Thomas Dickopf

Design and Analysis of a Lightweight Parallel Adaptive Scheme for the Solution of the Monodomain Equation

Numerical simulation of the nonlinear reaction-diffusion equations in computational electrocardiology requires locally high spatial resolution to capture the multiscale effects related to the electrical activation of the heart accurately, namely the strongly varying transmembrane potential. Here, we propose a novel lightweight adaptive algorithm which aims at combining the plainness of structured meshes with the resolving capabilities of unstructered adaptive meshes. Our “patchwise adaptive” approach is based on locally structured mesh hierarchies which are glued along their interfaces by a nonconforming mortar element discretization. To further increase the overall efficiency, we keep the spatial meshes constant over suitable time windows in which error indicators are accumulated. This approach facilitates strongly varying mesh sizes in neighboring patches as well as in consecutive time steps. For the transfer of the dynamic variables between different spatial approximation spaces we compare the

Towards a large-scale scalable adaptive heart model using shallow tree meshes

L^2

Poster: hybrid parallelization of a realistic heart model

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Numerical Study of the Almost Nested Case in a Multilevel Method Based on Non-nested Meshes

Electrophysiological heart models are sophisticated computational tools that place high demands on the computing hardware due to the high spatial resolution required to capture the steep depolarization front. To address this challenge, we present a novel adaptive scheme for resolving the deporalization front accurately using adaptivity in space. Our adaptive scheme is based on locally structured meshes. These tensor meshes in space are organized in a parallel forest of trees, which allows us to resolve complicated geometries and to realize high variations in the local mesh sizes with a minimal memory footprint in the adaptive scheme. We discuss both a non-conforming mortar element approximation and a conforming finite element space and present an efficient technique for the assembly of the respective stiffness matrices using matrix representations of the inclusion operators into the product space on the so-called shallow tree meshes.We analyzed the parallel performance and scalability for a two-dimensional ventricle slice as well as for a full large-scale heart model. Our results demonstrate that the method has good performance and high accuracy.

Monotone Multigrid Methods Based on Parametric Finite Elements

Heart failure is a major health problem, not only for the number of people affected (about five million in Europe alone) but also because of the direct and indirect costs for its treatment. A thorough understanding of the complex electrical activation system that triggers the mechanical contraction is a prerequisite for developing effective treatment strategies. Full-heart simulations are an indispensable tool to study the effect of molecular-level or tissue-level changes on clinical measurements [2]. Cardiac electrical activity originates in the millions of ion channels and pumps that are located in the outer membrane of each cardiac muscle cell. We denote the macroscopic ionic current density by Iion.

Domain Decomposition Methods in Science and Engineering XXII

Partial differential equations in complex domains are very flexibly discretized by finite elements with unstructured meshes. For such problems, the challenging task to construct coarse level spaces for efficient multilevel preconditioners can in many cases be solved by a semi-geometric approach, which is based on a hierarchy of non-nested meshes. In this paper, we investigate the connection between the resulting semi-geometric multigrid methods and the truly geometric variant more closely. This is done by considering a sufficiently simple computational domain and treating the geometric multigrid method as a special case in a family of almost nested settings. We study perturbations of the meshes and analyze how efficiency and robustness depend on a truncation of the interlevel transfer. This gives a precise idea of which results can be achieved in the general unstructured case.

Simultaneous Reduced Basis Approximation of Parameterized Eigenvalue Problems .

In this paper, a particular technique for the application of elementary multilevel ideas to problems with warped boundaries is studied in the context of the numerical simulation of elastic contact problems. Combining a general multilevel setting with a different perspective, namely an advanced geometric modeling point of view, we present a (monotone) multigrid method based on a hierarchy of parametric finite element spaces. For the construction, a full-dimensional parameterization of high order is employed which accurately represents the computational domain.The purpose of the volume parametric finite element discretization put forward here is two-fold. On the one hand, it allows for an elegant multilevel hierarchy to be used in preconditioners. On the other hand, it comes with particular advantages for the modeling of contact problems. After all, the long-term objective lies in an increased flexibility of h p-adaptive methods for contact problems.