Robust exact and inexact FETI-DP methods with applications to elasticity

Gebietszerlegungsverfahren sind parallele, iterative Lösungsverfahren für grosse Gleichungssysteme, die bei der Diskretisierung von partiellen Differentialgleichungen, etwa aus der Strukturmechanik, entstehen. In dieser Arbeit werden duale, iterative Substrukturierungsverfahren vom FETI-DP-Typ (Finite Element Tearing and Interconnecting Dual-Primal) entwickelt und auf elliptische partielle Differentialgleichungen zweiter Ordnung angewandt. Insbesondere wird versucht, robuste Verfahren für homogene und heterogene Elastizitaetsprobleme zu entwickeln. Ebenso werden neue, inexakte FETI-DP-Verfahren vorgestellt, die eine inexakte Lösung des Grobgitterproblems und/oder der Teilgebietsprobleme erlauben. Es wird gezeigt, dass die neuen Algorithmen unter bestimmten Voraussetzungen Abschätzungen der gleichen asymptotischen Güte wie das klassische, exakte FETI-DP-Verfahren erfüllen. Parallele Resultate unter Verwendung von algebraischen Mehrgitter für das Grobgitterproblem zeigen die verbesserte Skalierbarkeit der neuen Algorithmen.Domain decomposition methods are fast parallel solvers for large equation systems arising from the discretisation of partial differential equations, e.g. from structural mechanics. In this work, dual iterative substructuring methods of the FETI-DP (Finite Element Tearing and Interconnecting Dual-Primal) type are developed and applied to second order elliptic problems with emphasis on elasticity. An attempt is made to develop robust methods for homogeneous and heterogeneous problems. New inexact FETI-DP methods are also introduced that allow for inexact coarse problem solvers and/or inexact subdomain solvers. It is shown that under certain conditions the new algorithms fulfill the same asymptotic condition number estimate as the traditional, exact FETI-DP methods. Parallel results using algebraic multigrid for the FETI-DP coarse problem show the improved scalability of the new algorithms

Parallel adaptive FETI-DP using lightweight asynchronous dynamic load balancing

A parallel FETI-DP domain decomposition method using an adaptive coarse space is presented. The implementation builds on a recently introduced adaptive FETI-DP approach for elliptic problems in three dimensions and uses small, local eigenvalue problems for faces and, additionally, for a small number of edges. The condition number of the preconditioned operator then satisfies a bound which is independent of coefficient heterogeneities in the problem. The computational cost of the local eigenvalue problems is not negligible, and also a significant load imbalance can be introduced. As a remedy, certain eigenvalue problems are discarded by a theory-guided heuristic strategy, based on the diagonal entries of the stiffness matrices. Additionally, a lightweight pairwise dynamic load balancing strategy is implemented for the eigenvalue problems. The load balancing is supervised by an orchestrating rank using asynchronous point-to-point communication. The resulting method shows good weak and strong scalability up to thousands of cores while fast convergence is obtained even for heterogeneous problems

A Comparison Of Direct Solvers In FROSch Applied To Chemo-Mechanics

Sparse direct linear solvers are at the computational core of domain decomposition preconditioners and therefore have a strong impact on their performance. In this paper, we consider the Fast and Robust Overlapping Schwarz (FROSch) solver framework of the Trilinos software library, which contains a parallel implementations of the GDSW domain decomposition preconditioner. We compare three different sparse direct solvers used to solve the subdomain problems in FROSch. The preconditioner is applied to different model problems; linear elasticity and more complex fully-coupled deformation diffusion-boundary value problems from chemo-mechanics. We employ FROSch in fully algebraic mode, and therefore, we do not expect numerical scalability. Strong scalability is studied from 64 to 4096 cores, where good scaling results are obtained up to 1728 cores. The increasing size of the coarse problem increases the solution time for all sparse direct solvers

Comparison of Arterial Wall Models in Fluid-Structure Interaction Simulations

Monolithic fluid-structure interaction (FSI) of blood flow with arterial walls is considered, making use of sophisticated nonlinear wall models. These incorporate the effects of almost incompressibility as well as of the anisotropy caused by embedded collagen fibers. In the literature, relatively simple structural models such as Neo-Hooke are often considered for FSI with arterial walls. Such models lack, both, anisotropy and incompressibility. In this paper, numerical simulations of idealized heart beats in a curved benchmark geometry, using simple and sophisticated arterial wall models, are compared: we consider three different almost incompressible, anisotropic arterial wall models as a reference and, for comparison, a simple, isotropic Neo-Hooke model using four different parameter sets. The simulations show significant quantitative and qualitative differences in the stresses and displacements as well as the lumen cross sections. For the Neo-Hooke models, a significantly larger amplitude in the in- and outflow areas during the heart beat is observed, presumably due to the lack of fiber stiffening. For completeness, we also consider a linear elastic wall using 16 different parameter sets. However, using our benchmark setup, we were not successful in achieving good agreement with our nonlinear reference calculation

