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Efficient Implementation and Parallelization of Meshfree and Particle Methods—The Parallel Multilevel Partition of Unity Method

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Frontiers of Numerical Analysis

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Abstract

In these introductory notes, we focus on the efficient implementation and parallelization of meshfree methods. Even though there exist a large number of different meshfree methods, e.g. smoothed particle hydrodynamics (SPH), reproducing kernel particle methods (RKPM), element free Galerkin methods (EFGM), radial basis functions (RBF), generalized finite element methods (GFEM), and partition of unity methods (PUM), the computational challenges are very similar for many of these approaches.

Some of the key issues involved with meshfree Galerkin discretization techniques are the fast construction of the shape functions, the assembly of the stiffness matrix and the load vector, i.e. numerical integration, the treatment of essential boundary conditions, and the efficient solution of the arising linear systems. We shall consider these issues in the context of the PUM, however, the concepts presented are applicable to most meshfree Galerkin approaches.

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Authors and Affiliations

  1. Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms Universität Bonn, Wegelerstraße 6, Bonn, Germany

    Marc Alexander Schweitzer

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  1. Marc Alexander Schweitzer
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  1. Department of Mathematical Sciences, University of Durham, South Road DH1 3LE, Durham, UK

    James F. Blowey

  2. Department of Mathematical Sciences, University of Durham, South Road DH1 3LE, Durham, UK

    Alan W. Craig

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Schweitzer, M.A. (2005). Efficient Implementation and Parallelization of Meshfree and Particle Methods—The Parallel Multilevel Partition of Unity Method. In: Blowey, J.F., Craig, A.W. (eds) Frontiers of Numerical Analysis. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-28884-8_4

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