A Robust Solution Procedure for Hyperelastic Solids with Large Boundary Deformation

TL;DR

This paper introduces a robust and efficient solution procedure for simulating large boundary deformations in hyperelastic solids modeled by the Mooney-Rivlin theory, significantly improving speed and robustness over traditional methods.

Contribution

A novel geometric mesh untangling approach combined with safeguarded Newton iterations for faster, more robust simulation of large deformations in hyperelastic solids.

Findings

Algorithm is up to 70 times faster than standard methods.

Method tolerates larger deformations without mesh tangling.

Effective for extremely large boundary deformations with small steps.

Abstract

Compressible Mooney-Rivlin theory has been used to model hyperelastic solids, such as rubber and porous polymers, and more recently for the modeling of soft tissues for biomedical tissues, undergoing large elastic deformations. We propose a solution procedure for Lagrangian finite element discretization of a static nonlinear compressible Mooney-Rivlin hyperelastic solid. We consider the case in which the boundary condition is a large prescribed deformation, so that mesh tangling becomes an obstacle for straightforward algorithms. Our solution procedure involves a largely geometric procedure to untangle the mesh: solution of a sequence of linear systems to obtain initial guesses for interior nodal positions for which no element is inverted. After the mesh is untangled, we take Newton iterations to converge to a mechanical equilibrium. The Newton iterations are safeguarded by a line search similar to one used in optimization. Our computational results indicate that the algorithm is up to 70 times faster than a straightforward Newton continuation procedure and is also more robust (i.e., able to tolerate much larger deformations). For a few extremely large deformations, the deformed mesh could only be computed through the use of an expensive Newton continuation method while using a tight convergence tolerance and taking very small steps.