
||||
- Defer mesh quality diagnostics in Phase 1.
|
||||
- Map Abaqus `S4` to FESA `MITC4`; defer `S4R`.
|
||||
- Use the Planner -> Generator -> Evaluator harness from `docs/HARNESS_ENGINEERING.md` for nontrivial implementation and phase execution.
|
||||
|
||||
## Created Codex Agents
|
||||
The first recommended batch has been created under `.codex/agents/`.
|
||||
|
||||
1. `fem_literature_researcher`
|
||||
- File: `.codex/agents/fem-literature-researcher.toml`
|
||||
- Role: research FEM shell literature, MITC family background, locking behavior, and implementation implications.
|
||||
|
||||
2. `verification_benchmark_researcher`
|
||||
- File: `.codex/agents/verification-benchmark-researcher.toml`
|
||||
- Role: research benchmark cases, `references/` contracts, comparison methods, and acceptance criteria.
|
||||
|
||||
3. `mitc4_formulation_researcher`
|
||||
- File: `.codex/agents/mitc4-formulation-researcher.toml`
|
||||
- Role: research MITC4 formulation details and convert them into implementation requirements.
|
||||
|
||||
4. `abaqus_compatibility_researcher`
|
||||
- File: `.codex/agents/abaqus-compatibility-researcher.toml`
|
||||
- Role: research the Phase 1 Abaqus input subset and parser compatibility rules.
|
||||
|
||||
5. `harness_sprint_planner`
|
||||
- File: `.codex/agents/harness-sprint-planner.toml`
|
||||
- Role: convert PLAN.md tasks and phase steps into testable sprint contracts.
|
||||
|
||||
6. `implementation_generator`
|
||||
- File: `.codex/agents/implementation-generator.toml`
|
||||
- Role: implement exactly one accepted sprint contract using TDD.
|
||||
|
||||
7. `harness_sprint_evaluator`
|
||||
- File: `.codex/agents/harness-sprint-evaluator.toml`
|
||||
- Role: independently pass/fail sprint output against the contract, docs, tests, and reference artifacts.
|
||||
|
||||
## Recommended Agent Execution Order
|
||||
1. Run `fem_literature_researcher`.
|
||||
2. Run `verification_benchmark_researcher`.
|
||||
3. Run `mitc4_formulation_researcher`.
|
||||
4. Run `abaqus_compatibility_researcher`.
|
||||
|
||||
This order keeps theory, verification targets, element formulation, and input compatibility aligned before implementation planning.
|
||||
|
||||
Before asking `phase_planner` or `harness_sprint_planner` to create implementation steps, review `PROGRESS.md`, `PLAN.md`, `docs/README.md`, and `docs/HARNESS_ENGINEERING.md`; list any unchecked Implementation Readiness Checklist items in the planning output.
|
||||
|
||||
## Later Agent Candidates
|
||||
These agents should be created after the first four produce dossiers.
|
||||
|
||||
1. `solver_architecture_researcher`
|
||||
- Refines responsibilities among `Domain`, `AnalysisModel`, `AnalysisState`, `DofManager`, `Assembler`, and `LinearSolver`.
|
||||
|
||||
2. `sparse_solver_researcher`
|
||||
- Researches sparse pattern generation, CSR/COO assembly, MKL PARDISO integration, and TBB-safe assembly strategies.
|
||||
|
||||
3. `results_hdf5_schema_researcher`
|
||||
- Designs the HDF5 group/dataset schema for step/frame/field/history outputs.
|
||||
|
||||
4. `nonlinear_roadmap_researcher`
|
||||
- Researches geometric nonlinearity, Newton-Raphson, tangent stiffness, increments, and convergence criteria.
|
||||
|
||||
5. `thermal_coupling_researcher`
|
||||
- Researches heat transfer, thermal load generation, thermal strain, and thermal-stress coupling.
|
||||
|
||||
6. `architecture_guardrail_reviewer`
|
||||
- Reviews dossiers, phase plans, and future implementation against project rules and ADRs.
|
||||
|
||||
7. `reference_artifact_curator`
|
||||
- Reviews user-provided `references/` artifacts for CSV/manifest completeness, result schema consistency, source solver metadata, tolerance clarity, and comparison path validity.
|
||||
|
||||
8. `numerical_conventions_reviewer`
|
||||
- Reviews implementation plans for drift from DOF, unit, sign, precision, reaction recovery, and singular diagnostic rules.
|
||||
|
||||
## User-Provided Inputs Needed
|
||||
The following inputs should be requested from the user as the research matures:
|
||||
|
||||
1. Additional small Abaqus `.inp` files and solved result artifacts under `references/`.
|
||||
2. Reaction-force reference exports when available, preferably using a documented `*_reactions.csv` convention.
|
||||
3. Preferred numerical tolerances for reference comparison, or permission for agents to propose initial tolerances per benchmark.
|
||||
4. The Abaqus version used to generate each reference artifact when it is not evident from the `.inp` file header.
|
||||
5. Whether the first implementation plan should target CMake, another build system, or a project-specific build layout.
|
||||
6. The final default scale for artificial drilling stiffness, after formulation research.
|
||||
|
||||
## Seed Sources
|
||||
- Abaqus input syntax rules: https://abaqus-docs.mit.edu/2017/English/SIMACAEMODRefMap/simamod-c-inputsyntax.htm
|
||||
- Abaqus conventions for DOFs, units, coordinate systems, and stress/strain components: https://abaqus-docs.mit.edu/2017/English/SIMACAEMODRefMap/simamod-c-conventions.htm
|
||||
- Abaqus boundary keyword reference: https://abaqus-docs.mit.edu/2017/English/SIMACAEKEYRefMap/simakey-r-boundary.htm
|
||||
- Abaqus concentrated load keyword reference: https://abaqus-docs.mit.edu/2017/English/SIMACAEKEYRefMap/simakey-r-cload.htm
|
||||
- Abaqus shell section behavior: https://abaqus-docs.mit.edu/2017/English/SIMACAEELMRefMap/simaelm-c-shellsectionbehavior.htm
|
||||
- Abaqus conventional shell element library: https://abaqus-docs.mit.edu/2017/English/SIMACAEELMRefMap/simaelm-r-shellgeneral.htm
|
||||
- OpenSees ShellMITC4 manual: https://opensees.berkeley.edu/OpenSees/manuals/usermanual/640.htm
|
||||
- Dvorkin-Bathe four-node shell element paper: https://web.mit.edu/kjb/www/Publications_Prior_to_1998/A_Continuum_Mechanics_Based_Four-Node_Shell_Element_for_General_Nonlinear_Analysis.pdf
|
||||
- MITC3+/MITC4+ benchmark paper: https://web.mit.edu/kjb/www/Principal_Publications/Performance_of_the_MITC3%2B_and_MITC4%2B_shell_elements_in_widely_used_benchmark_problems.pdf
|
||||
- COMSOL Scordelis-Lo benchmark example: https://doc.comsol.com/5.6/doc/com.comsol.help.models.sme.scordelis_lo_roof/scordelis_lo_roof.html
|
||||
- NAFEMS nonlinear benchmark survey page: https://www.nafems.org/publications/pubguide/benchmarks/Page6/
|
||||
- HDF5 data model: https://docs.hdfgroup.org/documentation/hdf5/latest/_h5_d_m__u_g.html
|
||||
- HDF5 datasets: https://docs.hdfgroup.org/documentation/hdf5/latest/_h5_d__u_g.html
|
||||
- HDF5 attributes: https://portal.hdfgroup.org/documentation/hdf5/latest/_h5_a__u_g.html
|
||||
- Intel oneMKL `pardiso_64`: https://www.intel.com/content/www/us/en/docs/onemkl/developer-reference-c/2024-2/pardiso-64.html
|
||||
|
||||
## Non-Goals
|
||||
- Do not implement solver code.
|
||||
- Do not generate phase execution files until the user explicitly asks for implementation planning.
|
||||
- Do not require Abaqus execution in local validation.
|
||||
- Do not treat unsourced formulas or benchmark constants as final.
|
||||
- Do not add mesh quality diagnostics to Phase 1 planning unless the user changes the decision.
|
||||
@@ -1,235 +0,0 @@
|
||||
# Numerical Conventions
|
||||
|
||||
## Purpose
|
||||
This document defines the numerical conventions that must remain stable across the FESA solver, reference data, tests, and result files.
|
||||
|
||||
When a convention here conflicts with a lower-level implementation note, this document wins unless `docs/ADR.md` is updated.
|
||||
|
||||
## Source Basis
|
||||
- Abaqus defines translational DOFs 1-3 and rotational DOFs 4-6, with rotations expressed in radians: https://abaqus-docs.mit.edu/2017/English/SIMACAEMODRefMap/simamod-c-conventions.htm
|
||||
- Abaqus has no built-in unit system except rotation and angle measures; user data must be self-consistent: https://abaqus-docs.mit.edu/2017/English/SIMACAEMODRefMap/simamod-c-conventions.htm
|
||||
- Abaqus uses right-handed coordinate systems and defines shell local surface directions from the projected global axis and positive element normal: https://abaqus-docs.mit.edu/2017/English/SIMACAEMODRefMap/simamod-c-conventions.htm
|
||||
- Intel oneMKL provides `pardiso_64`, an ILP64-style PARDISO interface accepting `long long int` integer data for large sparse systems: https://www.intel.com/content/www/us/en/docs/onemkl/developer-reference-c/2024-2/pardiso-64.html
|
||||
|
||||
## Binding Decisions
|
||||
- MITC4 coordinate and strain details will be defined explicitly in `docs/MITC4_FORMULATION.md`.
|
||||
- Phase 1 shell nodes use 6 DOFs per node.
|
||||
- The drilling DOF is retained and receives a small artificial stiffness in Phase 1.
|
||||
- FESA does not enforce a unit system. Input values must be self-consistent, Abaqus-style.
|
||||
- Result sign conventions follow Abaqus conventions.
|
||||
- Essential boundary conditions are applied by constrained DOF elimination.
|
||||
- Reaction forces are recovered from the full system vectors, not from the reduced system alone.
|
||||
- Reference comparison uses stored Abaqus inputs and stored solved reference results under `references/`.
|
||||
- Initial displacement reference comparison uses Abaqus-exported `*_displacements.csv` files.
|
||||
- Mesh quality diagnostics are not part of Phase 1.
|
||||
- Singular system diagnostics are required.
|
||||
- Floating-point values use `double` by default.
|
||||
- Large-model ids and indices use signed 64-bit integers.
|
||||
|
||||
## Degrees of Freedom
|
||||
All Phase 1 shell nodes expose six structural DOFs:
|
||||
|
||||
| FESA Component | Abaqus DOF | Meaning | Unit |
|
||||
|---|---:|---|---|
|
||||
| `UX` | 1 | Translation in global or transformed x-direction | model length |
|
||||
| `UY` | 2 | Translation in global or transformed y-direction | model length |
|
||||
| `UZ` | 3 | Translation in global or transformed z-direction | model length |
|
||||
| `RX` | 4 | Rotation about x-axis | radian |
|
||||
| `RY` | 5 | Rotation about y-axis | radian |
|
||||
| `RZ` | 6 | Rotation about z-axis | radian |
|
||||
|
||||
Implementation rules:
|
||||
- Store DOF component ids using a compact enum or integer range `1..6`.
|
||||
- Store global node ids, element ids, equation ids, sparse indices, and nonzero counts with signed 64-bit integer types.
|
||||
- `DofManager` owns active DOF discovery, constrained/free partitioning, equation numbering, and sparse pattern inputs.
|
||||
- Do not store equation ids inside `Node` or `Element`.
|
||||
|
||||
## Precision and Numeric Types
|
||||
Recommended type aliases:
|
||||
|
||||
```cpp
|
||||
using Real = double;
|
||||
using GlobalId = std::int64_t;
|
||||
using LocalIndex = std::int64_t;
|
||||
using EquationId = std::int64_t;
|
||||
using SparseIndex = std::int64_t;
|
||||
```
|
||||
|
||||
Rules:
|
||||
- Use `double` for matrix entries, vector entries, coordinates, material constants, loads, displacements, rotations, and result values.
|
||||
- Use 64-bit integer indexing throughout the sparse assembly path so that MKL `pardiso_64` remains available for large models.
|
||||
- Avoid silent narrowing when passing ids or sparse arrays to external libraries.
|
||||
|
||||
## Units
|
||||
FESA follows the Abaqus convention: no unit system is built into the solver.
|
||||
|
||||
Rules:
|
||||
- The solver core treats all dimensional input as unitless numeric values.
|
||||
- The user and reference data must use a self-consistent unit system.
|
||||
- Rotations are always radians.
|
||||
- Result files may store an optional `unit_system_note` metadata string, but the solver must not convert units.
|
||||
|
||||
Examples of acceptable unit systems:
|
||||
- `N, mm, tonne, second`
|
||||
- `N, m, kg, second`
|
||||
- `lbf, inch, lbf*s^2/in, second`
|
||||
|
||||
## Coordinate Systems
|
||||
Global coordinates:
|
||||
- The default global system is right-handed Cartesian.
|
||||
- Phase 1 does not support user-defined transformed nodal coordinate systems unless explicitly added to `docs/ABAQUS_INPUT_SUBSET.md`.
|
||||
|
||||
Shell local directions:
|
||||
- FESA result signs follow Abaqus shell local direction conventions.
|
||||
- The exact MITC4 element basis construction must be defined in `docs/MITC4_FORMULATION.md`.
|
||||
- For Abaqus compatibility, the positive shell normal is tied to node ordering by the right-hand rule.
|
||||
- Local 1 and 2 directions must form a right-handed basis with the positive normal.
|
||||
|
||||
## Result Sign Convention
|
||||
FESA result sign conventions follow Abaqus unless a FESA-specific exception is documented.
|
||||
|
||||
Phase 1 output variables:
|
||||
- `U`: nodal displacement and rotation vector in DOF order `UX, UY, UZ, RX, RY, RZ`.
|
||||
- `RF`: nodal reaction force and reaction moment in the same component order.
|
||||
- `S`: shell stress components in local shell directions when available.
|
||||
- `E`: shell strain components in local shell directions when available.
|
||||
- `SF`: shell section force and moment resultants when available.
|
||||
|
||||
Stress and strain component ordering follows the Abaqus convention:
|
||||
- direct 1
|
||||
- direct 2
|
||||
- direct 3
|
||||
- shear 12
|
||||
- shear 13
|
||||
- shear 23
|
||||
|
||||
Shear strain output should be documented as engineering shear strain when matched to Abaqus-style output.
|
||||
|
||||
## Abaqus Displacement CSV Mapping
|
||||
The initial reference displacement CSV format maps Abaqus exported columns to FESA `U` components:
|
||||
|
||||
| CSV Column | FESA Component | Meaning |
|
||||
|---|---|---|
|
||||
| `Node Label` | `entity_id` | Node id, stored as int64 |
|
||||
| `U-U1` | `UX` | Translation in global 1/x direction |
|
||||
| `U-U2` | `UY` | Translation in global 2/y direction |
|
||||
| `U-U3` | `UZ` | Translation in global 3/z direction |
|
||||
| `UR-UR1` | `RX` | Rotation about global 1/x direction, radians |
|
||||
| `UR-UR2` | `RY` | Rotation about global 2/y direction, radians |
|
||||
| `UR-UR3` | `RZ` | Rotation about global 3/z direction, radians |
|
||||
|
||||
Rules:
|
||||
- CSV numeric values are parsed as `double`.
|
||||
- Node labels are matched to FESA result entity ids exactly.
|
||||
- Missing nodes, duplicate node labels, missing columns, or nonnumeric values are reference artifact errors.
|
||||
- CSV comparison uses the same absolute/relative tolerance policy as other reference artifacts.
|
||||
|
||||
## Abaqus Reaction CSV Mapping
|
||||
Reference reaction CSV files map Abaqus exported force and moment columns to FESA `RF` components:
|
||||
|
||||
| CSV Column | FESA Component | Meaning |
|
||||
|---|---|---|
|
||||
| `Node Label` | `entity_id` | Node id, stored as int64 |
|
||||
| `RF-RF1` | `RFX` | Reaction force in global 1/x direction |
|
||||
| `RF-RF2` | `RFY` | Reaction force in global 2/y direction |
|
||||
| `RF-RF3` | `RFZ` | Reaction force in global 3/z direction |
|
||||
| `RM-RM1` | `RMX` | Reaction moment about global 1/x direction |
|
||||
| `RM-RM2` | `RMY` | Reaction moment about global 2/y direction |
|
||||
| `RM-RM3` | `RMZ` | Reaction moment about global 3/z direction |
|
||||
|
||||
Rules:
|
||||
- Accepted reaction CSV filenames may use `*_reactionforces.csv` or `*_reactions.csv`.
|
||||
- CSV numeric values are parsed as `double`.
|
||||
- Node labels are matched to FESA result entity ids exactly.
|
||||
- Missing nodes, duplicate node labels, missing columns, or nonnumeric values are reference artifact errors.
|
||||
- Node-wise reaction comparison uses the full-vector FESA `RF` field, not reduced solver quantities.
|
||||
- Do not relax tolerances to pass a reaction comparison without documenting the numerical reason.
|
||||
|
||||
## Boundary Conditions
|
||||
Phase 1 uses constrained DOF elimination:
|
||||
|
||||
1. Build the full DOF list.
|
||||
2. Mark constrained DOFs from `*Boundary`.
|
||||
3. Partition DOFs into free and constrained sets.
|
||||
4. Assemble or project the reduced free-DOF system.
|
||||
5. Solve for free DOF unknowns.
|
||||
6. Reconstruct the full displacement vector.
|
||||
|
||||
Rules:
|
||||
- Phase 1 supports zero-valued fixed constraints as the primary path.
|
||||
- Nonzero prescribed displacements are not a Phase 1 requirement unless added to `docs/ABAQUS_INPUT_SUBSET.md`.
|
||||
- `DofManager` owns constrained/free mapping and must provide enough data to reconstruct full vectors.
|
||||
|
||||
## Reaction Recovery
|
||||
Reaction force and moment recovery uses the full system:
|
||||
|
||||
```text
|
||||
R_full = K_full * U_full - F_full
|
||||
```
|
||||
|
||||
Rules:
|
||||
- `K_full` means the assembled stiffness before constrained DOF elimination.
|
||||
- `U_full` means the solved displacement vector reconstructed with constrained DOF values.
|
||||
- `F_full` means the full external load vector in the original global DOF space.
|
||||
- `RF` is reported primarily for constrained DOFs, but full residual-like vectors may be retained for diagnostics.
|
||||
- Reference comparison should include at least one reaction balance test.
|
||||
|
||||
## Drilling DOF Policy
|
||||
Phase 1 retains `RZ` at each shell node and assigns a small artificial drilling stiffness.
|
||||
|
||||
Rules:
|
||||
- The value must be parameterized, not hard-coded deep inside the MITC4 kernel.
|
||||
- The Phase 1 default follows `docs/MITC4_FORMULATION.md`: `drilling_stiffness_scale = 1.0e-3` applied to the minimum positive physical local stiffness diagonal before the local-to-global six-DOF transform.
|
||||
- The artificial stiffness should be small relative to representative membrane or bending stiffness, but large enough to avoid a singular stiffness matrix for unconstrained drilling modes.
|
||||
- Reference comparisons should include sensitivity checks if the drilling stiffness materially changes displacement results.
|
||||
|
||||
## Singular System Diagnostics
|
||||
FESA must provide actionable diagnostics before or during linear solve failure.
|
||||
|
||||
Required Phase 1 checks:
|
||||
- No free DOFs exist after applying constraints.
|
||||
- At least one active element exists in the current `AnalysisModel`.
|
||||
- Every active element references existing nodes.
|
||||
- Every active element references an assigned property and material.
|
||||
- Every active property references an existing material.
|
||||
- Every load and boundary condition references an existing node or node set.
|
||||
- At least one load component is nonzero for ordinary static solve tests unless the test explicitly expects zero response.
|
||||
- Free DOFs exist that are not touched by any active element connectivity.
|
||||
- Rotational DOFs, especially drilling DOFs, may be unconstrained or weakly constrained.
|
||||
- The reduced system size and sparse nonzero count are nonzero before solve.
|
||||
- The sparse solver reports zero pivot, numerical factorization failure, or structural singularity.
|
||||
|
||||
Recommended diagnostic style:
|
||||
|
||||
```text
|
||||
FESA-SINGULAR-DOF-UNTOUCHED:
|
||||
Node 42 DOF RZ is free but has no stiffness contribution from active elements.
|
||||
Check boundary conditions, element connectivity, property assignment, or drilling stiffness settings.
|
||||
```
|
||||
|
||||
Phase 1 explicitly does not perform mesh quality diagnostics such as aspect ratio, warpage, skew, or element distortion checks.
|
||||
|
||||
## Reference Comparison Tolerances
|
||||
Reference comparison must use both absolute and relative tolerances:
|
||||
|
||||
```text
|
||||
pass if abs(actual - expected) <= abs_tol
|
||||
or abs(actual - expected) <= rel_tol * max(abs(expected), reference_scale)
|
||||
```
|
||||
|
||||
Initial defaults may be proposed per benchmark in `docs/VERIFICATION_PLAN.md`. Final tolerances should be tied to stored reference data and solver maturity.
|
||||
|
||||
## Implementation Gate
|
||||
Before implementing solver modules that depend on numerical conventions:
|
||||
- Define the `Real`, id, equation id, and sparse index aliases in one shared core header.
|
||||
- Define a single DOF enum or equivalent mapping for `UX`, `UY`, `UZ`, `RX`, `RY`, `RZ`.
|
||||
- Add tests proving Abaqus DOF numbers `1..6` map to the same internal components.
|
||||
- Add tests proving constrained/free vector reconstruction preserves original full-space DOF order.
|
||||
- Add at least one reaction recovery test that computes `K_full * U_full - F_full`.
|
||||
- Ensure singular diagnostics can reference node id, DOF component, element id, property id, and set name.
|
||||
|
||||
## Open Decisions
|
||||
- Stress/strain/resultant recovery locations and component labels after `U` and `RF` are stable.
|
||||
- Whether to parse Abaqus `*Orientation` or nodal shell normals.
|
||||
- The future nodal-director policy for truly non-flat or strongly warped quadrilateral elements.
|
||||
- Optional non-displacement CSV formats, such as reaction force or stress/resultant exports.
|
||||
-73
@@ -1,73 +0,0 @@
|
||||
# PRD: FESA
|
||||
|
||||
## 목표
|
||||
MITC4 Shell 요소를 사용해 구조 해석을 하는 유한요소 솔버를 개발
|
||||
|
||||
## 사용자
|
||||
구조해석을 원하는 엔지니어
|
||||
|
||||
## 제품 방향
|
||||
FESA는 Abaqus input subset을 읽고 `references/`에 저장된 Abaqus reference 결과와 비교 가능한 구조해석 결과를 생성하는 검증 중심 solver이다. Phase 1은 "많은 기능"보다 "작은 선형 정적 MITC4 모델들을 정확하게 풀고 반복 검증할 수 있는 기반"을 목표로 한다.
