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1. Wobker, Hilmar. Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems.

Degree: 2010, Technische Universität Dortmund

URL: http://dx.doi.org/10.17877/DE290R-497

In the simulation of realistic solid mechanical problems, linear equation systems with hundreds of million unknowns can arise. For the efficient solution of such systems, parallel multilevel methods are mandatory that are able to exploit the capabilities of modern hardware technologies. The finite element and solution toolbox FEAST, which is designed to solve scalar equations, pursues exactly this goal. It combines hardware-oriented implementation techniques with a multilevel domain decomposition method called ScaRC that achieves high numerical and parallel efficiency. In this thesis a concept is developed to solve multivariate elasticity problems based on the FEAST library. The general strategy is to reduce the solution of multivariate problems to the solution of a series of scalar problems. This approach facilitates a strict separation of 'low level' scalar kernel functionalities (in the form of the FEAST library) and 'high level' multivariate
application code (in the form of the elasticity problem), which is very attractive from a software-engineering point of view: All efforts to improve hardware-efficiency and adaptations to future technology trends can be restricted to scalar operations, and the multivariate application automatically benefits from these enhancements. In the first part of the thesis, substantial improvements of the scalar ScaRC solvers are developed, which are then used as essential building blocks for the efficient solution of multivariate elasticity problems. Extensive numerical studies demonstrate how the efficiency of the scalar FEAST library transfers to the multivariate solution process. The solver strategy is then applied to treat nonlinear problems of finite deformation elasticity. A line-search method is used to significantly increase the robustness of the Newton-Raphson method, and different strategies are compared how to choose the accuracy of the linear system solves within the nonlinear
iteration. In order to treat the important class of (nearly) incompressible material, a mixed displacement/pressure formulation is used which is discretised with stabilised bilinear finite elements (Q1/Q1). An enhanced version of the classical 'pressure Poisson' stabilisation is presented which is suitable for highly irregular meshes. Advantages and disadvantages of the Q1/Q1 discretisation are discussed, especially in the context of transient computations. Two solver classes for the resulting saddle point systems are described and compared: on the one hand various kinds of (accelerated) segregated solvers (Uzawa, pressure Schur complement methods, block preconditioners), and on the other hand coupled multigrid solvers with Vanka-smoothers. Efficient Schur complement preconditioners, which are required for the former class, are discussed for the static and the transient case. The main strategy to reduce the solution of multivariate systems to the solution of scalar systems is only
applicable in the case of segregated methods. It is shown that for the class of elasticity problems considered in this…
*Advisors/Committee Members: Turek, S. (advisor), Suttmeier, F.-T. (referee).*

Subjects/Keywords: Iterativer Löser; Multilevel; Mehrgitter; Gebietszerlegung; Mehrgitter-Krylov; Nicht-konformes Mehrgitter; ScaRC; Adaptive Grobgitterkorrektur; Minimale Überlappung; Sattelpunkt-Problem; Schurkomplement-Vorkonditionierer; Vanka; Gedämpftes Newton-Raphson; Globales Newton-Raphson; Inexaktes Newton-Raphson; Liniensuche; FEAST; Hardware-orientiert; Großskalig; Paralleles Rechnen; Parallele Effizienz; Finite-Elemente-Methode; Gemischte Formulierung; LBB Stabilisierung; Irreguläres Gitter; Festkörpermechanik; Strukturmechanik; Elastizität; Elastostatisch; Elastodynamisch; Zeitabhängig; Inkompressibles Material; Finite Deformation; Große Deformation; Volumenversteifung; Schubversteifung; Iterative solver; Multilevel; Multigrid; Domain decomposition; Multigrid-Krylov; Nonconforming multigrid; ScaRC; Adaptive coarse grid correction; Minimal overlap; Saddle point problem; Schur complement preconditioning; Vanka; Newton-Raphson; Damped Newton-Raphson; Global Newton-Raphson; Inexact Newton-Raphson; Line-search; FEAST; High performance computing; Hardware-oriented; Large-scale; Parallel computing; Parallel efficiency; Finite element method; Mixed formulation; LBB stabilisation; Equal-order finite elements; Irregular grids; Solid mechanics; Structural mechanics; Elasticity; Elastostatic; Elastodynamic; Transient; Incompressible material; Finite deformation; Large deformation; Volume locking; Shear locking; 510

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APA (6^{th} Edition):

Wobker, H. (2010). Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems. (Doctoral Dissertation). Technische Universität Dortmund. Retrieved from http://dx.doi.org/10.17877/DE290R-497

Chicago Manual of Style (16^{th} Edition):

Wobker, Hilmar. “Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems.” 2010. Doctoral Dissertation, Technische Universität Dortmund. Accessed April 14, 2021. http://dx.doi.org/10.17877/DE290R-497.

MLA Handbook (7^{th} Edition):

Wobker, Hilmar. “Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems.” 2010. Web. 14 Apr 2021.

Vancouver:

Wobker H. Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems. [Internet] [Doctoral dissertation]. Technische Universität Dortmund; 2010. [cited 2021 Apr 14]. Available from: http://dx.doi.org/10.17877/DE290R-497.

Council of Science Editors:

Wobker H. Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems. [Doctoral Dissertation]. Technische Universität Dortmund; 2010. Available from: http://dx.doi.org/10.17877/DE290R-497