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wiki/concepts/Isoparametric Linear Solid Elements.md

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+---
+type: concept
+title: "Isoparametric Linear Solid Elements"
+complexity: intermediate
+domain: computational-mechanics
+aliases:
+  - linear solid elements
+  - first-order solid elements
+  - isoparametric solid elements
+  - 3D solid elements
+created: 2026-05-28
+updated: 2026-05-28
+address: c-000049
+tags:
+  - concept
+  - finite-element-method
+  - solid-elements
+  - isoparametric-elements
+status: current
+related:
+  - "[[Solid Element Notes]]"
+  - "[[Isoparametric Finite Elements]]"
+  - "[[Displacement-Based Finite Element Formulation]]"
+  - "[[Solid Element Shape Functions]]"
+  - "[[Solid Element Strain-Displacement Matrix]]"
+  - "[[Solid Element Stiffness Integration]]"
+sources:
+  - "[[Solid Element Notes]]"
+---
+
+# Isoparametric Linear Solid Elements
+
+## Definition
+
+Isoparametric linear solid elements are first-order three-dimensional continuum finite elements that interpolate both geometry and displacement with the same nodal shape functions.
+
+## How They Work
+
+The source treats solid elements as volume elements with three translational displacement degrees of freedom per node: `u`, `v`, and `w`. They do not include rotational degrees of freedom, so connecting them directly to beam, plate, or shell elements can require care to avoid singular constraints.
+
+The physical position and displacement field are both interpolated from nodal values:
+
+```text
+x(xi) = sum N_i(xi) x_i
+u(xi) = sum N_i(xi) u_i
+```
+
+The covered topologies are 4-node tetrahedron, 5-node pyramid, 6-node wedge, and 8-node hexahedron. In each case, the element is defined in natural coordinates and mapped to physical space through the Jacobian.
+
+## Practical Notes
+
+- Solid elements are suited to three-dimensional volume response rather than beam or shell idealizations.
+- Aspect ratios close to one are preferred because distortion degrades the shape-function mapping and numerical integration quality.
+- The absence of rotational degrees of freedom is a modeling interface issue when solid elements meet structural elements.
+
+## Connections
+
+- [[Solid Element Shape Functions]] defines the natural-coordinate interpolation for each covered topology.
+- [[Solid Element Strain-Displacement Matrix]] converts the displacement interpolation into engineering strain components.
+- [[Solid Element Stiffness Integration]] assembles the stiffness matrix from `B`, `D`, and the Jacobian.
+
+## Sources
+
+- [[Solid Element Notes]]