TY - GEN A1 - Krause, Dorian A1 - Fackeldey, Konstantin A1 - Krause, Rolf T1 - A parallel multiscale simulation toolbox for coupling molecular dynamics and finite elements N2 - It is the ultimate goal of concurrent multiscale methods to provide computational tools that allow to simulation physical processes with the accuracy of micro-scale and the computational speed of macro-scale models. As a matter of fact, the efficient and scalable implementation of concurrent multiscale methods on clusters and supercomputers is a complicated endeavor. In this article we present the parallel multiscale simulation tool MACI which has been designed for efficient coupling between molecular dynamics and finite element codes. We propose a specification for a thin yet versatile interface for the coupling of molecular dynamics and finite element codes in a modular fashion. Further we discuss the parallelization strategy pursued in MACI, in particular, focusing on the parallel assembly of transfer operators and their efficient execution. T3 - ZIB-Report - 13-50 KW - Parallel KW - Atomistic to Continuum KW - Multiscale KW - MACI KW - Meshfree Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-42354 SN - 1438-0064 ER - TY - JOUR A1 - Fackeldey, Konstantin A1 - Krause, Dorian A1 - Krause, Rolf A1 - Lenzen, Christoph T1 - Coupling Molecular Dynamics and Continua with Weak Constraints JF - SIAM Multiscale Model. Simul. Y1 - 2011 U6 - https://doi.org//10.1137/100782097 N1 - also as MATHEON Preprint VL - 9 SP - 1459 EP - 1494 ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Krause, Dorian A1 - Krause, Rolf ED - Griebel, Michael ED - Schweitzer, M. T1 - Numerical Validation of Constraints Based Multiscale Methods T2 - Lecture Notes in Computational Science and Engineering Y1 - 2010 SP - 141 EP - 154 ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Krause, Dorian A1 - Krause, Rolf T1 - Quadrature and Implementation of the Weak Coupling Method T2 - MMM 2008: Proceedings of the 4th International Conference on Multiscale Materials Modelling, 27-31 October 2008, Tallahassee, Fla. Y1 - 2008 SP - 62 EP - 65 ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Krause, Dorian A1 - Krause, Rolf T1 - Weak Coupling Algorithms in Multiscale Simulations T2 - Proceedings of the third Inter. Confer. on Comp. Meth. for Coupled Problems in Science and Engrg Y1 - 2009 SP - 1023pp ER - TY - JOUR A1 - Krause, Dorian A1 - Fackeldey, Konstantin A1 - Krause, Rolf ED - Griebel, Michael T1 - A Parallel Multiscale Simulation Toolbox for Coupling Molecular Dynamics and Finite Elements JF - Singular Phenomena and Scaling in Mathematical Models Y1 - 2014 U6 - https://doi.org/10.1007/978-3-319-00786-1_14 SP - 327 EP - 346 PB - Springer International Publishing ER - TY - GEN A1 - Sander, Oliver A1 - Krause, Rolf T1 - Automatic Construction of Boundary Parametrizations for Geometric Multigrid Solvers N2 - We present an algorithm that constructs parametrizations of boundary and interface surfaces automatically. Starting with high-resolution triangulated surfaces describing the computational domains, we iteratively simplify the surfaces yielding a coarse approximation of the boundaries with the same topological type. While simplifying we construct a function that is defined on the coarse surface and whose image is the original surface. This function allows access to the correct shape and surface normals of the original surface as well as to any kind of data defined on it. Such information can be used by geometric multigrid solvers doing adaptive mesh refinement. Our algorithm runs stable on all types of input surfaces, including those that describe domains consisting of several materials. We have used our method with success in different fields and we discuss examples from structural mechanics and biomechanics. T3 - ZIB-Report - 03-02 KW - boundary parametrization KW - surface simplification KW - multigrid KW - geometry Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7240 ER - TY - GEN A1 - Deuflhard, Peter A1 - Krause, Rolf A1 - Ertel, Susanne T1 - A Contact-Stabilized Newmark Method for Dynamical Contact Problems N2 - The numerical integration of dynamical contact problems often leads to instabilities at contact boundaries caused by the non-penetration condition between bodies in contact. Even a recent energy dissipative modification due to Kane et al. (1999), which discretizes the non-penetration constraints implicitly, is not able to circumvent artificial oscillations. For this reason, the present paper suggests a contact stabilization which avoids artificial oscillations at contact interfaces