An exact non reflecting boundary condition for 2D time-dependent wave equation problems

Falletta, Silvia; Monegato, Giovanni

doi:10.1016/j.wavemoti.2013.06.001

Cited by 31 publications

(56 citation statements)

References 38 publications

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“…If we assume M ∼ N , we obtain an optimal complexity (up to logarithmic terms) with respect to the total number of freedoms. We note in passing that boundary integral equations can be used to define transparent transmission conditions at artificial boundaries for wave propagation problems; the above mentioned CQ-BEM method has been proposed in [11] for an efficient discretization of such conditions. The new method we propose here also allows for a sparse realization of such exact non-local transmission conditions in log-linear complexity.…”

Section: Introductionmentioning

confidence: 99%

The panel-clustering method for the wave equation in two spatial dimensions

Falletta

Sauter

2016

Journal of Computational Physics

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We consider the numerical solution of the wave equation in a two-dimensional domain and start from a boundary integral formulation for its discretization. We employ the convolution quadrature (CQ) for the temporal and a Galerkin boundary element method (BEM) for the spatial discretization. Our main focus is the sparse approximation of the arising sequence of boundary integral operators by panel clustering. This requires the definition of an appropriate admissibility condition such that the arising kernel functions can be efficiently approximated on admissible blocks.The resulting method has log-linear complexity O N (N + M ) q 4+s , s ∈ {0, 1}, where N is the number of time points, M denotes the dimension of the boundary element space, and q = O (log N + log M ) is the order of the panelclustering expansion.Numerical experiments will illustrate the efficiency and accuracy of the proposed CQ-BEM method with panel clustering.

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Section: Introductionmentioning

confidence: 99%

The panel-clustering method for the wave equation in two spatial dimensions

Falletta

Sauter

2016

Journal of Computational Physics

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“…For a review, see for example [5], [6], [7]. All these papers, except for [12], Sections 5.5, 5.6, [8], [9], [2], deal with the construction of NRBC with the property of absorbing only outgoing waves, not waves that are either outgoing or incoming. Therefore, known sources must necessarily be included in the computational domain.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we aim at using the NRBC proposed in [2] as a transparent condition coupled with a fictitious domain method. In particular, we consider the scattering of a wave by an ob-stacle O ⊂ R 2 having a smooth boundary Γ.…”

Section: Introductionmentioning

confidence: 99%

“…By using the fictitious domain approach, we artificially extend the solution inside the obstacle O and we impose the boundary conditions on Γ in the weak form, by means of Lagrange multipliers. Then, we truncate the infinite external domain by an artificial boundary B where we impose the transparent boundary conditions proposed in [2]. We solve the original problem in the finite computational domain Ω that includes O and is bounded by the artificial boundary B.…”

Section: Introductionmentioning

confidence: 99%

“…Then we solve the original problem in the simpler domain that includes O and that is artificially bounded by a perfectly transparent boundary B, delimiting the region of interest. We prescribe on B a Non Reflecting Boundary Condition (NRBC) which is based on a space-time integral equation and that defines a relationship that the solution of the differential problem and its normal derivative must satisfy on B (see [2]). Further, we enforce the Dirichlet boundary condition on Γ in a weak sense, by means of Lagrange multipliers.…”

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A Fictitious Domain Approach for Wave Propagation Problems in Unbounded Domains

Falletta¹,

Monegato²

2015

Proceedings of the 5th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. We consider the scattering of waves by an obstacle O ⊂ R 2 having a sufficiently smooth boundary Γ. By using a fictitious domain approach, we artificially extend the solution inside the obstacle O. Then we solve the original problem in the simpler domain that includes O and that is artificially bounded by a perfectly transparent boundary B, delimiting the region of interest. We prescribe on B a Non Reflecting Boundary Condition (NRBC) which is based on a space-time integral equation and that defines a relationship that the solution of the differential problem and its normal derivative must satisfy on B (see [2]). Further, we enforce the Dirichlet boundary condition on Γ in a weak sense, by means of Lagrange multipliers. The NRBC is discretized on B by combining a special second order (in time) convolution quadrature and a standard collocation method in space. In the enlarged domain, this discretization is coupled with an unconditionally stable ODE time integrator and a FEM in space. The constraint on Γ is imposed by a matrix B h that represents a discrete trace operator.A particularly useful application of this approach is the scattering of a wave by rotating rigid bodies. In this case the method avoids the complexity of constructing at each time step a new finite element computational mesh and requires only the construction of the discrete trace operator B h . We present some results we have obtained for rotating (even multiple) scatterers and non trivial data. 959

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Time‐Dependent Problems with the Boundary Integral Equation Method

Costabel

2017

Encyclopedia of Computational Mechanics Second Edition

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Time‐dependent problems that are modeled by initial‐boundary value problems for parabolic or hyperbolic partial differential equations can be treated with the boundary integral equation (BIE) method. The ideal situation is when the right‐hand side in the partial differential equation and the initial conditions vanish, the data are given only on the boundary of the domain, the equation is linear with constant coefficients, and the domain does not depend on time. In this situation, the transformation of the problem to a BIE follows the same well‐known lines as for the case of stationary or time‐harmonic problems modeled by elliptic boundary value problems. The same main advantages of the reduction to the boundary prevail: reduction of the dimension by one, and reduction of an unbounded exterior domain to a bounded boundary. There are, however, specific difficulties due to the additional time dimension. Apart from the practical problems of increased complexity related to the higher dimension, there can appear new stability problems. In the stationary case, one often has unconditional stability for reasonable approximation methods, and this stability is closely related to variational formulations based on the ellipticity of the underlying boundary value problem. In the time‐dependent case, instabilities have been observed in practice, but due to the absence of ellipticity, the stability analysis is more difficult and fewer theoretical results are available. In this chapter, the mathematical principles governing the construction of BIE methods for time‐dependent problems are presented. We describe some of the main numerical algorithms that are used in practice and have been analyzed in the mathematical literature.

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An exact non reflecting boundary condition for 2D time-dependent wave equation problems

Cited by 31 publications

References 38 publications

The panel-clustering method for the wave equation in two spatial dimensions

The panel-clustering method for the wave equation in two spatial dimensions

A Fictitious Domain Approach for Wave Propagation Problems in Unbounded Domains

Time‐Dependent Problems with the Boundary Integral Equation Method

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