201 lines
8.5 KiB
Markdown
201 lines
8.5 KiB
Markdown
# MITC4 Formulation
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## Purpose
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This document defines the baseline MITC4 formulation target for FESA Phase 1.
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It is intentionally a formulation contract, not implementation code. Exact formulas should be added and reviewed before coding the MITC4 element.
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## Source Basis
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- Dvorkin and Bathe's four-node shell element paper presents a continuum-mechanics-based, non-flat, general quadrilateral shell element for thin and thick shells and nonlinear analysis: https://web.mit.edu/kjb/www/Publications_Prior_to_1998/A_Continuum_Mechanics_Based_Four-Node_Shell_Element_for_General_Nonlinear_Analysis.pdf
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- The paper identifies transverse shear locking as a key problem in simple 4-node shell interpolation and motivates modified transverse shear treatment: https://web.mit.edu/kjb/www/Publications_Prior_to_1998/A_Continuum_Mechanics_Based_Four-Node_Shell_Element_for_General_Nonlinear_Analysis.pdf
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- OpenSees describes `ShellMITC4` as a bilinear isoparametric shell element with modified shear interpolation, four counter-clockwise nodes, and six DOFs per node: https://opensees.berkeley.edu/wiki/index.php/Shell_Element
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- The MITC benchmark paper states that the MITC method is used to remedy shell locking and that the standard MITC4 employs MITC treatment for transverse shear strains; it also notes that Abaqus S4 uses Dvorkin-Bathe transverse shear interpolation: https://web.mit.edu/kjb/www/Principal_Publications/Performance_of_the_MITC3%2B_and_MITC4%2B_shell_elements_in_widely_used_benchmark_problems.pdf
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- Abaqus finite-strain shell theory documentation provides useful comparison context for S4/S4R geometry, interpolation, orientation update, and transverse shear treatment, but FESA Phase 1 is linear static: https://abaqus-docs.mit.edu/2017/English/SIMACAETHERefMap/simathe-c-finitestrainshells.htm
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## Phase 1 Target
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Phase 1 implements a clear MITC4 baseline formulation and passes reference benchmarks before performance optimization.
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Scope:
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- 4-node quadrilateral shell.
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- Linear static analysis.
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- Linear isotropic elastic material.
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- Homogeneous shell section.
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- 6 DOFs per node.
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- Small-strain formulation for Phase 1.
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- Transverse shear interpolation based on MITC4 assumptions.
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- Abaqus-compatible result signs.
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Non-scope:
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- S4R reduced-integration behavior.
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- Hourglass control.
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- Composite sections.
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- Material nonlinearity.
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- Geometric nonlinearity.
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- Pressure loads.
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- Thermal-stress coupling.
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- Mesh quality diagnostics.
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## Nodal DOFs
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Each node has:
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```text
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UX, UY, UZ, RX, RY, RZ
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```
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Rules:
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- Translational DOFs are global translations.
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- Rotational DOFs are rotations about global or transformed axes following Abaqus component convention.
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- `RZ` is retained as a drilling DOF.
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- Drilling stiffness is artificial in Phase 1 and must be parameterized.
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## Element Input Contract
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`MITC4Element` requires:
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- four node ids in Abaqus S4 order.
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- four node coordinates.
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- shell thickness.
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- linear elastic material constants `E` and `nu`.
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- drilling stiffness parameter.
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- element id and property id for diagnostics and output.
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Node ordering:
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- Input node order follows Abaqus S4 convention.
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- Positive normal follows the right-hand rule around the nodes.
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- FESA maps Abaqus `TYPE=S4` to `MITC4`.
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- Abaqus `TYPE=S4R` is not supported in Phase 1.
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## Coordinate Frames
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The exact local basis construction must be completed before MITC4 implementation.
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Minimum requirements:
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- Define a local shell normal from the quadrilateral geometry.
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- Define local in-plane axes `e1` and `e2` so that `e1`, `e2`, and normal form a right-handed basis.
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- Preserve Abaqus-compatible output signs.
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- Document behavior for non-planar quadrilaterals.
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- Use the same convention consistently for stiffness, stress/strain recovery, and result output.
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Recommended Phase 1 convention:
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- Use the element midsurface geometry to compute an average normal.
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- Use a projected global axis to define the local 1-direction when possible, matching Abaqus convention conceptually.
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- Fall back to a stable element-edge-based direction when the projected global axis is nearly parallel to the normal.
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- Record the final algorithm in this document before coding.
