Parallel multilevel solvers for the cardiac electro-mechanical coupling

Abstract

We develop a parallel solver for the cardiac electro-mechanical coupling. The electric model consists of two non-linear parabolic partial differential equations (PDEs), the socalled Bidomain model, which describes the spread of the electric impulse in the heart muscle. The two PDEs are coupled with a non-linear elastic model, where the myocardium is considered as a nearly-incompressible transversely isotropic hyperelastic material. The discretization of the whole electro-mechanical model is performed by Q1 finite elements in space and a semi-implicit finite difference scheme in time. This approximation strategy yields at each time step the solution of a large scale ill-conditioned linear system deriving from the discretization of the Bidomain model and a non-linear system deriving from the discretization of the finite elasticity model. The parallel solver developed consists of solving the linear system with the Conjugate Gradient method, preconditioned by a Multilevel Schwarz preconditioner, and the non-linear system with a Newton-KrylovAlgebraic Multigrid solver. Three-dimensional parallel numerical tests on a Linux cluster show that the parallel solver proposed is scalable and robust with respect to the domain deformations induced by the cardiac contraction.

Newton–Krylov-BDDC solvers for nonlinear cardiac mechanics

“…to the varying ratio H/h in the condition number bound (21) that we conjectured to hold in Remark 1. We finally remark that these results outperform those obtained with an Algebraic Multigrid solver, presented in our previous publication [13]. slab domains ellipsoidal domains…”

“…[11,22,43,45,55,44,51,59,61,53,54,66,69,70] and the recent monograph [12], a few studies have focused on the development of efficient solvers for the quasi-static cardiac mechanical model, see [48,67] for parallel GMRES solvers and [57,30,31,29] for parallel direct solvers. In our previous work [13], we have developed an Algebraic Multigrid solver for the cardiac mechanical model.…”

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.…”

A Parallel Implementation of a Two-Level Overlapping Schwarz Method with Energy-Minimizing Coarse Space Based on Trilinos

“…A hybrid multilevel version of this Schwarz preconditioner has scaled to 2 048 cores in Scacchi [38,39] using up to 5 levels. Recently, a multilevel Schwarz preconditioner has also been applied to cardiac electro-mechanical coupling in [20]; therein, strong scalability using 4 levels for up to 512 cores has been reported. Recently, in [34], a three-level overlapping Schwarz method has been shown to be strongly scalable for up to 10 240 cores of an IBM cluster for a three-dimensional linear elasticity problem.…”