Follow nodal unknowns through interpolation, physical gradients, element integration and global assembly.
01 Model and scope
The element basis approximates the field, the geometry map transforms gradients, quadrature evaluates local contributions and connectivity assembles them. An error at any one of these stages can survive a successful linear solve.
Route assumptions and calculation checks
- Geometry mapping and displacement interpolation are distinct choices even when they share a basis.
- Transform gradients and evaluate the material law at integration points.
- Assemble through the declared connectivity and degree-of-freedom order.
Useful foundations and prerequisites
02 Essential equations and symbols
Definitions and conventions
- Displacement interpolation and compatible strain operator.
- Geometry-map Jacobian.
- Global-to-element degree-of-freedom gather operator.
- Quadrature weight and volume scale.
03 Step by step derivation
Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.
1 · Interpolate geometry and displacement
In an isoparametric element the same basis interpolates geometry and displacement, but their coefficients are coordinates and unknown displacements respectively. Connectivity and local node order are part of the contract.
2 · Transform gradients
Compute the Jacobian in the declared convention and use the chain rule. Geometry checks must match the particular element and sampling rule. A positive determinant at sampled points is not a general proof for curved geometry.
3 · Integrate the element contribution
Evaluate the strain operator and constitutive matrix at each quadrature point. Exactness depends on the mapped integrand and integration rule, not only on the polynomial degree of the reference basis.
4 · Assemble before enforcing the selected constraints
Gather each local vector from the global unknowns, then scatter its force and matrix contribution back. Apply essential or multipoint constraints using the declared reduction method. Recover reactions in the original coordinates.
Solid, shell, beam and discrete entities use different local degrees of freedom and strain operators.
04 Algorithm and implementation
Follow the route sheets below to see how this shared mathematics is implemented.
05 Verification and limitations
FOUNDATION · displacement finite elements
The formulation and its explanatory checks do not by themselves establish a qualified application domain. Consult the capability matrix and the applicable recorded case.