|
||||
|
||||
## 핵심 기능
|
||||
1. MITC4 Shell 요소를 사용한 구조해석
|
||||
2. Parallel 연산을 통한 계산 성능 향상
|
||||
3. Abaqus Input format 사용을 통해 다른 상용 소프트웨어와 호환성 높음
|
||||
4. step/frame/history 기반 해석 결과 관리
|
||||
5. `references/*.inp`와 `references/*_displacements.csv` 등 reference 모델 결과 비교를 통한 정확도 검증
|
||||
6. singular system 진단을 통한 해석 실패 원인 추적
|
||||
|
||||
## 개발 계획
|
||||
1. Phase 1
|
||||
- MITC4 Shell 요소
|
||||
- 선형 탄성 재료
|
||||
- 절점하중
|
||||
- 고정 경계조건
|
||||
- Abaqus input subset: *Node, *Element, *Nset, *Elset, *Material, *Elastic, *Shell Section, *Boundary, *Cload, *Step
|
||||
- 선형 정적 해석
|
||||
- step/frame 기반 결과 저장의 최소 구조
|
||||
- `references/`의 Abaqus input/result artifact 기반 reference 모델 결과 비교 테스트
|
||||
- 6자유도 shell node: UX, UY, UZ, RX, RY, RZ
|
||||
- drilling 자유도에는 작은 인공 강성 적용
|
||||
- constrained DOF 제거 방식
|
||||
- full vector 기반 reaction force 계산
|
||||
- double precision 및 int64 id/index/equation numbering
|
||||
- singular system 진단
|
||||
- Abaqus 실행 없이 저장된 reference artifact와 비교
|
||||
2. Phase 2
|
||||
- 압력하중
|
||||
- RBE2, RBE3 경계조건
|
||||
- Newton-Raphson 비선형 알고리즘
|
||||
- 기하비선형 중심의 비선형 정적해석
|
||||
3. Phase 3
|
||||
- HHT 시간 적분 알고리즘
|
||||
- time dependent 하중
|
||||
- 비선형 동적 해석
|
||||
4. Phase 4
|
||||
- Heat transfer 해석
|
||||
- 절점 온도에 대한 열전도 요소 행렬 계산
|
||||
- 온도 하중 계산
|
||||
- thermal-stress coupling 확장
|
||||
5. Phase 5
|
||||
- 1D, 3D 요소 구현
|
||||
- 기타 하중, 경계조건 구현
|
||||
|
||||
## Phase 1 성공 기준
|
||||
- Phase 1 Abaqus input subset을 파싱하여 `Domain`과 `AnalysisModel`을 일관되게 구성할 수 있다.
|
||||
- `DofManager`가 6자유도 shell node, constrained/free mapping, full/reduced vector reconstruction을 검증 가능한 방식으로 처리한다.
|
||||
- 최소 MITC4 element-level test가 통과한다: shape function, stiffness symmetry, rigid body behavior, drilling stiffness sensitivity.
|
||||
- 선형 정적 해석에서 `U`와 `RF`를 결과 schema에 맞게 저장한다.
|
||||
- `RF = K_full * U_full - F_full` 기반 reaction recovery가 reference 또는 평형 테스트로 검증된다.
|
||||
- singular system negative test가 원인을 설명하는 진단 메시지를 낸다.
|
||||
- 최소 3개 stored reference case가 통과한다: single-element case, simple multi-element plate/shell case, curved shell benchmark. 초기 자동 displacement 비교 형식은 `*_displacements.csv`이다.
|
||||
|
||||
## Phase 1 제외 범위
|
||||
- Abaqus `S4R` 및 hourglass control
|
||||
- pressure load
|
||||
- RBE2/RBE3
|
||||
- nonzero prescribed displacement
|
||||
- geometric/material nonlinearity
|
||||
- dynamic analysis
|
||||
- heat transfer 및 thermal-stress coupling
|
||||
- composite shell section
|
||||
- mesh quality diagnostics
|
||||
@@ -1,558 +0,0 @@
|
||||
# <sup>쉘</sup> 구조물의 유한요소해석에 대하여
|
||||
|
||||
On the Finite Element Analysis of Shell Structures
|
||||
|
||||
이필승\*·노혁천\*\*
|
||||
|
||||
Lee, Phill-Seung·Noh, Hyuk-Chun ····························································································································································································································· 1. 서 론
|
||||
|
||||
Based on recent research works, important concepts on the finite element analysis of shell structures and the relations among them are presented in this paper. We review the basic shell mathematical model, which is the underlying mathematical model of the continuum mechanics based shell finite elements. The asymptotic theory of shell structures then is reviewed and we present how to evaluate the asymptotic behavior in finite element solutions. S-norm is introduced as an error measure of finite element solutions and we show "locking" in the convergence curves of shell finite element solutions. We discuss the concept of Keywords : shell structures, finite elements, asymptotic behavior, uniform optimal convergence, benchmark tests ····························································································································································································································· 요 지 본 논문에서는 최근 주요 연구들을 토대로 쉘 구조물의 유한요소해석에 대하여 중요한 개념들과 그 연관관계를 고찰한다. 감절점 쉘 유한요소의 수학모델인 기본쉘수학모델을 살펴본다. 쉘 구조물의 두께가 얇아짐에 따라 일어나는 쉘 구조문제의 세가지 극한거동들(휨지배거동, 막지배거동, 혼합지배거동)에 대한 쉘의 점근거동 이론을 소개하고 점근거동을 유한요소해석 을 통해 찾아내는 방법을 알아본다. 유한요소해의 오차를 s-norm으로 평가하는 방법을 소개하고 이를 이용하여 쉘 유한요소 의 잠김현상이 유한요소해의 수렴곡선에 어떻게 나타나는지 살펴본다. 쉘 구조물의 유한요소해석에서 균일최적수렴의 개념을 논의한다. 마지막으로 이상적인 쉘 유한요소의 조건을 알아보고 쉘 유한요소의 성능평가를 위한 방법론을 제시한다.
|
||||
|
||||
핵심용어 : 쉘 구조물, 유한요소해석, 점근거동, 균일최적수렴, 성능시험
|
||||
|
||||
ments and propose how to perform benchmark tests of shell finite elements. 계란의 외피가 얼마나 큰 외력에 견딜 수 있는가 하는 것
|
||||
|
||||
은 일반 대중들에게도 잘 알려져 있는 사실이다. 그 밖에도 우리는 자연 속에서 갑각류의 표피나 조개 껍질과 같은 여 러 종류의 쉘 구조물들(shell structures)을 접할 수 있다. 이 런 자연의 예들은 쉘 구조가 매우 이상적이고 효과적인 구 조물의 한 형태라는 것을 간접적으로 증명하고 있다. 인간이 만들어낸 쉘 구조물들도 셀 수 없이 다양하고 많으며 우리 는 쉘 구조물들 속에서 살고 있다고 해도 과언이 아니다. 유한요소법(finite element method)은 쉘 구조물의 선형 및 비선형 해석에 가장 널리 쓰이는 방법으로 지난 수십 년 동안 쉘 구조물의 유한요소해석에 대한 연구가 활발하게 이 루어져 오고 있다(Bathe, 1996; 최창근, 2002; Chapelle and Bathe, 2003; Noh, 2006). 그러나 쉘 구조물은 쉘의 형상, 경계조건, 하중 등에 따라 아주 다양한 거동을 보이며
|
||||
|
||||
····························································································································································································································· 기 때문에 쉘 구조물의 유한요소해석에 앞서 근본적인 쉘 이론 및 물리적 거동에 대한 이해가 필수적이다(Chapelle and Bathe, 1998; Lee and Bathe, 2002; Chapelle and
|
||||
|
||||
Bathe, 2003). 이러한 쉘 구조물의 거동에 대한 이해의 핵 심은 쉘 구조물의 두께가 얇아짐에 따라 나타나는 점근거동 (asymptotic behavior)을 연구하는 것이다(Lee and Bathe, 2002; Bathe, Chapelle and Lee, 2003). 쉘 구조물의 점근 거동은 일반적으로 휨지배(bending dominated)거동, 막지배 (membrane dominated)거동, 혼합(mixed)지배거동 등으로 나 누어진다. 일반적으로 공학(engineering)에서 주어진 문제를 풀기 위 한 접근방법은 실험(experiment)과 관찰(investigation)을 통 하여 물리적 거동을 살펴본 후 그 물리문제의 중요한 특성/ 인자를 찾아내고 그에 따른 여러 가지 가정들(assumptions) 을 사용하여 문제를 단순화시켜 수학모델(mathematical model)을 만든다. 수학모델은 주어진 물리문제를 이론적으로 접근할 수 있는 방법을 제공해 준다. 수학모델의 해는 이론
|
||||
|
||||
특히 쉘의 두께가 얇을 경우 거동의 특성이 매우 민감해지
|
||||
|
||||
<sup>\*</sup>정회원 · 교신저자 · 삼성중공업(주) 건설부문 과장 (E-mail : phillseung.lee@samsung.com)
|
||||
|
||||
적인 방법으로 구해질 수 있으나 풀고자 하는 문제가 매우 복잡할 때는 해를 구하기가 거의 불가능하다. 수치해석 (numerical analysis)을 이용하면 복잡한 물리문제의 근사해 (approximation)를 구할 수 있고 그 결과를 실험/관찰 결과 와 비교하면 수학모델이나 수치해석의 적정성을 평가할 수 있다.
|
||||
|
||||
이와 같은 일련의 작업들, 즉, 물리적 거동, 수학모델, 수 치해석은 서로 밀접하게 연관되어 있다. 그러므로, 쉘 구조 물의 유한요소해석을 명확하게 이해하기 위해서는 쉘 구조 물의 물리적 거동에 대한 이해, 쉘의 수학모델(mathematical shell model)에 대한 이해와 쉘 유한요소에 대한 이해가 동 시에 체계적이고 심도 있게 이루어져야 한다. 세가지 모두에 대한 통합적인 이해가 없을 경우 쉘 구조물의 유한요소해석 에 있어서 중대한 오류를 범할 수 있다. 본 논문의 목적은 이 각각의 세가지 부분에 대한 이해와 이들이 서로 어떻게 유기적으로 관계를 맺고 있는지를 최근 주요 연구들(Lee and Bathe, 2002; Chapelle and Bathe, 2003; Bathe, Chapelle and Lee, 2003; Hiller and Bathe, 2003; Lee and Bathe, 2005)을 중심으로 정리하여 고찰해 보고 이상적 인 쉘 유한요소의 성질과 쉘 유한요소의 성능평가 방법을 제시하는 것이다. 특히 쉘 구조물의 설계를 위하여 유한요소 해석을 수행하는 기술자들(engineers)이나 유한요소법을 공부 /연구하는 학생/연구자들의 "쉘 구조물의 유한요소해석에 대 한 이해"를 돕고자 하는데 글의 초점을 맞추었다.
|
||||
|
||||
유한요소법에 의해 쉘 구조물을 효과적으로 해석하기 위해 서는 신뢰할 만한 쉘 유한요소를 사용해야 한다는 것은 자 명하다. 일반적으로 변위법에 의해 정식화(displacement based formulation)된 쉘 유한요소는 사용된 근사함수 (interpolation function)의 차수(order)에 상관없이 휨지배 및 혼합지배거동을 하는 쉘 구조물에 대하여 과도한 강성을 나 타낸다(Bathe, 1996; Chapelle and Bathe, 2003). 이를 잠 김현상(locking phenomenon)이라 하며 효과적인 쉘 유한요 소의 개발에 있어서 극복해야 할 어려운 문제들 중의 하나 이다. 이상적인 쉘 유한요소는 여러 가지 점근거동과 다양한 형상의 쉘 구조문제에 있어서 균일최적수렴(uniform optimal convergence)을 보여야 하며 이상적인 쉘 유한요소를 개발하 는 것은 매우 어려운 일이다.
|
||||
|
||||
본 논문에서는 먼저 쉘의 대표적인 수학모델을 설명하고 쉘 구조물의 점근거동에 대한 기본이론을 살펴본 후 임의의 쉘 구조물의 점근거동을 어떻게 알아낼 수 있는지를 알아본 다. 또한 쉘 유한요소해의 오차(error)를 평가하는 규준을 살 펴보고 이를 바탕으로 잠김현상 발생 시의 쉘 유한요소의 수렴과 균일최적수렴에 대해 알아본다. 마지막으로 이상적인 쉘 유한요소와 쉘 유한요소의 성능평가에 대하여 논한다. 앞 으로 논의하는 내용은 등방성재료(isotropic material)에 대한 선형탄성(linear elastic)이론에 국한된다.
|
||||
|
||||
# 2. 쉘의 수학모델(mathematical shell model)
|
||||
|
||||
기본쉘수학모델(basic shell mathematical model)은 쉘 구 조물의 휨(bending)거동, 면내(membrane)거동, 면외전단 (transverse shearing)거동과 그 상호연관(coupling)관계를 모 두 표현할 수 있는 가장 일반적인 쉘 수학모델로 3차원 연 속체 역학으로부터 유도된 감절점 쉘 유한요소(Ahmad, Irons and Zienkiewicz, 1970; Bathe, 1996)와 동일한 변형 률 항들을 가지고 있다(Chapelle and Bathe, 1998; Chapelle and Bathe, 2003; Lee and Bathe, 2005). 즉, 기본쉘수학 모델은 감절점 쉘 유한요소의 수학모델인 것이다. 본 장에서 는 미분기하학(differential geometry)을 이용하여 쉘의 형상 (geometry)과 변형거동(kinematics)을 살펴보고 기본쉘수학모 델의 유도를 보여준다.
|
||||
|
||||
# 2.1 쉘의 형상 (shell geometry)
|
||||
|
||||
쉘은 두께(thickness)가 얇은 3차원 구조물이다. 두께가 얇 다는 특징 때문에 쉘의 형상은 쉘의 중심면으로 이루어진 2 차원 영역과 두께에 의하여 정의 될 수 있다. 여기서는 기 본쉘수학모델에서 쓰이는 쉘의 형상에 관한 개념을 미분기 하학을 통하여 보여준다.
|
||||
|
||||
|
||||
|
||||
그림 1. 쉘의 형상
|
||||
|
||||
λ
|
||||
|
||||
φ
|
||||
|
||||
$$\vec{a}_{\alpha} = \frac{\partial \vec{\phi}(\xi^1, \xi^2)}{\partial \xi^{\alpha}} \tag{1}$$
|
||||
|
||||
$$\vec{a}^{\alpha} \cdot \vec{a}_{\beta} = \delta^{\alpha}_{\beta} \tag{2}$$
|
||||
|
||||
미분기하학의 정의들을 유도하기 위하여 아인슈타인 합표 시규약(Einstein summation convention)을 사용한다. α, β, , µ는 1에서 2까지 변하며 i, j, k는 1에서 3까지 변하는 첨자(index)들이다. 쉘의 중심면(midsurface)은 그림 1에서 보이는 2차원 영역 ω에서 S로의 사상(mapping)을 나타내는 함수 에 의하여 정의 된다. 중심면의 공변기저벡터(covariant base vector)는 다음과 같이 나타내어진다. (1) 위의 식의 공변(covariant)기저벡터에 대응하는 반변기저벡터 (contravariant base vector)는 다음의 관계에 의하여 얻어진다. (2) 여기서 는 α와 β가 같을 때 1이고 다를 때 0인 δ β α
|
||||
|
||||
$$\vec{a}_3 = \frac{\vec{a}_1 \times \vec{a}_2}{\left|\vec{a}_1 \times \vec{a}_2\right|} \tag{3}$$
|
||||
|
||||
$$\overrightarrow{\Phi}(\xi^{l}, \xi^{2}, \xi^{3}) = \overrightarrow{\phi}(\xi^{l}, \xi^{2}) + \xi^{3}\overrightarrow{a}_{3}(\xi^{l}, \xi^{2}) \tag{4}$$
|
||||
|
||||
$$\Omega = \left\{ (\xi^{1}, \xi^{2}, \xi^{3}) \in \Re^{3} | (\xi^{1}, \xi^{2}) \in \omega, \xi^{3} \in \left[ -\frac{t(\xi^{1}, \xi^{2})}{2}, \frac{t(\xi^{1}, \xi^{2})}{2} \right] \right\}$$
|
||||
(5)
|
||||
|
||||
Kronecker symbol이다. 쉘의 중심면에 수직인 벡터는 중심 면의 공변(covariant)기저벡터들의 벡터곱(vector product)으로 정의 된다. (3) 쉘의 3차원 기하형상은 다음 식으로 표현된다. (4) 여기서 변수 ξ , ξ , ξ 의 영역은 다음과 같다. (5) 여기서 t는 쉘의 두께이다.
|
||||
|
||||
$$a_{\alpha\beta} = \vec{a}_{\alpha} \cdot \vec{a}_{\beta} \tag{6}$$
|
||||
|
||||
$$a^{\alpha\beta} = \vec{a}^{\alpha} \cdot \vec{a}^{\beta} \tag{7}$$
|
||||
|
||||
위의 정의들을 이용하여 쉘의 중심면에서 surface 텐서들 (tensors)을 정의 할 수 있다. 첫 번째 기본텐서는 2D metric 텐서로 공변형(covariant type)은 다음 식과 같다. (6) 위 식의 반변형(contravariant type)은 다음과 같이 나타난다. (7) 두 번째 기본텐서는 곡률(curvature)텐서로서 쉘 중심면의 곡
|
||||
|
||||
$$b_{\alpha\beta} = \vec{a}_3 \cdot \vec{a}_{\alpha,\beta} \tag{8}$$
|
||||
|
||||
$$b^{\alpha}_{\beta} = a^{\alpha\lambda}b_{\lambda\beta} \tag{9}$$
|
||||
|
||||
$$c_{\alpha\beta} = b_{\alpha}^{\lambda} b_{\lambda\beta} \tag{10}$$
|
||||
|
||||
률에 관한 정보를 담고 있다. 공변형(covariant type)은 다음 과 같다. (8) 또한 위 식의 합성형(mixed)텐서는 다음 식과 같다. (9) 세 번째 기본텐서는 다음과 같이 정의 된다. (10) 쉘의 중심면에서의 벡터를 라고 하면 이 벡터의 공변미분 w
|
||||
|
||||
$$w_{\alpha|\beta} = w_{\alpha,\beta} - \Gamma^{\lambda}_{\alpha\beta} w_{\lambda} \tag{11}$$
|
||||
|
||||
Γαβ λ
|
||||
|
||||
$$\Gamma^{\lambda}_{\alpha\beta} = \vec{a}_{\alpha,\beta} \cdot \vec{a}^{\lambda} \tag{12}$$
|
||||
|
||||
$$\vec{g}_i = \frac{\partial \vec{\Phi}(\xi^1, \xi^2, \xi^3)}{\partial \xi^i} \tag{13}$$
|
||||
|
||||
$$\vec{g}_{\alpha} = \vec{a}_{\alpha} - \xi^3 b_{\alpha}^{\lambda} \vec{a}_{\lambda} \tag{14}$$
|
||||
|
||||
$$\vec{g}_3 = \vec{a}_3 \tag{15}$$
|
||||
|
||||
$$\vec{g} \cdot \vec{g}_j = \delta_j^i \tag{16}$$
|
||||
|
||||
# 2.2 쉘의 변형거동(shell kinematics)
|
||||
|
||||
$$\overrightarrow{U}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}}, \xi^{\mathsf{3}}) = \overrightarrow{u}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}}) + \xi^{\mathsf{3}} \theta_{\lambda}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}}) \overrightarrow{a}^{\lambda}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}})$$
|
||||
|
||||
$$\tag{17}$$
|
||||
|
||||
(11) 여기서 는 면에서의 Christoffel symbol이다. (12) 식 (4)로부터 3차원 공변(covariant)기저벡터를 얻을 수 있다. (13) 그리고 위 식으로부터 다음 식들이 얻어진다. (14) (15) 3차원 반변(contravariant)기저벡터는 다음과 같이 정의 된다. (16) 쉘의 변형거동에 있어서 기본이 되는 가정은 변형전 쉘의 중심면에 수직인 직선은 변형중에도 직선을 유지하며 늘어 나거나 줄어들지 않는다는 것이고, 이때 쉘의 변위는 다음과 같이 표현된다. (17) 여기서 는 쉘의 중심면의 미소변위(infinitesimal translation)를 나타내고 는 쉘의 중심면에 수직인 직선의 미소회전(infinitesimal rotation)을 나타낸다. 는 회전 벡터 이고 는 그 회전 벡터에 의한 변위를 <sup>u</sup> ξ 1 ,ξ 2 ( ) <sup>θ</sup> <sup>λ</sup> ξ 1 ,ξ 2 ( ) <sup>θ</sup> <sup>λ</sup><sup>a</sup> λ θ ξ 3 <sup>θ</sup> <sup>λ</sup><sup>a</sup> λ u a 1 2 <sup>a</sup> <sup>3</sup> a3
|
||||
|
||||
$$e_{ij} = \frac{1}{2} (\vec{g}_i \cdot \vec{U}_{,j} + \vec{g}_j \cdot \vec{U}_{,i})$$
|
||||
(18)
|
||||
|
||||
$$\vec{a}^2$$
|
||||
, $\vec{a}^3$ (= $\vec{a}_3$ )에 의하여 주어진다.
|
||||
선형해석을 위한 3차원 Green-Lagrange 공변변형률
|
||||
(covariant strain) 텐서의 선형부분은 다음과 같이 정의 된다.
|
||||
$e_{ij} = \frac{1}{2}(\vec{g}_i \cdot \vec{U}_{,j} + \vec{g}_{j} \cdot \vec{U}_{,i})$ (18)
|
||||
여기서
|
||||
$\vec{U}_{,i} = \frac{\partial \vec{U}(\xi^1, \xi^2, \xi^3)}{\partial \xi^i}$ . (19)
|
||||
식 (14), (15)와 (17)을 식 (18)에 대입시키면 변형률 텐서의 공변(covariant)항들을 얻을 수 있다.
