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wiki/concepts/Incompatible Mode Solid Elements.md

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+---
+type: concept
+title: "Incompatible Mode Solid Elements"
+complexity: advanced
+domain: computational-mechanics
+aliases:
+  - incompatible mode solid element
+  - nonconforming solid element mode
+  - internal mode solid element
+  - static condensation of incompatible modes
+created: 2026-05-28
+updated: 2026-05-28
+address: c-000053
+tags:
+  - concept
+  - finite-element-method
+  - solid-elements
+  - locking
+status: current
+related:
+  - "[[Solid Element Notes]]"
+  - "[[Mixed Finite Element Formulations]]"
+  - "[[Solid Element Stiffness Integration]]"
+  - "[[Shell Locking Phenomenon]]"
+sources:
+  - "[[Solid Element Notes]]"
+---
+
+# Incompatible Mode Solid Elements
+
+## Definition
+
+Incompatible mode solid elements add internal, non-nodal displacement modes to a solid element to enrich its deformation field and reduce locking or overly stiff behavior.
+
+## How It Works
+
+The source introduces extra internal degrees of freedom for selected 6-node wedge and 8-node hexahedral elements. The standard displacement interpolation is augmented by additional mode functions, producing an expanded strain-displacement matrix:
+
+```text
+B_tot = [ B  B_inc ]
+```
+
+The resulting stiffness is assembled from the expanded matrix. Because the incompatible mode degrees of freedom are internal to the element and are not global nodal unknowns, they are eliminated by static condensation before the global solve.
+
+## Why It Matters
+
+Low-order displacement-based solid elements can be too stiff in deformation states their interpolation cannot represent well. Incompatible modes are a local enrichment strategy: they improve element flexibility without changing the global mesh topology or adding nodal unknowns.
+
+## Connections
+
+- [[Solid Element Stiffness Integration]] is modified because the `B` matrix is augmented before condensation.
+- [[Mixed Finite Element Formulations]] provides the broader reliability pattern of adding or altering fields to avoid locking and constraints.
+- [[Shell Locking Phenomenon]] is a shell-specific locking case; this page records the analogous solid-element enrichment strategy from the source.
+
+## Sources
+
+- [[Solid Element Notes]]