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A fully implicit finite element method for bidomain models of cardiac electromechanics. (English) Zbl 1297.74077
Summary: We propose a novel, monolithic, and unconditionally stable finite element algorithm for the bidomain-based approach to cardiac electromechanics. We introduce the transmembrane potential, the extracellular potential, and the displacement field as independent variables, and extend the common two-field bidomain formulation of electrophysiology to a three-field formulation of electromechanics. The intrinsic coupling arises from both excitation-induced contraction of cardiac cells and the deformation-induced generation of intra-cellular currents. The coupled reaction-diffusion equations of the electrical problem and the momentum balance of the mechanical problem are recast into their weak forms through a conventional isoparametric Galerkin approach. As a novel aspect, we propose a monolithic approach to solve the governing equations of excitation-contraction coupling in a fully coupled, implicit sense. We demonstrate the consistent linearization of the resulting set of non-linear residual equations. To assess the algorithmic performance, we illustrate characteristic features by means of representative three-dimensional initial-boundary value problems. The proposed algorithm may open new avenues to patient specific therapy design by circumventing stability and convergence issues inherent to conventional staggered solution schemes.

MSC:
74L15 Biomechanical solid mechanics
92C99 Physiological, cellular and medical topics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
74S05 Finite element methods applied to problems in solid mechanics
92-08 Computational methods for problems pertaining to biology
Software:
FEAP
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