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.
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