---
type: concept
title: "Solid Element Stiffness Integration"
complexity: advanced
domain: computational-mechanics
aliases:
  - solid element stiffness matrix
  - solid element Gauss integration
  - 3D element quadrature
created: 2026-05-28
updated: 2026-05-29
address: c-000052
tags:
  - concept
  - finite-element-method
  - solid-elements
  - numerical-integration
status: current
related:
  - "[[Solid Element Notes]]"
  - "[[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]]"
source_refs:
  - source: "[[Solid Element Notes]]"
    raw_path: ".raw/SolidElement/"
    raw_files:
      - "SolidElement_001.md"
    md_indices:
      - 1
    match: "heuristic-heading-keyword"
    confidence: low
  - source: "[[Abaqus Theory Manual]]"
    raw_path: ".raw/AbaqusTheoriesManual/"
    raw_files:
      - "AbaqusTheoriesManual_029.md"
      - "AbaqusTheoriesManual_032.md"
      - "AbaqusTheoriesManual_index2.md"
      - "AbaqusTheoriesManual_027.md"
    md_indices:
      - 29
      - 32
      - 87
      - 27
    match: "heuristic-heading-keyword"
    confidence: high
---

# Solid Element Stiffness Integration

## Definition

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.

## How It Works

The source uses the standard displacement-based stiffness form:

```text
K = integral_V B^T D B dV
  = integral B^T D B |J| dxi deta dzeta
```

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.

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.

## Connections

- [[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]]