Scalable domain decomposition solvers for 3D cardiac electro-mechanical simulations are constructed and studied. The cardiac electro-mechanical model considered consists of four coupled components: a) the quasi-static transversely isotropic finite elasticity equations for the deformation of the cardiac tissue; b) the active tension model for the intracellular calcium dynamics and cross-bridge binding; c) the anisotropic Bidomain model for the electrical current flow through the deforming cardiac tissue; d) the membrane model of ventricular myocytes, including stretch-activated channels. This complex nonlinear model poses great theoretical and numerical challenges. At the theoretical level, the well-posedness of the cardiac electro-mechanical coupling model is still an open problem, as well as the convergence of its finite element approximation. At the numerical level, the approximation and simulation of the cardiac electro-mechanical coupling model are very demanding and expensive tasks, because of the very different space and time scales associated with the electrical and mechanical models, as well as their nonlinear and multiphysics interactions. The most efficient high-performance solvers for these complex cardiac models are parallel iterative methods, such as the Preconditioned Conjugate Gradient method (PCG) and Generalized Minimal Residual Method (GMRES), accelerated by proper scalable preconditioners. Our parallel solver employs Multilevel Additive Schwarz preconditioners for the solution of the discretized Bidomain equations and Newton-Krylov methods with AMG or BDDC preconditioners for the solution of the discretized nonlinear finite elasticity equations.The results of several parallel simulations show the scalability of both linear and nonlinear solvers and their application to the study of the physiological excitation-contraction cardiac dynamics and of re-entrant waves in the presence of different mechano-electrical feedbacks.
This is joint work with Piero Colli Franzone (University of Pavia, Italy), Simone Scacchi (University of Milano, Italy) and Stefano Zampini (KAUST, Saudi Arabia).