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The numerical simulation of cardiac electrophysiology is a highly challenging problem in scientific computing. The Bidomain system is the most complete mathematical model of cardiac bioelectrical activity. It consists of an elliptic and a parabolic partial differential equation (PDE), of reaction–diffusion type, describing the spread of electrical excitation in the cardiac tissue. The two PDEs are coupled with a stiff system of ordinary differential equations (ODEs), representing ionic currents through the cardiac membrane. Developing efficient and scalable preconditioners for the linear systems arising from the discretization of such computationally challenging model is crucial in order to reduce the computational costs required by the numerical simulations of cardiac electrophysiology. In this work, focusing on the Bidomain system as a model problem, we have benchmarked two popular implementations of the Algebraic Multigrid (AMG) preconditioner embedded in the PETSc library and we have studied the performance on the calibration of specific parameters. We have conducted our analysis on modern HPC architectures, performing scalability tests on multi-core and multi-GPUs settings. The results have shown that, for our problem, although scalability is verified on CPUs, GPUs are the optimal choice, since they yield the best performance in terms of solution time. [ABSTRACT FROM AUTHOR]

Motivated by the inability of classical computational plasticity to fully exploit modern scientific computing, a multifield formulation for finite strain plasticity is presented. This avoids a local integration of the elastoplastic model. In the multifield approach, the balance of linear momentum, the flow relation and the Karush–Kuhn–Tucker constraints are collectively cast in a variational format. In addition to the deformation, both the plastic strain and the consistency parameter are global degrees of freedom in the resulting spatially discrete problem. The ensuing proliferation of global degrees of freedom in the multifield approach is addressed by exploiting the block sparse structure of the algebraic system together with a tailored block matrix solver which can utilise emerging hardware architectures. A series of numerical problems demonstrate the validity, capability and efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

This work introduces a novel, fully robust and highly-scalable, h -adaptive aggregated unfitted finite element method for large-scale interface elliptic problems. The new method is based on a recent distributed-memory implementation of the aggregated finite element method atop a highly-scalable Cartesian forest-of-trees mesh engine. It follows the classical approach of weakly coupling nonmatching discretisations at the interface to model internal discontinuities at the interface. We propose a natural extension of a single-domain parallel cell aggregation scheme to problems with a finite number of interfaces; it straightforwardly leads to aggregated finite element spaces that have the structure of a Cartesian product. We demonstrate, through standard numerical analysis and exhaustive numerical experimentation on several complex Poisson and linear elasticity benchmarks, that the new technique enjoys the following properties: well-posedness, robustness with respect to cut location and material contrast, optimal (h-adaptive) approximation properties, high scalability and easy implementation in large-scale finite element codes. As a result, the method offers great potential as a useful finite element solver for large-scale interface problems modelled by partial differential equations. • Novel approach enabling large-scale simulations for unfitted/embedded interfaces. • Methodology based on the approach of weakly coupling interface-overlapping meshes. • AgFEM is endowed with optimal (h-adaptive) approximation properties. • AgFEM shows remarkable robustness to cut location and material contrast. • Weak scaling tests up to 90M degrees of freedom and 2K CPU cores on complex domains. [ABSTRACT FROM AUTHOR]

The aggregated unfitted finite element method (AgFEM) is a methodology recently introduced in order to address conditioning and stability problems associated with embedded, unfitted, or extended finite element methods. The method is based on removal of basis functions associated with badly cut cells by introducing carefully designed constraints, which results in well-posed systems of linear algebraic equations, while preserving the optimal approximation order of the underlying finite element spaces. The specific goal of this work is to present the implementation and performance of the method on distributed-memory platforms aiming at the efficient solution of large-scale problems. In particular, we show that, by considering AgFEM, the resulting systems of linear algebraic equations can be effectively solved using standard algebraic multigrid preconditioners. This is in contrast with previous works that consider highly customized preconditioners in order to allow one the usage of iterative solvers in combination with unfitted techniques. Another novelty with respect to the methods available in the literature is the problem sizes that can be handled with the proposed approach. While most of previous references discussing linear solvers for unfitted methods are based on serial non-scalable algorithms, we propose a parallel distributed-memory method able to efficiently solve problems at large scales. This is demonstrated by means of a weak scaling test defined on complex 3D domains up to 300M degrees of freedom and one billion cells on 16K CPU cores in the Marenostrum-IV platform. The parallel implementation of the AgFEM method is available in the large-scale finite element package FEMPAR. [ABSTRACT FROM AUTHOR]