M. Diligenti

NUMERICAL INTEGRATION SCHEMES FOR THE BEM SOLUTION OF HYPERSINGULAR INTEGRAL EQUATIONS

In this paper we consider singular and hypersingular integral equations associated with 2D boundary value problems defined on domains whose boundaries have piecewise smooth parametric representations. In particular, given any (polynomial) local basis, we show how to compute efficiently all integrals required by the Galerkin method. The proposed numerical schemes require the user to specify only the local polynomial degrees; therefore they are quite suitable for the construction of p- and h-p versions of Galerkin BEM. Copyright

Hypersingular kernel integration in 3D Galerkin boundary element method

We consider Cauchy singular and Hypersingular boundary integral equations associated with 3D potential problems defined on polygonal domains, whose solutions are approximated with a Galerkin boundary element method, related to a given triangulation of the boundary. In particular, for constant and linear shape functions, the most frequently used basis functions, we give explicit results of the analytical inner integrations and suggest suitable quadrature schemes to evaluate the outer integrals required to form the Galerkin matrix elements. These numerical indications are given after an analysis of the singularities arising in the whole integration process, which is valid also for shape functions of higher degrees.

Integral evaluation in the BEM solution of (hyper)singular integral equations. 2D problems on polygonal domains

Abstract The formulation of certain classes of boundary value problems in terms of hypersingular integral equations is currently gaining increasing interest. In this paper we consider such type of equations on 2D polygonal domains, and assume we have to solve them by a collocation or a Galerkin BEM. In particular, given any (polynomial) local basis, we show how to compute efficiently, using a very low number of points, all integrals required by these methods. These integrals have kernels of the type log r , r −1 and r −2 . The quadrature rules we propose to compute the above-mentioned integrals require the user to specify only the local polynomial degrees; therefore, they are quite suitable for the construction of a p or h − p version of the BEM.

Isogemetric analysis and symmetric Galerkin BEM

Isogeometric approach applied to Boundary Element Methods is an emerging research area (see e.g. Simpson et?al. (2012) 33). In this context, the aim of the present contribution is that of investigating, from a numerical point of view, the Symmetric Galerkin Boundary Element Method (SGBEM) devoted to the solution of 2D boundary value problems for the Laplace equation, where the boundary and the unknowns on it are both represented by B-splines (de Boor (2001) 9). We mainly compare this approach, which we call IGA-SGBEM, with a curvilinear SGBEM (Aimi et?al. (1999) 2), which operates on any boundary given by explicit parametric representation and where the approximate solution is obtained using Lagrangian basis. Both techniques are further compared with a standard (conventional) SGBEM approach (Aimi et?al. (1997) 1), where the boundary of the assigned problem is approximated by linear elements and the numerical solution is expressed in terms of Lagrangian basis. Several examples will be presented and discussed, underlying benefits and drawbacks of all the above-mentioned approaches.

On the energetic Galerkin boundary element method applied to interior wave propagation problems

We consider two-dimensional interior wave propagation problems with vanishing initial and mixed boundary conditions, reformulated as a system of two boundary integral equations with retarded potential. These latter are then set in a weak form, based on a natural energy identity satisfied by the solution of the differential problem, and discretized by the related energetic Galerkin boundary element method. Numerical results are presented and discussed.

Implementation of a symmetric boundary element method in transient heat conduction with semi-analytical integrations

A time-convolutive variational hypersingular integral formulation of transient heat conduction over a 2-D homogeneous domain is considered. The adopted discretization leads to a linear equation system, whose coefficient matrix is symmetric, and is generated by double integrations in space and time. Assuming polynomial shape functions for the boundary unknowns, a set of compact formulae for the analytical time integrations is established. The spatial integrations are performed numerically using very efficient formulae just recently proposed. The competitiveness from the computational point of view of the symmetric boundary integral equation approach proposed herein is investigated on the basis of an original computer implementation. Copyright

Restriction matrices for numerically exploiting symmetry

In this paper we develop a technique for exploiting symmetry in the numerical treatment of boundary value problems (BVP) and eigenvalue problems which are invariant under a finite group

Efficient assembly based on B-spline tailored quadrature rules for the IgA-SGBEM

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Energetic BEM-FEM coupling for the numerical solution of the damped wave equation

of congruences of

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

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