Mathematical aspects of variational boundary integral equations for time dependent wave propagation

2017 ◽  
Vol 29 (1) ◽  
pp. 137-187 ◽  
Author(s):  
Patrick Joly ◽  
Jerónimo Rodríguez
Keyword(s):  
Wave Propagation ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Time Dependent

scholarly journals Runge–Kutta convolution coercivity and its use for time-dependent boundary integral equations

2018 ◽  
Vol 39 (3) ◽  
pp. 1134-1157
Author(s):  
Lehel Banjai ◽  
Christian Lubich
Keyword(s):  
Integral Equations ◽  
Boundary Integral Equations ◽  
Scattering Problem ◽  
Time Discretization ◽  
Boundary Integral ◽  
Time Dependent ◽  
Stability And Convergence ◽  
Linear Wave ◽  
Convolution Quadrature ◽  
Runge Kutta

Abstract A coercivity property of temporal convolution operators is an essential tool in the analysis of time-dependent boundary integral equations and their space and time discretizations. It is known that this coercivity property is inherited by convolution quadrature time discretization based on A-stable multistep methods, which are of order at most 2. Here we study the question as to which Runge–Kutta-based convolution quadrature methods inherit the convolution coercivity property. It is shown that this holds without any restriction for the third-order Radau IIA method, and on permitting a shift in the Laplace domain variable, this holds for all algebraically stable Runge–Kutta methods and hence for methods of arbitrary order. As an illustration the discrete convolution coercivity is used to analyse the stability and convergence properties of the time discretization of a nonlinear boundary integral equation that originates from a nonlinear scattering problem for the linear wave equation. Numerical experiments illustrate the error behaviour of the Runge–Kutta convolution quadrature time discretization.

Modeling of Wave Propagation in the Unsaturated Soils Using Boundary Element Method

2017 ◽  
Vol 743 ◽  
pp. 158-161
Author(s):  
Andrey Petrov ◽  
Sergey Aizikovich ◽  
Leonid A. Igumnov
Keyword(s):  
Integral Equation ◽  
Wave Propagation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equation ◽  
Boundary Integral Equations ◽  
Water Pressure ◽  
Boundary Integral ◽  
Element Method

Problems of wave propagation in poroelastic bodies and media are considered. The behavior of the poroelastic medium is described by Biot theory for partially saturated material. Mathematical model is written in term of five basic functions – elastic skeleton displacements, pore water pressure and pore air pressure. Boundary element method (BEM) is used with step method of numerical inversion of Laplace transform to obtain the solution. Research is based on direct boundary integral equation of three-dimensional isotropic linear theory of poroelasticity. Green’s matrices and, based on it, boundary integral equations are written for basic differential equations in partial derivatives. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. To approximate the boundary consider its decomposition to a set of quadrangular and triangular 8-node biquadratic elements, where triangular elements are treated as singular quadrangular. Every element is mapped to a reference one. Interpolation nodes for boundary unknowns are a subset of geometrical boundary-element grid nodes. Local approximation follows the Goldshteyn’s generalized displacement-stress matched model: generalized boundary displacements are approximated by bilinear elements whereas generalized tractions are approximated by constant. Integrals in discretized boundary integral equations are calculated using Gaussian quadrature in combination with singularity decreasing and eliminating algorithms.

scholarly journals Energetic BEM for the numerical analysis of 2D Dirichlet damped wave propagation exterior problems

2017 ◽  
Vol 8 (1) ◽  
pp. 103-127
Author(s):  
A. Aimi ◽  
M. Diligenti ◽  
C. Guardasoni
Keyword(s):  
Wave Propagation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Hyperbolic Partial Differential Equations ◽  
Weak Formulation ◽  
Exterior Problems ◽  
Element Method

Abstract Time-dependent problems modeled by hyperbolic partial differential equations can be reformulated in terms of boundary integral equations and solved via the boundary element method. In this context, the analysis of damping phenomena that occur in many physics and engineering problems is a novelty. Starting from a recently developed energetic space-time weak formulation for 1D damped wave propagation problems rewritten in terms of boundary integral equations, we develop here an extension of the so-called energetic boundary element method for the 2D case. Several numerical benchmarks, whose numerical results confirm accuracy and stability of the proposed technique, already proved for the numerical treatment of undamped wave propagation problems in several dimensions and for the 1D damped case, are illustrated and discussed.

scholarly journals A numerical study of energetic BEM-FEM applied to wave propagation in 2D multidomains

2014 ◽  
Vol 96 (110) ◽  
pp. 5-22 ◽  
Author(s):  
A. Aimi ◽  
L. Desiderio ◽  
M. Diligenti ◽  
C. Guardasoni
Keyword(s):  
Wave Propagation ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Numerical Study ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Weak Formulation ◽  
Multilayered Media ◽  
The Stability ◽  
Coupling Algorithm

Starting from a recently developed energetic space-time weak formulation of boundary integral equations related to wave propagation problems defined on single and multidomains, a coupling algorithm is presented, which allows a flexible use of finite and boundary element methods as local discretization techniques, in order to efficiently treat unbounded multilayered media. Partial differential equations associated to boundary integral equations will be weakly reformulated by the energetic approach and a particular emphasis will be given to theoretical and experimental analysis of the stability of the proposed method.

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