김경종
91 lines
3.3 KiB
Markdown
91 lines
3.3 KiB
Markdown
---
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type: concept
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title: "Solid Element Stiffness Integration"
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complexity: advanced
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domain: computational-mechanics
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aliases:
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- solid element stiffness matrix
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- solid element Gauss integration
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- 3D element quadrature
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created: 2026-05-28
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updated: 2026-05-29
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address: c-000052
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tags:
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- concept
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- finite-element-method
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- solid-elements
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- numerical-integration
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status: current
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related:
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- "[[Solid Element Notes]]"
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- "[[Solid Element Strain-Displacement Matrix]]"
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- "[[Isoparametric Finite Elements]]"
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- "[[Displacement-Based Finite Element Formulation]]"
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- "[[Abaqus Element Library]]"
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- "[[Reduced Integration and Hourglass Control]]"
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- "[[Hybrid Incompressible Elements]]"
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sources:
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- "[[Solid Element Notes]]"
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- "[[Abaqus Theory Manual]]"
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source_refs:
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- source: "[[Solid Element Notes]]"
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raw_path: ".raw/SolidElement/"
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raw_files:
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- "SolidElement_001.md"
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md_indices:
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- 1
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match: "heuristic-heading-keyword"
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confidence: low
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- source: "[[Abaqus Theory Manual]]"
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raw_path: ".raw/AbaqusTheoriesManual/"
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raw_files:
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- "AbaqusTheoriesManual_029.md"
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- "AbaqusTheoriesManual_032.md"
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- "AbaqusTheoriesManual_index2.md"
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- "AbaqusTheoriesManual_027.md"
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md_indices:
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- 29
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- 32
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- 87
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- 27
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match: "heuristic-heading-keyword"
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confidence: high
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---
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# Solid Element Stiffness Integration
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## Definition
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Solid element stiffness integration evaluates the element stiffness matrix for a three-dimensional continuum element by numerically integrating `B^T D B` over the element volume.
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## How It Works
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The source uses the standard displacement-based stiffness form:
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```text
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K = integral_V B^T D B dV
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= integral B^T D B |J| dxi deta dzeta
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```
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Here `B` is the [[Solid Element Strain-Displacement Matrix]], `D` is the three-dimensional Hooke-law constitutive matrix, and `|J|` is the determinant of the Jacobian that maps the natural-coordinate integration region to physical volume.
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The notes list quadrature schemes for the first-order solid topologies: one-point integration for the 4-node tetrahedron, eight-point integration for the 5-node pyramid, six-point integration for the 6-node wedge, and eight-point integration for the 8-node hexahedron.
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[[Abaqus Element Library]] adds the broader element-library tradeoff: full, reduced, selective, and hybrid integration choices affect locking, hourglass modes, cost, and incompressible material behavior.
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## Why It Matters
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The stiffness integral is where interpolation, material law, element distortion, and numerical quadrature meet. Incorrect quadrature or a poor Jacobian can produce inaccurate stiffness, spurious mechanisms, or poor convergence even when the symbolic formulation is correct.
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## Connections
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- [[Isoparametric Finite Elements]] supplies the natural-coordinate integration framework.
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- [[Solid Element Shape Functions]] and [[Solid Element Strain-Displacement Matrix]] define the integrand.
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- [[Incompatible Mode Solid Elements]] modifies the displacement field and therefore expands the stiffness matrix before static condensation.
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- [[Reduced Integration and Hourglass Control]] and [[Hybrid Incompressible Elements]] describe two common responses to stiffness and constraint pathologies.
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## Sources
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- [[Solid Element Notes]]
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- [[Abaqus Theory Manual]]
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