Using surrogate models to accelerate load step methods for nonlinear finite element problems in hyperelasticity

Finite element methods for displacement problems in hyperelasticity lead to systems of nonlinear equations. These equations are usually solved with Newton's method or a related method. The convergence of Newton's method depends heavily on the proximity of the initial guess to the numerical solution. Load step methods overcome problems with divergence by applying the load in increments, leading to a sequence of sub‐problems with initial guesses closer to the numerical solution of each sub‐problem, supporting the convergence. The downside of this approach is the high computational effort needed to solve the load steps. Based on a benchmark problem in high‐order FEM, we extend traditional load step methods to a new approach exploiting the hierarchical basis used for the spatial discretization of the problem and saving up to 50% of computation time (vs. benchmark).