|
||||
$e_{\alpha\beta} = \gamma_{\alpha\beta}(\vec{u}) + \xi^3 \chi_{\alpha\beta}(\vec{u},\vec{\theta}) - (\xi^3)^2 \kappa_{\alpha\beta}(\vec{\theta})$ (20a)
|
||||
$e_{\alpha3} = \zeta_{\alpha}(\vec{u},\vec{\theta})$ (20b)
|
||||
$e_{33} = 0$ (20c)
|
||||
여기서
|
||||
$\gamma_{\alpha\beta}(\vec{u}) = \frac{1}{2}(u_{\alpha|\beta} + u_{\beta|\alpha}) - b_{\alpha\beta}u_3$ (21a)
|
||||
$\chi_{\alpha\beta}(\vec{u},\vec{\theta}) = \frac{1}{2}(\theta_{\alpha|\beta} + \theta_{\beta|\alpha} - b_{\beta}^{\lambda}u_{\lambda|\alpha} - b_{\alpha}^{\lambda}u_{\lambda|\beta}) + c_{\alpha\beta}u_3$ (21b)
|
||||
$\kappa_{\alpha\beta}(\vec{\theta}) = \frac{1}{2}(b_{\beta}^{\lambda}\theta_{\lambda|\alpha} - b_{\alpha}^{\lambda}\theta_{\lambda|\beta})$ (21c)
|
||||
$\zeta_{\alpha}(\vec{u},\vec{\theta}) = \frac{1}{2}(\theta_{\alpha} + u_{3,\alpha} + b_{\alpha}^{\lambda}u_{\lambda})$ (21d)
|
||||
등방성재료(isotropic material)에 대한 평면응력조건(plane
|
||||
|
||||
. (19)
|
||||
|
||||
$$e_{\alpha\beta} = \gamma_{\alpha\beta}(\vec{u}) + \xi^3 \chi_{\alpha\beta}(\vec{u}, \vec{\theta}) - (\xi^3)^2 \kappa_{\alpha\beta}(\vec{\theta})$$
|
||||
(20a)
|
||||
|
||||
$$e_{\alpha\beta} = \zeta_{\alpha}(\vec{u}, \vec{\theta}) \tag{20b}$$
|
||||
|
||||
$$e_{33} = 0$$
|
||||
(20c)
|
||||
|
||||
의 공변(covariant)항들을 얻을 수 있다.
|
||||
$$e_{\alpha\beta} = \gamma_{\alpha\beta}(\vec{u}) + \xi^3 \chi_{\alpha\beta}(\vec{u},\vec{\theta}) - (\xi^3)^2 \kappa_{\alpha\beta}(\vec{\theta}) \qquad (20a)$$
|
||||
|
||||
$$e_{\alpha3} = \zeta_{\alpha}(\vec{u},\vec{\theta}) \qquad (20b)$$
|
||||
|
||||
$$e_{33} = 0 \qquad (20c)$$
|
||||
역기서
|
||||
$$\gamma_{\alpha\beta}(\vec{u}) = \frac{1}{2}(u_{\alpha|\beta} + u_{\beta|\alpha}) - b_{\alpha\beta}u_3 \qquad (21a)$$
|
||||
|
||||
$$\chi_{\alpha\beta}(\vec{u},\vec{\theta}) = \frac{1}{2}(\theta_{\alpha|\beta} + \theta_{\beta|\alpha} - b_{\beta}^{\lambda}u_{\lambda|\alpha} - b_{\alpha}^{\lambda}u_{\lambda|\beta}) + c_{\alpha\beta}u_3 \qquad (21b)$$
|
||||
|
||||
$$\kappa_{\alpha\beta}(\vec{\theta}) = \frac{1}{2}(b_{\beta}^{\lambda}\theta_{\lambda|\alpha} - b_{\alpha}^{\lambda}\theta_{\lambda|\beta}) \qquad (21c)$$
|
||||
|
||||
$$\zeta_{\alpha}(\vec{u},\vec{\theta}) = \frac{1}{2}(\theta_{\alpha} + u_{3,\alpha} + b_{\alpha}^{\lambda}u_{\lambda}) \qquad (21d)$$
|
||||
등방성재료(isotropic material)에 대한 평면응력조건(plane)
|
||||
|
||||
$$\chi_{\alpha\beta}(\vec{u}, \vec{\theta}) = \frac{1}{2}(\theta_{\alpha|\beta} + \theta_{\beta|\alpha} - b_{\beta}^{\lambda} u_{\lambda|\alpha} - b_{\alpha}^{\lambda} u_{\lambda|\beta}) + c_{\alpha\beta} u_{3}$$
|
||||
(21b)
|
||||
$$\kappa_{\alpha\beta}(\vec{\theta}) = \frac{1}{2}(b_{\beta}^{\lambda} \theta_{\lambda|\alpha} - b_{\alpha}^{\lambda} \theta_{\lambda|\beta})$$
|
||||
(21c)
|
||||
$$\zeta_{\alpha}(\vec{u}, \vec{\theta}) = \frac{1}{2}(\theta_{\alpha} + u_{3,\alpha} + b_{\alpha}^{\lambda} u_{\lambda})$$
|
||||
(21d)
|
||||
동방성재료(isotropic material)에 대한 평면응력조건(plane
|
||||
|
||||
$$\kappa_{\alpha\beta}(\vec{\theta}) = \frac{1}{2}(b_{\beta}^{\lambda}\theta_{\lambda|\alpha} - b_{\alpha}^{\lambda}\theta_{\lambda|\beta})$$
|
||||
(21c)
|
||||
$$\zeta_{\alpha}(\vec{u},\vec{\theta}) = \frac{1}{2}(\theta_{\alpha} + u_{3,\alpha} + b_{\alpha}^{\lambda}u_{\lambda})$$
|
||||
(21d)
|
||||
한방성재료(isotropic material)에 대한 평면응력조건(plane
|
||||
|
||||
$$\zeta_{\alpha}(\vec{u}, \vec{\theta}) = \frac{1}{2}(\theta_{\alpha} + u_{3,\alpha} + b_{\alpha}^{\lambda}u_{\lambda})$$
|
||||
(21d)
|
||||
통방성재료(isotropic material)에 대한 평면응력조건(plane
|
||||
|
||||
(21d) 등방성재료(isotropic material)에 대한 평면응력조건(plane
|
||||
|
||||
$$\sigma^{\alpha\beta} = C^{\alpha\beta\lambda\mu} e_{\lambda\mu} \tag{22a}$$
|
||||
|
||||
$$\sigma^{\alpha\beta} = \frac{1}{2} D^{\alpha\lambda} e_{\lambda\beta} \tag{22b}$$
|
||||
|
||||
$$C^{\alpha\beta\lambda\mu} = \frac{E}{2(1-v)} \left( g^{\alpha\lambda} g^{\beta\mu} + g^{\alpha\mu} g^{\beta\lambda} + \frac{2v}{1+v} g^{\alpha\beta} g^{\lambda\mu} \right)$$
|
||||
(23a)
|
||||
|
||||
$$D^{\alpha\lambda} = \frac{2E}{1+\nu}g^{\alpha\lambda} \tag{23b}$$
|
||||
|
||||
g αβ g αβ<sup>=</sup><sup>g</sup> <sup>α</sup> g <sup>β</sup> ( ) <sup>⋅</sup>
|
||||
|
||||
V U
|
||||
|
||||
$$\int_{\Omega} C^{\alpha\beta\lambda\mu} e_{\alpha\beta}(\overrightarrow{U}) e_{\lambda\mu}(\overrightarrow{V}) dV + \int_{\Omega} D^{\alpha\lambda} e_{\alpha\beta}(\overrightarrow{U}) e_{\lambda\beta}(\overrightarrow{V}) dV$$
|
||||
|
||||
$$= \int_{\Omega} \overrightarrow{F} \cdot \overrightarrow{V} dV \qquad (24)$$
|
||||
|
||||
F
|
||||
|
||||
$$\vec{V}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}}, \xi^{\mathsf{3}}) = \vec{v}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}}) + \xi^{\mathsf{3}} \eta_{\lambda}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}}) \vec{a}^{\lambda}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}})$$
|
||||
(25)
|
||||
|
||||
# 3. 쉘 구조물의 점근거동
|
||||
|
||||
stress condition)을 적용하면 면에 수직인 응력은 0이며 (σ33 = 0) 이때 응력과 변형률의 상관관계는 다음과 같다. (22a) (22b) 위의 식 (22)에서 (23a) (23b) 여기서 E는 재료의 탄성계수(elastic modulus)이고 v는 포 아손비(Poisson's ratio)이며 는 식 (14)의 3차원 반변 (contravariant)기저벡터로 정의된 metric텐서이다 . 쉘 구조물에 강체운동(rigid body motion)이 일어나지 않 도록 적절한 변위경계조건이 주어지면 식 (20)에서 (23)까지 를 이용하여 기본쉘수학모델(basic shell mathematical model) 의 지배변분식(variational equation)을 얻을 수 있다. 해를 구하는 과정은 모든 임의의 시험함수(test function) 에 대하여 다음 식 (24)와 변위 경계조건을 만족시키는 미지변위 를 찾는 것으로 나타내어질 수 있다. (24) 여기서 는 쉘 구조물에 작용하는 외력(external loading) 이며 시험함수는 변위경계조건을 만족시켜야 한다. (25) 휨(bending), 막(membrane), 면외전단(transverse shearing) 작용들은 쉘 구조물이 하중을 지지하는 기본적인 원리 (mechanism)이다. 그러므로 하중 재하 시 쉘 구조물은 휨, 막, 전단 에너지를 그 내부에 저장하게 된다. 쉘의 두께가 얇아짐에 따라 쉘의 전단 에너지는 무시할 만큼 작아지므로 쉘 구조물의 에너지는 주로 휨 에너지와 막 에너지에 의해 구성된다고 할 수 있다. 쉘은 그 두께가 얇아짐에 따라 특정한 한계거동 - 휨지배 (bending dominated)거동, 막지배(membrane dominated)거동, 혼합(mixed)지배거동 - 을 보이게 되며 이를 쉘의 점근거동 (asymptotic behavior)이라 한다. 쉘의 두께가 얇아짐에 따라 쉘 구조물이 주로 휨 거동에 의해 하중을 지탱할 경우 그
|
||||
|
||||
3.1 점근거동의 분류
|
||||
|
||||
U∈Ψ
|
||||
|
||||
$$\varepsilon^3 A_b(\vec{U}, \vec{V}) + \varepsilon A_m(\vec{U}, \vec{V}) = \vec{F}(\vec{V}), \forall \vec{V} \in \vec{\Psi}$$
|
||||
(26. 여기서 $\varepsilon$ 는 쉘의 두께와 전체 쉘 구조물 크기의 비 $(t/L)$ $\varepsilon^3 A_b(\cdot,\cdot)$ 는 휨 에너지, $\varepsilon$ $A_b(\cdot,\cdot)$ 은 막 및 전단 에너지에 대응하는 겹선형식들(bilinear forms)이며, $\vec{U}$ 는 변위장의 해 $\vec{V}$ 는 시험함수, $\vec{\Psi}$ 는 Sobolev 공간(space) $^1$ , $\vec{F}(\cdot)$ 는 외력에 대응하는 선형식(linear form)을 나타낸다. 일반적으로 쉴의 두께가 얇을 때 전단 에너지는 막 에너지에 비해 매우작으므로 $\varepsilon$ $A_m$ 을 막 에너지에 대응하는 항이라고 부를 수 있다
|
||||
|
||||
ε<sup>ρ</sup>
|
||||
|
||||
$$\vec{F}(\vec{V}) = \varepsilon^{\rho} \vec{G}(\vec{V}) \tag{27}$$
|
||||
|
||||
G Ψ′ Ψ
|
||||
|
||||
$$1 \le \rho \le 3 \tag{28}$$
|
||||
|
||||
$$\overrightarrow{\Psi}_0 = \left\{ \overrightarrow{V} \in \overrightarrow{\Psi} \middle| A_m(\overrightarrow{V}, \overrightarrow{V}) = 0 \right\}$$
|
||||
(29)
|
||||
|
||||
얇아짐에 따라 쉘 구조물이 휨과 막거동 두 가지 모두에 의 해 외력을 지지할 경우 혼합(mixed)지배 쉘 구조물이라 한 다. 쉘 구조물의 점근거동은 쉘의 형상(geometry), 경계조건 (boundary condition), 하중(loading)에 따라 달라진다(Lee and Bathe, 2002; Bathe, Chapelle and Lee, 2003; Chapelle and Bathe, 2003). 식 (24)에서 구해진 선형 쉘 이론의 변분식(variational form)을 두께(t)에 대하여 정리하여 t의 고차항을 제거하면 다음과 같이 간략화하여 나타낼 수 있다. Find such that (26) 여기서 ε 는 쉘의 두께와 전체 쉘 구조물 크기의 비(t/L), Ab(·,·)는 휨 에너지, ε Ab(·,·)은 막 및 전단 에너지에 대응 하는 겹선형식들(bilinear forms)이며, 는 변위장의 해, 는 시험함수, 는 Sobolev 공간(space)1 , 는 외력 에 대응하는 선형식(linear form)을 나타낸다. 일반적으로 쉘 의 두께가 얇을 때 전단 에너지는 막 에너지에 비해 매우 작으므로 ε Am을 막 에너지에 대응하는 항이라고 부를 수 있다. ε이 작아짐에 따른 쉘의 점근거동을 살펴 보기 위해 (ρ = load scaling factor)가 곱해진 외력(scaled loading)을 사용한다. (27) 여기서 는 ( 의 dual space2 )에 속하며 ρ는 실수이 다. 식 (26)의 좌변의 각 항은 ε 과 ε에 비례하므로 ρ는 1보다 크거나 같고 3보다 작거나 같은 실수임을 알 수 있다. (28) 다음의 공간(space)은 쉘의 점근거동에 중요한 역할을 한다. (29) 공간은 순수 휨(pure bending)을 나타내는 변위의 공간 이며 막 및 전단 에너지를 0으로 만들 수 있는 모든 변위 의 형태들(patterns)을 포함한다. 이 공간(space)이 단지 모든 변위를 0으로 하는 변위의 형태들만을 가지고 있을 때 쉘 구조물에서 순수 휨거동이 구속되었다고 하며 그러한 쉘을 순수 휨이 구속된 쉘(inhibited shell)이라 한다. 반면에 쉘이 모든 변위가 0인 형태(pattern)가 아닌 순수 휨 모드(mode) 를 가지고 있을 경우 그 쉘 구조물을 순수 휨이 구속되지 않은 쉘(non-inhibited shell) 구조물이라 한다. 쉘의 점근거 Ψ0
|
||||
|
||||
탄성문제에서 변위경계조건을 만족시키는 Sobolev 공간은 으로 표기되며 정의는 다음과 같다. 1
|
||||
|
||||
쉘 구조물을 휨지배 쉘(bending dominated shell) 구조물이 라 부르며 막거동에 의해 하중을 지탱할 경우 막지배 쉘 (membrane dominated shell) 구조물이라 한다. 또한 두께가 여기서 이다(Bathe, 1996). H0 H0 1 ( ) <sup>Ω</sup> = V: V L<sup>2</sup> ∈ Ω( ); ∂Vi ∂xj ------- <sup>L</sup><sup>2</sup> ∈ Ω( ); Vi =0 at prescribed displacement boundary ⎩ ⎭ ⎨ ⎬ ⎧ ⎫ ( ) <sup>Ω</sup> = V: <sup>Ω</sup> ∫ 3 <sup>∑</sup> Vi ( )<sup>2</sup> <sup>d</sup>Ω<+<sup>∞</sup>
|
||||
|
||||
2 공간위의 모든 선형식 들의 공간을 의 dual space(쌍대공간)라고 부른며 로 표기한다. i = 1 <sup>Ψ</sup> <sup>F</sup> <sup>Ψ</sup> Ψ′
|
||||
|
||||
| 표 1. 쉘의 점근거동의 분류 | | | | |
|
||||
|----------------------|--------------------------------------|-------------------------|--|--|
|
||||
| | | | | |
|
||||
| 경우 | 하중<br>순수 휨을 유발하는 외력 | 분류 | | |
|
||||
| 순수 휨이 구속되지 | ∃<br>Ψ V∈ 0<br>such that<br>G V( )≠0 | (i) 휨지배거동 | | |
|
||||
| 않은 쉘 구조물,<br>Ψ0≠{ }0 | 순수 휨을 유발하지 않는 외력 | (ii) 불안정한 막지배 또는 혼합지배거동 | | |
|
||||
| | ,<br>G V( ) = 0<br>∀<br>Ψ V∈ 0 | | | |
|
||||
| | Admissible membrane loading | | | |
|
||||
| 순수 휨이 구속된 | G∈Ψm′ | (iii) 막지배거동 | | |
|
||||
| 쉘 구조물,<br>Ψ0 = 0{ } | Non-admissible membrane loading | | | |
|
||||
| | G∉Ψm′ | (iv) 혼합지배거동 | | |
|
||||
|
||||
<sup>Ψ</sup>0≠{ }<sup>0</sup>
|
||||
|
||||
U ∈Ψ<sup>0</sup>
|
||||
|
||||
$$A_{b}(\overrightarrow{U}, \overrightarrow{V}) = \overrightarrow{G}(\overrightarrow{V}), \ \forall \overrightarrow{V} \in \overrightarrow{\Psi}_{0}$$
|
||||
|
||||
$$\tag{30}$$
|
||||
|
||||
<sup>Ψ</sup>0= 0{ } Ψm Ψ
|
||||
|
||||
U ∈Ψ<sup>m</sup>
|
||||
|
||||
|
||||
|
||||
그림 2. 쉘의 변형거동
|
||||
|
||||
$$A_{m}(\overrightarrow{U}^{m}, \overrightarrow{V}) = \overrightarrow{G}(\overrightarrow{V}), \forall \overrightarrow{V} \in \overrightarrow{\Psi}_{m}$$
|
||||
|
||||
$$(31)$$
|
||||
|
||||
<sup>G</sup> <sup>Ψ</sup><sup>m</sup> <sup>G</sup>∈Ψm′
|
||||
|
||||
$$\left| \overrightarrow{G}(\overrightarrow{V}) \right| \le c_{\sqrt{A_m(\overrightarrow{V}, \overrightarrow{V})}}, \ \forall \overrightarrow{V} \in \overrightarrow{\Psi}$$
|
||||
(32)
|
||||
|
||||
<sup>G</sup>∈Ψm′
|
||||
|
||||
(31) (32)
|
||||
|
||||
|
||||
|
||||
그림 3. 쉘의 변형형상과 유효응력분포에 나타난 경계층(boundary layer)
|
||||
|
||||
3.2 쉘의 층과 특성길이 쉘의 응력/변형률/변위장들은 그 변화가 매끄러운 영역들 (smooth areas)과 그렇지 않은 여러 종류의 층들(layers)로 나누어진다. 층(layer)은 곡률(curvature)이나 두께의 변화와 같은 쉘의 형상의 변화, 적합하지 않은 변위경계조건 (incompatible boundary condition), 불규칙한 하중(irregular loading) 등에 의하여 유발된다(Lee and Bathe, 2002). 층 (layer)에서는 응력/변형률/변위 등이 매우 급하게 변하며 변 형에너지의 집중이 일어난다. 층의 특성길이(Lc, characteristic length)는 쉘 구조물의 두께에 따라 변하며 쉘의 두께(t) 와 전체 쉘 구조물 길이(L)의 함수로 나타나다.
|
||||
|
||||
$$L_c = ct^{1-l}L^l \tag{33}$$
|
||||
|
||||
여기서 c는 상수이며 l은 양의 실수 이다.
|
||||
|
||||
참고문헌(Lee and Bathe, 2002)는 쉘의 두께가 얇아짐에 따라 나타나는 Scodelis-Lo roof shell problem에서의 경계 층(boundary layer)과 쌍곡포물면(hyperbolic paraboloid) 쉘 구조문제의 내부층(inner layer)을 보여준다. 참고문헌(Bathe, Chapelle and Lee, 2003)에서는 또 다른 형태의 경계층이 쉘의 두께가 얇아짐에 따라 변화하는 것을 보여준다. 그림 3은 쉘의 변형형상(deformed shape)과 유효응력(effective stress)분포에 나타난 경계층(boundary layer)의 예를 보여주 고 있다.
|
||||
|
||||
3.3 점근거동의 해석 이론적인 방법으로 일반적인 쉘 구조물의 점근거동을 알아 내려는 시도는 Lovadina를 비롯한 많은 연구자들에 의해 수 행되었으나 쉬운 일이 아니었다(Lovadina, 2001). 근래에 수 치적인 방법에 의해 점근거동을 알아내는 연구가 진행되었 으며 유한요소해석을 통한 여러 가지 방법이 Lee와 Bathe에 의해 제안되었다(Lee and Bathe, 2002; Bathe, Chapelle and Lee, 2003). 본 논문에서는 가장 손쉬운 한가지 방법을 소 개한다.
|
||||
|
||||
쉘 구조물의 ρ(load scaling factor) 값을 구하게 되면 그 구조물의 점근거동을 알아낼 수 있다. 다음은 Lee와 Bathe 에 의해 제안된 ρ의 근사 값을 구하는 식이다.
|
||||
|
||||
$$\rho \cong \frac{\log E(\varepsilon + \Delta \varepsilon) - \log E(\varepsilon)}{\log(\varepsilon + \Delta \varepsilon) - \log \varepsilon}$$
|
||||
(34)
|
||||
|
||||
여기서 E는 ε(=t/L)에 대한 유한요소해석으로부터 구해진 쉘 구조물의 변형에너지(strain energy) 값이다. 주어진 ε<sup>에</sup> 대하여 유한요소해의 정확도가 높을수록 보다 정확한 ρ값을 얻을 수 있다. 계산된 ρ의 값이 1일 경우 구조물은 막지배 거동을, 3일 경우 휨지배거동을, 1과 3의 중간 값일 경우 막과 휨의 혼합지배거동을 한다.
|
||||
|
||||
Lovadina는 보간이론(interpolation theory)을 사용하여 쉘 구조물의 점근거동을 연구하였으며 ρ값과 쉘 구조물에 저장 되는 에너지들과의 관계를 나타내는 흥미로운 식을 제안하 였다(Lovadina, 2001).
|
||||
|
||||
$$\lim_{\varepsilon \to 0} R(\varepsilon) = \frac{\rho - 1}{2} \tag{35}$$
|
||||
|
||||
여기서 R(ε)는 쉘 구조물의 전체 변형에너지에 대한 휨 변
|
||||
|
||||
형에너지의 비이다.
|
||||
|
||||
$$R(\varepsilon) = \frac{\varepsilon^3 A_b(\vec{U}, \vec{U})}{\varepsilon^3 A_b(\vec{U}, \vec{U}) + \varepsilon A_m(\vec{U}, \vec{U})}$$
|
||||
(36)
|
||||
|
||||
역으로 쉘 구조물의 R(ε)를 계산하면 또한 쉘의 점근거동을 찾을 수 있는 ρ값을 알 수 있다.
