김경종
60 lines
2.2 KiB
Markdown
60 lines
2.2 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: 2026-05-28
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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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sources:
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- "[[Solid Element Notes]]"
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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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## 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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## Sources
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- [[Solid Element Notes]]
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