and is also energy dissipative. The key idea of this contact stabilization is an additional $L^2$-projection at contact interfaces, which can easily be added to any existing time integration scheme. In case of a lumped mass matrix, this projection can be carried out completely locally, thus creating only negligible additional numerical cost. For the new scheme, an elementary analysis is given, which is confirmed by numerical findings in an illustrative test example (Hertzian two body contact). T3 - ZIB-Report - 06-42 KW - Numerical problems in dynamical systems KW - nonlinear stabilities Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9361 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Krause, Rolf T1 - Classical and Cascadic Multigrid - A Methodical Comparison N2 - Using the full multigrid method {\em without} any coarse grid correction steps but with an a posteriori control of the number of smoothing iterations was shown by Bornemann and Deuflhard [1996] to be an optimal iteration method with respect to the energy norm. They named this new kind of multigrid iteration the {\em cascadic multigrid method}. However, numerical examples with {\em linear} finite elements raised serious doubts whether the cascadic multigrid method can be made optimal with respect to the {\em $L^2$-norm}. In this paper we prove that the cascadic multigrid method cannot be optimal for linear finite elements and show that the case might be different for higher order elements. We present a careful analysis of the two grid variant of the cascadic multigrid method providing a setting where one can understand the methodical difference between the cascadic multigrid method and the classical multigrid $V$-cycle almost immediately. As a rule of thumb we get that whenever the cascadic multigrid works the classical multigrid will work too but not vice versa. T3 - ZIB-Report - SC-96-25 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2368 ER - TY - JOUR A1 - Deuflhard, Peter A1 - Krause, Rolf A1 - Ertel, Susanne T1 - A Contact-Stabilized Newmark Method for Dynamical Contact Problems JF - International Journal for Numerical Methods in Engineering Y1 - 2007 VL - 73 IS - 9 SP - 1274 EP - 1290 ER - TY - CHAP A1 - Krause, Rolf A1 - Weiser, Martin T1 - Multilevel augmented Lagrangian solvers for overconstrained contact formulations T2 - ESAIM: ProcS N2 - Multigrid methods for two-body contact problems are mostly based on special mortar discretizations, nonlinear Gauss-Seidel solvers, and solution-adapted coarse grid spaces. Their high computational efficiency comes at the cost of a complex implementation and a nonsymmetric master-slave discretization of the nonpenetration condition. Here we investigate an alternative symmetric and overconstrained segment-to-segment contact formulation that allows for a simple implementation based on standard multigrid and a symmetric treatment of contact boundaries, but leads to nonunique multipliers. For the solution of the arising quadratic programs, we propose augmented Lagrangian multigrid with overlapping block Gauss-Seidel smoothers. Approximation and convergence properties are studied numerically at standard test problems. Y1 - 2021 U6 - https://doi.org/10.1051/proc/202171175 VL - 71 SP - 175 EP - 184 ER - TY - CHAP A1 - Chegini, Fatemeh A1 - Kopanicakova, Alena A1 - Weiser, Martin A1 - Krause, Rolf T1 - Quantitative Analysis of Nonlinear MultifidelityOptimization for Inverse Electrophysiology T2 - Domain Decomposition Methods in Science and Engineering XXVI N2 - The electric conductivity of cardiac tissue determines excitation propagation and is important for quantifying ischemia and scar tissue and for building personalized models. Estimating conductivity distributions from endocardial mapping data is a challenging inverse problem due to the computational complexity of the monodomain equation, which describes the cardiac excitation. For computing a maximum posterior estimate, we investigate different optimization approaches based on adjoint gradient computation: steepest descent, limited memory BFGS, and recursive multilevel trust region methods, which are using mesh hierarchies or heterogeneous model hierarchies. We compare overall performance, asymptotic convergence rate, and pre-asymptotic progress on selected examples in order to assess the benefit of our multifidelity acceleration. Y1 - 2022 SP - 65 EP - 76 PB - Springer ER - TY - JOUR A1 - Chegini, Fatemeh A1 - Kopanicakova, Alena A1 - Krause, Rolf A1 - Weiser, Martin T1 - Efficient Identification of Scars using Heterogeneous Model Hierarchies JF - EP Europace N2 - Aims. Detection