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.

中文翻译：

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心脏机电双域模型的全隐式有限元方法


我们提出了一种新颖的、整体的、无条件稳定的有限元算法，用于基于双域的心脏机电方法。我们引入跨膜电位、细胞外电位和位移场作为自变量，并将电生理学的常见双域公式扩展到机电的三场公式。内在耦合是由兴奋引起的心肌细胞收缩和变形引起的细胞内电流产生的。通过传统的等参伽辽金方法，将电气问题的耦合反应扩散方程和机械问题的动量平衡重新转换为其弱形式。作为一个新颖的方面，我们提出了一种整体方法，以完全耦合、隐式的方式求解激励-收缩耦合的控制方程。我们证明了非线性残差方程组结果的一致线性化。为了评估算法性能，我们通过代表性的三维初始边值问题来说明特征。所提出的算法可以通过规避传统交错解决方案固有的稳定性和收敛问题，为患者特定治疗设计开辟新途径。