Classical and All-floating FETI Methods with Applications to Biomechanical Models

This work deals with domain decomposition solvers, more precisely the finite element tearing and interconnecting (FETI) approach, to simulate the elastic behavior of cardiovascular tissues, such as the myocardium or the artery. These biological materials are characterized by anisotropic and nonlinear material properties due to preferential orientations of collagen and muscle fibers in the tissue. The high complexity of the underlying nonlinear equations as well as fine geometrical structures of the cardiovascular components demand fast solving algorithms, where FETI is an efficient choice. This approach shows high performance and enables a natural parallelization to solve the nonlinear elasticity problem. The strategy of the FETI method is to decompose the computational domain into a finite number of non-overlapping subdomains. Therein the corresponding local problems can be handled efficiently by direct solvers. The reduced global system, that is related to discrete Lagrange multipliers on the interface of the subdomains, is then solved with a parallel Krylov space method to compute the desired solution. This is, in the case of elasticity, the stress and subsequently, in a postprocessing step, we deduce the displacement locally. For the global iterative method suitable preconditioning is a substantial factor. Besides a simple lumped preconditioner and an optimal Dirichlet preconditioner a novel BEM based preconditioner, formed by local hypersingular boundary integral operators, is considered. The idea behind this preconditioner is the approximation of Steklov–Poincaré operators, the basis for the optimal Dirichlet preconditioning, by computationally less expensive hypersingular operators. Another innovative aspect is the usage of all-floating FETI, a variant of classical FETI, for nonlinear soft tissue mechanics. This approach, where the Dirichlet boundary acts as a part of the interface, shows significant advantages in the implementation and in the convergence of the global iterative method which is evidenced by numerical examples. As realistic and clinically relevant applications we present passive inflation experiments, comparable to stenting or angioplasty procedures, using anatomically detailed geometries of arteries and the myocardium.