|
||||
|
||||
Lee와 Bathe는 쉘 유한요소의 성능을 평가(benchmark)하 기 위한 쉘 문제로 널리 알려져 있는 Scodelis-Lo roof shell problem을 대상으로 하여 ρ값을 계산하였으며 그 쉘 구조물의 점근거동을 보여주었다(Lee and Bathe, 2002). 그 후에 Lovadina는 이론적인 방법을 이용해 같은 문제의 <sup>ρ</sup><sup>값</sup> 을 구하였고 두 결과는 일치하였다. 쉘 구조물의 휨지배거동 과 막지배거동에 대한 보다 자세한 예들을 참고문헌(Lee and Bathe, 2002)에서 볼 수 있으며, 쉘 두께의 변화에 따 <sup>라</sup>ρ값이 변동(fluctuation)하는 민감한 쉘 구조물의 점근거 동에 대한 해석이 참고문헌(Bathe, Chapelle and Lee, 2003)에 제시되어 있다.
|
||||
|
||||
# 4. 쉘 유한요소의 잠김현상과 균일최적수렴
|
||||
|
||||
쉘 구조물의 유한요소해석에서 직면한 가장 큰 어려움은 잠김현상(locking phenomenon)이다. 잠김현상은 쉘 구조물의 두께가 얇을 수록 유한요소해의 오차(error)를 매우 빠르게 증가시킨다. 본 장에서는 쉘의 점근거동에 대한 이해를 바탕 으로 쉘 유한요소의 잠김현상과 균일최적수렴에 대하여 살 펴본다.
|
||||
|
||||
4.1 쉘 유한요소의 종류와 잠김현상 쉘 유한요소의 종류는 크게 평면 쉘(flat shell) 유한요소 그룹과 감절점 쉘(degenerated shell or continuum mechanics based shell) 유한요소 그룹으로 나뉘어진다(Ahmad, Iron and Zienkiewicz, 1970; Bathe, 1996; Choi, Lee and Park, 1999; 최창근, 2002; Chapelle and Bathe, 2003). 평면 쉘 유한요소는 평판 유한요소와 평면응력 유한요소의 결합에 의 해 만들어진다. 곡면의 쉘 구조물을 여러 개의 평면으로 나 누어 표현한다는 물리적 개념에서 출발하였다. 장점은 유한 요소 정식화가 쉽고 면내회전자유도(drilling degrees of freedom)를 쉽게 도입하여 절점 당 6개의 자유도를 가질 수 있으며 그로 인해 절점 당 6개의 자유도를 갖는 빔(beam) 과 같은 유한요소들과 결합이 편리하다는 것이다. 그러나, 기 본적으로 요소자체의 형상이 평면이기 때문에 복잡한 곡면 형태의 쉘의 형상과 그 거동을 정밀하게 표현하기 어려워 곡면 쉘 구조물의 해석 시 수렴이 느리며 정확해로의 수렴 에 실패하는 경우가 발생하는 단점을 가지고 있다. 반면 3 차원 연속체역학에 근거한 감절점 쉘 요소는 절점 당 5개의 자유도를 가지고 있어 빔과 같은 유한요소들과의 결합 시 특별한 인위적인 방법이 필요하지만 쉘 고유의 복잡한 곡면 형상과 그 거동을 잘 표현해 낼 수 있고 정확해로의 수렴속 도가 빠르다. 또한, 2장에서 설명된 바와 같이 가장 일반적 인 쉘 수학모델(mathematical shell model)인 기본쉘수학모 델에 근거한다.
|
||||
|
||||
쉘 유한요소를 유한요소의 정식화(formulation)방법에 따라
|
||||
|
||||
분류하면 크게 변위법(displacement based method)에 의한
|
||||
|
||||
Ψ0
|
||||
|
||||
쉘 유한요소와 혼합법(mixed method)에 의한 쉘 유한요소로 나눌 수 있다. 변위법에 근거한 쉘 유한요소(displacement based shell finite element)는 막지배 구조물들의 해석에 있 어서는 이상적인 거동을 보인다. 그러나 요소의 종류와 근사 차수(interpolation order)에 상관없이 두께가 얇은 휨지배 (bending dominated) 또는 혼합지배(mixed) 쉘 구조물들의 해석에 있어서 유한요소해석의 해가 매우 느리게 수렴하는 치명적인 단점을 가지고 있는데 이를 잠김(locking)현상이라 고 한다. 주어진 요소망(mesh)에 대하여 유한요소해의 정확 성이 쉘의 두께가 얇아짐에 따라 급속히 나빠지는 현상을 말하며 최악의 경우 쉘의 두께가 얇아짐에 따라 변위장이 0 에 수렴하게 된다. 잠김현상은 막잠김(membrane locking)과 전단잠김(shear locking)현상으로 나누어질 수 있으며 잠김현상이 일어나는 근본적인 이유는 변위법에 의한 쉘 유한요소가 식 (29)의 순수 휨 변위공간(space) 을 충분히 근사할 수 없기 때 문이다. 즉, 잠김현상은 쉘 구조물의 점근거동과 직접적인 연 관관계를 가지고 있다. 쉘 구조물의 점근거동이 쉘의 형상, 경계조건, 하중 등에 따라 변하기 때문에 잠김현상 또한 쉘 의 형상, 경계조건, 하중에 의존적이다. 막잠김현상은 곡률을 가진 쉘 구조물에서만 발생하고 평면형상의 쉘 구조물에서 는 발생하지 않으나 전단잠김현상은 곡률에 관계없이 발생 한다. 잠김현상은 휨지배 및 혼합지배거동 시 식 (21a)와 (21d) 에 있는 변형률항들로부터 발생한다. 참고문헌(Bathe, Lee and Hiller, 2003)과 (Lee and Bathe, 2005)는 잠김현상이 유한요소해에 어떻게 나타나는지를 보여주고 잠김현상이 제 거되기 어려운 근본적인 이유를 잘 설명하고 있다. 실제 쉘 유한요소해석에 있어서 잠김현상은 과대한 강성 또는 그로 인한 과소한 변위, 응력, 변형률 및 변형에너지로 나타난다. 그러므로 두께가 얇은 쉘 구조물의 설계 시 충분 하게 조밀하지 못한 유한요소망을 사용하여 해석한 결과를 이용할 경우(강성이 과대평가되어) 과소설계의 오류를 범할 수 있다. 이와 같이 유한요소해의 오차 또는 정확도가 쉘 구조물의 두께에 따라 변하는 현상인 잠김현상은 쉘 구조물 의 유한요소해석에 있어서 빈번하게 일어나기 때문에 쉘 구 조물의 해석에 앞서 잠김현상에 대한 이해는 필수적이다. 4.2 유한요소해의 오차측정 신뢰할 만한 쉘 유한요소의 해는 사용된 요소의 수가 증 가함에 따라 2장에서 설명된 쉘 유한요소의 수학모델 (mathematical shell model)의 정확해(exact solution)에 수 렴하여야 하며 잠김현상을 알아보기 위해서는 쉘 유한요소 해의 수렴곡선(convergence curve)을 관찰해야 한다. 여기서 쉘 유한요소의 수렴을 측정하기 위하여 적절한 규준(norm) 을 사용하는 것이 매우 중요하다. 그 규준(norm)은 한 점에 서 특정 물리량 값(point-wise value)의 수렴이 아니라 쉘 구조물 전체 영역에서의 해의 수렴을 고려할 수 있어야 한 다. 일반적으로 한 점에서의 변위/응력/변형률 등을 가지고 수렴을 측정하는 방법이 많이 사용되어왔는데 이는 전체영 역에서의 수렴을 대표하지 않으므로 적절하지 못하다. 변형
|
||||
|
||||
⋅ s
|
||||
|
||||
$$\left| \overrightarrow{U} - \overrightarrow{U}_h \right|_s^2 = \int_{\Omega} \Delta \overrightarrow{\varepsilon}^T \Delta \overrightarrow{\sigma} d\Omega \tag{37}$$
|
||||
|
||||
<sup>U</sup> Uh <sup>ε</sup> <sup>σ</sup>
|
||||
|
||||
$$\overleftarrow{\mathcal{E}} = \left[ \varepsilon_{xx}, \ \varepsilon_{yy}, \ \varepsilon_{zz}, \ 2\varepsilon_{xy}, \ 2\varepsilon_{yz}, \ 2\varepsilon_{zx} \right]^{T}$$
|
||||
(38a)
|
||||
|
||||
$$\vec{\sigma} = [\sigma_{xx}, \sigma_{yy}, \sigma_{zz}, \sigma_{xy}, \sigma_{yz}, \sigma_{zx}]^T$$
|
||||
(38b)
|
||||
|
||||
$$\Delta \vec{\varepsilon} = \vec{\varepsilon} - \vec{\varepsilon}_h = \vec{\varepsilon}(\vec{x}) - \mathbf{B}_h \vec{x}_h) \vec{U}_h \tag{39a}$$
|
||||
|
||||
$$\Delta \vec{\sigma} = \vec{\sigma} - \vec{\sigma}_h = \vec{\sigma}(\vec{x}) - \mathbf{C}_h(\vec{x}_h) \mathbf{B}_h(\vec{x}_h) \vec{U}_h$$
|
||||
(39b)
|
||||
|
||||
에너지의 산술적 차이 또한 규준으로 사용할 수 있지만 변 위법 정식화에 근거한 유한요소가 아닌 경우에는 일반적으 로 사용될 수 없기 때문에 정식화의 방법에 관계없이 일반 적으로 쓰일 수 있는 규준이 필요하다. Hiller와 Bathe에 의해 제안된 s-norm( )은 쉘 구조물 전체의 거동을 반영할 수 있을 뿐만 아니라 물리적인 개념 으로부터 유도되었기 때문에 정식화 방법에 관계없이 사용 될 수 있다(Hiller and Bathe, 2003). 여기서 는 정확해이며, 는 유한요소해이다. 와 는 전체직각좌표계(global Cartesian coordinate system)에서 정 의된 변형률 벡터와 응력 벡터이다. 식 (37)에서 변형률과 응력에 대하여 정확해와 유한요소해의 차이(difference)는 다음과 같이 구해질 수 있다. x xh
|
||||
|
||||
$$\vec{x} = \Pi(\vec{x}_h) \tag{40}$$
|
||||
|
||||
(37) (38a) (38b) (39a) (39b) 여기서 C는 재료의 응력-변형률 관계 행렬(matrix)이고 B는 변형률-변위(strain-displacement)관계 연산자(operator)이 다. 위치벡터 와 는 각각 원래 쉘 구조물의 영역 (domain)과 이산화된 유한요소 모델의 영역에 대응된다. 두 벡터의 관계는 일대일 사상(injective mapping, Π)에 의하여 정의할 수 있다. (40) 일반적인 쉘 구조문제에 있어서 이론적인 방법을 이용하여 정확해를 찾아내는 것은 거의 불가능하므로 매우 조밀한 유 Uref
|
||||
|
||||
$$\left| \overrightarrow{U}_{ref} - \overrightarrow{U}_h \right|_s^2 = \int_{\Omega_{ref}} \Delta \overrightarrow{\varepsilon}^T \Delta \overrightarrow{\sigma} d\Omega_{ref}, \tag{41}$$
|
||||
|
||||
여기서
|
||||
|
||||
$$\Delta \vec{\xi} = \vec{\xi}_{ref} - \vec{\xi}_h = \mathbf{B}_{ref}(\vec{x}_{ref}) \vec{U}_{ref} - \mathbf{B}_h(\vec{x}_h) \vec{U}_h, \qquad (42a)$$
|
||||
|
||||
$$\Delta \overrightarrow{\sigma} = \overrightarrow{\sigma}_{ref} - \overrightarrow{\sigma}_h = \mathbf{C}_{ref}(\overrightarrow{x}_{ref}) \mathbf{B}_{ref}(\overrightarrow{x}_{ref}) \overrightarrow{U}_{ref} - \mathbf{C}_h(\overrightarrow{x}_h) \mathbf{B}_h(\overrightarrow{x}_h) \overrightarrow{U}_h,$$
|
||||
(42b)
|
||||
|
||||
$$\vec{x}_{ref} = \Pi(\vec{x}_h) \tag{42c}$$
|
||||
|
||||
한요소망을 사용하여 계산된 해를 정확해로 고려하여 snorm을 계산하는 것이 훨씬 실용적이다. 정확해로 고려된 유한요소해를 라고 하면 위식의 s-norm은 다음과 같이 나타내어질 수 있다. , (41) , (42a) , (42b) (42c) 다양한 쉘 구조문제와 여러 경우의 쉘 두께에 대하여 유한 요소해의 수렴을 상호 비교 가능하게 하기 위해서는 상대오
|
||||
|
||||
차(relative error)를 사용하여야 한다. 상대오차(
|
||||
$$E_h$$
|
||||
)는 다음과
|
||||
같이 정의된다.
|
||||
$$E_h = \frac{|\overrightarrow{U}_{ref} - \overrightarrow{U}_h|_s^2}{||\overrightarrow{U}_{ref}||_s^2} \tag{43}$$
|
||||
|
||||
|
||||
|
||||
그림
|
||||
|
||||
|
||||
|
||||
그림
|
||||
|
||||
유한요소해의 이론적인 수렴은 다음과 같다.
|
||||
|
||||
$$E_h \cong ch^{2k} \tag{44}$$
|
||||
|
||||
여기서 c는 상수, h는 사용된 유한요소의 크기, k는 유한 요소의 변위 근사함수(displacement interpolation function) 차수이다. 예를 들면 선형(linear) 근사함수를 사용하는 3절 점 및 4절점 쉘 유한요소에 대하여 k = 1이며, 2차의 (quadratic) 근사함수를 사용하는 6절점 및 9절점 쉘 유한요 소에 대하여 k = 2이다. S-norm은 수치적인 방법에 의해 계 산될 수 있다(Lee, Noh and Bathe, 2007). 을 때-MITC4 쉘 요소
|
||||
|
||||
4.3 수렴곡선과 균일최적수렴 다음으로 2개의 휨지배 쉘 구조문제들의 예를 통하여 잠 김현상이 s-norm을 이용하여 구해진 수렴곡선에 어떻게 나 타나는지를 보여주고 균일최적수렴(uniform optimal convergence)에 대하여 알아본다.
|
||||
|
||||
판(plate) 구조는 곡률이 영인 쉘 구조물로서 쉘 구조의 가장 단순한 형태이다. 첫 번째 예로 그림 4(a)에 보여진 네 변이 완전히 구속된 평판 휨(plate bending) 구조문제에 대하여 쉘 유한요소해의 수렴곡선을 살펴보자. 사용된 탄성 계수는 1.7472×107 , 포아손비는 0.3, 길이 L=1.0, 하중은 단위 면적당 90이 <sup>−</sup>Z 방향으로 작용한다. 구조물의 형상, 하중, 변위경계조건의 대칭성으로 인해 그림 4(a)의 색칠된 부분만 해석되었다. 5. 두께의 변화에 따른 4절점 쉘 유한요소들의 수렴곡선: (a) 잠김현상이 일어날 때-QUAD4 쉘 요소 (b) 잠김현상이 일어나지 않
|
||||
|
||||
주어진 평판 휨 문제를 풀기 위하여 변위법(displacement based formulation)에 의한 4절점 감절점 쉘 유한요소 (QUAD4)와 혼합법(mixed formulation)에 의한 4절점 감절 점 쉘 유한요소(MITC43 )를 사용하였다(Ahmad, Irons and
|
||||
|
||||
MITC(Mixed Interpolation of Tensorial Components) 방법을 이용하여 개발된 n절점 쉘 유한요소를 MITCn 유한요소라고 부른다. 즉, MITC4는 4절점, MITC6는 6절점 쉘 유한요소이다.
|
||||
|
||||
|
||||
|
||||
그림
|
||||
|
||||
Zienkiewicz, 1970; Bathe and Dvorkin, 1989; Bathe, 1996). 각각의 경우 두께 변화에 따른 수렴곡선에서 잠김현상이 어 떻게 나타나는지를 살펴보자.
|
||||
|
||||
그림 5는 두 가지 4절점 쉘 유한요소들에 대하여 쉘 두 께의 변화(t/L=1/10, 1/100, 1/1000, 1/10000)에 따른 수렴 곡선들(convergence curves)을 보여준다. 여기서 h는 사용된 유한요소의 크기를 나타내며 상대오차를 계산하기 위하여 snorm을 사용하였다. h와 상대오차에 log를 취함으로써 식 (44)의 이론적인 수렴률과 구해진 수렴곡선의 수렴률을 비교 할 수 있다. 그림 5의 각각의 그래프에 이론적인 수렴곡선 의 기울기(2k)가 두께가 굵은 직선으로 그려져 있다.
|
||||
|
||||
그림 5(a)는 잠김현상이 일어날 때의 전형적인 수렴곡선들 이다. 쉘의 두께가 얇아짐에 따라 상대오차(relative error)가 급증하는 것을 알 수 있다. 반면에 그림 5(b)의 수렴곡선들 에서는 상대오차가 쉘 두께(t)에 영향을 받지 않고 오직 요 소의 크기 h에만 관계함을 알 수 있다. 또한 그림 5(b)에서 는 수렴곡선들이 이론적인 기울기와 같다. 그림 5(b)와 같은 형태의 쉘 유한요소해의 수렴형태를 균일최적수렴(uniform optimal convergence)이라고 한다. 때 - MITC6 쉘 요소
|
||||
|
||||
두 번째 예제로 그림 4(b)는 한 변이 구속된 쌍곡포물면 (hyperbolic paraboloid) 쉘 구조문제이다. 쉘의 중심면 (midsurface)은 다음과 같이 정의된다.
|
||||
|
||||
$$\begin{pmatrix} X \\ Y \\ Z \end{pmatrix} = L \begin{pmatrix} \xi^1 \\ \xi^2 \\ (\xi^1)^2 - (\xi^2)^2 \end{pmatrix}; (\xi^1, \xi^2) \in \left[ -\frac{1}{2}, \frac{1}{2} \right]^2$$
|
||||
(45)
|
||||
|
||||
경계조건은 X=−0.5인 변을 따라 완전히 구속되며 자중(selfweight)이 <sup>−</sup>Z방향으로 작용한다. 구조물의 형상과 변위 및 경계조건이 Y=0인 면을 따라 대칭이므로 그림 4(b)의 색칠 된 부분만 해석되었다. 사용된 탄성계수는 2.0×1011, 포아손 비(Poisson's ratio)는 0.3, 길이 L=1.0이며 자중은 단위면적 당 80이다.
|
||||
|
||||
주어진 휨지배문제를 풀기 위해 변위법(displacement
|
||||
|
||||
based formulation)에 의한 6절점 삼각형 쉘 유한요소 (QUAD6)와 혼합법(mixed formulation)에 의한 6절점 삼각 형 쉘 유한요소(MITC6)를 사용하였다(Ahmad, Irons and Zienkiewicz, 1970; Bathe, 1996; Lee and Bathe, 2004).
|
||||
|
||||
그림 6에서 t/L이 1/100, 1/1000, 1/10000인 세가지 경우 에 대하여 6절점 삼각형 쉘 유한요소들의 수렴곡선들 (convergence curves)을 보여준다. 각각의 그래프에 이론적인 수렴곡선의 기울기(2k)가 두께가 굵은 직선으로 그려져 있다. 그림 6(a)는 QUAD6 유한요소가 쉘의 두께가 얇아짐에 따 라 계산된 해의 오차가 늘어나는 경향, 즉, 잠김현상을 유발 한다는 것을 보여준다. 그림 6(b)에서는 MITC6 유한요소를 사용했을 때 잠김현상이 완전히 제거되지는 않으나 QUAD6 유한요소에 비하여 상당히 완화되었음을 나타낸다.
|
||||
|
||||
여기서 우리는 두 가지 쉘 구조문제들을 대상으로 쉘 유 한요소해의 수렴곡선과 잠김현상에 대하여 알아보았다. 중요 한 점은 어떤 쉘 유한요소가 몇 가지 쉘 구조문제들에 대하 여 잠김현상을 일으키지 않는다고 해서 다른 쉘 구조문제들 에 대하여도 잠김현상을 유발시키지 않는다고 말할 수 없다 는 것이다. 즉, 몇 가지 쉘 구조문제들에 대하여 균일최적수 렴을 보이며 잠김현상을 일으키지 않는 유한요소도 다른 쉘 구조문제들에 대해서 잠김현상을 보일 수 있으며 모든 쉘 구조문제들에 대하여 잠김현상을 일으키지 않는 쉘 유한요 소를 개발하는 것은 극히 어렵다. 보다 다양한 쉘 구조문제 들에 대한 유한요소해의 수렴곡선에 대한 예는 참고문헌에 나와있는 Lee와 Bathe의 논문들에서 찾을 수 있다. 6. 두께의 변화에 따른 6절점 쉘 유한요소들의 수렴곡선: (a) 잠김현상이 일어날 때-QUAD6 쉘 요소 (b) 잠김현상을 완화시켰을
|
||||
|
||||
쉘 유한요소의 잠김현상을 알아보기 위해서는 다양한 형상 을 가진 휨 및 혼합지배 쉘 구조문제에 대하여 그림 5와 6 에 보여진 것과 같이 쉘의 두께를 변화시키며 수렴을 시험 하여야 한다. 또한 특정 쉘 요소가 휨지배문제에 좋은 거동 을 보인다고 해서 막지배문제에 대해서도 좋은 거동을 보이 는 것은 아니다. 결론적으로 다양한 형상의 쉘 구조문제들을 고려하여 휨과 막지배라는 두 가지 양단의 거동에서 이상적 인 수렴을 보이는 쉘 유한요소가 가장 바람직하고 이런 쉘 유한요소는 혼합지배 쉘 구조문제에 대하여도 좋은 거동을 보일 것이다. 즉, 이상적인 쉘 유한요소는 여러 가지 점근적 인 거동을 보이는 다양한 형상의 쉘 구조문제들에 대하여 균일최적수렴을 보여 주어야 한다.
|
||||
|
||||
# 5. 쉘 유한요소의 성능평가에 대하여
|
||||
|
||||
나날이 셀 수 없을 만큼 많은 쉘 유한요소들이 개발되고 있지만 쉘 유한요소의 성능평가는 아직도 고전적인 방법을 통하여 이루어지고 있다. 본 장에서는 이상적인 쉘 유한요소 의 조건과 잠김현상을 완화시키는 방법들을 정리하고 쉘 유 한요소의 성능평가를 위한 방법론을 제시한다.
|
||||
|
||||
5.1 이상적인 쉘 유한요소 유한요소 구조해석 문제는 다음과 같은 변분식(variational form)으로 나타내어질 수 있다.
