On improving the numerical convergence of highly nonlinear elasticity problems

TL;DR

This paper introduces a novel transformation method for discretized equations in highly nonlinear elasticity problems, significantly improving convergence rates of Newton solvers and enabling larger load steps in simulations.

Contribution

A new formulation that reduces nonlinearity in elasticity problems, enhancing convergence and allowing larger load steps, applicable to various nonlinearities.

Findings

Reduced iteration counts for convergence

Successful application to soft tissue and compression problems

Convergence achieved with 10-100 times larger load steps

Abstract

Finite elasticity problems commonly include material and geometric nonlinearities and are solved using various numerical methods. However, for highly nonlinear problems, achieving convergence is relatively difficult and requires small load step sizes. In this work, we present a new method to transform the discretized governing equations so that the transformed problem has significantly reduced nonlinearity and, therefore, Newton solvers exhibit improved convergence properties. We study exponential-type nonlinearity in soft tissues and geometric nonlinearity in compression, and propose novel formulations for the two problems. We test the new formulations in several numerical examples and show significant reduction in iterations required for convergence, especially at large load steps. Notably, the proposed formulation is capable of yielding convergent solution even when 10 to 100 times larger load steps are applied. The proposed framework is generic and can be applied to other types of nonlinearities as well.