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## Shape Functions
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Baseline quadrilateral bilinear interpolation:
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```text
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N1 = 0.25 * (1 - r) * (1 - s)
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N2 = 0.25 * (1 + r) * (1 - s)
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N3 = 0.25 * (1 + r) * (1 + s)
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N4 = 0.25 * (1 - r) * (1 + s)
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```
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where `r, s` are natural coordinates in `[-1, 1]`.
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Implementation requirements:
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- Compute shape function derivatives with respect to natural coordinates.
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- Build the surface Jacobian and local derivatives.
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- Detect invalid or near-zero Jacobian as a singular/invalid element diagnostic, not as a mesh quality metric.
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## Strain Treatment
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The baseline element separates:
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- membrane strain terms.
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- bending curvature terms.
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- transverse shear strain terms.
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- artificial drilling stabilization.
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MITC4 requirement:
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- Use standard displacement interpolation for membrane and bending terms in Phase 1.
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- Use MITC transverse shear interpolation to alleviate shear locking.
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- Do not replace MITC4 with plain full-integration Reissner-Mindlin Q4.
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The exact tying point equations and shear interpolation formula must be added from the selected primary source before implementation.
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## Numerical Integration
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Initial Phase 1 plan:
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- In-plane integration: 2x2 Gauss for membrane/bending/shear stiffness unless the final formulation requires a different split.
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- Thickness integration: homogeneous linear elastic section may be integrated analytically or with a documented simple rule.
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- Benchmark literature commonly reports 2x2 in-plane Gauss integration for S4/MITC4-style 4-node elements and 2-point thickness integration in comparative shell studies.
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Rules:
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- Do not introduce reduced integration or hourglass control for S4R behavior in Phase 1.
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- Do not optimize integration before reference benchmarks pass.
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- Integration point ordering for output must be documented before stress/strain reference comparisons.
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## Drilling DOF Stabilization
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Decision:
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- Phase 1 uses small artificial drilling stiffness.
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Requirements:
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- Expose a parameter such as `drilling_stiffness_scale`.
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- Provide a deterministic default.
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- Make the default small enough not to dominate physical response.
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- Include benchmark sensitivity checks if reference results are sensitive to the value.
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- Report the value in result metadata.
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Open default proposal:
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```text
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k_drill = alpha * representative_element_stiffness
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```
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where `alpha` should be selected only after reviewing formulation sources and early reference cases.
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## Element Outputs
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Phase 1 minimum:
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- element stiffness matrix.
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- element equivalent nodal internal force for full-vector residual/reaction recovery.
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- optional stress/strain output after displacement benchmarks are stable.
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Future output:
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- local shell stresses.
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- local shell strains.
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- section forces and moments.
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- integration point and section point data.
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## Required Element-Level Tests
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Before integration with the global solver:
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- shape functions sum to one.
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- derivatives satisfy expected bilinear identities.
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- element stiffness dimensions are `24 x 24`.
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- stiffness is symmetric for linear elastic Phase 1.
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- rigid body translations produce near-zero internal strain energy.
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- rigid body rotations do not create physical membrane/bending stiffness beyond documented drilling effects.
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- constant membrane patch behavior.
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- bending-dominated sanity case.
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- drilling stiffness sensitivity check.
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## Reference Benchmarks
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MITC4 baseline acceptance should include:
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- single-element membrane test.
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- single-element bending test.
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- cantilever shell strip.
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- simply supported square plate.
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- Scordelis-Lo roof.
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- pinched cylinder.
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- twisted beam.
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Distorted mesh tests should be added after the baseline passes, but Phase 1 does not implement general mesh quality diagnostics.
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## Future Extensions
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Geometric nonlinearity:
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- Add updated geometry, current frame handling, tangent stiffness, and Newton-Raphson integration.
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- Preserve `AnalysisState` element/internal state extension points.
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Thermal-stress coupling:
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- Add temperature field state.
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- Add thermal strain contribution.
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- Add material expansion data.
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- Add result fields for temperature and thermal strain/stress.
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S4R:
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- Add only after a separate ADR and formulation document update.
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- Requires reduced integration and hourglass control decisions.
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## Open Decisions Before Coding
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- Exact MITC4 transverse shear tying point formula.
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- Exact element local basis for warped quads.
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- Exact drilling stiffness default.
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- Exact stress/strain recovery locations.
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- Whether Phase 1 reports `S`, `E`, and `SF`, or only `U` and `RF`.
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- Whether local coordinate transforms from Abaqus input are deferred or rejected.
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