Fast MATLAB evaluation of nonlinear energies using FEM in 2D and 3D: Nodal elements

Nonlinear energy functionals appearing in the calculus of variations can be discretized by finite element (FE) method and formulated as a sum contributions from local elements. A fast evaluation containing first order gradient terms is central part this contribution. We describe vectorized implementation using simplest linear nodal (P1) elements which all are evaluated at once without loop over triangular or tetrahedral Furthermore, connection to first-order optimization methods, discrete functional assembled way that components degrees freedom once. The key ingredient vectorization exact approximate gradients patches. It leads time-efficient higher memory-cost. Provided codes MATLAB related 2D/3D hyperelasticity 2D p-Laplacian problem available for download structured it easily extended other types vector scalar forms energies.