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

Abstract

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.

A time‐domain boundary element method for the 3D dissipative wave equation: Case of Neumann problems

“…The emphasis of this study is on the realisation of the 3D case, following the 2D case investigated by the previous researches. 8,11,13 Specifically, the present TDBEM is formulated in truly time domain by employing the time-dependent Green's function. [1][2][3][4] The corresponding OBIE in (4) is solved by the same methodology as the prior studies 5 on the 3D TDBEM for the classical/lossless wave equation, that is, the triangle-shaped piecewise-constant spatial basis, the piecewise-linear temporal basis, the collocation method, and the (not fast but conventional) MOT algorithm.…”

“…Another related work was completed by Aimi et al 13 regarding the 2D DWE expressed by c 2 △ u(x, t) − Pu(x, t) − 2D𝜕 t u(x, t) − 𝜕 2 t u(x, t) = 0 for R 2 ∈ Γ and t > 0 ( 2 ) with the inhomogeneous Neumann boundary condition and the null initial state, where Γ denotes a open arc or crack in IR 2 and P and D are the material damping and viscous coefficients, respectively. Equation ( 2) is essentially the same as the present one in (3a).…”

Boundary element method for contactless estimation of spatially varying internal heat transfer coefficient in circular pipes

“…The local steady state energy balance equation in the wall domain is formulated in cartesian coordinates as [2,3]:…”

“…Assuming a discretization of the boundary in M = Mint + Mext straight elements e1,…., eM (as in Fig. 2) and a class of finite dimensional functional subspaces VM such that dim(VM) = M [2,3]:…”

“…Concerning IHCPs, many solution techniques have been proposed under both the parameter estimation and function estimation approach [1] using both numerical and analytical models. Between the numerical methods, a decidedly appealing technique for solving problems where the boundary is of principal relevance, as for instance in wave scattering by bounded obstacles in unbounded domains [2], where high precision is demanded, is the Boundary Element Method (BEM) [3,4]. BEM shows some benefits over the more classical domain approaches, such as the dimensionality reduction since there is no need for discretizing the domain under consideration, the necessity of only the boundary data as inputs, the limited meshing effort, the smaller dimension of linear system matrices, the consequent lower computational cost, and the higher achievable accuracy.…”