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During the contraction of the ventricles, the ventricles interact with the atria as well as with the pericardium and the surrounding tissue in which the heart is embedded. The atria are stretched, and the atrioventricular plane moves toward the apex. The atrioventricular plane displacement (AVPD) is considered to be a major contributor to the ventricular function, and a reduced AVPD is strongly related to heart failure. At the same time, the epicardium slides almost frictionlessly on the pericardium with permanent contact. Although the interaction between the ventricles, the atria and the pericardium plays an important role for the deformation of the heart, this aspect is usually not considered in computational models. In this work, we present an electromechanical model of the heart, which takes into account the interaction between ventricles, pericardium and atria and allows to reproduce the AVPD. To solve the contact problem of epicardium and pericardium, a contact handling algorithm based on penalty formulation was developed, which ensures frictionless and permanent contact. Two simulations of the ventricular contraction were conducted, one with contact handling of pericardium and heart and one without. In the simulation with contact handling, the atria were stretched during the contraction of the ventricles, while, due to the permanent contact with the pericardium, their volume increased. In contrast to that, in the simulations without pericardium, the atria were also stretched, but the change in the atrial volume was much smaller. Furthermore, the pericardium reduced the radial contraction of the ventricles and at the same time increased the AVPD.

Mathematical models of the human heart are evolving to become a cornerstone of precision medicine and support clinical decision making by providing a powerful tool to understand the mechanisms underlying pathophysiological conditions. In this study, we present a detailed mathematical description of a fully coupled multi-scale model of the human heart, including electrophysiology, mechanics, and a closed-loop model of circulation. State-of-the-art models based on human physiology are used to describe membrane kinetics, excitation-contraction coupling and active tension generation in the atria and the ventricles. Furthermore, we highlight ways to adapt this framework to patient specific measurements to build digital twins. The validity of the model is demonstrated through simulations on a personalized whole heart geometry based on magnetic resonance imaging data of a healthy volunteer. Additionally, the fully coupled model was employed to evaluate the effects of a typical atrial ablation scar on the cardiovascular system. With this work, we provide an adaptable multi-scale model that allows a comprehensive personalization from ion channels to the organ level enabling digital twin modeling.

The objective of this paper is to develop and analyse a multigrid algorithm for the system of equations arising from the mortar nite element discretization of second order elliptic boundary value problems. In order to establish the inf-sup condition for the saddle point formulation and to motivate the subsequent treatment of the discretizations we revisit rst brie y the theoretical concept of the mortar nite element method. Employing suitable mesh-dependent norms we verify the validity of the LBB condition for the resulting mixed method and prove an L 2 error estimate. This is the key for establishing a suitable approximation property for our multigrid convergence proof via a duality argument. In fact, we are able to verify optimal multigrid e ciency based on a smoother which is applied to the whole coupled system of equations. We conclude with several numerical tests of the proposed scheme which con rm the theoretical results and show the e ciency and the robustness of the method even in situations not covered by the theory.

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We review the classical return algorithm for incremental plasticity in the context of nonlinear programming, and we discuss the algorithmic realization of the SQP method for infinitesimal perfect plasticity. We show that the radial return corresponds to an orthogonal projection onto the convex set of admissible stresses. Inserting this projection into the equilibrium equation results in a semismooth equation which can be solved by a generalized Newton method. Alternatively, an appropriate linearization of the projection is equivalent to the SQP method, which is shown to be more robust as the classical radial return. This is illustrated by a numerical comparison of both methods for a benchmark problem.

Abstract:We introduce a space-time discretization for linear first-order hyperbolic evolution systems using a discontinuous Galerkin approximation in space and a Petrov-Galerkin scheme in time. We show wellposedness and convergence of the discrete system. Then we introduce an adaptive strategy based on goaloriented dual-weighted error estimation. The full space-time linear system is solved with a parallel multilevel preconditioner. Numerical experiments for the linear transport equation and the Maxwell equation in 2D underline the efficiency of the overall adaptive solution process.

We investigate a representative model of continuum infinitesimal gradient plasticity. The formulation is an extension of classical rate‐independent infinitesimal plasticity based on the additive decomposition of the symmetric strain tensor into elastic and plastic parts. It is assumed that dislocation processes contribute to the storage of energy in the material whereby the curl of the plastic distortion appears in the thermodynamic potential and leads to an additional nonlocal backstress tensor. The formulation is cast into a numerical framework by a saddle point approximation of the corresponding minimization problem in each incremental loading step. This allows one to reformulate the (nonlocal) dissipation inequality to a point‐wise flow rule and yields a solution scheme, which is a direct extension of the standard approach in classical plasticity. Our numerical results show the regularizing effects of the additional physically motivated terms. Copyright © 2008 John Wiley & Sons, Ltd.