|
||||
|
||||
Find such that
|
||||
|
||||
$$A_h(\overrightarrow{U}_h, \overrightarrow{V}_h) = \overrightarrow{F}(\overrightarrow{V}_h), \ \forall \overrightarrow{V}_h \in \overrightarrow{\Psi}_h, \tag{46}$$
|
||||
|
||||
선형식(bilinear form)이고 는 유한요소 변위장의 Sobolev 공간(space)이다. 물론 이 성립한다. Ψh <sup>Ψ</sup>h⊂Ψ
|
||||
|
||||
$$A_{h}(\overrightarrow{U}_{h}, \overrightarrow{V}_{h}) = \overrightarrow{V}_{h}^{T} \left( \int_{\Omega_{h}} \mathbf{B}_{h}^{T} \mathbf{C}_{h} \mathbf{B}_{h} d\Omega_{h} \right) \overrightarrow{U}_{h}$$
|
||||
|
||||
$$(47)$$
|
||||
|
||||
여기서 는 유한요소해(finite element solution), 는 유한요소 시험함수, 는 유한요소 변위장의 공간, 는 외력을 나타내는 선형식(linear form)이다. 쉘 유한요소해 석일 경우 식 (46)은 식 (26)의 형태로 표현될 수 있다. Ψh
|
||||
|
||||
일반적인 쉘 구조물의 효과적인 유한요소해석에 쓰일 수 있는 이상적인 쉘 유한요소의 개발은 매우 어려운 일이다. 이상적인 쉘 유한요소의 조건을 다음과 같이 정리할 수 있다.
|
||||
|
||||
여기서 Ah(·,·)는 유한요소법으로 이산화(discretization)된 겹 ● 첫째, 쉘 유한요소는 거짓영에너지모드(spurious zero energy mode)를 갖지 말아야 한다. 변위경계조건이 주어지 지 않은 임의의 형상을 갖는 개개의 쉘 유한요소는 물리적 인 강체운동에 대응하는 6개의 영에너지모드(zero energy mode)만을 가져야 한다. 이 조건을 ellipticity(타원율)라 하 며 다음과 같이 정의된다. Uh∈Ψh Ah( ) Uh, Vh = F V( )h , ∀Vh∈Ψh Ah( ) Uh, Vh = Vh dΩh ⎛ ⎞Uh
|
||||
|
||||
$$\exists \alpha > 0 \text{ such that } \forall \overrightarrow{U}_h \in \overrightarrow{\Psi}_h, \ A_h(\overrightarrow{U}_h, \overrightarrow{U}_h) \ge \alpha |\overrightarrow{U}_h|_1^2$$
|
||||
(48)
|
||||
|
||||
여기서 α는 상수이며 는 1차 Sobolev norm4 이다. ⋅ 1
|
||||
|
||||
이 조건은 쉘 유한요소 강성행렬(stiffness matrix)의 고유 치들(eigenvalues) 중 0인 개수와 그에 대응하는 고유벡터들 (eigenvectors)을 살펴봄으로써 시험될 수 있다. 탄성체는 강 체운동이 아닌 변위에 대하여 변형에너지를 저장한다. 식 (48)의 조건을 만족시키지 못하는 유한요소는 강체운동이 아 닌 변위에 대하여 변형에너지를 저장할 수 없으며 이는 물 리적으로 적합하지 못하다. Uh Vh F( ) ⋅ Uh∈Ψh Ah( ) Uh, Uh α Uh 1
|
||||
|
||||
● 둘째, 쉘 유한요소는 쉘 수학모델에 근거하였기 때문에 쉘
|
||||
|
||||
유한요소해석의 해는 주어진 쉘 구조문제에 대하여 사용된 증가함에 따라 쉘 수학모델의 정확해에 수렴해야 한다. 이 조건을 consistency(무모순성 또는 정합성)라 부르며 다음과 같이 정의된다.
|
||||
|
||||
$$\lim_{h \to 0} \overrightarrow{U}_h = \overrightarrow{U} \quad \text{or} \quad \lim_{h \to 0} A_h(\overrightarrow{U}_h, \overrightarrow{U}_h) = A(\overrightarrow{U}, \overrightarrow{U})$$
|
||||
(49)
|
||||
|
||||
bilinear form)이며 는 정확해(exact solution)이다.
|
||||
|
||||
이 조건이 만족되지 않을 경우 쉘 유한요소의 해는 이론 해에 수렴하지 못 하므로 신뢰할 만한 결과를 줄 수 없다.
|
||||
|
||||
요소의 크기(h)가 줄어듦에 따라 또는 사용된 요소의 수가 여기서 Ah(·,·)는 쉘 수학모델의 정확한 겹선형식(exact ● 셋째, 쉘 유한요소는 모든 종류의 휨 및 혼합지배 쉘 구조 문제에 대하여 균일최적수렴(uniform optimal convergence) 을 보여야 한다. 이 조건을 만족시키는 쉘 유한요소는 비 로소 쉘의 두께와 상관 없이 전단잠김과 막잠김으로부터 자유로운 유한요소가 된다. 이는 본 논문의 4장에서 설명 된 방법에 의해 시험될 수 있다. 혼합법에 의해 정식화 (mixed formulation)된 쉘 유한요소에 대하여 이 조건을 "inf-sup condition"이라 부른다(Bathe, 1996; Bathe, Iosilevich and Chapelle, 2000b). Uh = U h→0 Ah( ) Uh, Uh = A U( ) , U U
|
||||
|
||||
위에서 언급된 세가지 조건을 모두 만족하는 이상적이 쉘 유한요소를 개발하는 것은 극히 어렵다. 실용적인 쉘 유한요 소해석에서는 보다 완화된 다음의 조건들을 만족시키는 쉘 요소의 사용이 권장된다(Lee and Bathe, 2004).
|
||||
|
||||
- <sup>−</sup> 거짓영에너지모드(spurious zero energy mode) 없음 (ellipticity조건 만족)
|
||||
- <sup>−</sup> Consistency조건 만족
|
||||
- <sup>−</sup> 판 문제의 해석에 있어서 전단잠김 없음
|
||||
- − 막지배거동 쉘 구조물에 대하여 균일최적수렴
|
||||
- 범위(1/10~1/10000)에서 신뢰할 수 있는 결과
|
||||
- <sup>−</sup> 비선형 해석에 있어서 효율적인 정식화
|
||||
|
||||
5.2 잠김현상의 제어 지난 수십 년 동안 쉘 유한요소의 잠김현상을 제어하기 위해 많은 방법들이 고안되어왔다. 그 방법들은 크게 세가지 로 나누어질 수 있다.
|
||||
|
||||
- <sup>−</sup> 식 (47)의 변형률-변위 관계 연산자 Bh를 변형하여 전 단 및 막 변형률장의 차수를 줄여주는 방법, (예) reduced integration, ANS method, MITC method
|
||||
- <sup>−</sup> 식 (47)의 Bh에 새로운 항을 추가하여 전단 및 막 변 형률장의 공간(space)을 늘려주는 방법, (예) EAS method
|
||||
- <sup>−</sup> 식 (46)에서 유한요소 변위장 의 공간( )을 늘려주 는 방법, (예) non-conforming method
|
||||
|
||||
− 휨 및 혼합지배거동 쉘 구조물에 대하여 실용적인 t/L의 쉘 유한요소의 잠김현상을 완화시키기 위한 대부분의 방법 들은 위의 세 분류들(categories)에 속하며 세 방법들을 복합 적으로 이용한 예들도 있다. Uh Ψh
|
||||
|
||||
가장 쉬운 방법은 감차적분(reduced integration)을 사용하
|
||||
|
||||
$$\|\vec{v}\|_{1}^{2} = \int_{\Omega} \sum_{i=1}^{3} (V_{i})^{2} d\Omega + \int_{\Omega} \sum_{i,j=1}^{3} \left(\frac{\partial V_{i}}{\partial x_{j}}\right)^{2} d\Omega$$
|
||||
|
||||
<sup>1</sup>차 Sobolev norm의 제곱은 다음과 같이 정의된다(Bathe, 1996).
|
||||
|
||||
는 방법이다(Bathe, 1996). 그러나 이 방법은 거짓영에너지 모드(spurious zero energy mode)를 유발시키는 치명적인 단점을 지니고 있다. 이점을 극복하기 위하여 각종 안정화 (stabilization)기법이 사용된다.
|
||||
|
||||
비적합모드(non-conforming or incompatible mode)를 추 가함으로써 요소의 휨 모드를 복원하여 쉘 유한요소의 잠김 현상을 완화할 수 있다. 이 방법은 요소간 변위의 적합성 (inter-elemental compatibility)을 만족시키지 못하는 단점과 최종 강성행렬을 구하기 위해 정적응축(static condensation) 을 사용하기 때문에 비선형 해석으로 확장 시 식이 복잡해 지는 단점을 가지고 있다(Choi, Lee and Park, 1999; 최창 근, 2002).
|
||||
|
||||
변위와 변형률을 각각 따로 근사하는 혼합법(mixed formulation)에 근거한 방법들은 쉘 유한요소의 잠김현상을 완화시키는 알려진 방법들 중 가장 우수하다고 평가된다. 그 중 MITC(Mixed Interpolation of Tensorial Components) 방법은 이론적으로 잘 확립되어 있으며 다양한 수치실험으 로 잠김현상의 제어에 매우 효과적임이 입증되었다(Bathe and Dvorkin, 1989; Bathe, 1996; Bathe, Iosilevich and Chapelle, 2000a; Hiller and Bathe, 2003). 최근의 연구들 은 MITC방법에 의해 만들어진 사각형 쉘 유한요소들이 이 상적인 쉘 유한요소에 상당히 접근해 있음을 보여준다(Hiller and Bathe, 2003; Bathe, Lee and Hiller, 2003).
|
||||
|
||||
MITC 방법에서는 변위법에 근거한 쉘 유한요소의 특정한 위치들(tying points)에서 공변변형률들(covariant strains)을 이용하여 원래 공변(covariant)변형률의 근사차수보다 낮은 차수로 변형률장을 근사(interpolation)한다. 일반적으로 근사 함수는 낮은 차수일수록 잠김현상 제어에 더 효과적이지만 너무 낮은 근사차수는 막지배거동을 하는 쉘 구조문제를 풀 때 유한요소해가 이론해에 수렴하지 못하는(즉, consistency 조건을 만족시키지 못하는) 현상을 유발시킨다. 심한 경우에 는 거짓영에너지모드를 발생시켜 ellipticity조건까지 만족시 킬 수 없게 만든다. 그러므로 MITC방법의 핵심은 휨 및 혼합지배거동 시 잠김현상을 줄여주면서 막지배거동 시 수 렴성을 유지하는(즉, consistency를 만족시키는) 균형 잡힌 변형률의 근사장을 찾아내는 것이다.
|
||||
|
||||
대체변형률장(assumed strain field)을 사용하는 ANS (Assumed Natural Strain)방법은 MITC방법과 유사하며, EAS(Extended Assumed Strain)방법은 ANS 또는 MITC방 법에 추가적인 변형률장을 도입하여 변형률장이 표현 가능 한 형태들(patterns)의 수를 늘려주는 방법이다. EAS방법은 비적합모드를 사용하는 방법과 비슷하게 추가된 변형률장의 자유도를 제거하기 위해 정적응축을 필요로 한다는 단점을 가지고 있으며 기존의 ANS나 MITC방법에 비하여 쉘 유한 요소의 수렴성을 개선시킬 수 있지만 그 효과가 크지는 않 다고 알려져 있다.
|
||||
|
||||
다른 여러 가지 방법을 사용하여 휨 및 혼합지배거동 쉘 구조문제에 대하여 보다 유연(flexible)한 거동을 하는 쉘 유 한요소를 개발하는 것은 어렵지 않디. 그러나 그와 동시에 consistency와 ellipticity조건들을 모두 만족시키는 것은 쉽지 않다. 앞으로 여러 종류의 개발된 쉘 유한요소에 대하여 쉘 이론에 바탕을 둔 심도 있는 성능시험(benchmark test)과 연구가 필요하다.
|
||||
|
||||
5.3 쉘 유한요소의 성능평가 일반적으로 유한요소법을 사용하여 쉘 구조물을 해석하는 대부분의 기술자들(engineers)은 구하여진 해의 오차(error)에 대한 평가 없이 해석결과를 받아들이기 때문에 쉘 유한요소 를 개발하는 연구자들은 개발된 쉘 유한요소를 실제 해석에 사용하기에 앞서 성능을 평가하고 그 결과를 보고하여야 한 다. 개발된 쉘 유한요소의 오차특성이 명확하게 알려진다면 사용자들은 주어진 쉘 유한요소를 어떻게 올바르게 사용할 수 있을지를 판단할 수 있게 된다. 쉘 유한요소의 성능평가 는 다음에 열거된 사항들을 고려하여 이루어져야 한다.
|
||||
|
||||
# ● 기본시험
|
||||
|
||||
쉘 유한요소는 표 2에 정리되어있는 기본시험들(basic tests)을 통과하여야 한다. 기본시험들을 통과하지 못하는 쉘 유한요소의 사용은 바람직하지 못하다.
|
||||
|
||||
# ● 성능평가방법
|
||||
|
||||
쉘 유한요소의 성능을 평가하기 위해서는 다양한 쉘 구조 문제들이 사용되어야 한다. 쉘 유한요소들의 성능을 비교평 가하기 위하여 오래 전부터 많은 쉘 해석문제들이 제안되어 왔다. 현재까지 가장 널리 쓰이는 성능평가방법은 1985년에 MacNeal와 Harder에 의해 정리된 것으로 두께가 정해진 몇 가지 쉘 구조문제들을 해석하여 정해진 위치에서 구해진 변 위 및 응력/변형률 등의 결과치들을 유한요소망을 조밀화 하 면서 비교하는 것이다. 이미 언급한 바와 같이 몇몇 점들에 서의 해의 수렴을 측정하는 것은 전체 유한요소해의 거동을 올바르게 반영할 수 없다. 변위형상이나 응력/변형률의 분포 는 유한요소해의 전체적인 수렴정도를 보여줄 수 있으며 이 것들을 비교하는 것은 매우 좋은 보완방법이다. 그러나 이 방법으로 과연 유한요소해가 어느 정도 수렴했는지를 측정 하여 그 정도를 한 개의 값으로 보여주기는 대단히 어렵다. 그러므로, 응력과 변형률의 오차분포로부터 구해진 s-norm은 전체 유한요소해를 반영하는 좋은 오차측정의 규준이 된다. 또한 4.3절에서 설명한 바와 같이 두께의 변화를 고려한 성
|
||||
|
||||
| 표 2. 쉘 유한요소의 기본시험 | | | |
|
||||
|----------------------------------------------------------------------|--------------------------|------------------------------------|--|
|
||||
| 시험 | 대상 | 참고문헌 | |
|
||||
| 영에너지 시험 (Zero<br>energy mode test) | 사각형 쉘 유한요소<br>삼각형 쉘 유한요소 | Bathe, 1996 | |
|
||||
| 조각 시험 (Patch tests)<br>− Membrane patch test<br>− Bending patch test | 사각형 쉘 유한요소<br>삼각형 쉘 유한요소 | Bathe, 1996<br>Lee and Bathe, 2004 | |
|
||||
| 요소 등방성 시험 (Element isotropy test) | 삼각형 쉘 유한요소 | Lee and Bathe, 2004 | |
|
||||
|
||||
|
||||
|
||||
그림 7. Gaussian곡률에 따른 곡면의 종류: (a) Positive Gaussian curvature, (b) Zero Gaussian curvature, (c) Negative Gaussian curvature
|
||||
|
||||
<sup>표</sup> 3. 쉘 유한요소의 성능평가를 위한 쉘 구조문제의 예들(Bathe, Iosilevich and Chapelle, 2000; Lee and Bathe, 2002; Bathe, Chapelle and Lee, 2003; Bathe, Lee and Hiller, 2003; Chapelle and Bathe, 2003; Hiller and Bathe, 2003; Lee and Bathe, 2004; Lee and Bathe, 2005; Lee, Noh and Bathe, 2007)
|
||||
|
||||
| 쉘 구조문제 (shell<br>problems) | Gaussian 곡률 | 점근거동 (ρ) |
|
||||
|----------------------------------------------------|-------------|------------------|
|
||||
| Fully clamped plate problem | Zero | 휨지배 (ρ = 3.0) |
|
||||
| Scodelis-Lo roof shell problem | Zero | 혼합지배 (ρ = 1.75) |
|
||||
| Modified Scodelis-Lo roof shell problem | Zero | 막지배 (ρ = 1.0) |
|
||||
| Free cylindrical shell problem | Zero | 휨지배 (ρ = 3.0) |
|
||||
| Fixed cylindrical shell problem | Zero | 막지배 (ρ = 1.0) |
|
||||
| Clamped hemispherical cap problem | Positive | 막지배 (ρ = 1.0) |
|
||||
| Monster shell problem | Positive | Not well-defined |
|
||||
| Partly clamped hyperbolic paraboloid shell problem | Negative | 휨지배 (ρ = 3.0) |
|
||||
| Free hyperboloid shell problem | Negative | 휨지배 (ρ = 3.0) |
|
||||
| Fixed hyperboloid shell problem | Negative | 막지배 (ρ = 1.0) |
|
||||
|
||||
능평가방법을 사용하는 것이 바람직하다. ● 층(layer) 쉘의 응력/변형률/변위장들이 급격히 변하는 층(layer)의 폭, 즉, 특성길이(characteristic length)는 쉘의 두께에 따라 변화 한다. 일반적으로 특성길이는 쉘의 두께가 얇아짐에 따라 식 (33)에 의하여 급격히 줄어든다. 층에서의 에너지 집중현상 때문에 이런 층이 발생하는 쉘 구조문제를 풀 때는 균일한 유한요소망을 사용하여 균일최적수렴을 얻기 힘들다. 각각 층의 특성길이를 반영한 유한요소망을 사용하여야 한다. 즉, 층이 발생하는 영역에서 보다 조밀한 유한요소망의 사용이 요구된다(Bathe, Iosilevich and Chapelle, 2000a). 이러한 유한요소망을 "graded mesh"라 부른다. ● 중심면의 곡률 쉘 구조물의 중심면은 곡률(curvature)을 가지고 있다. 곡 면은 Gaussian곡률의 부호에 따라 세가지로 나누어질 수 있 다. 특히 Gaussian곡률이 음인 곡면을 가지는 쉘 구조물은 유한요소해석에 있어서 상당한 어려움이 뒤따른다(Lee and Bathe 2004). 쉘 유한요소의 성능을 평가하기 위한 해석시 험문제들(benchmark test set)은 다양한 곡률을 고려하여 구 성되어야 한다. 이는 어떤 쉘 유한요소가 특정한 곡률을 가 지는 쉘 구조문제에서 좋은 수렴성을 보인다고 하여 다른 곡률을 가지는 구조문제에 대하여도 좋은 수렴을 보이는 것 은 아니기 때문이다. 그림 7은 Gaussian곡률에 따른 곡면의 예들을 보여주고 있다.
|
||||
|
||||
# ● 점근거동의 종류
|
||||
|
||||
3장에서 우리는 쉘 구조물의 두께가 얇아짐에 따라 나타나
|
||||
|
||||
는 3가지 점근거동(휨지배, 막지배, 혼합지배거동)을 살펴 보 았다. 각각의 점근거동을 모두 시험할 수 있도록 해석시험문 제들을 구성해 주어야 한다. 특히, 휨지배 및 혼합지배거동 쉘 구조문제들에서는 잠김현상이 일어나는지를 시험하여야 하며 막지배거동 쉘 구조문제들에서는 5.1절에서 언급한 consistency조건이 만족되는지를 살펴보아야 한다. 표 3은 쉘 유한요소의 성능평가를 위한 쉘 구조문제의 예들을 보여주 고 있다. ● 요소망(mesh)의 형태 유한요소해석의 해는 유한요소망을 어떻게 구성하는지에 따라 그 수렴특성이 변화하게 된다. 요소형상의 찌그러짐에 민감하지 않고 좋은 수렴성을 유지하는 쉘 유한요소를 개발 하는 것은 쉽지 않은 일이다. 따라서 해석시험문제들은 다양
|
||||
|
||||
# 6. 결 론
|
||||
|
||||
한 유한요소망에 따른 쉘 유한요소의 수렴특성을 반영해 주 어야 한다. 특히, 비등방성(non-isotropic) 삼각형 쉘 유한요 소의 시험에서는 유한요소망의 형태뿐만 아니라 요소의 방 향에 따라 수렴특성이 변하므로 요소의 방향성 또한 고려되 어야 한다(Lee, Noh and Bathe, 2007). 머리말에서 언급된 바와 같이 쉘 구조물의 유한요소해석을 명확하게 이해하기 위해서는 쉘 구조물의 물리적 거동, 수학 모델 및 쉘 유한요소해석에 대한 이해가 동시에 체계적이고 심도 있게 이루어져야 한다. 본 논문에서는 이 세가지 부분 에 대한 이해와 이들이 서로 어떻게 유기적으로 관계를 맺 고 있는지를 최근 주요 연구들을 토대로 정리하여 고찰하였 고 이상적인 쉘 유한요소의 성질과 쉘 유한요소의 성능평가 방법을 제시하였다.
|
||||
|
||||
본 논문에서는 대표적인 쉘 수학모델과 휨지배거동, 막지 배거동, 혼합지배거동 등으로 나누어지는 쉘 구조물의 점근 거동에 대한 기본적인 이론과 그 점근거동을 수치적으로 알 아내는 방법을 알아보았다. 또한 휨지배 및 혼합지배거동에 서 나타나는 쉘 유한요소의 잠김현상을 두께의 변화에 따른 수렴곡선을 통하여 고찰하였다. 마지막으로 이상적인 쉘 유 한요소의 조건과 잠김현상의 제어하는 방법을 알아보았고 쉘 유한요소의 성능평가방법을 제안하였다.
|
||||
|
||||
쉘 구조물의 수학모델과 점근거동은 쉘의 물리적 거동을 이해하는데 핵심사항으로 쉘 구조물을 설계하는 기술자나 쉘 유한요소해석을 연구하는 연구자들이 명확하게 알아야 할 매 우 중요한 부분이다. 유한요소법을 이용하여 쉘 구조물을 해 석하기에 앞서 쉘 구조물의 점근거동과 그와 관련된 쉘 유 한요소의 감김현상에 대한 이해는 필수적이다. 통합적인 이 해의 바탕이 있을 때 신뢰할 만한 쉘 유한요소의 개발이 이 루어질 수 있으며 쉘 구조물의 유한요소해석을 통하여 얻어 진 결과를 정확하게 이해할 수 있다.
|
||||
|
||||
# 감사의 글
|
||||
|
||||
글을 맺으며 본 논문에 소개된 기본개념들을 정립하고 정 리하는데 많은 가르침을 주신 MIT(Massachusetts Institute of Technology)의 Klaus-Jürgen Bathe 교수님과 KAIST(한 국과학기술원)의 최창근 교수님께 깊은 감사드립니다.
|
||||
|
||||
# 참고문헌
|
||||
|
||||
- 최창근 (2002) 유한요소법. 테크노프레스.
|
||||
- Ahmad, S., Irons, B.M., and Zienkiewicz, O.C. (1970) Analysis of thick and thin shell structures by curved finite elements. International Journal for Numerical Methods and Engineering, Vol. 2, pp. 419-451.