and quantification of myocardial scars are helpful both for diagnosis of heart diseases and for building personalized simulation models. Scar tissue is generally charac­terized by a different conduction of electrical excitation. We aim at estimating conductivity-related parameters from endocardial mapping data, in particular the conductivity tensor. Solving this inverse problem requires computationally expensive monodomain simulations on fine discretizations. Therefore, we aim at accelerating the estimation using a multilevel method combining electrophysiology models of different complexity, namely the mono­domain and the eikonal model. Methods. Distributed parameter estimation is performed by minimizing the misfit between simulated and measured electrical activity on the endocardial surface, subject to the mono­domain model and regularization, leading to a constrained optimization problem. We formulate this optimization problem, including the modeling of scar tissue and different regularizations, and design an efficient iterative solver. We consider monodomain grid hierarchies and monodomain-eikonal model hierarchies in a recursive multilevel trust-region method. Results. From several numerical examples, both the efficiency of the method and the estimation quality, depending on the data, are investigated. The multilevel solver is significantly faster than a comparable single level solver. Endocardial mapping data of realistic density appears to be just sufficient to provide quantitatively reasonable estimates of location, size, and shape of scars close to the endocardial surface. Conclusion. In several situations, scar reconstruction based on eikonal and monodomain models differ significantly, suggesting the use of the more accurate but more expensive monodomain model for this purpose. Still, eikonal models can be utilized to accelerate the computations considerably, enabling the use of complex electrophysiology models for estimating myocardial scars from endocardial mapping data. Y1 - 2021 U6 - https://doi.org/10.1093/europace/euaa402 VL - 23 SP - i113 EP - i122 ER - TY - CHAP A1 - Gander, Lia A1 - Krause, Rolf A1 - Weiser, Martin A1 - Costabal, Francisco A1 - Pezzuto, Simone T1 - On the Accuracy of Eikonal Approximations in Cardiac Electrophysiology in the Presence of Fibrosis T2 - Functional Imaging and Modeling of the Heart. FIMH 2023. N2 - Fibrotic tissue is one of the main risk factors for cardiac arrhythmias. It is therefore a key component in computational studies. In this work, we compare the monodomain equation to two eikonal models for cardiac electrophysiology in the presence of fibrosis. We show that discontinuities in the conductivity field, due to the presence of fibrosis, introduce a delay in the activation times. The monodomain equation and eikonal-diffusion model correctly capture these delays, contrarily to the classical eikonal equation. Importantly, a coarse space discretization of the monodomain equation amplifies these delays, even after accounting for numerical error in conduction velocity. The numerical discretization may also introduce artificial conduction blocks and hence increase propagation complexity. Therefore, some care is required when comparing eikonal models to the discretized monodomain equation. Y1 - 2023 U6 - https://doi.org/10.1007/978-3-031-35302-4_14 VL - 13958 PB - Springer, Cham ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Krause, Rolf A1 - Schweitzer, M. ED - Griebel, Michael ED - Schweitzer, M. T1 - Stability of Energy Transfer in the Weak Coupling Method T2 - Meshfree Methods for Partial Differential Equations IV Y1 - 2008 VL - 65 SP - 111 EP - 121 ER - TY - JOUR A1 - Fackeldey, Konstantin A1 - Krause, Rolf T1 - Multiscale Coupling in Function Space - Weak Coupling between Molecular Dynamics and Continuum Mechanics JF - International Journal for Numerical Methods in Engineering Y1 - 2009 U6 - https://doi.org/10.1002/nme.2626 VL - 79 IS - 12 SP - 1517 EP - 1535 ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Krause, Rolf ED - Widlund, Olof T1 - Solving Frictional Contact Problems with Multigrid Efficiency T2 - Proc.of the 16th International Conference on Domain Decomposition Methods Y1 - 2007 VL - 50 SP - 547 EP - 554 ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Krause, Rolf T1 - CM/MD Coupling - A Function Space Oriented Multiscale-Coupling Approach T2 - Proc. Appl. Math. Mech. Y1 - 2008 VL - 8 IS - 1 SP - 10495pp ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Krause, Rolf T1 - Weak coupling in function space T2 - Proc. Appl. Math. Mech. Y1 - 2007 VL - 7 IS - 1 SP - 2020113pp ER -