---
type: source
title: "Solid Element Notes"
source_type: study-notes
created: 2026-05-28
updated: 2026-05-28
address: c-000048
aliases:
  - SolidElement
  - "등매개 선형 솔리드 요소"
tags:
  - source
  - finite-element-method
  - solid-elements
  - isoparametric-elements
status: current
confidence: high
raw_path: ".raw/SolidElement/"
source_files:
  markdown_files: 1
  image_files: 65
related:
  - "[[Isoparametric Linear Solid Elements]]"
  - "[[Solid Element Shape Functions]]"
  - "[[Solid Element Strain-Displacement Matrix]]"
  - "[[Solid Element Stiffness Integration]]"
  - "[[Incompatible Mode Solid Elements]]"
  - "[[Isoparametric Finite Elements]]"
---

# Solid Element Notes

## Summary

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.

The local source is a converted Markdown/image extraction: one Markdown file and 65 extracted images under `.raw/SolidElement/`.

## Coverage Map

| Section | Topic |
|---|---|
| 1 | [[Isoparametric Linear Solid Elements]] and practical modeling notes |
| 2.1 | Position and displacement interpolation with shape functions |
| 2.1 | [[Solid Element Shape Functions]] for 4-node tetrahedron, 5-node pyramid, 6-node wedge, and 8-node hexahedron |
| 2.2 | [[Solid Element Strain-Displacement Matrix]] and Jacobian derivative mapping |
| 2.3 | Three-dimensional Hooke-law stress-strain matrix |
| 2.4 | [[Solid Element Stiffness Integration]] using `B^T D B` and element-specific Gauss points |
| 2.5 | [[Incompatible Mode Solid Elements]] and static condensation of internal mode degrees of freedom |

## Key Takeaways

- Linear solid elements model three-dimensional volume response with three translational degrees of freedom per node and no rotational degrees of freedom.
- The same shape functions interpolate both geometry and displacement, making the notes a direct extension of [[Isoparametric Finite Elements]] into 3D continuum elements.
- The Jacobian maps derivatives from natural coordinates to physical coordinates, allowing the `B` matrix to be assembled in global coordinates.
- The element stiffness follows the standard displacement-based form `K = integral B^T D B dV`, evaluated in natural coordinates with the Jacobian determinant.
- Incompatible modes add internal displacement modes to selected wedge and hexahedral elements, then eliminate those extra degrees of freedom by static condensation.

## Concepts Introduced

- [[Isoparametric Linear Solid Elements]]
- [[Solid Element Shape Functions]]
- [[Solid Element Strain-Displacement Matrix]]
- [[Solid Element Stiffness Integration]]
- [[Incompatible Mode Solid Elements]]

## Source Notes

- Source path: `.raw/SolidElement/`
- Composite source hash recorded in `.raw/.manifest.json`.
- The note contains a few OCR/math transcription issues in the quadrature expressions, but the formulation sequence and element topology coverage are usable.