Numerical integration schemes for space–time hypersingular integrals in energetic Galerkin BEM

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

doi:10.1007/s11075-010-9371-3

Cited by 9 publications

(6 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Further, we observe that the Heaviside function

H [Δ_{hk}^{i} MathClass-bin- r]

in Equation and the function

\sqrt{{(Δ_{hk}^{i})}^{2} MathClass-bin- r^{2}}

in the kernel

{scriptS}_{i} (r MathClass-punc, t_{h} MathClass-punc, t_{k})

, give rise to other different type of troubles, which have to be properly faced, exactly as described in for the case of monodomain wave propagation problems. Hence, the numerical treatment of Equation has been done through quadrature schemes widely used in the context of Galerkin BEM coming from elliptic problems , coupled with a suitable regularization technique , after a careful subdivision of the integration domain due to the presence of the Heaviside function.…”

Section: Galerkin Boundary Element Methods Discretizationmentioning

confidence: 99%

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

Aimi

Gazzola

Guardasoni

2012

Math Methods in App Sciences

View full text Add to dashboard Cite

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.

show abstract

“…Further, we observe that the Heaviside function

H [Δ_{hk}^{i} MathClass-bin- r]

in Equation and the function

\sqrt{{(Δ_{hk}^{i})}^{2} MathClass-bin- r^{2}}

in the kernel

{scriptS}_{i} (r MathClass-punc, t_{h} MathClass-punc, t_{k})

Section: Galerkin Boundary Element Methods Discretizationmentioning

confidence: 99%

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

Aimi

Gazzola

Guardasoni

2012

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…In the case of the undamped wave equation, the analysis of the space hypersingularity O(1/r 2 ) of the integrand function has been performed in detail in [20]; furthermore, the presence in the kernels of the Heaviside distribution H [c (t − τ ) − r ], which represents the wavefront, and of the square root c 2 (t − τ ) 2 − r 2 can cause numerical troubles that in [28] have been solved by suitable splitting of the outer integral over Γ and using quadrature schemes which regularize integrand functions with mild singularities. We stress the fact that all the above-recalled space singularities appear after analytical integration in time variables.…”

Section: Handling Hypersingular Kernel Space-time Integration In Matrmentioning

confidence: 99%

“…where r = r (s, z) andw i ,w j define one of the Lagrangian basis functions in local space variables over the elements e i and e j , respectively. Due to the hypersingularity O(1/r 2 ) for r → 0, the evaluation of double integrals of type (29) is troublesome when e i ≡ e j and when e i , e j are consecutive, and it has been performed similarly as in [28]. Anyway, for reader's convenience, in the following we describe the double integration over coincident elements of the boundary mesh, i.e., i = j, which is the the most difficult case since hypersingularity appears at any point of e i .…”

Section: Proposition 4 Using Piece-wise Linear Time Basis Functions mentioning

confidence: 99%

Energetic boundary element method for accurate solution of damped waves hard scattering problems

2021

Self Cite

View full text Add to dashboard Cite

The paper deals with the numerical solution of 2D wave propagation exterior problems including viscous and material damping coefficients and equipped by Neumann boundary condition, hence modeling the hard scattering of damped waves. The differential problem, which includes, besides diffusion, advection and reaction terms, is written as a space–time boundary integral equation (BIE) whose kernel is given by the hypersingular fundamental solution of the 2D damped waves operator. The resulting BIE is solved by a modified Energetic Boundary Element Method, where a suitable kernel treatment is introduced for the evaluation of the discretization linear system matrix entries represented by space–time quadruple integrals with hypersingular kernel in space variables. A wide variety of numerical results, obtained varying both damping coefficients and discretization parameters, is presented and shows accuracy and stability of the proposed technique, confirming what was theoretically proved for the simpler undamped case. Post-processing phase is also taken into account, giving the approximate solution of the exterior differential problem involving damped waves propagation around disconnected obstacles and bounded domains.

show abstract

“…The boundary integral method typically requires the construction of ad hoc quadrature formulas to handle weak, strong and hyper singularities that appear in the boundary integral equations (see for example [24]). But, when applied to this context, only weak singularities arise and no particular quadrature rules have been implemented for the evaluation of integrals in the system entries and in the postprocessing.…”

Section: Remarks About Numerical Integrationmentioning

confidence: 99%

Semi-Analytical method for the pricing of barrier options in case of time-dependent parameters (with Matlab^® codes)

Guardasoni

2018

Communications in Applied and Industrial Mathematics

View full text Add to dashboard Cite

A Semi-Analytical method for pricing of Barrier Options (SABO) is presented. The method is based on the foundations of Boundary Integral Methods which is recast here for the application to barrier option pricing in the Black-Scholes model with time-dependent interest rate, volatility and dividend yield. The validity of the numerical method is illustrated by several numerical examples and comparisons.

show abstract

Numerical integration schemes for space–time hypersingular integrals in energetic Galerkin BEM

Abstract: Here we consider exterior Neumann wave propagation problems reformulated in terms of space-time hypersingular boundary integral equations. We deal with quadrature schemes required, in the discretization phase, by the energetic Galerkin boundary element method.

Cited by 9 publications

References 12 publications

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

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

Energetic boundary element method for accurate solution of damped waves hard scattering problems

Semi-Analytical method for the pricing of barrier options in case of time-dependent parameters (with Matlab^® codes)

Contact Info

Product

Resources

About

Numerical integration schemes for space–time hypersingular integrals in energetic Galerkin BEM

Abstract: Here we consider exterior Neumann wave propagation problems reformulated in terms of space-time hypersingular boundary integral equations. We deal with quadrature schemes required, in the discretization phase, by the energetic Galerkin boundary element method.

Cited by 9 publications

References 12 publications

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

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

Energetic boundary element method for accurate solution of damped waves hard scattering problems

Semi-Analytical method for the pricing of barrier options in case of time-dependent parameters (with Matlab® codes)

Contact Info

Product

Resources

About

Semi-Analytical method for the pricing of barrier options in case of time-dependent parameters (with Matlab^® codes)