|
||||
- Bathe, KJ. (1996) Finite Element Procedures. Prentice Hall: New Jersey.
|
||||
- Bathe, K.J., Chapelle, D., and Lee, P.S. (2003) A shell problem 'highly sensitive' to thickness changes. International Journal for Numerical Methods and Engineering, Vol. 57, pp. 1039-1052.
|
||||
- Bathe, K.J. and Dvorkin, E.N. (1989) A formulation of general
|
||||
|
||||
- shell elements the use of mixed interpolation of tensorial components. International Journal for Numerical Methods and Engineering, Vol. 22, pp. 697-722.
|
||||
- Bathe, K.J., Iosilevich, A., and Chapelle, D. (2000a) An evaluation of the MITC shell elements. Computers & Structures, Vol. 75, pp. 1-30.
|
||||
- Bathe, K.J., Iosilevich, A., and Chapelle, D. (2000b) An inf-sup test for shell finite elements. Computers & Structures, Vol. 75, pp. 439-456.
|
||||
- Bathe, K.J., Lee, P.S., and Hiller, J.F. (2003) Towards improving the MITC9 shell element. Computers & Structures, Vol. 81, pp. 477-489.
|
||||
- Chapelle, D. and Bathe, K.J. (1998) Fundamental considerations for the finite element analysis of shell structures. Computers & Structures, Vol. 66, pp. 19-36, pp. 711-712.
|
||||
- Chapelle, D. and Bathe, K.J. (2003) The finite element analysis of shells? fundamentals. Berlin:Springer-Verlag.
|
||||
- Choi, C.K., Lee, P.S., and Park, Y.M. (1999) Defect-free 4-node flat shell element: NMS-4F element. Structural Engineering and Mechanics, Vol. 8, pp. 207-231.
|
||||
- Hiller, J.F. and Bathe, K.J. (2003) Measuring convergence of mixed finite element discretizations: an application to shell structures. Computers & Structures, Vol. 81, pp. 639-654.
|
||||
- Lee, P.S. and Bathe, K.J. (2002) On the asymptotic behavior of shell structures and the evaluation in finite element solutions. Computers & Structures, Vol. 80, pp. 235-255.
|
||||
- Lee, P.S. and Bathe, K.J. (2004) Development of MITC isotropic triangular shell finite elements. Computers & Structures, Vol. 82, pp. 945-962.
|
||||
- Lee, P.S. and Bathe, K.J. (2005) Insight into finite element shell discretizations by use of the basic shell mathematical model. Computers & Structures, Vol. 83, pp. 69-90.
|
||||
- Lee, P.S. Noh, H.C., and Bathe, K.J. (2007) Insight into 3-node triangular shell finite elements: the effects of element isotropy and mesh patterns. Computers & Structures, Vol. 85, pp. 404- 418.
|
||||
- Lovadina, C. (2001) Energy estimates for linear elastic shells. Computational Fluid and Solid Mechanics (Bathe KJ ed.), pp. 330- 331, Elsevier Science.
|
||||
- MacNeal, R.H. and Harder, R.L. (1985) A proposed standard set of problems to test finite element accuracy. Finite Element in Analysis and Design, Vol. 1, pp. 3-20.
|
||||
- Noh, H.C. (2006) Nonlinear behavior and ultimate load bearing capacity of reinforced concrete natural draught cooling tower shell. Engineering Structures, Vol. 28, pp. 399-410.
|
||||
|
||||
(접수일: 2006.7.18/심사일: 2007.1.16/심사완료일: 2007.3.27)
|
||||
File diff suppressed because it is too large
Load Diff
Binary file not shown.
|
Before Width: | Height: | Size: 51 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 42 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 250 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 204 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 57 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 251 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 60 KiB |
-447
@@ -1,447 +0,0 @@
|
||||
# **A continuum mechanics based four-node shell element for general nonlinear analysis**
|
||||
|
||||
Eduardo N. Dvorkin and Klaus-Jürgen Bathe
|
||||
|
||||
*Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Received December 1983)*
|
||||
|
||||
# ABSTRACT
|
||||
|
||||
A new four-node (non-flat) general quadrilateral shell element for geometric and material non-linear analysis is presented. The element is formulated using threedimensional continuum mechanics theory and it is applicable to the analysis of thin and thick shells. The formulation of the element and the solutions to various test and demonstrative example problems are presented and discussed.
|
||||
|
||||
### INTRODUCTION
|
||||
|
||||
The finite element analysis of general shell structures has been a very active field of research for a large number of years14,29 . However, despite the fact that many different shell elements have already been proposed, the search for a shell element capable of representing the general nonlinear behaviour of shells with arbitrary geometry and loading conditions in an effective and reliable manner is still continuing very actively.
|
||||
|
||||
During recent years it has become apparent that two approaches for the development of shell elements are very appropriate: (1) the use of simple elements, based on the discrete-Kirchhoff approach for the analysis of thin shells2,5-9; (2) the use of degenerated isoparametric elements in which fully three-dimensional stress and strain conditions are degenerated to shell behaviour2,3,5,7,17,19,24,29 .
|
||||
|
||||
The latter approach has the advantage of being independent of any particular shell theory, and this approach was used by Bathe and Bolourchi<sup>3</sup> to formulate a general shell element for geometric and material non-linear analysis. This element has been employed very successfully when used with 9 or, in particular, 16 nodes. However, the 16-node element is quite expensive, and although it is possible to use in some analyses only a few elements to represent the total structure (see later examples) in other analyses still a fairly large number of elements need by employed<sup>5</sup> .
|
||||
|
||||
Considering general shell analyses, much emphasis has been placed onto the development of a versatile, reliable and cost-effective 4-node shell element<sup>16</sup>' 17,22,28 . Such element would complement the above high-order 16-node element and may be more effective in certain analyses. The difficulties in the development of such element lie in that the element should be applicable in a reliable manner to
|
||||
|
||||
thin and thick shells of arbitrary geometries for general non-linear analysis.
|
||||
|
||||
The objective in this paper is to present a simple 4-node general shell element with the following properties: the element is formulated using three-dimensional stress and strain conditions without use of a shell theory; the element is applicable to thin and thick shells and can be employed to model arbitrary geometries; the element is applicable to the conditions of large displacements and rotations but small strains, and can be used effectively in materially non-linear analysis.
|
||||
|
||||
The formulation of the element is quite simple and transparent, and the element has good predictive capability without containing spurious zero energy modes.
|
||||
|
||||
In the next section of the paper we discuss some basic considerations with respect to the assumptions used, and in the following section we present the element formulation for non-linear analysis. The results obtained in numerical solutions that demonstrate the properties of the element are given in the final section.
|
||||
|
||||
## BASIC CONSIDERATIONS
|
||||
|
||||
The formulation of the 4-node shell element represents an extension of the shell element discussed previously2,3 , and we therefore use the same notation as in those references. Also, to focus attention onto some key issues of the formulation, we consider in this section only linear analysis conditions.
|
||||
|
||||
The geometry of the element (see *Figure 1)* is described using<sup>2</sup> :
|
||||
|
||||
$${}^{l}x_{i} = \sum_{k=1}^{4} h_{k}{}^{l}x_{i}^{k} + \frac{r_{3}}{2} \sum_{k=1}^{4} a_{k}h_{k}{}^{l}V_{ni}^{k}$$
|
||||
(1)
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
where the *hk(r<sup>l</sup> ,r2)* are the two-dimensional interpolation functions corresponding to node *k;* the *r<sup>i</sup>* are the natural coordinates; and 1 *xi =* Cartesian coordinates of any point in the element; = Cartesian coordinates of nodal point :=components of director vector at node *k* (which is not necessarily normal to the midsurface of the element); and *ak* is the shell thickness at node *k,* measured along the vector The left superscript is zero for the initial geometry of the element and is equal to 1 for the deformed element geometry. Note that the thickness of the element varies and the element is in general non-flat.
|
||||
|
||||
The displacements of any particle with natural coordinates *r<sup>i</sup>* of the shell element in the stationary Cartesian coordinate system are:
|
||||
|
||||
$$u_{i} = \sum_{k=1}^{4} h_{k} u_{i}^{k} + \frac{r_{3}}{2} \sum_{k=1}^{4} a_{k} h_{k} (-{}^{0}V_{2i}^{k} \alpha_{k} + {}^{0}V_{1i}^{k} \beta_{k})$$
|
||||
(2)
|
||||
|
||||
where the are the nodal point displacements into the Cartesian coordinate directions, and the *ak* and *Bk* are the rotations of the director vector about the and axes (see *Figure 1).*
|
||||
|
||||
A basic problem inherent in the use of the above interpolation of the displacements, and the derivation of the strain-displacement matrices therefrom, is that the element 'locks' when it is thin. This is due to the fact that with these interpolations the transverse shear strains cannot vanish at all points in the element, when it is subjected to a constant bending moment. Hence, although the basic continuum mechanics assumptions contain the Kirchhoff shell assumptions, the finite element discretization is not able to represent these assumptions rendering the element not applicable to the analysis of thin plates or shells2,5,7 . To solve this deficiency, various remedies based on selective and reduced integration have been proposed17,22,23 but there is still much room for more effective and reliable elements for general non-linear analysis.
|
||||
|
||||
Considering our element formulation - because the problem lies in the representation of the transverse shear strains - we proceed to not evaluate these shear strains from the displacements in (2), but to introduce separate interpolations for these strain components. Since we consider non-flat shell elements, the separate interpolations are performed effectively in a convected coordinate system†.
|
||||
|
||||
The choice of the interpolation for the transverse shear strain components is the key assumption in our element formulation, because adequate coupling between the element displacements and rotations must be introduced and the element should not exhibit any spurious zero energy modes. For our element we use (see *Figure 2):*
|
||||
|
||||
$$\tilde{\varepsilon}_{13} = \frac{1}{2}(1+r_2)\tilde{\varepsilon}_{13}^{A} + \frac{1}{2}(1-r_2)\tilde{\varepsilon}_{13}^{C}
|
||||
\tilde{\varepsilon}_{23} = \frac{1}{2}(1+r_1)\tilde{\varepsilon}_{23}^{D} + \frac{1}{2}(1-r_1)\tilde{\varepsilon}_{23}^{B}$$
|
||||
(3)
|
||||
|
||||
Since the kinematic relations for the above shear strains are not satisfied using (3), we impose them using Lagrange multipliers2,27 to obtain,
|
||||
|
||||
$$\Pi^* = \frac{1}{2} \int_{V} \tilde{\tau}^{ij} \tilde{\varepsilon}_{ij} \, dV + \int_{V} \lambda^{13} (\tilde{\varepsilon}_{13} - \tilde{\varepsilon}_{13}^{DI}) \, dV + \int_{V} \lambda^{23} (\tilde{\varepsilon}_{23} - \tilde{\varepsilon}_{23}^{DI}) \, dV - \mathcal{H}^{c} \tag{4}$$
|
||||
|
||||
where the are the contravariant components of the Cauchy stress tensor13,15 , the are the covariant components of the infinitesimal strain tensor, the are the Lagrange multipliers, the are the transverse shear strains evaluated using the displacement interpolations in (2), and is the potential of the external loads. For the Lagrange multipliers we choose the following interpolations,
|
||||
|
||||
$$\lambda^{13} = \lambda^{A} \delta(r_1) \delta(1 - r_2) + \lambda^{C} \delta(r_1) \delta(1 + r_2)$$
|
||||
|
||||
$$\lambda^{23} = \lambda^{D} \delta(r_2) \delta(1 - r_1) + \lambda^{B} \delta(r_2) \delta(1 + r_1)$$
|
||||
(5)
|
||||
|
||||
where <5(...) is the Dirac-delta function. This represents a weakening of the Lagrange multiplier constraint in (4)<sup>10</sup> . Substituting from (5) into (4) and invoking that gives the distinct constrains:
|
||||
|
||||
$$\tilde{\varepsilon}_{13}\Big|_{\text{at A}} = \tilde{\varepsilon}_{13}^{\text{DI}}\Big|_{\text{at A}} \qquad \tilde{\varepsilon}_{13}\Big|_{\text{at C}} = \tilde{\varepsilon}_{13}^{\text{DI}}\Big|_{\text{at C}}
|
||||
\tilde{\varepsilon}_{23}\Big|_{\text{at D}} = \tilde{\varepsilon}_{23}^{\text{DI}}\Big|_{\text{at D}} \qquad \tilde{\varepsilon}_{23}\Big|_{\text{at B}} = \tilde{\varepsilon}_{23}^{\text{DI}}\Big|_{\text{at B}} \tag{6}$$
|
||||
|
||||
Hence, the complete element stiffness matrix is calculated using the functional:
|
||||
|
||||
$$\Pi^* = \frac{1}{2} \int \tilde{\tau}^{ij} \tilde{\epsilon}_{ij} \, \mathrm{d}V - \mathcal{H}^{\hat{r}} \tag{7}$$
|
||||
|
||||
<sup>†</sup> Note that in refs. 2 and 3, the shell element formulation is discussed in the global stationary coordinate system, because all displacement components are interpolated in the same way. To emphasize that we use here stress and strain measures in the convected coordinate system, we place a tilde (~) over these quantities.
|
||||
|
||||
|
||||
|
||||
with stress and strain components in convected coordinates and (1) and (2) to evaluate the strain components (3) to evaluate the strain components ; and (6) to express the variables in terms of the nodal point displacements and rotations of (2).
|
||||
|
||||
Considering the representation that we have chosen for the transverse shear strains, we can make the following three important observations:
|
||||
|
||||
- (1) *The element is able to represent the six rigid body modes.* The element contains the rigid body modes because zero strains are calculated in the formulation when the element nodal point displacements and rotations correspond to an element rigid body displacement. This can be verified by using (1) to (6) to evaluate the strains, but more easily we can use the fact that the 4-node shell element of reference 3 satisfies the rigid body mode criterion. Hence, for a rigid body displacement the are zero, from which it follows that also the shear strains in (3) are zero, and the rigid body mode criterion is satisfied.
|
||||
- (2) *The element can approximate the Kirchhoff-Love hypothesis of negligible shear deformation effects and can be used for thin shells.* Various demonstrative solutions are given in the fourth section.
|
||||
- (3) *Based on our studies the element does not contain any spurious zero energy modes* (*using a 'full' numerical integration).* We reach this observation by studying the strains along the element sides. If the element were to contain a spurious zero energy mode, the strains along every side should vanish for a displacement pattern (to be identified) other than the displacements corresponding to a true rigid body mode. However, such displacement pattern could not be identified.
|
||||
|
||||
Considering the practical use of the element the interpolation employed for the transverse shear strains shows that is constant with *r*1 and in general discontinuous at (between elements), and similarly is constant with r2 and in general discontinuous at As a consequence, the accuracy with which transverse shear stresses are predicted depends to a significant degree on the mesh used and the geometric distortions of the elements. However, our experience is that the bending stress predictions are relatively little affected by element distortions (see examples).
|
||||
|
||||
To employ (7), we also need to use the appropriate constitutive relations:
|
||||
|
||||
$$\tilde{\tau}^{ij} = \tilde{C}^{ijkl} \tilde{\varepsilon}_{kl} \tag{8}$$
|
||||
|
||||
where is the fourth-order contravariant constitutive tensor in the convected coordinates r*<sup>i</sup>* . The constitutive law is known in the local Cartesian system of orthonormal base vectors *i =* 1,2,3, with the condition equal to zero<sup>2</sup> , (see *Figure 3).* Denoting this constitutive tensor by the constitutive tensor for (8) is obtained using the transformation:
|
||||
|
||||
$$\tilde{C}^{ijkl} = (\mathbf{g}^i \cdot \hat{\mathbf{e}}_m)(\mathbf{g}^j \cdot \hat{\mathbf{e}}_n)(\mathbf{g}^k \cdot \hat{\mathbf{e}}_0)(\mathbf{g}^l \cdot \hat{\mathbf{e}}_n)\hat{C}^{mnop} \tag{9}$$
|
||||
|
||||
where the g<sup>i</sup> are the contravariant base vectors of the convected coordinates *r<sup>i</sup> .* These vectors are calculated, using the covariant base vectors g<sup>i</sup> , where:
|
||||
|
||||
$$\mathbf{g}_i = \frac{\partial^0 \mathbf{x}}{\partial r_i} \tag{10}$$
|
||||
|
||||
$$g_{ij} = \mathbf{g}_i \cdot \mathbf{g}_j \tag{11}$$
|
||||
|
||||
$$\mathbf{g}^{i} = g^{ij}\mathbf{g}_{j}$$
|
||||
|
||||
$$g^{ij} = \frac{D^{ij}}{\Pi^{2}}$$
|
||||
(12)
|
||||
|
||||
where *Dij* is the cofactor of the term *gij* in the matrix of the metric tensor and |J| is the determinant of the Jacobian matrix at the point considered.
|
||||
|
||||
# TOTAL LAGRANGIAN FORMULATION
|
||||
|
||||
The large displacement formulation of the shell element is based on the derivation given in ref. 2 (Section 6.3.5), and the concepts and interpolations presented in the previous section.
|
||||
|
||||
The geometry of the element at any time *t* is defined as in (1) but using the nodal point coordinates, and director vectors
|
||||
|
||||
$${}^{t}x_{i} = h_{k}{}^{t}x_{i}^{k} + \frac{r_{3}}{2}a_{k}h_{k}{}^{t}V_{ni}^{k}$$
|
||||
(13)
|
||||
|
||||
where we imply summation over *k.* The displacements, and incremental displacements, *u<sup>i</sup>* , of a particle of the element at time *t* are hence given by:
|
||||
|
||||
$${}^{i}u_{i} = h_{k}{}^{i}u_{i}^{k} + \frac{r_{3}}{2}a_{k}h_{k}({}^{i}V_{ni}^{k} - {}^{0}V_{ni}^{k})$$
|
||||
|
||||
$$u_{i} = h_{k}u_{i}^{k} + \frac{r_{3}}{2}a_{k}h_{k}(-{}^{i}V_{2i}^{k}\alpha_{k} + {}^{i}V_{1i}^{k}\beta_{k})$$
|
||||
|
||||
$$(14)$$
|
||||
|
||||
where the are the nodal point displacements at time *t*, the are the incremental nodal point displacements from the configuration at time *t,* and the variables and *βk* are defined as in (2) but referred to the configuration at time *t.*
|
||||
|
||||
This kinematic description implies the following hy-
|
||||
|
||||
<sup>†</sup> Note that the superscript *t* on a variable denotes the configuration at time *t* in the incremental solution, and does not imply a dynamic analysis.
|
||||
|
||||
potheses: the director vectors remain straight during the deformations; the 'thickness' of the element measured along the director vectors remains constant during the deformations; hence only small strain conditions are considered.
|
||||
|
||||
Using the assumptions in (13) and (14) the geometric and material non-linear response is analysed using an incremental formulation<sup>2</sup> , in which the configuration is sought for time (load step) '*t* +∆*t',* when the configuration for time *t* is known. The basis of this incremental formulation is the use of the virtual work principle applied to the configuration at time *t + ∆t.* In essence, two approaches can be employed leading to the updated Lagrangian and the total Lagrangian formulations. These approaches are, from a continuum mechanics point of view, equivalent, and in the following we develop the governing finite element relations for the total Lagrangian formulation.
|
||||
|
||||
The principle of virtual work applied to the configuration at time *t+At* is:
|
||||
|
||||
$$\int_{0V}^{t+\Delta t} \tilde{S}^{ij} \delta^{t+\Delta t} \tilde{e}_{ij} \,^{0} \mathrm{d}V = {}^{t+\Delta t} \mathcal{R}$$
|
||||
(15)
|
||||
|
||||
where the are the contra variant components of the second Piola-Kirchhoff stress tensor at time and referred to the configuration at time 0, and the are the covariant components of the Green-Lagrange strain tensor at time and referred to time 0. Both sets of tensor components are measured in the convected coordinate system / = 1,2,3. The external virtual work is given by and includes the work due to the applied surface tractions and body forces.
|
||||
|
||||
For the incremental solution, the stresses and strains are decomposed into the known quantities, and unknown increments, so that
|
||||
|
||||
$${}^{i+\Delta i}_{\alpha}\tilde{S}^{\bar{i}j} = {}^{i}_{\alpha}\tilde{S}^{\bar{i}j} + {}_{\alpha}\tilde{S}^{\bar{i}j} \tag{16}$$
|
||||
|
||||
$${}^{t+\Delta t}\circ\tilde{\varepsilon}_{i,i} = {}^{t}\circ\tilde{\varepsilon}_{i,i} + {}^{\circ}\circ\tilde{\varepsilon}_{i,i} \tag{17}$$
|
||||
|
||||
In addition, the strain increment can be written as a linear part, and a non-linear part, hence
|
||||
|
||||
$$_{0}\tilde{\varepsilon}_{ii} = _{0}\tilde{e}_{ii} + _{0}\tilde{\eta}_{ii} \tag{18}$$
|
||||
|
||||
Substituting from (16) to (18) into (15) and using the linearized expressions we obtain the linearized equation of motion:
|
||||
|
||||
$$\int_{\delta_{V}} {0} \tilde{C}^{ijkl} {0} \tilde{e}_{kl} \delta_{0} \tilde{e}_{ij} {0} dV + \int_{\delta_{V}} {0} \tilde{S}^{ij} \delta_{0} \tilde{\eta}_{ij} {0} dV$$
|
||||
|
||||
$$= {0 \choose k} {0} \tilde{S}^{ij} \delta_{0} \tilde{e}_{ij} {0} dV$$
|
||||
(19)
|
||||
|
||||
This equation is the basic equilibrium relation employed to develop the governing finite element matrices. For the actual solution of problems it is frequently important to use equilibrium iterations, but the finite element matrices and vectors used in these iterations can be derived directly from the matrices obtained using (19*) 2 .* Note that is now obtained using (9) with the condition =0, which implies the more natural condition only in the small strain case.
