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wiki/concepts/Solid Element Shape Functions.md

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+---
+type: concept
+title: "Solid Element Shape Functions"
+complexity: intermediate
+domain: computational-mechanics
+aliases:
+  - solid element interpolation functions
+  - linear solid shape functions
+  - tetrahedral wedge pyramid hexahedral shape functions
+created: 2026-05-28
+updated: 2026-05-28
+address: c-000050
+tags:
+  - concept
+  - finite-element-method
+  - solid-elements
+  - interpolation
+status: current
+related:
+  - "[[Solid Element Notes]]"
+  - "[[Isoparametric Linear Solid Elements]]"
+  - "[[Isoparametric Finite Elements]]"
+  - "[[Solid Element Strain-Displacement Matrix]]"
+sources:
+  - "[[Solid Element Notes]]"
+---
+
+# Solid Element Shape Functions
+
+## Definition
+
+Solid element shape functions interpolate three-dimensional element geometry and displacement from nodal values in natural coordinates.
+
+## Covered Topologies
+
+The notes give first-order interpolation for four common solid element shapes:
+
+- 4-node tetrahedron with barycentric-style coordinates.
+- 5-node pyramid connecting a quadrilateral base to an apex.
+- 6-node wedge, or triangular prism, using triangular interpolation through a two-node thickness direction.
+- 8-node hexahedron with trilinear interpolation in `xi`, `eta`, and `zeta`.
+
+## Why They Matter
+
+Shape functions are the starting point for every later element calculation. They define the displacement approximation, the geometry mapping, the Jacobian, the derivative transformation, and ultimately the strain-displacement matrix. Because the same functions interpolate geometry and field variables, the source is a concrete example of [[Isoparametric Finite Elements]].
+
+## Modeling Implications
+
+Low-order solid shape functions are economical but sensitive to distortion and limited in bending-dominated response. This is why element aspect ratio and topology selection matter before any solver choice is considered.
+
+## Connections
+
+- [[Isoparametric Linear Solid Elements]] gives the element-level context.
+- [[Solid Element Strain-Displacement Matrix]] differentiates these functions after Jacobian mapping.
+- [[Solid Element Stiffness Integration]] integrates the resulting `B^T D B` expression over the element volume.
+
+## Sources
+
+- [[Solid Element Notes]]