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2026-05-29 17:05:53 +09:00
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wiki/concepts/Solid Element Stiffness Integration.md
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@@ -8,7 +8,7 @@ aliases:
- solid element Gauss integration
- 3D element quadrature
created: 2026-05-28
updated: 2026-05-28
updated: 2026-05-29
address: c-000052
tags:
- concept
@@ -21,8 +21,12 @@ related:
- "[[Solid Element Strain-Displacement Matrix]]"
- "[[Isoparametric Finite Elements]]"
- "[[Displacement-Based Finite Element Formulation]]"
- "[[Abaqus Element Library]]"
- "[[Reduced Integration and Hourglass Control]]"
- "[[Hybrid Incompressible Elements]]"
sources:
- "[[Solid Element Notes]]"
- "[[Abaqus Theory Manual]]"
---
# Solid Element Stiffness Integration
@@ -44,6 +48,8 @@ Here `B` is the [[Solid Element Strain-Displacement Matrix]], `D` is the three-d
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.
[[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.
## Why It Matters
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.
@@ -53,7 +59,9 @@ The stiffness integral is where interpolation, material law, element distortion,
- [[Isoparametric Finite Elements]] supplies the natural-coordinate integration framework.
- [[Solid Element Shape Functions]] and [[Solid Element Strain-Displacement Matrix]] define the integrand.
- [[Incompatible Mode Solid Elements]] modifies the displacement field and therefore expands the stiffness matrix before static condensation.
- [[Reduced Integration and Hourglass Control]] and [[Hybrid Incompressible Elements]] describe two common responses to stiffness and constraint pathologies.
## Sources
- [[Solid Element Notes]]
- [[Abaqus Theory Manual]]