An energy approach to space–time Galerkin BEM for wave propagation problems

Aimi, A.; Diligenti, M.; Guardasoni, C.; Mazzieri, Ilario; Panizzi, Stefano

doi:10.1002/nme.2660

Cited by 58 publications

(88 citation statements)

References 28 publications

(28 reference statements)

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“…The energetic weak formulation of problem (8) is defined similarly as in [29] and it can be deduced observing that, multiplying the PDE (1) by u t , integrating over [0, T ] × (R 2 \ Γ) and using integration by parts in space, one obtains that the energy E(u, T ) of the solution u at the final time of analysis T , defined by…”

Section: Model Problem and Its Weak Boundary Integral Formulationmentioning

confidence: 99%

“…Remark.The theoretical analysis of the quadratic form coming from the left-hand side of (11) was carried out for P = D = 0 in [29] where, under suitable hypothesis, coercivity was proved with some technicalities. This allowed us to deduce stability and convergence of the related Galerkin approximate solution, which in this paper, for the case of non-trivial damping coefficients, will be verified from a numerical point of view.…”

Section: Model Problem and Its Weak Boundary Integral Formulationmentioning

confidence: 99%

“…for P = D = 0, in [29]; in particular the presence of the Heaviside distribution H[c (t − τ ) − x − ξ ], which represents the wavefront, and of the square root c 2 (t − τ ) 2 − x − ξ 2 can cause a lot of numerical troubles that in [29] have been solved by suitable splitting of the outer integral over Γ and using quadrature schemes which regularize integrand functions with mild singularities for the second nested integral in space variable [36]. These singularities appear after the inner time integration.…”

Section: Multiple Integration Issuesmentioning

confidence: 99%

“…The time integral of the first term is instead performed numerically, using a modified Gaussian rule [36]. At last, the outer integrals over Γ are numerically treated by suitable quadrature schemes as in [29], to which the interested reader is referred. In particular, we use 32 quadrature nodes for the space cell and from 32 to 128 quadrature nodes for the time cell, depending on the accuracy we need to catch the highly oscillatory behavior of the approximate solutions, as it will be clear in the next Section.…”

Section: Multiple Integration Issuesmentioning

confidence: 99%

“…The mathematical background of time-dependent boundary integral equations is summarized by M. Costabel in [26]. For the numerical solution of the damped wave equation in 1D unbounded media, we have already considered in [17,18,27] the extension of the so-called energetic BEM, introduced for the undamped wave equation in several space dimensions [28][29][30]. The analysis carried out for 1D damped wave propagation problems allowed to fully understand the approximation technique for what concerns marching on time, avoiding space integration with BEM singular kernels and it was considered as a touchstone for the extension to higher space dimensions, which is done here for the 2D case.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Aimi

Diligenti

Guardasoni

2017

Communications in Applied and Industrial Mathematics

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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.

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Section: Model Problem and Its Weak Boundary Integral Formulationmentioning

confidence: 99%

Section: Model Problem and Its Weak Boundary Integral Formulationmentioning

confidence: 99%

Section: Multiple Integration Issuesmentioning

confidence: 99%

Section: Multiple Integration Issuesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Aimi

Diligenti

Guardasoni

2017

Communications in Applied and Industrial Mathematics

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Energetic boundary element method analysis of wave propagation in 2D multilayered media

Aimi

Gazzola

Guardasoni

2012

Math Methods in App Sciences

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The analysis of scalar wave propagation in 2D zonewise homogeneous media with vanishing initial and mixed boundary conditions is carried out. The problem is formulated in terms of time-dependent boundary integral equations, and then it is set in a weak form, based on a natural energy identity satisfied by the differential problem solution. Several numerical results have been obtained by means of the related energetic Galerkin boundary element method showing accuracy and stability of the method.A. AIMI, S. GAZZOLA AND C. GUARDASONI precise continuity and coerciveness properties [26,27]. Consequently, it can be discretized by unconditionally stable schemes with well-behaved stability constants even for large times.We remark that the idea of introducing a space-time weak formulation for the transient wave problem based on the energy identity is not new. In fact, in [16], it was exploited to get a satisfactory stability result for the acoustic wave equation with the aid of absorbing boundary condition. Unfortunately, the cases of Dirichlet and Neumann boundary conditions on a bounded time interval, as we have always considered, are much more difficult to be theoretically treated, and they have been analyzed in [26,27].The proposed procedure can be generalized to multi-domain wave propagation BIE problems as introduced in [28] within the limit of bi-domain cases. Here, the extension to n-domain case is proposed and further numerical simulations are presented and discussed: through comparisons with available (when possible) literature results [13,29], the stability and accuracy of the energetic approach are shown even in this framework. Plenty of examples, not easily findable in mathematical and even engineering papers on the subject, and numerical results included in this work illustrate the applicability and potentialities of this technique, even if theoretical analysis of stability and convergence for the multi-domain case is still to be done.H 1=2 .S/ :D fu 2 H 1=2 : i .U/ D uj @ i , i D 1, 2^ .U/ D u for some U 2 H 1 . /g, Figure 1. Example of bi-domain.

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A fast boundary element method using the Z‐transform and high‐frequency approximations for large‐scale three‐dimensional transient wave problems

Mavaleix-Marchessoux

Bonnet

Chaillat

et al. 2020

Numerical Meth Engineering

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Three-dimensional (3D) rapid transient acoustic problems are difficult to solve numerically when dealing with large geometries, because numerical methods based on geometry discretization (mesh), such as the boundary element method (BEM) or the finite element method (FEM), often require to solve a linear system (from the spacial discretization) for each time step. We propose a numerical method to efficiently deal with 3D rapid transient acoustic problems set in large exterior domains. Using the -transform and the convolution quadrature method, we first present a straightforward way to reframe the problem to the solving of a large amount (the number of time steps, M) of frequency-domain BEMs. Then, taking advantage of a well-designed high-frequency approximation, we drastically reduce the number of frequency-domain BEMs to be solved, with little loss of accuracy. The complexity of the resulting numerical procedure turns out to be O(1) in regard to the time discretization and O(N log N) for the spacial discretization, the latter being prescribed by the complexity of the used fast BEM solver. Examples of applications are proposed to illustrate the efficiency of the procedure in the case of fluid-structure interaction: the radiation of an acoustic wave into a fluid by a deformable structure with prescribed velocity and the scattering of an abrupt wave by simple and realistic geometries.

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An energy approach to space–time Galerkin BEM for wave propagation problems

Cited by 58 publications

References 28 publications

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

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

Energetic boundary element method analysis of wave propagation in 2D multilayered media

A fast boundary element method using the Z‐transform and high‐frequency approximations for large‐scale three‐dimensional transient wave problems

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