---
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]]"
source_refs:
  - source: "[[Solid Element Notes]]"
    raw_path: ".raw/SolidElement/"
    raw_files:
      - "SolidElement_001.md"
    md_indices:
      - 1
    match: "heuristic-heading-keyword"
    confidence: low
---

# 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]]