Analytic force/moment balance and exact constant-strain fields expose sign, load integration, and assembly errors.
01 Model and scope
A constant-strain patch gives a known exact displacement field for an affine map; body force and face pressure have analytically checkable resultants. Comparing those quantities isolates interpolation, load integration, signs and assembly before attempting a convergence study.
Route assumptions and calculation checks
- The one-element JSON is the shortest CLI/API smoke test.
- Pressure and body-force fixtures compare resultants and moments independently.
- The patch test verifies affine reproduction; refinement is a separate convergence study.
Useful foundations and prerequisites
02 Essential equations and symbols
Definitions and conventions
- bV
- Resultant body force for uniform density-free body load b over volume V.
- t̄A
- Resultant of uniform face traction over area A.
- Affine displacement field.
- sym(H)
- Exact constant strain for that field.
03 Step by step derivation
Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.
1 · Predict resultants before the solve
For constant body-force density b, the total force is bV. A uniform traction over one triangular face has resultant t̄A and shares it equally among the three linear face nodes. These closed-form totals test load integration independently of displacement response.
2 · Prescribe an affine displacement field
Set nodal values from u(x)=Hx+c. Linear barycentric interpolation reproduces this field exactly within one nondegenerate TET4, so the strain is constant and equals the symmetric part of H. This is a direct check of interpolation, gradients, and B assembly.
3 · Audit assembly with equilibrium and work
Compare assembled force and moment totals with the applied resultants, inspect the constant strain/stress, and verify the linear energy identity. Then refine the mesh separately; a successful patch test establishes affine consistency, not convergence for every boundary-value problem.
04 Algorithm and implementation
This sheet provides explanatory references rather than an inline source excerpt. Use the labeled sources to inspect the precise boundary.
05 Verification and limitations
SMALL REPRODUCIBLE CHECKS · not a convergence claim
Each reference supports its own model, observable and analysis route. Read the original contract and limitations before transferring a conclusion.