김경종
72 lines
3.1 KiB
Markdown
72 lines
3.1 KiB
Markdown
---
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type: source
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title: "Solid Element Notes"
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source_type: study-notes
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created: 2026-05-28
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updated: 2026-05-28
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address: c-000048
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aliases:
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- SolidElement
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- "등매개 선형 솔리드 요소"
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tags:
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- source
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- finite-element-method
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- solid-elements
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- isoparametric-elements
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status: current
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confidence: high
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raw_path: ".raw/SolidElement/"
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source_files:
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markdown_files: 1
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image_files: 65
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related:
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- "[[Isoparametric Linear Solid Elements]]"
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- "[[Solid Element Shape Functions]]"
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- "[[Solid Element Strain-Displacement Matrix]]"
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- "[[Solid Element Stiffness Integration]]"
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- "[[Incompatible Mode Solid Elements]]"
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- "[[Isoparametric Finite Elements]]"
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---
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# Solid Element Notes
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## Summary
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These local notes derive linear isoparametric solid elements for three-dimensional continuum modeling. They cover nodal displacement interpolation, common first-order solid element topologies, natural-coordinate shape functions, the strain-displacement matrix, Hooke-law material stiffness, Gaussian integration for element stiffness, and incompatible modes for locking relief.
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The local source is a converted Markdown/image extraction: one Markdown file and 65 extracted images under `.raw/SolidElement/`.
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## Coverage Map
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| Section | Topic |
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|---|---|
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| 1 | [[Isoparametric Linear Solid Elements]] and practical modeling notes |
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| 2.1 | Position and displacement interpolation with shape functions |
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| 2.1 | [[Solid Element Shape Functions]] for 4-node tetrahedron, 5-node pyramid, 6-node wedge, and 8-node hexahedron |
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| 2.2 | [[Solid Element Strain-Displacement Matrix]] and Jacobian derivative mapping |
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| 2.3 | Three-dimensional Hooke-law stress-strain matrix |
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| 2.4 | [[Solid Element Stiffness Integration]] using `B^T D B` and element-specific Gauss points |
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| 2.5 | [[Incompatible Mode Solid Elements]] and static condensation of internal mode degrees of freedom |
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## Key Takeaways
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- Linear solid elements model three-dimensional volume response with three translational degrees of freedom per node and no rotational degrees of freedom.
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- The same shape functions interpolate both geometry and displacement, making the notes a direct extension of [[Isoparametric Finite Elements]] into 3D continuum elements.
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- The Jacobian maps derivatives from natural coordinates to physical coordinates, allowing the `B` matrix to be assembled in global coordinates.
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- The element stiffness follows the standard displacement-based form `K = integral B^T D B dV`, evaluated in natural coordinates with the Jacobian determinant.
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- Incompatible modes add internal displacement modes to selected wedge and hexahedral elements, then eliminate those extra degrees of freedom by static condensation.
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## Concepts Introduced
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- [[Isoparametric Linear Solid Elements]]
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- [[Solid Element Shape Functions]]
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- [[Solid Element Strain-Displacement Matrix]]
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- [[Solid Element Stiffness Integration]]
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- [[Incompatible Mode Solid Elements]]
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## Source Notes
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- Source path: `.raw/SolidElement/`
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- Composite source hash recorded in `.raw/.manifest.json`.
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- The note contains a few OCR/math transcription issues in the quadrature expressions, but the formulation sequence and element topology coverage are usable.
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