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In this work, a parallel three-dimensional solver for numerical simulations in computational electrocardiology is introduced and studied. The solver is based on the anisotropic Bidomain cardiac model, consisting of a system of two degenerate parabolic reaction–diffusion equations describing the intra and extracellular potentials of the myocardial tissue. This model includes intramural fiber rotation and anisotropic conductivity coefficients that can be fully orthotropic or axially symmetric around the fiber direction. The solver also includes the simpler anisotropic Monodomain model, consisting of only one reaction–diffusion equation. These cardiac models are coupled with a membrane model for the ionic currents, consisting of a system of ordinary differential equations that can vary from the simple FitzHugh–Nagumo (FHN) model to the more complex phase-I Luo–Rudy model (LR1). The solver employs structured isoparametric Q1finite elements in space and a semi-implicit adaptive method in time. Parallelization and portability are based on the PETSc parallel library. Large-scale computations with up to O(107) unknowns have been run on parallel computers, simulating excitation and repolarization phenomena in three-dimensional domains.

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SUMMARYA model for the active deformation of cardiac tissue considering orthotropic constitutive laws is introduced and studied. In particular, the passive mechanical properties of the myocardium are described by the Holzapfel-Ogden relation, whereas the activation model is based on the concept of active strain. There, an incompatible intermediate configuration is considered, which entails a multiplicative decomposition between active and passive deformation gradients. The underlying Euler-Lagrange equations for minimizing the total energy are written in terms of these deformation factors, where the active part is assumed to depend, at the cell level, on the electrodynamics and on the specific orientation of the cardiomyocytes. The active strain formulation is compared with the classical active stress model from both numerical and modeling perspectives. The well-posedness of the linear system derived from a generic Newton iteration of the original problem is analyzed, and different mechanical activation functions are considered. Taylor-Hood and MINI finite elements are used in the discretization of the overall mechanical problem. The results of several numerical experiments show that the proposed formulation is mathematically consistent and is able to represent the main features of the phenomenon, while allowing savings in computational costs. Copyright

A Balancing Domain Decomposition by Constraints (BDDC) preconditioner for Isogeometric Analysis of scalar elliptic problems is constructed and analyzed by introducing appropriate discrete norms. A main result of this work is the proof that the proposed isogeometric BDDC preconditioner is scalable in the number of subdomains and quasi-optimal in the ratio of subdomain and element sizes. Another main result is the numerical validation of the theoretical convergence rate estimates by carrying out several two- and three-dimensional tests on serial and parallel computers. These numerical experiments also illustrate the preconditioner performance with respect to the polynomial degree and the regularity of the NURBS basis functions, as well as its robustness with respect to discontinuities of the coefficient of the elliptic problem across subdomain boundaries.