GPU Accelerated Explicit Time Integration Methods for Electro-Quasistatic Fields

📝 Abstract

Electro-quasistatic field problems involving nonlinear materials are commonly discretized in space using finite elements. In this paper, it is proposed to solve the resulting system of ordinary differential equations by an explicit Runge-Kutta-Chebyshev time-integration scheme. This mitigates the need for Newton-Raphson iterations, as they are necessary within fully implicit time integration schemes. However, the electro-quasistatic system of ordinary differential equations has a Laplace-type mass matrix such that parts of the explicit time-integration scheme remain implicit. An iterative solver with constant preconditioner is shown to efficiently solve the resulting multiple right-hand side problem. This approach allows an efficient parallel implementation on a system featuring multiple graphic processing units.

💡 Analysis

Electro-quasistatic field problems involving nonlinear materials are commonly discretized in space using finite elements. In this paper, it is proposed to solve the resulting system of ordinary differential equations by an explicit Runge-Kutta-Chebyshev time-integration scheme. This mitigates the need for Newton-Raphson iterations, as they are necessary within fully implicit time integration schemes. However, the electro-quasistatic system of ordinary differential equations has a Laplace-type mass matrix such that parts of the explicit time-integration scheme remain implicit. An iterative solver with constant preconditioner is shown to efficiently solve the resulting multiple right-hand side problem. This approach allows an efficient parallel implementation on a system featuring multiple graphic processing units.

📄 Content

0018-9464 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 1 GPU Accelerated Explicit Time Integration Methods for Electro-Quasistatic Fields

Christian Richter1, Sebastian Schöps2 and Markus Clemens1, Senior Member, IEEE

1Chair of Electromagnetic Theory, University of Wuppertal, 42119 Wuppertal, Germany, christian.richter@uni-wuppertal.de 2Graduate School of Computational Engineering and Institut für Theorie Elektromagnetischer Felder, Technische Universität Darmstadt, 64285 Darmstadt, Germany, schoeps@gsc.tu-darmstadt.de

Electro-quasistatic field problems involving nonlinear materials are commonly discretized in space using finite elements. In this paper, it is proposed to solve the resulting system of ordinary differential equations by an explicit Runge-Kutta-Chebyshev time-integration scheme. This mitigates the need for Newton-Raphson iterations, as they are necessary within fully implicit time integration schemes. However, the electro-quasistatic system of ordinary differential equations has a Laplace-type mass matrix such that parts of the explicit time-integration scheme remain implicit. An iterative solver with constant preconditioner is shown to efficiently solve the resulting multiple right-hand side problem. This approach allows an efficient parallel implementation on a system featuring multiple graphic processing units. Index Terms—algebraic multigrid, electro-quasistatic, parallelism, GPUs, explicit time-integration

I. INTRODUCTION LECTRICAL field grading using microvaristor materials embedded in polymeric materials is a field of ongoing research for high-voltage applications as for example insulators, bushings and surge arrestors. The materials change their electrical conductivity depending on the local electrical field with it rising by several orders of magnitude at a switching point. For proper use and design complex 3D models as shown in Fig. 1 must be numerically analyzed. The electro-quasistatic (EQS) approximation of Maxwell’s equation is applied to simultaneously consider capacitive and resistive effects. For solving these problems, they must be discretized in space and time. Space discretization is typically carried out by the Finite Element Method (FEM) and time discretization by sequential time-integration schemes. The most common approaches are based on implicit time integration schemes, like the implicit Euler scheme or the Singly Diagonal Implicit Runge-Kutta (SDIRK) method [1,2]. To solve the nonlinear problem in each time step, an iterative linearization method, as e.g. the Newton-Raphson scheme, is applied and thus many linear algebraic systems need to be solved [3]. The time-consuming computation of large-scale models often exceeds multiple days. However, it can be accelerated by using graphic processor units (GPUs) [4]. Particularly, iterative linear solvers based on sparse matrix-vector operations highly benefit from GPUs [5]. For example, algebraic multigrid (AMG) preconditioners are often used [6] due to their (almost) optimal asymptotic complexity. On the other hand, already medium sized FEM models hit the limits of a contemporary single GPU’s global memory. Therefore, multi-GPU AMG- preconditioned conjugate gradients (AMG-CG) solvers have been presented [7,8]. These solvers allow fast solutions of the linear equation systems, but the repeated construction of the preconditioner for the Jacobian and its upload to the GPUs is still a bottleneck of these schemes.
Therefore, this paper deals with the application of explicit time-integration schemes such as the explicit Euler method or the more sophisticated Runge-Kutta-Chebyshev method [1,9]. In the case of EQS, the resulting scheme is not entirely explicit since the mass matrix is a discretization of the electrostatic Laplacian operator. Nonetheless, due to constant permittivity, the explicit scheme leads to a multiple right-hand side (MRHS) problem. This favors the parallelization of the linear algebra since communication costs required for the (nonlinear) Jacobian matrix updates are avoided. Especially the acceleration by GPUs compensates for the increased effort due the time-step stability restriction of the explicit schemes. The paper is structured as follows: Section II introduces the formulation of the EQS problem and an explicit time-integration scheme is described. A GPU-based combined matrix assembly and sparse matrix vector multiplication is discussed in Section III. Efficient approaches for the MRHS problem are described in Section IV. The methodology is validated by numerical examples in Section V and the paper closes with conclusions in Section VI. E Corresponding author: C. Richter (e-mail: christian.richter@uni- wuppertal.de).

Fig. 1: CAD model and electrical field of the ca

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