Mechanics of deformable solids Definition

Patch test

Builds on Finite element method in structural mechanics Finite element method

Definition

Let an element formulation of the finite element method in structural mechanics for linear elasticity be given. A patch is a mesh of a few such elements, with at least one node in the interior, in general with distorted element shapes. Let $u^*(x) = c + A x$ be a displacement field with a constant vector $c$ and a constant matrix $A$ . Its strain $\tfrac12(A + A^T)$ is constant, and with no body force it is an exact solution of the equilibrium equations. Solve the finite element problem on a patch with the boundary nodal displacements set equal to $u^*$ . The element passes the patch test if, for every patch and every such $u^*$ , this solution:

equals $u^*$ at every interior node;
has strain equal to $\tfrac12(A + A^T)$ everywhere in the patch.

Remarks

Example: the linear triangle (constant strain triangle) passes. Its trial space contains every linear field, the discrete problem is the Galerkin projection onto a space that contains $u^*$ , and the exact solution $u^*$ is therefore reproduced. The same argument works for every conforming element whose trial space contains the linear fields.

Non-example: the incompatible quadrilateral element of Wilson adds internal displacement modes that are discontinuous between elements. It fails the test on patches of quadrilaterals that are not parallelograms: the computed strain is not constant, so clause 2 fails. Modified versions of this element were designed to pass it.

Other versions prescribe tractions instead of displacements on the boundary, or test higher-order fields for higher-order elements.

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Finite element method concept
Finite element method in structural mechanics concept
Patch test definition

Tag	`MFJ7DPE6EE`
URI	`m2://MFJ7DPE6EE`
Kind	definition
Writing	W0 stub
MSC	74S05, 65N30
Defines	Patch test
Uses	Finite element method in structural mechanics, Finite element method
Changed	3 November 2026

Patch test

Definition

Remarks

Reading path

Further reading

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