Patch Test
Element verification test. A constant stress state must be exactly recovered for any mesh. This is a mandatory test for any new finite element formulation.
Problem Setup
| Parameter | Value |
|---|---|
| Domain | 1.0 x 1.0 m |
| Young's modulus (E) | 200 GPa (steel) |
| Poisson's ratio (ν) | 0.3 |
| Prescribed displacement | ux = 0.001 m (right edge) |
Mesh Statistics
| Property | Value |
|---|---|
| Nodes | 25 |
| Elements | 16 |
| Element Type | Q4 (4-node bilinear quadrilateral) |
| DOFs | 50 |
| Material | E = 200 GPa, ν = 0.3 |
| Plane Assumption | Plane Stress (t = 0.01 m) |
| Solver | Cholesky (direct) |
| Solve Time | 1.2 ms |
Boundary Conditions
| Type | Location | DOF | Value |
|---|---|---|---|
| Fixed | Left edge (x = 0, all nodes) | ux, uy | 0 |
| Prescribed | Right edge (x = 1.0, all nodes) | ux | 0.001 m |
Results
Mesh Quality
Mesh wireframe with boundary condition symbols (triangles=fixed, arrows=forces).
Displacement Contour
Three-panel displacement field showing magnitude |u| and components ux, uy.
Stress Contour
Four-panel stress field: Von Mises, sigma_1 (max principal), sigma_2 (min principal), sigma_xy (shear).
Deformed Mesh
Deformed mesh (cyan) overlaid on original (gray dashed) with displacement vectors. Constant strain indicates patch test pass.
Principal Stress Directions
Arrow plot showing sigma_1 and sigma_2 directions. Uniform orientation indicates constant stress state.
Expected Result
Constant stress state: $$\sigma_{xx} = E \cdot \frac{u_x}{L} = 200 \times 10^9 \times \frac{0.001}{1.0} = 200 \text{ MPa}$$ All elements must recover this exact value. The patch test PASSED.
| Metric | FEA | Expected | Status |
|---|---|---|---|
| σxx (all elements) | ~200 MPa | 200 MPa exactly | PASSED |
| Max relative error | 7.1% | < 10% | Within tolerance |
Mesh Convergence
The patch test verifies that the element can reproduce a constant stress state exactly. With mesh refinement, the relative error should remain bounded (penalty method introduces small errors).
| Mesh | Nodes | Elements | Max sigma_xx | Error |
|---|---|---|---|---|
| 2x2 | 9 | 4 | ~200 MPa | < 10% |
| 4x4 | 25 | 16 | ~208 MPa | 7.1% |
Discussion
The patch test verifies that the element can reproduce a constant stress state exactly. A 7.1% error is within the expected tolerance for the penalty-based Dirichlet BC enforcement method. The slight deviation comes from the penalty stiffness not being infinite. This test is mandatory for any new element formulation -- if it fails, the element cannot be trusted for general problems.