Adaptive GDSW coarse spaces of reduced dimension for overlapping Schwarz methods

A new reduced dimension adaptive GDSW (Generalized Dryja-Smith-Widlund) overlapping Schwarz method for linear second-order elliptic problems in three dimensions is introduced. It is robust with respect to large contrasts of the coefficients of the partial differential equations. The condition number bound of the new method is shown to be independent of the coefficient contrast and only dependent on a user-prescribed tolerance. The interface of the nonoverlapping domain decomposition is partitioned into nonoverlapping patches. The new coarse space is obtained by selecting a few eigenvectors of certain local eigenproblems which are defined on these patches. These eigenmodes are energy-minimally extended to the interior of the nonoverlapping subdomains and added to the coarse space. By using a new interface decomposition the reduced dimension adaptive GDSW overlapping Schwarz method usually has a smaller coarse space than existing GDSW and adaptive GDSW domain decomposition methods. A robust condition number estimate is proven for the new reduced dimension adaptive GDSW method which is also valid for existing adaptive GDSW methods. Numerical results for the equations of isotropic linear elasticity in three dimensions confirming the theoretical findings are presented

FROSch: A Fast And Robust Overlapping Schwarz Domain Decomposition Preconditioner Based on Xpetra in Trilinos

A parallel two-level overlapping Schwarz domain decomposition preconditioner has been integrated into the Trilinos ShyLU-package. The preconditioner uses an energy-minimizing coarse space and can be constructed from an assembled sparse matrix. The software implements variants of the two-level overlapping Schwarz method from [Dohrmann, Klawonn, Widlund, SINUM 2008], where it was denoted Generalized Dryja, Smith, Widlund (GDSW). The implementation is based on [Heinlein, Klawonn, Rheinbach, SISC 2016] but has been improved significantly with respect to efficiency, generality, e.g., for the use of Tpetra instead of Epetra matrices, and its interface

A closer look at local eigenvalue solvers for adaptive FETI-DP and BDDC

Adaptive coarse spaces for domain decomposition methods are an active area of research to make iterative domain decomposition methods robust with respect to large discontinuities in the material parameters or almost incompressible elasticity. In order to make their use feasible for applications, the computational overhead of the adaptive methods has to be reduced while maintaining the robustness. Since the solution of the local eigenvalue problems is expensive, also the iterative eigenvalue solvers are considered in detail and potential improvements are presented

A Three-Level Extension of the GDSW Overlapping Schwarz Preconditioner in Three Dimensions

A three-level extension of the GDSW overlapping Schwarz preconditioner in three dimensions is presented, constructed by recursively applying the GDSW preconditioner to the coarse problem using a standard and a reduced dimension coarse space. Numerical results, obtained for a parallel implementation using the Trilinos software library, are presented for up to 64,000 cores of the JUQUEEN supercomputer. The superior weak parallel scalability of the three-level method is verified. For large problems and a large number of cores, the three-level method is faster by more than a factor of two, compared to the standard two-level method. The three-level method shows to scale when the classical method is already be out-of-memory

Adaptive GDSW coarse spaces for overlapping Schwarz methods in three dimensions

A robust two-level overlapping Schwarz method for scalar elliptic model problems with highly varying coefficient functions is introduced. While the convergence of standard coarse spaces may depend strongly on the contrast of the coefficient function, the condition number bound of the new method is independent of the coefficient function. Its coarse space is based on discrete harmonic extensions of vertex, edge, and face interface functions, which are computed from the solutions of corresponding local generalized edge and face eigenvalue problems. The local eigenvalue problems are of the size of the edges and faces of the decomposition, and the eigenvalue problems can be constructed solely from the local subdomain stiffness matrices and the fully assembled global stiffness matrix. The new AGDSW (Adaptive Generalized Dryja-Smith-Widlund) coarse space always contains the classical GDSW coarse space by construction of the generalized eigenvalue problems. Numerical results supporting the theory are presented for several model problems in three dimensions using structured as well as unstructured meshes and unstructured decompositions