A Robust Solution Procedure for Hyperelastic Solids with Large Boundary Deformation

TL;DR

This work tackles robustly solving static nonlinear elasticity for compressible Mooney-Rivlin hyperelastic solids under large boundary deformations, where mesh tangling (inversions with $J=\det(F)\le0$) thwarts standard Newton methods. The proposed UBN method first untangles the mesh via iterative stiffening in a linear-elastic surrogate, then solves the full nonlinear problem with safeguarded Newton iterations that include a line search to maintain $J>0$. Compared to Newton continuation (load stepping), UBN delivers substantially greater robustness and efficiency, up to about 70× faster in 2D and 3D tests, and tolerates much larger deformations before failure. The approach is extendable to incompressible regimes and higher-order discretizations, and offers practical benefits for simulations of soft tissues, polymers, and other hyperelastic materials under large boundary deformations.

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.

Figures (4)

• Figure 1: The annulus mesh used for testing in this section.
• Figure 2: Deformed annulus meshes resulting from rotating the exterior boundary circle of the mesh shown in Fig. \ref{['fig:annmesh']} by $f$ radians and moving the inner boundary circle by a factor $f$ closer to the outer boundary. The deformed meshes are for (a) $f=0.1$, (b) $f=0.3$, (c) $f=0.6$, and (d) $f=0.7$.
• Figure 3: The top row line diagrams $(a)$ and $(b)$ show the undeformed Hook body from two different viewpoints. Dirichlet boundary conditions were applied to two of the boundary surfaces to yield deformed meshes. In particular, the asterisks mark the zero-displacement boundary, while the $\times$'s mark fixed displacement. The bottom row diagrams $(c)$ and $(d)$ show Hook after the maximum deformation of 40 is applied.
• Figure 4: The top row line diagrams $(a)$ and $(b)$ show the undeformed Foam5 body from two different viewpoints. Dirichlet boundary conditions were applied to two of the boundary surfaces to yield deformed meshes. In particular, the asterisks mark the zero-displacement boundary, while the $\times$'s mark fixed displacement. The bottom row diagrams $(c)$ and $(d)$ show Foam5 after the maximum deformation of 5 is applied.