scholarly journals Grid independent convergence using multilevel circulant preconditioning: Poisson’s equation

2021 ◽  
Author(s):  
Henrik Brandén
Keyword(s):  
Boundary Integral Equations ◽  
Linear Equations ◽  
Iterative Solution ◽  
Boundary Integral ◽  
Poisson’S Equation ◽  
Dirichlet Boundary Conditions ◽  
Poisson's Equation ◽  
Spatial Dimensions ◽  
Preconditioned Iterative ◽  
Circulant Preconditioning

AbstractWe consider the iterative solution of the discrete Poisson’s equation with Dirichlet boundary conditions. The discrete domain is embedded into an extended domain and the resulting system of linear equations is solved using a fixed point iteration combined with a multilevel circulant preconditioner. Our numerical results show that the rate of convergence is independent of the grid’s step sizes and of the number of spatial dimensions, despite the fact that the iteration operator is not bounded as the grid is refined. The embedding technique and the preconditioner is derived with inspiration from theory of boundary integral equations. The same theory is used to explain the behaviour of the preconditioned iterative method.

scholarly journals Application of Energetic BEM to 2D Elastodynamic Soft Scattering Problems

2019 ◽  
Vol 10 (1) ◽  
pp. 182-198
Author(s):  
A. Aimi ◽  
L. Desiderio ◽  
M. Diligenti ◽  
C. Guardasoni
Keyword(s):  
Wave Propagation ◽  
Dirichlet Boundary ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Dirichlet Boundary Conditions ◽  
Benchmark Problems ◽  
Weak Formulation ◽  
Scattering Problems ◽  
Propagation Analysis ◽  
First Time

Abstract Starting from a recently developed energetic space-time weak formulation of the Boundary Integral Equations related to scalar wave propagation problems, in this paper we focus for the first time on the 2D elastodynamic extension of the above wave propagation analysis. In particular, we consider elastodynamic scattering problems by open arcs, with vanishing initial and Dirichlet boundary conditions and we assess the efficiency and accuracy of the proposed method, on the basis of numerical results obtained for benchmark problems having available analytical solution.

Dynamic Crack Analysis in Layered Piezoelectric Composites under Time-Harmonic Loading

2013 ◽  
Vol 577-578 ◽  
pp. 449-452
Author(s):  
Michael Wünsche ◽  
Felipe García-Sánchez ◽  
Chuan Zeng Zhang ◽  
Andrés Sáez
Keyword(s):  
Frequency Domain ◽  
Boundary Integral Equations ◽  
Piezoelectric Materials ◽  
Iterative Solution ◽  
Boundary Integral ◽  
Dynamic Crack ◽  
Crack Face ◽  
Piezoelectric Composites ◽  
Crack Analysis ◽  
Time Harmonic

In this Paper, Time-Harmonic Dynamic Crack Analysis in Two-Dimensional (2D), Layered and Linear Piezoelectric Composites is Presented. A Frequency-Domain Symmetric Galerkin Boundary Element Method (SGBEM) is Developed for this Purpose. the Piecewise Homogeneous Sub-Layers of the Piezoelectric Composites are Modeled by the Multi-Domain BEM Formulation. the Frequency-Domain Dynamic Fundamental Solutions for Linear Piezoelectric Materials are Applied in the Present BEM. the Boundary Integral Equations are Solved Numerically by a Galerkin-Method Using Quadratic Elements. an Iterative Solution Algorithm is Implemented to Consider the Non-Linear Semi-Permeable Electrical Crack-Face Boundary Conditions. Numerical Examples will be Presented and Discussed to Show the Influences of the Location and Size of the Crack, the Material Combination of the Sub-Layers, the Piezoelectric Effect and the Time-Harmonic Dynamic Loading on the Dynamic Intensity Factors.

Hypersingular Boundary Integral Equations: Some Applications in Acoustic and Elastic Wave Scattering

1990 ◽  
Vol 57 (2) ◽  
pp. 404-414 ◽  
Author(s):  
G. Krishnasamy ◽  
L. W. Schmerr ◽  
T. J. Rudolphi ◽  
F. J. Rizzo
Keyword(s):  
Boundary Integral Equations ◽  
Acoustic Scattering ◽  
Boundary Integral ◽  
Three Dimensions ◽  
Finite Part ◽  
Important Special Case ◽  
Hypersingular Integrals ◽  
Hadamard Finite Part ◽  
Spatial Dimensions ◽  
Regular Line

The properties of hypersingular integrals, which arise when the gradient of conventional boundary integrals is taken, are discussed. Interpretation in terms of Hadamard finite-part integrals, even for integrals in three dimensions, is given, and this concept is compared with the Cauchy Principal Value, which, by itself, is insufficient to render meaning to the hypersingular integrals. It is shown that the finite-part integrals may be avoided, if desired, by conversion to regular line and surface integrals through a novel use of Stokes’ theorem. Motivation for this work is given in the context of scattering of time-harmonic waves by cracks. Static crack analysis of linear elastic fracture mechanics is included as an important special case in the zero-frequency limit. A numerical example is given for the problem of acoustic scattering by a rigid screen in three spatial dimensions.

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