A stable 3D energetic Galerkin BEM approach for wave propagation interior problems

Aimi, A.; Diligenti, M.; Frangi, Attilio; Guardasoni, C.

doi:10.1016/j.enganabound.2012.06.003

Cited by 19 publications

(28 citation statements)

References 18 publications

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“…In a first set of analyses, where c 1 = c 2 = 1, the inner surface is subjected to a uniform traction

\overset{p}{true} MathClass-rel= H [t]

. The analytical solution is endowed with spherical symmetry and can be obtained following the same scheme as in , obtaining

\begin{matrix} leftalign \\ rightalign-odd u MathClass-open(r MathClass-close) & align-even = \frac{R_{C}^{2}}{r} (1 - e^{MathClass-open(r - R_{C} - t MathClass-close) ∕ R_{C}}), t > r - R_{C} & rightalign-label (58) \end{matrix}

\begin{matrix} leftalign \\ rightalign-odd p MathClass-open(r MathClass-close) & align-even = e^{MathClass-open(r - R_{C} - t MathClass-close) ∕ R_{C}} (\frac{R_{C}^{2}}{r^{2}} - \frac{R_{C}}{r}) - \frac{R_{C}^{2}}{r^{2}}, t > r - R_{C} . & rightalign-label (59) \end{matrix}

A spherical wave travels with unit speed from the inner surface towards infinity and traverses the interface at time t = R I − R C . The time interval of the analysis has been set to [0,40].…”

Section: Numerical Resultsmentioning

confidence: 99%

“…At every time instant, the unknowns are p 1 over the interface and u 2 in N 2 . Further, let us note that in the particular case 1 D 2 and c 1 D c 2 , the first equation in (20) can be regarded as a weak ABC to reduce an unbounded domain to a computational bounded one, avoiding spurious incoming reflections: this property will be shown in Section 7 by means of several numerical results.…”

Section: Energetic Weak Formulation For the Couplingmentioning

confidence: 98%

“…With the same choice for the second equations in (20), which involves all the problem unknowns, and remembering (15), the following relation appears 381…”

Section: Energetic Weak Formulation For the Couplingmentioning

confidence: 99%

“…Owing to this procedure, we can construct and store only the blocks E .0/ , , E .N t 1/ , with a considerable reduction of computational cost and memory requirement. Now, referring to the left-hand sides of the weak problem (20), after an analytic integration in the time variable, we have the following results.…”

mentioning

confidence: 96%

“…0, which are typical of weakly singular kernels related to 3D elliptic problems, and also the presence of the Heaviside function, which analytically represents the wave front propagation. We refer the interested reader to [20,21] for the description of the accurate and fast evaluation of such types of double integrals on the interface . Further, it is simple to prove that, because of the chosen time basis functions, expressions (31) and (32) vanish for`such that t` d iam [ For what concerns matrix elements coming from < P u 2j , q 1 >, one has…”

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confidence: 99%

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Energetic BEM–FEM coupling for wave propagation in 3D multidomains

Aimi

Diligenti

Frangi

et al. 2013

Numerical Meth Engineering

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SUMMARYStarting from a recently developed energetic space–time weak formulation of boundary integral equations for wave propagation problems, a coupling algorithm is presented, which allows a flexible use of FEMs and BEMs as local discretization techniques. Emphasis is given to theoretical and experimental analysis of the stability of the proposed method. Several numerical results on model problems are presented and discussed, showing that both bounded and unbounded three‐dimensional domains can be efficiently addressed. Copyright © 2013 John Wiley & Sons, Ltd.

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“…In a first set of analyses, where c 1 = c 2 = 1, the inner surface is subjected to a uniform traction

\overset{p}{true} MathClass-rel= H [t]

. The analytical solution is endowed with spherical symmetry and can be obtained following the same scheme as in , obtaining

\begin{matrix} leftalign \\ rightalign-odd u MathClass-open(r MathClass-close) & align-even = \frac{R_{C}^{2}}{r} (1 - e^{MathClass-open(r - R_{C} - t MathClass-close) ∕ R_{C}}), t > r - R_{C} & rightalign-label (58) \end{matrix}

\begin{matrix} leftalign \\ rightalign-odd p MathClass-open(r MathClass-close) & align-even = e^{MathClass-open(r - R_{C} - t MathClass-close) ∕ R_{C}} (\frac{R_{C}^{2}}{r^{2}} - \frac{R_{C}}{r}) - \frac{R_{C}^{2}}{r^{2}}, t > r - R_{C} . & rightalign-label (59) \end{matrix}

A spherical wave travels with unit speed from the inner surface towards infinity and traverses the interface at time t = R I − R C . The time interval of the analysis has been set to [0,40].…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Energetic Weak Formulation For the Couplingmentioning

confidence: 98%

“…With the same choice for the second equations in (20), which involves all the problem unknowns, and remembering (15), the following relation appears 381…”

Section: Energetic Weak Formulation For the Couplingmentioning

confidence: 99%

mentioning

confidence: 96%

mentioning

confidence: 99%

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Energetic BEM–FEM coupling for wave propagation in 3D multidomains

Aimi

Diligenti

Frangi

et al. 2013

Numerical Meth Engineering

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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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