|
||||
|
||||
The basic problem of the finite element discretization of (19) lies in expressing the strain terms of (19) in terms of the finite element interpolations. Using the definition of the Green-Lagrange strain components:
|
||||
|
||||
$${}_{0}^{t}\tilde{\varepsilon}_{i,i} = \frac{1}{2} ({}^{t}\mathbf{g}_{i} \cdot {}^{t}\mathbf{g}_{i} - {}^{0}\mathbf{g}_{i} \cdot {}^{0}\mathbf{g}_{i}) \tag{20}$$
|
||||
|
||||
$${}_{0}\tilde{e}_{ii} = h_{k,i}{}^{t}\mathbf{g}_{i} \cdot \mathbf{u}_{k} + \frac{r_{3}}{2}a_{k}h_{k,i}(-\alpha_{k}{}^{t}\mathbf{g}_{i} \cdot {}^{t}\mathbf{V}_{2}^{k} + \beta_{k}{}^{t}\mathbf{g}_{i} \cdot {}^{t}\mathbf{V}_{1}^{k})$$
|
||||
(21a)
|
||||
|
||||
$${}_{0}\tilde{\eta}_{ii} = \frac{1}{2}h_{k,i}h_{p,i}\mathbf{u}_{k} \cdot \mathbf{u}_{p} + \frac{r_{3}}{2}h_{k,i}h_{p,i}a_{p}(-\alpha_{p}{}^{t}\mathbf{V}_{2}^{p} \cdot \mathbf{u}_{k} + \beta_{p}{}^{t}\mathbf{V}_{1}^{p} \cdot \mathbf{u}_{k}) +$$
|
||||
|
||||
$$\frac{(r_{3})^{2}}{8}h_{k,i}h_{p,i}a_{k}a_{p}(-\alpha_{k}{}^{t}\mathbf{V}_{2}^{k}+\beta_{k}{}^{t}\mathbf{V}_{1}^{k})\cdot(-\alpha_{p}{}^{t}\mathbf{V}_{2}^{p}+\beta^{p}{}^{t}\mathbf{V}_{1}^{p})$$
|
||||
(i = 1,2)
|
||||
(21b)
|
||||
|
||||
$${}_{0}e_{12} = \frac{1}{2} [h_{k,2} \mathbf{g}_{1} \cdot \mathbf{u}_{k} + h_{k,1} \mathbf{g}_{2} \cdot \mathbf{u}_{k} + \frac{r_{3}}{2} h_{k,2} J_{k} (-\alpha_{k}^{i} \mathbf{V}_{2}^{k} \cdot {}^{i} \mathbf{g}_{1} + \beta_{k}^{i} \mathbf{V}_{1}^{k} \cdot {}^{i} \mathbf{g}_{1}) +$$
|
||||
|
||||
$$\frac{r_3}{2}h_{k,1}a_k(-\alpha_k{}^t\mathbf{V}_2^k \cdot {}^t\mathbf{g}_2 + \beta_k{}^t\mathbf{V}_1^k \cdot {}^t\mathbf{g}_2)]$$
|
||||
(22a)
|
||||
|
||||
$$\frac{r_3}{2}h_{k,1}h_{p,2}a_p(-\alpha_p{}^t\mathbf{V}_2^p \cdot \mathbf{u}_k + \beta_p{}^t\mathbf{V}_1^p \cdot \mathbf{u}_k) +$$
|
||||
|
||||
$$\frac{r_3}{2}h_{k,1}h_{p,2}a_k(-\alpha_k{}^t\mathbf{V}_2^k\cdot\mathbf{u}_p+\beta_k{}^t\mathbf{V}_1^k\cdot\mathbf{u}_p)+$$
|
||||
|
||||
$$\frac{(r_3)^2}{4}h_{k,1}h_{p,2}a_ka_p(-\alpha_k{}^t\mathbf{V}_2^k+\beta_k{}^t\mathbf{V}_1^k) \cdot (-\alpha_p{}^t\mathbf{V}_2^p+\beta_p{}^t\mathbf{V}_1^p)]$$
|
||||
|
||||
Further, we obtain for the transverse shear strains, using (3) and (6):
|
||||
|
||||
$$\begin{split} &_{0}\tilde{e}_{13} = \frac{1}{8}(1+r_{2})\left[{}^{t}g_{3i}^{A}(u_{i}^{1}-u_{i}^{2}) + \right. \\ &_{2}{}^{t}g_{1i}^{A}(-\alpha_{1}a_{1}{}^{t}V_{2i}^{1} + \beta_{1}a_{1}{}^{t}V_{1i}^{1} - \alpha_{2}a_{2}{}^{t}V_{2i}^{2} + \beta_{2}a_{2}{}^{t}V_{1i}^{2})\right] + \\ &_{3}{}^{t}(1-r_{2})\left[{}^{t}g_{3i}^{C}(u_{i}^{4}-u_{i}^{3}) + \frac{1}{2}{}^{t}g_{1i}^{C}(-\alpha_{4}a_{4}{}^{t}V_{2i}^{4} + \beta_{4}a_{4}{}^{t}V_{1i}^{4} - \alpha_{3}a_{3}{}^{t}V_{2i}^{3} + \beta_{3}a_{3}{}^{t}V_{1i}^{3})\right] \end{split}$$
|
||||
|
||||
$$\tilde{\eta}_{13} = \frac{1}{32}(1 + r_2) [(-\alpha_1 a_1^{\ t} V_{2i}^1 + \beta_1 a_1^{\ t} V_{1i}^1 - \alpha_2 a_2^{\ t} V_{2i}^2 + \beta_2 a_2^{\ t} V_{1i}^2)(u_i^1 - u_i^2)] + \frac{1}{32}(1 - r_2) [(-\alpha_4 a_4^{\ t} V_{2i}^4 + \beta_4 a_4^{\ t} V_{1i}^4 - \alpha_3 a_3^{\ t} V_{2i}^3 + \beta_3 a_3^{\ t} V_{1i}^3)]$$
|
||||
(23b)
|
||||
|
||||
$$\begin{split} & _{0}\tilde{e}_{23} = \frac{1}{8}(1+r_{1})[^{t}g_{3i}^{D}(u_{i}^{1}-u_{i}^{4}) + \\ & \frac{1}{2}^{t}g_{2i}^{D}(-\alpha_{1}a_{1}^{t}V_{2i}^{1} + \beta_{1}a_{1}^{t}V_{1i}^{1} - \alpha_{4}a_{4}^{t}V_{2i}^{4} + \beta_{4}a_{4}^{t}V_{1i}^{4})] + \\ & \frac{1}{8}(1-r_{1})[^{t}g_{3i}^{B}(w_{i}^{2}-w_{i}^{3}) + \frac{1}{2}^{t}g_{2i}^{B}(-\alpha_{2}\alpha_{2}^{t}V_{2i}^{2} + \\ & \beta_{2}a_{2}^{t}V_{1i}^{2} - \alpha_{3}a_{3}^{t}V_{2i}^{3} + \beta_{3}a_{3}^{t}V_{1i}^{3})] \end{split}$$
|
||||
|
||||
$${}_{0}\tilde{\eta}_{23} = \frac{1}{32}(1+r_{1})[(-\alpha_{1}a_{1}{}^{t}V_{2i}^{1} + \beta_{1}a_{1}{}^{t}V_{1i}^{1} - \alpha_{4}a_{4}{}^{t}V_{2i}^{4} + \beta_{4}a_{4}{}^{t}V_{1i}^{4})(u_{i}^{1} - u_{i}^{4})] + \frac{1}{32}(1-r_{1})[(-\alpha_{2}a_{2}{}^{t}V_{2i}^{2} + \beta_{2}a_{2}{}^{t}V_{1i}^{2} - \alpha_{3}a_{3}{}^{t}V_{2i}^{3} + \beta_{3}a_{3}{}^{t}V_{1i}^{3})(u_{i}^{2} - u_{i}^{3})]$$
|
||||
|
||||
$$(24b)$$
|
||||
|
||||
Note that, since we assume the thickness of the shell to be constant, the strain through the element thickness is zero.
|
||||
|
||||
The expressions in (21) to (24) are substituted into (19) which in the standard manner yields the linear strain incremental stiffness matrix the non-linear strain (or geometric) incremental stiffness matrix and the nodal point force vector in the finite element incremental equilibrium relations<sup>2</sup> ,
|
||||
|
||||
$$\binom{t}{0}\mathbf{K}_{t} + \binom{t}{0}\mathbf{K}_{Nt}\mathbf{u} = t + \Delta t\mathbf{R} - \binom{t}{0}\mathbf{F} \tag{25}$$
|
||||
|
||||
The element matrices in (25) correspond to five degrees of freedom per node (see *Figure 1)* but in some applications it is convenient to use instead of three rotations about the global coordinate axes (see examples). In this case, we simply transform the matrices of (25) in the standard manner<sup>2</sup> .
|
||||
|
||||
# NUMERICAL TESTS AND EXAMPLE SOLUTIONS
|
||||
|
||||
We have implemented our shell element in the ADINA computer program and have performed various numerical tests to study the predictive capabilities of the element. The following solutions were all obtained using 2x2 Gauss integration in the r3 =0 surface of the element, and 2 and 4 point Gauss integration in the r3 direction, for elastic and elastoplastic analyses, respectively.
|
||||
|
||||
## *Some simple tests*
|
||||
|
||||
As a first step to test the element, the eigenvalues of the stiffness matrices of undistorted and distorted elements were calculated. In all cases, as expected, the element displayed the six rigid body modes and no spurious zero energy modes.
|
||||
|
||||
*Patch tests.* For the patch test2,18 the mesh shown in *Figure 4a* was used. In the first analysis *(Figure 4b)* the mesh was loaded with the constant moment indicated and a constant curvature (linear distribution of rotations) was obtained for both plate thicknesses in the two plate directions. The transverse displacements predicted by the model were, as expected, those of Kirchhoff-Love plate theory at nodes 7 and 8.
|
||||
|
||||
In the second analysis *(Figure 4c)* the rotational degrees of freedom were deleted and the mesh was subjected to shear forces. As expected, for both plate thicknesses a linear distribution of transverse displacements was obtained.
|
||||
|
||||
In the third analysis *(Figure 4d)* the mesh was subjected to an external twisting moment. In the thin plate analysis, constant curvatures were obtained in both plate directions and the transverse displacements agreed with the analytical thin plate theory solution. In the thick plate analysis, a slight non-symmetry in the displacement response (the third digit) was obtained due to the unsymmetric representation of the transverse shear deformations. This non-symmetry is not observed, if the shear deformations are suppressed (which corresponds to thin
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
plate theory) by choosing a large value for the shear correction factor *k* (or when using rectangular elements in the mesh)<sup>2</sup> .
|
||||
|
||||
Finally, it should be noted that the patch test is of course passed for the three membrane stress states (Τ1 1 , *τ22* andτT12 constants).
|
||||
|
||||
*Cantilever linear analyses.* A cantilever of unit width, thickness 0.1 and lengths lOand 100 was subjected to a tip bending moment. The structure was modelled using one single element and two distorted elements as shown in *Figure 5.* The results obtained in these analyses for the displacements and rotations at the cantilever tip and the stresses were those of Bernoulli beam theory.
|
||||
|
||||
Next, the cantilever in *Figure 6a* was analysed for the transverse tip load shown. Using 4 equal size elements to idealize the cantilever, again good results were obtained when compared with beam theoretical results (see *Figure 6b* and *Table 1).*
|
||||
|
||||
Finally, the elements modelling the cantilever were distorted as shown in *Figure 6c* for a thin and a thick cantilever. The results given in *Figure 6d* and *Table 2* show that the transverse displacements and normal bending stresses are almost insensitive to the element distortions. However, the calculated transverse shear stresses (not shown in the Figure) are not accurate.
|
||||
|
||||
*Linear analyses of a simply-supported plate.* A simplysupported plate was considered for a static and a frequency analysis using a consistent mass matrix. To model one quarter of the plate the 4 x 4 mesh of equal elements *(Figure 7a)* was used. *Figure 7b* and *Tables 3* and *4* give a comparison of the numerically and analytically predicted results. The same plate was also analysed using the distorted element mesh also shown in *Figure 7a* and the results of *Figure 7b* and *Tables 3* and *4* were obtained.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
*Table 1* Cantilever tip transverse displacement: non-distorted meshes of *N* elements
|
||||
|
||||
| 0.750 |
|
||||
|-------|
|
||||
| 0.984 |
|
||||
| |
|
||||
|
||||
*Table 2* Cantilever tip transverse displacements
|
||||
|
||||
| Thickness | ηIpoint<br>B | ηIpoint<br>A | |
|
||||
|-----------|--------------|--------------|--|
|
||||
| 0.1 | 0.989 | 0.996 | |
|
||||
| 2.0 | 1.0013 | 0.995 | |
|
||||
|
||||
*η*) = (*u*3 distorted mesh)/(*u*3 non-distorted mesh)
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
*Table 3* Non-dimensional displacements at centre of simplysupported plate: distorted and non-distorted meshes
|
||||
|
||||
| Model | FEM/u3<br>thin plate<br>u3<br>at centre | | | |
|
||||
|--------------------|-----------------------------------------|--|--|--|
|
||||
| non-dist.<br>dist. | 0.995<br>0.992 | | | |
|
||||
| | | | | |
|
||||
|
||||
*Table 4* Non-dimensional frequencies *f* (cycles/sec) for a simplysupported plate: distorted and non-distorted meshes
|
||||
|
||||
| Mode shape | FEM/fthin plat e<br>f | | | |
|
||||
|------------|-----------------------|--|--|--|
|
||||
| 1-1 | 1.02 | | | |
|
||||
| 1-3 | 1.18 | | | |
|
||||
| 3-3 | 1.17 | | | |
|
||||
|
||||
*Analysis of a rhombic cantilever.* The rhombic cantilever shown in *Figure 8,* fixed at one side and subjected to constant pressure was analysed using a 4x 4 element mesh. In *Table 5,* the results for the transverse displacements at six locations are compared against the solutions obtained using the DKT triangular element<sup>6</sup> , experimental measurements<sup>1</sup> and using the 16-node isoparametric element (with 4x4x 2 Gauss integration). In all cases a one step geometric non-linear analysis with equilibrium iterations was performed. Good correspondence between the experimental results and the solution obtained using our new 4-node element is observed.
|
||||
|
||||
# *Linear analysis of a cylindrical (Scordelis-Lo) shell*
|
||||
|
||||
The shell structure shown in *Figure 9a* has frequently been used to test the performance of shell elements<sup>12</sup> . *Figure 9b* shows the solutions obtained with our elements. In each of the solutions uniform meshes with equal sized elements were employed over one-quarter of the shell. Solutions obtained using the 3-node DKT triangular element<sup>25</sup> and the 16-node isoparametric element<sup>25</sup> are also shown.
|
||||
|
||||
### *Linear analysis of a pinched cylinder*
|
||||
|
||||
The pinched cylinder problem shown in *Figure 10a* was also frequently analysed to test shell elements. *Figure 10b* and *Tables 6* and 7 show the convergence behaviour obtained with our new element, when comparing the finite element solutions<sup>11</sup>' 21 . Note that using the isoparametric shell element<sup>3</sup> also a fairly large number of degrees of freedom are required to predict the response of the cylinder accurately.
|
||||
|
||||
# *Large deflection analysis of a cantilever*
|
||||
|
||||
The cantilever shown in *Figure 11a* was analysed for its large displacement and large rotation response. This is a typical problem considered to test the geometric nonlinear behaviour of beam and shell elements<sup>25</sup> . *Figure 11a* also shows the models used in the analysis.
|
||||
|
||||
The first two models are single element, cubic and parabolic isoparametric degenerate shell element models. Model I predicts the response of the cantilever very accurately, whereas model II yields an accurate response solution in linear analysis but locks once the element is curved in the non-linear response solution. This observation is in accordance with the results reported elsewhere<sup>5</sup> .
|
||||
|
||||
The same nodal point layouts were next employed for models IN and IV using our new 4-node shell element. *Figures 11b-11d* give the results obtained with these models. It is seen that model III yields an accurate large displacement response prediction, and even model IV yields quite accurate results up to about 60 degrees of rotation. The computer time required in these analyses were only little different using models I, III and IV.
|
||||
|
||||
Another important result is shown in *Table 8.* As reported earlier<sup>5</sup> , the cubic shell element is sensitive to 'in-plane' distortions, and hence it is interesting to study the effect of using a distorted element mesh in the analysis of the cantilever (see *Figures 12a* and *12b). Table 8* summarizes the results obtained using the one cubic element and three 4-node elements with a nodal layout that corresponds to distorting the elements. It is seen that the predictive capability of our new 4-node element is considerably less sensitive to the element distortions.
|
||||
|
||||
|
||||
|
||||
*Figure 8* Response of rhombic cantilever subjected to constant pressure. q = 0.26066; £=10.5x10<sup>6</sup> ; thickness = 0.125; v = 0.3
|
||||
|
||||
*Table 5*
|
||||
|
||||
| Element | | CPU time | Deflection at location | | | | | | |
|
||||
|---------------|------|-----------------|------------------------|-------|-------|-------|-------|-------|--|
|
||||
| | Mesh | CPU time of DKT | 1 | 2 | 3 | 4 | 5 | 6 | |
|
||||
| DKT | 4x4 | 1.00 | 0.293 | 0.196 | 0.114 | 0.118 | 0.055 | 0.024 | |
|
||||
| 4-node | 4 x4 | approx. 2 | 0.272 | 0.183 | 0.106 | 0.102 | 0.046 | 0.019 | |
|
||||
| 16-node | 2x2 | approx. 61/2 | 0.266 | 0.182 | 0.110 | 0.105 | 0.048 | 0.019 | |
|
||||
| Experimental1 | | | 0.297 | 0.204 | 0.121 | 0.129 | 0.056 | 0.022 | |
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
*Geometric non-linear response of a shallow spherical shell*
|
||||
|
||||
*Figure 13a* shows the spherical shell that was also analysed<sup>3</sup> with one cubic shell element, modelling onequarter of the shell. To test our new 4-node shell element, the same nodal point layout was used<sup>3</sup> , giving a mesh of nine elements. *Figure 13b* shows the response calculated, including the post-buckling response (not reported in ref. 3) with the automatic load stepping algorithm<sup>4</sup> . Good correspondence with the analytical solution of Leicester<sup>2</sup> . 0 and the solution of Horrigmoe<sup>16</sup> was obtained. The solution with the 16-node element was almost twice as expensive as the 4-node element solution (using in both cases the same parameters for the automatic step-by-step solution algorithm).
|
||||
|
||||
*Linear buckling analysis and large deflection response of a simply-supported stiffened plate*
|
||||
|
||||
The stiffened plate shown in *Figure 14a* was analysed for its buckling reesponse. Since we expect the buckling mode to be symmetric<sup>26</sup> only one-quarter of the plate is modelled using symmetry boundary conditions. The model consists of nine 4-node shell elements and three 2 node isoparametric beam elements. At the nodes where a shell element connects to a beam element, three rotational degrees of freedom aligned with the global axes are considered for the shell element. In order to avoid locking of the isoparametric beam elements, one point Gauss integration along the beam axes was used. This does not introduce spurious zero energy modes in the model although the bending stiffness of the beam is underestimated.
|
||||
|
||||
The linearized buckling problem was solved as described in reference 4(37) and we obtained:
|
||||
|
||||
$$\frac{\sigma_{\rm cr}(\text{finite element solution})}{\sigma_{\rm cr}(\text{analytical solution})} = 1.02$$
|
||||
|
||||
84 Eng. Comput., 1984, Vol. 1, March
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
*Table 6* Convergence study for 4-node element: pinched cylinder
|
||||
|
||||
| Mesh for 1/8th of shell | Number of d.o.f. | FEM/Ŵc<br>analyt<br>Ŵc |
|
||||
|-------------------------|------------------|------------------------|
|
||||
| 5x5 | 130 | 0.51 |
|
||||
| 10x10 | 510 | 0.83 |
|
||||
| 20x20 | 2020 | 0.96 |
|
||||
|
||||
*Table* 7 Comparison between displacements for 4-node and 16 node elements: pinched cylinder
|
||||
|
||||
| Element | Mesh for 1/8th | Number of | FEM/Ŵcanalyt | | |
|
||||
|---------|----------------|-----------|--------------|--|--|
|
||||
| | of shell | d.o.f. | Ŵc | | |
|
||||
| 4-node | 20x20 | 2020 | 0.96 | | |
|
||||
| 16-node | 10x10 | 4530 | 0.98 | | |
|
||||
|
||||
Next, an initial imperfection with the shape of the first buckling mode and a maximum amplitude of 1/5 of the plate thickness was introduced. *Figure 14b* shows the large deflection response of this model as calculated using the automatic load stepping scheme of reference 4 with a tight energy convergence tolerance.
|
||||
|
||||
*Analysis of elastoplastic response of a circular plate*
|
||||
|
||||
The thin circular plate shown in *Figure 15a* was analysed for its elastoplastic response, when subjected to a concentrated load at its centre. The plate is simply-supported with its edges restrained from moving in its plane.
|
||||
|
||||
In a first solution, the plate model shown in *Figure 15a* was used to analyse the plate assuming small displacements (materially-non-linear-only conditions). *Figure 15c* shows that the theoretical collapse load is overestimated, but for the coarse mesh used, the predicted response is quite reasonable.
|
||||
|
||||
In a second solution, large displacements and elastoplastic conditions were assumed and in this case the stiffening behaviour of the plate shown in *Figure 15c* was predicted. In order to have a comparison, also the model of five axisymmetric 8-node elements shown in *Figure 15b* was solved. *Figure 15c* shows that both models predict in essence the same response; however, in this case relatively little plasticity was developed for the range of displacements considered.
|
||||
|
||||
#### CONCLUSIONS
|
||||
|
||||
A new four-node non-flat general non-linear shell element has been presented with the following important element properties: (1) the element is formulated using threedimensional continuum mechanics theory; hence the use of the element is not restricted by application of a specific shell theory; (2) the element is reliable and has good predictive capability in the analysis of thick and thin shells; (3) the amount of computations required to calculate the element stiffness matrix are very closely those that are used in standard isoparametric formulations. The computer time used could be reduced considerably in elastic analysis by using analytical integration through the element thickness.
|
||||
|
||||
In this paper we have presented the formulation and some applications of the element. The solution results obtained are most encouraging, but a formal mathematical convergence study of the element would be very valuable, and we are currently pursuing such research.
|
||||
|
||||
Finally, it should be noted that the element presented here provides a very attractive basic formulation that could be extended to large strain analysis and analysis of composite shells. Also, the concepts applied here to formulate a 4-node element could equally well be employed in an effective manner to formulate higher-order shell elements.
|
||||
|
||||
#### ACKNOWLEDGEMENTS
|
||||
|
||||
We are grateful for the financial support by the U.S. Army contract no. DAAK11-82-K-0005 and the ADINA users group for this work.
|
||||
|
||||
*Note added in proof* — We have just learned — and regret not to have known of it earlier — that R. H. MacNeal [J. *Nucl. Eng. Design,* 70, 3-12 (1982)] proposed a plate element for linear analysis that is very close to the element presented above.
