TY - JOUR
T1 - Mixed Kirchhoff stress–displacement–pressure formulations for incompressible hyperelasticity
AU - Farrell, Patrick E.
AU - Gatica, Luis F.
AU - Lamichhane, Bishnu P.
AU - Oyarzúa, Ricardo
AU - Ruiz-Baier, Ricardo
PY - 2021/2/1
Y1 - 2021/2/1
N2 - The numerical approximation of hyperelasticity must address nonlinear constitutive laws, geometric nonlinearities associated with large strains and deformations, the imposition of the incompressibility of the solid, and the solution of large linear systems arising from the discretisation of 3D problems in complex geometries. We adapt the three-field formulation for nearly incompressible hyperelasticity introduced in Chavan et al. (2007) to the fully incompressible case. The mixed formulation is of Hu–Washizu type and it differs from other approaches in that we use the Kirchhoff stress, displacement, and pressure as principal unknowns. We also discuss the solvability of the linearised problem restricted to neo-Hookean materials, illustrating the interplay between the coupling blocks. We construct a family of mixed finite element schemes (with different polynomial degrees) for simplicial meshes and verify its error decay through computational tests. We also propose a new augmented Lagrangian preconditioner that improves convergence properties of iterative solvers. The numerical performance of the family of mixed methods is assessed with benchmark solutions, and the applicability of the formulation is further tested in a model of cardiac biomechanics using orthotropic strain energy densities. The proposed methods are advantageous in terms of physical fidelity (as the Kirchhoff stress can be approximated with arbitrary accuracy and no locking is observed) and convergence (the discretisation and the preconditioners are robust and computationally efficient, and they compare favourably at least with respect to classical displacement–pressure schemes).
AB - The numerical approximation of hyperelasticity must address nonlinear constitutive laws, geometric nonlinearities associated with large strains and deformations, the imposition of the incompressibility of the solid, and the solution of large linear systems arising from the discretisation of 3D problems in complex geometries. We adapt the three-field formulation for nearly incompressible hyperelasticity introduced in Chavan et al. (2007) to the fully incompressible case. The mixed formulation is of Hu–Washizu type and it differs from other approaches in that we use the Kirchhoff stress, displacement, and pressure as principal unknowns. We also discuss the solvability of the linearised problem restricted to neo-Hookean materials, illustrating the interplay between the coupling blocks. We construct a family of mixed finite element schemes (with different polynomial degrees) for simplicial meshes and verify its error decay through computational tests. We also propose a new augmented Lagrangian preconditioner that improves convergence properties of iterative solvers. The numerical performance of the family of mixed methods is assessed with benchmark solutions, and the applicability of the formulation is further tested in a model of cardiac biomechanics using orthotropic strain energy densities. The proposed methods are advantageous in terms of physical fidelity (as the Kirchhoff stress can be approximated with arbitrary accuracy and no locking is observed) and convergence (the discretisation and the preconditioners are robust and computationally efficient, and they compare favourably at least with respect to classical displacement–pressure schemes).
KW - Augmented Lagrangian preconditioning
KW - Incompressible hyperelasticity
KW - Kirchhoff stress formulation
KW - Mixed finite element methods
UR - http://www.scopus.com/inward/record.url?scp=85097640655&partnerID=8YFLogxK
U2 - 10.1016/j.cma.2020.113562
DO - 10.1016/j.cma.2020.113562
M3 - Article
AN - SCOPUS:85097640655
VL - 374
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
SN - 0045-7825
M1 - 113562
ER -