Augmented Lagrange methods for quasi-incompressible materials--applications to soft biological tissue

Arterial walls in the healthy physiological regime are characterized by quasi-incompressible, anisotropic, hyperelastic material behavior. Polyconvex material functions representing such materials typically incorporate a penalty function to account for the incompressibility. Unfortunately, the penalty will affect the conditioning of the stiffness matrices. For high penalty parameters, the performance of iterative solvers will degrade, and when direct solvers are used, the quality of the solutions will deteriorate. In this paper, an augmented Lagrange approach is used to cope with the quasi-incompressibility condition. Here, the penalty parameter can be chosen much smaller, and as a consequence, the arising linear systems of equations have better properties. An improved convergence is then observed for the finite element tearing and interconnecting-dual primal domain decomposition method, which is used as an iterative solver. Numerical results for an arterial geometry obtained from ultrasound imaging are presented.