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Performance of the multigrid method with time-stepping to solve 1D and 2D wave equations

International Journal for Computational methods in engineering Science and Mechanics 2022年第1期23卷 45-56页

作者： Malacarne, Maicon F. Pinto, Marcio A. V. Franco, Sebastião R. Graduate Program on Numerical Methods in Engineering–PPGMNE Federal University of Parana–UFPR PR Curitiba Brazil Department of Mechanical Engineering Federal University of Parana–UFPR PR Curitiba Brazil Department of Mathematics State University of Centro-Oeste–UNICENTRO PR Irati Brazil

This work aims to discuss a proposed solution for wave equations that utilize discretization by means of the finite difference method, weighted by a parameter η, with sweeping done according to the time-stepping method. The multigrid method is employed to speed up the convergence in obtaining the solution of the system of equations resulting from the discretization. To validate the proposed model, the discretization errors, effective and apparent orders, convergence factor, orders of complexity, and the computational time were assessed. A comparison between the singlegrid and multigrid methods was carried out in order to determine the most advantageous one. © 2021 Taylor & Francis Group, LLC.

关键词： Wave equations

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Subdomain Method in Time with Waveform Relaxation in Space Applied to the Wave Equation Combined with the Multigrid Method

SSRN

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SSRN 2022年

作者： Malacarne, Maicon F. Pinto, Marcio A.V. Franco, Sebastião R. Graduate Program in Numerical Methods in Engineering - PPGMNE Federal University of Parana - UFPR Brazil Cel. Francisco H. dos Santos Ave 100 Jardim das Américas Curitiba81530-000 Brazil Department of Mechanical Engineering Federal University of Parana - UFPR Marcio Brazil Department of Mathematics State University of Centro Oeste - UNICENTRO Brazil Professora Maria Roza Zanon de Almeida St - Engenheiro Gutierrez Irati84505-677 Brazil

This work aims to evaluate a parallelizable approximate solution model for the wave equation. For that, we used the temporal sweep known as the Waveform Relaxation method to guarantee the parallelization in space. However, this technique has limitations for this class of problems. Therefore, we combined the Subdomain method in a non-conventional way in time with the Multigrid method, intending to reduce computational time and improve the convergence factors. In this work, we present the mathematical analysis of the stability of the discretization model, which uses the Central Finite Difference method with weighting at each time step. As an application of the proposed method, in addition to a problem with a known analytical solution, we solved a wave propagation problem with reflection and phase inversion. © 2022, The Authors. All rights reserved.

关键词： Finite difference method

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Analysis of A 2D cathodic protection system for a buried slender structure with the dual boundary element method

Analysis of A 2D cathodic protection system for a buried sle...

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European Congress on Computational methods in Applied Sciences and engineering, ECCOMAS 2004

作者： De Lacerda, Luiz A. De Oliveira, Paulo A. Lázaris, José Da Silva, José M. LACTEC - Institute of Technology for Development Materials Department Centro Politécnico Cx. P. 19067 Curitiba Brazil PPGMNE - Post-Graduate Program on Numerical Methods in Engineering UFPR - Federal University of Paraná Curitiba Brazil

In this paper a boundary element formulation is developed and used for the analysis of cathodic protection systems of buried slender structures, such as foundations of electric transmission line towers. The foundation is basically a truss structure and the slenderness of the pieces composing it, brings numerical difficulties into the classical boundary element method. To avoid this problem, the dual boundary element method is implemented: combination of standard and hypersingular integral equations to form a mathematically well-posed system. A simple experiment is carried out where a galvanized metallic sheet is buried alongside two Copper electrodes (anodes) in parallel, with the objective of simulating a two-dimensional problem. The soil resistivity properties are measured along the depth and the non-linear polarization curves describing the relation between the current density and the electrochemical potential at the metallic sheet is investigated. Two situations are analysed: a) current is injected into the ground through both electrodes;b) current is injected through only one of the electrodes. In the numerical implementation, the soil is modeled as a homogeneous layered half-space with a different property for each layer. The validity of the proposed dual approach is discussed and compared with the measurements of potential at the ground surface.

关键词： Boundary element method

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