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

TL;DR

This paper presents a vectorized MATLAB implementation for fast evaluation of nonlinear energy functionals in 2D and 3D using finite element methods with nodal elements, optimized for first-order gradient computations.

Contribution

It introduces a highly efficient, vectorized MATLAB approach for evaluating nonlinear energies and their gradients in finite element discretizations, applicable to various problems.

Findings

Significantly reduces computation time for energy evaluation.

Enables simultaneous computation of energy and gradients over all degrees of freedom.

Provides accessible MATLAB codes for hyperelasticity and p-Laplacian problems.

Abstract

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