FaCSI: A block parallel preconditioner for fluid–structure interaction in hemodynamics

Abstract

Modeling Fluid-Structure Interaction (FSI) in the vascular system is mandatory to reliably compute mechanical indicators in vessels undergoing large deformations. In order to cope with the computational complexity of the coupled 3D FSI problem after discretizations in space and time, a parallel solution is often mandatory. In this paper we propose a new block parallel preconditioner for the coupled linearized FSI system obtained after space and time discretization. We name it FaCSI to indicate that it exploits the factorized form of the linearized FSI matrix, the use of static condensation to formally eliminate the interface degrees of freedom of the fluid equations, and the use of a SIMPLE preconditioner for saddle-point problems. FaCSI is built upon a block Gauss-Seidel factorization of the FSI Jacobian matrix and it uses ad-hoc preconditioners for each physical component of the coupled problem, namely the fluid, the structure and the geometry. In the fluid subproblem, after operating static condensation of the interface fluid variables, we use a SIMPLE preconditioner on the reduced fluid matrix. Moreover, to efficiently deal with a large number of processes, FaCSI exploits efficient single field preconditioners, e.g., based on domain decomposition or the multigrid method. We measure the parallel performances of FaCSI on a benchmark cylindrical geometry and on a problem of physiological interest, namely the blood flow through a patient-specific femoropopliteal bypass. We analyse the dependence of the number of linear solver iterations on the cores count (scalability of the preconditioner) and on the mesh size (optimality).

A Parallel Newton Multigrid Framework for Monolithic Fluid-Structure Interactions

“…Recently, a block-preconditioned parallel GMRES iteration was presented [25] and showed good performance on various 2d and 3d test cases. A Gauss-Seidel decoupling with highly efficient and massively parallel preconditioners based on the SIMPLE scheme for the fluid and multigrid for a linear elasticity problem is presented in [13].…”

“…Monolithic approaches all give rise to strongly coupled, usually very large and nonlinear algebraic systems of equations. Although there has been substantial progress in designing efficient numerical schemes for tackling the nonlinear problems [23,21,16] (usually by Newton's method) and the resulting linear systems [19,36,32,28,2,11,13], the computational effort is still immense and numerically accurate results for 3d problems are still rare.…”

Geometric multiscale modeling of the cardiovascular system, between theory and practice

“…In the former case, the complete non-linear system arising after the space discretization is assembled and solved with a suitable preconditioned Krylov [15,86], domain-decomposition [44,50] or multigrid [13,75] methods. In the partitioned case the successive solution of the fluid and solid subproblems in an iterative framework is carried out (see, e.g., [9,11,39,46,55,105,134]).…”

Numerical approximation of the electromechanical coupling in the left ventricle with inclusion of the Purkinje network

“…Preconditioning for cardiac electromechanical solvers is an active field of research, see Pavarino et al and Franzone et al In this work, the block Gauss‐Seidel preconditioning strategy proposed in Deparis et al in the context of fluid‐structure interaction problems (FaCSI preconditioning), and then extended in for cardiac electromechanical problems with the monodomain model, is further extended for the Jacobian matrix in Equation 22, with the modifications required by the extra blocks , and , coming from the bidomain model.…”