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wiki/concepts/Solid Element Stiffness Integration.md

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+---
+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: 2026-05-28
+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]]"
+sources:
+  - "[[Solid Element Notes]]"
+---
+
+# 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.
+
+## 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.
+
+## Sources
+
+- [[Solid Element Notes]]