|
||||
|
||||
|
||||
|
||||
# REFERENCES
|
||||
|
||||
- 1 Adini, A. Analysis of shell structures by the finite element method, *PhD Dissertation,* Department of Civil Engineering, University of California, Berkeley (1961)
|
||||
- 2 Bathe, K. J. *Finite Element Procedures in Engineering Analysis,* Prentice-Hall, Englewood Cliffs, New Jersey (1982)
|
||||
- 3 Bathe, K. J. and Bolourchi, S. A geometric and material nonlinear plate and shell element, *J. Comput. Struct.,* 11, 23-48 (1979)
|
||||
- 4 Bathe, K. J. and Dvorkin, E. N. On the automatic solution of nonlinear finite element equations, *J. Comput. Struct.* 17, (5-6), 871-879 (1983)
|
||||
- 5 Bathe, K. J.,Dvorkin, E. N. and Ho, L. W. Our discrete-Kirchhoff and isoparametric shell elements for nonlinear analysis - an assessment, *J. Comput. Struct.,* 16, (1-4), 89-98 (1983)
|
||||
|
||||
86 Eng. Comput., 1984, Vol. 1, March
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
- 6 Bathe, K. J. and Ho, L. W. A simple and effective element for analysis of general shell structures, *J. Comput. Struct.,* 13, 673-382 (1980)
|
||||
- 7 Bathe, K. J. and Ho, L. W. Some results in the analysis of thin shell structures, *Nonlinear Finite Element Analysis in Structural Mechanics,* (Ed. W. Wunderlich *et al.).* Springer-Verlag, Berlin (1981)
|
||||
- 8 Batoz, J. L., Bathe, K, J. and Ho, L. W. A study of three-node triangular plate bending elements, *Int. J, Num. Meth. Eng.,* 15, 1771-1812(1980)
|
||||
- 9 Batoz, J. L. and Ben Tahar, M. Evaluation of a new quadrilateral plate bending element, *Int. J, Num. Meth. Eng.,* 18, 1655-1677 (1982)
|
||||
- 10 Bercovier, M., Hasbgni, V., Gilon, Y, and Bathe, K, J, On a finite element procedure for nonlinear incompressible elasticity, *Hybrid and Mixed Finite Element Methods,* (Ed, S. M. Atluri *et al.),* John Wiley, New York (1983)
|
||||
- 11 Flügge, W. *Stresses in Shells,* 2nd edn, Springer-Verlag, Berlin (1973)
|
||||
- 12 Forsberg, K. and Hartung, R. An evaluation of finite difference and finite element techniques for analysis of general shells, *Symp. High Speed Computing of Elastic Structures,* IUTAM, Liege (1970)
|
||||
- 13 Fung, Y. C. *Foundations of Solid Mechanics,* Prentice-Hall, Englewood Cliffs, New Jersey (1965)
|
||||
- 14 Gallagher, R. H. Problems and progress in thin shell finite element analysis, *Finite Elements in Thin Shells and Curved Members,* (Ed. D. G. Ashwell and R. H. Gallagher), John Wiley, New York (1976)
|
||||
- 15 Green, A. E. and Zerna, W. *Theoretical Elasticity,* 2nd edn, Oxford University Press (1968)
|
||||
- 16 Horrigmoe, G. Finite element instability analysis of free-form shells, *Report 77-2,* Division of Structural Mechanics, The Norwegian Institute of Technology, University of Trondheim, Norway (1977)
|
||||
- 17 Hughes, T. J. R. and Liu, W. K. Nonlinear finite element analysis of shells: Part I, Three-dimensional shells, *J. Comput. Meth. Appl. Mech. Eng., 26,* 331-362 (1981)
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
- 19 Krakeland, B. Nonlinear analysis of shells using degenerate isoparametric elements, *Finite Elements in Nonlinear Mechanics,* Vol. 1, (Ed. P. G. Bergan *et al*.), Tapir Publishers (Norwegian Institute of Technology, Trondheim, Norway) (1978)
|
||||
- 20 Leicester, R. H. Finite deformations of shallow shells, *Proc. Am. Soc. Civil Eng.,* 94, (EM6), 1409-1423 (1968)
|
||||
- 21 Lindberg, G. M., Olson, M. D. and Cowper, G. R. New developments in the finite element analysis of shells, *Q. Bull. Div. Mech. Eng. and the National Aeronautical Establishment,* National Research Council of Canada, Vol. 4 (1969)
|
||||
- 22 MacNeal, R. H. A simple quadrilateral shell element, *J. Comput. Struct.* 8, 175-183 (1978)
|
||||
- 23 Noor, A. K. and Peters, J. M. Mixed models and reduced/selec-
|
||||
|
||||
|
||||
|
||||
tive integration displacement models for nonlinear analysis of curved beams, *Int. J. Num. Meth. Eng.,* 17, 615-631 (1981)
|
||||
|
||||
- 24 Ramm, E. and Sattele, J. M. Elasto-plastic large deformation shell analysis using degenerated elements, *Nonlinear Finite Element Analysis of Plates and Shells,* (Ed. T. J. R. Hughes), AMD-Vol. 48, Am. Soc. Mech. Eng., New York (1981)
|
||||
- 25 *Report AE 83-5,* ADINA System Verification Manual, ADINA Engineering, Vasteras, Sweden and Watertown, Mass. (1983)
|
||||
- 26 Timoshenko, S. P. and Gere, J. M. *Theory of Elastic Stability,* 2nd edn, McGraw-Hill, New York (1961)
|
||||
- 27 Washizu, K. *Variational Methods in Elasticity and Plasticity,* Pergamon Press, Oxford and New York (1968)
|
||||
- 28 Wempner, G., Talaslidis, D. and Hwang, C.-M. A simple and efficient approximation of shells via finite quadrilateral elements, *J. Appl. Mech.,* 49, 115-120 (1982)
|
||||
- 29 Zienkiewicz, O. C. *The Finite Element Method,* McGraw-Hill, New York (1977)
|
||||
-1002
File diff suppressed because it is too large
Load Diff
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 52 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 26 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 38 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 33 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 22 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 32 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 17 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 23 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 21 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 42 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 37 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 30 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 19 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 17 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 17 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 16 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 17 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 13 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 8.8 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 17 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 14 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 48 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 15 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 12 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 58 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 26 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 23 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 39 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 20 KiB |
BIN
Binary file not shown.
|
Before Width: | Height: | Size: 161 KiB |
-171
@@ -1,171 +0,0 @@
|
||||
# Four-Node Quadrilateral Shell Element MITC4
|
||||
|
||||
DVORˇ AKOV ´ A Edita ´ 1 ,a, ∗ , PATZAK Bo ´ ˇrek1, b
|
||||
|
||||
1 Department of Mechanics, Faculty of Civil Engineering, CTU in Prague, Thakurova 7, ´ 160 00 Prague, Czech Republic
|
||||
|
||||
a edita.dvorakova@fsv.cvut.cz, b borek.patzak@fsv.cvut.cz
|
||||
|
||||
Keywords: Finite Elements; MITC4; Scordelis-Lo Shell; Shear Locking; Shell Structures.
|
||||
|
||||
Abstract. Four-node quadrilateral element MITC4 applicable to both thick and thin shells is presented. The element formulation starts from three-dimensional continuum description degenerated to shell behavior. Shear locking, which is common problem in analysis of thin shells, is overcome by the use of MITC (Mixed Interpolation of Tensorial Components) approach. Element has been implemented into finite element code OOFEM and its performance is demonstrated on Scordelis-Lo shell, a benchmark problem frequently used in the evaluation of shell elements.
|
||||
|
||||
### Introduction
|
||||
|
||||
Shell structures are widely used in structural engineering for their load-carrying efficiency. The finite element method appears to be the most powerful tool in analysis of shell structures, however the accuracy of the method is significantly influenced by the choice of the particular element. Many shell elements have been developed over the years. Existing elements can be divided into two main groups. One group consists of elements based on some particular shell theory, the other group of the elements is based on three-dimensional analysis degenerated to shell behavior.
|
||||
|
||||
Presented quadrilateral element has been proposed by Dvorkin and Bathe [1] and is based on the second approach, therefore it is independent of any particular shell theory. The authors aimed to formulate element applicable to both thick and thin shells of arbitrary geometries with all degrees of freedom concentrated to the vertices of the element. However high-order elements with 9 and 16 nodes have been already successfully employed by Bathe and Bolourchi [2], this 4-node element suffers from the deficiency of shear locking. To prevent the element from the shear locking the use of MITC approach has been proposed.
|
||||
|
||||
# Element Formulation
|
||||
|
||||
The element is shown in Fig. 1 and its geometry is described using
|
||||
|
||||
$$
|
||||
x_{i}=\sum_{k=1}^{4}h_{k}x_{i}^{k}+\frac{r_{3}}{2}\sum_{k=1}^{4}a_{k}h_{k}V_{ ni}^{k}, \tag{1}
|
||||
$$
|
||||
|
||||
where hk(r1, r2) are the two-dimensional interpolation functions corresponding to node k, r i are the natural coordinates, x i are the Cartesian coordinates of any point in the element, x k i are the Cartesian coordinates of node k, V k ni are the components of director vector at node k and a k is the thickness of structure measured along the director vector. The displacement of any point in the element can be described using
|
||||
|
||||
$$
|
||||
u_{i}=\sum_{k=1}^{4}h_{k}u_{i}^{k}+\frac{r_{3}}{2}\sum_{k=1}^{4}a_{k}h_{k}(-V_ {2i}^{k}\alpha_{k}+V_{1i}^{k}\beta_{k}), \tag{2}
|
||||
$$
|
||||
|
||||
|
||||
|
||||
where `u_i` are the displacements in direction of Cartesian coordinates and `\alpha_k` and `\beta_k` are the rotations of director vector at node k around `V_1^k` and `V_2^k` , where
|
||||
|
||||
$$
|
||||
V_{1}^{k}=\frac{\mathbf{e}_{2}\times V_{n}^{k}}{\|\mathbf{e}_{2}\times V_{n}^{k}\|}, \tag{3}
|
||||
$$
|
||||
|
||||
$$
|
||||
V_{2}^{k}=V_{n}^{k}\times V_{1}^{k}. \tag{4}
|
||||
$$
|
||||
|
||||
Vector `e_2` is the basis vector of Cartesian coordinate system. Considering linear interpolation of displacement field, we introduce the vector of unknown displacements
|
||||
|
||||
$$
|
||||
\boldsymbol{r}_e = \{u_1^1, u_2^1, u_3^1, \alpha_1, \beta_1, u_1^2, u_2^2, u_3^2, \alpha_2, \beta_2, u_1^3, u_2^3, u_3^3, \alpha_3, \beta_3, u_1^4, u_2^4, u_3^4, \alpha_4, \beta_4\}^T.
|
||||
$$
|
||||
|
||||
The biggest drawback of this formulation of the element is that the element locks when the thickness of shell is small. Using the interpolation (2), non-zero transverse shear strains are obtained even when the structure is subjected to the constant bending moment. The remedy is to construct an assumed transverse strain, such that
|
||||
|
||||
$$
|
||||
\tilde{\varepsilon}_{13} = \frac{1}{2}(1+r_2)\tilde{\varepsilon}_{13}^A + \frac{1}{2}(1-r_2)\tilde{\varepsilon}_{13}^C,
|
||||
\tilde{\varepsilon}_{23} = \frac{1}{2}(1+r_1)\tilde{\varepsilon}_{23}^D + \frac{1}{2}(1-r_1)\tilde{\varepsilon}_{23}^B,
|
||||
$$
|
||||
|
||||
where $\sim$ over the quantities emphasizes the formulation in convected coordinate system and components `\tilde{\varepsilon}_{13}^A` , `\tilde{\varepsilon}_{23}^B` , `\tilde{\varepsilon}_{13}^C` and `\tilde{\varepsilon}_{23}^D` are equal to the values of corresponding strains in the middle of element edges calculated from displacements. This assumed transversal strain $\tilde{\varepsilon}_{13}$ is constant along `r_1` direction and linear along `r_2` direction.
|
||||
|
||||
To transform formulas for shear strain components to Cartesian coordinate system, it is convenient to use covariant and contravariant bases in convected coordinate system of `r_1, r_2` and `r_3` . For the element with midsurface in plane x, y the covariant basis vectors `\mathbf{g}_i` (i = 1, 2, 3) are given by
|
||||
|
||||
$$
|
||||
\mathbf{g}_{i}=\frac{\partial\mathbf{x}}{\partial r_{i}}. \tag{7}
|
||||
$$
|
||||
|
||||
Contravariant basis vectors $\mathbf{g}^{j}$ (j = 1, 2, 3) can be then calculated using
|
||||
|
||||
$$
|
||||
\mathbf{g}_{i}\mathbf{g}^{j}=\delta_{i}^{j}, \tag{8}
|
||||
$$
|
||||
|
||||
where `\delta_i^j` is Kronecker delta, `\delta_i^j = 1` for i = j and `\delta_i^j = 0` otherwise. Components $\tilde{\varepsilon}_{ij}$ can be evaluated using
|
||||
|
||||
$$
|
||||
\tilde{\varepsilon}_{ij}=\frac{1}{2}\left(\frac{\partial\mathbf{u}}{\partial r_{ i}}\mathbf{g}_{j}+\frac{\partial\mathbf{u}}{\partial r_{j}}\mathbf{g}_{i}\right), \tag{9}
|
||||
$$
|
||||
|
||||
$$
|
||||
\tilde{\varepsilon}_{ij}=\frac{1}{2}\left(\frac{\partial\mathbf{u}}{\partial r_{ i}}\frac{\partial\mathbf{x}}{\partial r_{j}}+\frac{\partial\mathbf{u}}{\partial r_{j}} \frac{\partial\mathbf{x}}{\partial r_{i}}\right). \tag{10}
|
||||
$$
|
||||
|
||||
By substituting from (1) and (2) into equation (10) and evaluating at points A, B, C, D the relations for `\tilde{\varepsilon}_{13}^A` , `\tilde{\varepsilon}_{23}^B` , `\tilde{\varepsilon}_{13}^C` and `\tilde{\varepsilon}_{23}^D` can be obtained
|
||||
|
||||
$$
|
||||
\tilde{\varepsilon}_{13}^{A} = \frac{1}{8} \left[ (\mathbf{u}^{1} - \mathbf{u}^{2}) \cdot \frac{1}{2} (a_{1}V_{n}^{1} + a_{2}V_{n}^{2}) + (\mathbf{x}^{1} - \mathbf{x}^{2}) \cdot \frac{1}{2} (a_{1}(-V_{2}^{1}\alpha_{1} + V_{1}^{1}\beta_{1}) + a_{2}(-V_{2}^{2}\alpha_{2} + V_{1}^{2}\beta_{2})) \right],
|
||||
\tilde{\varepsilon}_{13}^{C} = \frac{1}{8} \left[ (\mathbf{u}^{4} - \mathbf{u}^{3}) \cdot \frac{1}{2} (a_{3}V_{n}^{3} + a_{4}V_{n}^{4}) + (\mathbf{x}^{4} - \mathbf{x}^{3}) \cdot \frac{1}{2} (a_{3}(-V_{2}^{3}\alpha_{3} + V_{1}^{3}\beta_{3}) + a_{4}(-V_{2}^{4}\alpha_{4} + V_{1}^{4}\beta_{4})) \right],
|
||||
\tilde{\varepsilon}_{23}^{B} = \frac{1}{8} \left[ (\mathbf{u}^{1} - \mathbf{u}^{4}) \cdot \frac{1}{2} (a_{1}V_{n}^{1} + a_{4}V_{n}^{4}) + (\mathbf{x}^{1} - \mathbf{x}^{4}) \cdot \frac{1}{2} (a_{1}(-V_{2}^{1}\alpha_{1} + V_{1}^{1}\beta_{1}) + a_{4}(-V_{2}^{4}\alpha_{4} + V_{1}^{4}\beta_{4})) \right],
|
||||
\tilde{\varepsilon}_{23}^{D} = \frac{1}{8} \left[ (\mathbf{u}^{2} - \mathbf{u}^{3}) \cdot \frac{1}{2} (a_{2}V_{n}^{2} + a_{3}V_{n}^{3}) + (\mathbf{x}^{2} - \mathbf{x}^{3}) \cdot \frac{1}{2} (a_{2}(-V_{2}^{2}\alpha_{2} + V_{1}^{2}\beta_{2}) + a_{3}(-V_{2}^{3}\alpha_{3} + V_{1}^{3}\beta_{3})) \right].
|
||||
$$
|
||||
|
||||
Final equations for $\tilde{\varepsilon}_{13}$ and $\tilde{\varepsilon}_{23}$ are then derived by substituting (11) back into the equations (6). Transformation to the Cartesian coordinates is performed using
|
||||
|
||||
$$
|
||||
\tilde{\varepsilon}_{ij}\mathbf{g}^{i}\mathbf{g}^{j}=\varepsilon_{kl}\mathbf{e}_{k}\mathbf{e}_ {l}, \tag{12}
|
||||
$$
|
||||
|
||||
where $\varepsilon_{kl}$ are components of strain tensor in Cartesian coordinates with basis vectors `e_k` and `e_l` . By considering the symmetry of strain tensor, the shear components of strain vector $\gamma_{xz}$ and $\gamma_{yz}$ are obtained
|
||||
|
||||
$$
|
||||
\gamma_{xy} = 2\tilde{\varepsilon}_{13}(\boldsymbol{g}^{1} \cdot \boldsymbol{e}_{1})(\boldsymbol{g}^{3} \cdot \boldsymbol{e}_{3}) + 2\tilde{\varepsilon}_{23}(\boldsymbol{g}^{2} \cdot \boldsymbol{e}_{1})(\boldsymbol{g}^{3} \cdot \boldsymbol{e}_{3}),
|
||||
\gamma_{yz} = 2\tilde{\varepsilon}_{13}(\boldsymbol{g}^{1} \cdot \boldsymbol{e}_{2})(\boldsymbol{g}^{3} \cdot \boldsymbol{e}_{3}) + 2\tilde{\varepsilon}_{23}(\boldsymbol{g}^{2} \cdot \boldsymbol{e}_{2})(\boldsymbol{g}^{3} \cdot \boldsymbol{e}_{3})
|
||||
$$
|
||||
|
||||
Substituting from (6) and (11) into (13) the final formulas for transverse shear strains are derived. The remaining components of strain vector are calculated directly from the displacements.
|
||||
|
||||
|
||||
|
||||
Stiffness matrix is evaluated using
|
||||
|
||||
$$
|
||||
\mathbf{K}=\int\limits_{V}\mathbf{B}^{T}\mathbf{D}\mathbf{B}\ dV, \tag{14}
|
||||
$$
|
||||
|
||||
where B is a strain-displacement matrix resulting from derived formulas for strain components and D is a material matrix for three-dimensional problem degenerated to shell behaviour, where the condition of σ z = 0 is enforced.
|
||||
|
||||
# Scordelis-Lo Shell Problem
|
||||
|
||||
Element has been implemented into finite element code OOFEM [3, 4] and implementation has been verified using classical patch tests. It has been proven, that the patch tests are passed for the case of pure bending, pure shear and pure twist and also for three membrane stress states.
|
||||
|
||||
To test the element performance on more complex structure and to compare its results with other available elements and with the analytical solution the analysis of Scordelis-Lo Shell [1] is presented. The structure is loaded by its dead weight and is shown in Fig. 2. Due to the symmetry only one quarter of the structure has been analyzed. Deformed shape of the structure is shown in Fig. 3, obtained results are shown in Fig. 4. Solution obtained using the element composed of plane-stress element with rotational degrees of freedom and Discrete Kirchhoff Triangle plate element, is also shown for reference, labeled as RDKT.
|
||||
|
||||
#### Summary
|
||||
|
||||
Element for shells has been implemented into existing finite element code OOFEM and its performance has been verified. Shear locking, which can cause highly inaccurate results in analysis of thin shells, has been overcome by MITC approach. This approach seems to be sufficient as appropriate results were obtained while testing the element implementation. Numerical results has shown that the element is competitive with the performance of thin-plate elements. Possibility of use of MITC4 element for both thick and thin shells can be seen as advantage over the RDKT and other shell theory based elements, which are usually limited by thickness/span ratio to either thin or thick shells. The element shows very good convergence to the reference solution [1], even outperforming the planar shell element composed of thin Kirchhoff plate element and membrane element.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
## Acknowledgement
|
||||
|
||||
The financial support of this research by the Grant Agency of the Czech Technical University in Prague (SGS project No. 15/031/OHK1/1T/11) and by the Technology Agency of the Czech Republic (TACR project No. TA02011196) is gratefully acknowledged. ˇ
|
||||
|
||||
## References
|
||||
|
||||
- [1] K. J. Bathe, E. Dvorkin, A continuum mechanics based four-node shell element for general nonlinear analysis, Eng. Comput., Vol.1, March (1984) 77-88.
|
||||
- [2] K. J. Bathe, S. Bolourchi, A geometric ad material nonlinear plate and shell element, Computers & Structures, Vol.11, (1980) 23-48.
|
||||
- [3] B. Patzak, Z. Bittnar, Design of object oriented finite element code, Advances in Engineering ´ Software 32 (10-11) (2001) 759-767.
|
||||
- [4] B. Patzak, OOFEM project home page, http://www.oofem.org, 2014. ´
|
||||
- [5] K. J. Bathe, Finite Element Procedures, Prentice Hall International, Inc., 1996.
|
||||
- [6] M. L. Bucalem, K. J. Bathe, Finite element analysis of shell structures, Archives of Computational Methods in Engineering, Vol.4, 1 (1997) 3-61.
|
||||
|
||||
#### Modern Methods of Experimental and Computational Investigations in Area of Construction
|
||||
|
||||
10.4028/www.scientific.net/AMM.825
|
||||
|
||||
#### Four-Node Quadrilateral Shell Element MITC4
|
||||
|
||||
10.4028/www.scientific.net/AMM.825.99
|
||||
|
||||
## DOI References
|
||||
|
||||
[1] K. J. Bathe, E. Dvorkin, A continuum mechanics based four-node shell element for general nonlinear analysis, Eng. Comput., Vol. 1, March (1984) 77-88.
|
||||
|
||||
10.1108/eb023562
|
||||
|
||||
[2] K. J. Bathe, S. Bolourchi, A geometric ad material nonlinear plate and shell element, Computers & Structures, Vol. 11, (1980) 23-48.
|
||||
|
||||
10.1016/0045-7949(80)90144-3
|
||||
|
||||
[3] B. Patz´ak, Z. Bittnar, Design of object oriented finite element code, Advances in Engineering Software 32 (10-11) (2001) 759-767.
|
||||
|
||||
10.1016/s0965-9978(01)00027-8
|
||||
|
||||
[5] K. J. Bathe, Finite Element Procedures, Prentice Hall International, Inc., (1996).
|
||||
|
||||
10.1002/nme.1620190115
|
||||
|
||||
[6] M. L. Bucalem, K. J. Bathe, Finite element analysis of shell structures, Archives of Computational Methods in Engineering, Vol. 4, 1 (1997) 3-61.
|
||||
|
||||
10.1007/bf02818930
|
||||
Binary file not shown.
|
Before Width: | Height: | Size: 34 KiB |
Some files were not shown because too many files have changed in this diff Show More
Reference in New Issue
Block a user