A State Space Boundary Element Method with Analytical Formulas for Nearly Singular Integrals

The boundary element method (BEM) is an efficient numerical method that approximates solutions of various boundary value problems. Despite its success little research has been performed on the conditioning of the linear systems that appear in the BEM. For a Laplace equation with Dirichlet boundary conditions a remarkable phenomenon is observed; the corresponding boundary integral equation (BIE) is singular for a certain critical size of the 2D domain. As a consequence the discrete counterpart of the BIE, the linear system, is singular too, or at least ill-conditioned. This is reflected by the condition number of the system matrix, which is infinitely large, or at least very large. When the condition number of the BEM-matrix is large, the linear system is difficult to solve and the solution of the system is very sensible to perturbations in the boundary data. For a Laplace equation with mixed boundary conditions a similar phenomenon is observed. The corresponding BEM-matrix consists of two blocks; one block originates from the BEM-matrix belonging to the Dirichlet problem, the other block originates from the BEM-matrix belonging to the Neumann problem. The composite matrix inherits the solvability problems from the Dirichlet block. In other words, for the Laplace equation with mixed boundary conditions there exists also a critical size of the 2D domain for which the BEM-matrix has an infinitely large condition number. Hence the size and shape of the domain affects the solvability of the BEM problem. The critical size of the domain for which the BIE becomes singular is related to the logarithmic capacity of the domain. The logarithmic capacity is a positive real number that is a function of the size and shape of the domain. If this logarithmic capacity is equal to one, the domain is a critical domain, and for this domain the BIE becomes singular. Thus by computing the logarithmic capacity we can a-priori determine whether the BIE will be singular or not. The logarithmic capacity depends linearly on the scale of the domain, and thus a domain with logarithmic capacity equal to one can always be found by rescaling the domain. Unfortunately the logarithmic capacity can only be computed analytically for a few simple domains; for more involved domains the logarithmic capacity can be estimated though. There are several possibilities to avoid large condition numbers, i.e. singular BIEs that appear at critical domains. The first option is to rescale the domain such that the logarithmic capacity is unequal to one. One can also add a supplementary condition to the BIE and the linear system. A drawback of this option is that the linear system has more equations than unknowns and different techniques are required to solve the system. A third option is to slightly modify the fundamental solution of the Laplace operator. This fundamental solution directly appears in the BIE and it can be shown that a suitable modification yields BIEs that do not become singular. The critical domains for which the BIEs become singular do not restrict to Laplace equations only. Also for BIEs applied to the biharmonic equation or the elastostatic equations and the Stokes equations such critical domains exist. As the last two equations are vectorial equations, also the corresponding BIE consists of two equations. As a consequence two critical domains can be found for which these BIEs become singular. To obtain nonsingular BIEs techniques similar to the Laplace case can be used. Unfortunately we cannot a-priori determine the sizes for which the BIEs becomes singular, and thus do not know to what size we should rescale the domain to obtain nonsingular BIEs. The existence of critical domains is in essence caused by the logarithmic term in the fundamental solutions for the elliptic boundary value problems in 2D. This logarithmic term does not depend linearly on the size of the domain. When a domain is scaled, i.e. multiplied by a scale factor, the argument of the logarithm is also multiplied by this scale factor, but the logarithm turns this into an additive term. Thus the logarithm transforms multiplication into addition. This affects the BIEs in such a way that critical domains can appear. The fundamental solutions of boundary value problems in 3D do not contain a logarithmic term. Hence scaling of the domain does not affect the fundamental solution, and consequently also the BIE is not affected. Hence we may safely rescale 3D domains without the risk to encounter a critical domain. An example in which a domain takes many different sizes and shapes is the blowing problem. In this problem a viscous fluid is blown to a desired shape. Typically the time is discretised into a set of discrete time steps, and at each step the shape of the fluid is computed by solving the Stokes equations. When attempting to simulate this problem in 2D, we meet a large number of 2D domains, and we risk that one of these domains is equal to or approaches a critical domain. In such a case the BEM will have difficulties with solving the Stokes equations for that particular domain. When simulating the blowing problem in 3D, no critical domains are encountered. It turns out that the BEM is a very efficient numerical method for this particular 3D problem with a moving boundary. As we are merely interested in the shape of the fluid, we only need to know the flow of its boundary. The BEM does exactly that; it does not compute the flow at the interior of the fluid. Furthermore it is rather easy to include other effects from the blowing problem in the model, such as gravity, surface tension and friction from the contact of the fluid with a wall. As only the boundary of the fluid is discretised, the system matrices that appear in the BEM are smaller than the system matrices that appear when solving the problem with a finite element method, for example. Though the BEM-matrices are dense, while the finite element matrices are sparse, the computational effort for the BEM is relatively low. In short, the BEMis a very appropriate numerical method when solving blowing problems.

Abstract: In the boundary element method, treatment of all the possible singular integrals is very important for the correctness and accuracy of the solutions. Generally, the directional derivative of a weakly singular integral is computed by an integral in the sense of Cauchy principle value if the directional derivative of the weakly singular integral kernel is strongly singular or in the sense of Hadamard finite part integral if the the directional derivative of the weakly singular integral kernel is hypersingular. We will try to discover how the strongly singular and hypersingular integrals are generated and propose a method to avoid the appearance of such kind of strongly singular and hypersingular integrals. Here, using some examples, we will show that with an ’exact derivation’ the directional derivative of a weakly singular integral will still be a weakly singular integral or a normal integral. That is none strongly or hypersingular integrals are generated in such a process. And therefore, Cauchy principle value and Hadamard finite part integral are not indispensable. The result will help us to form a new approach to overcome the so-called strongly and hypersingular integrals occurring in the boundary element method.

In the boundary element method (BEM) for stress analysis, it is well known that thermal loads give rise to an additional volume integral in the primary form of the boundary integral equation (BIE). This volume integral needs to be further transformed to surface ones in order to retain the characteristic of the BEM as a boundary solution technique. In this study of the BEM for 3D thermoelasticity in general anisotropy, the fundamental solutions are expressed as Fourier series with coefficients calculated using an explicit-form Green’s function. In the exact volume-to-surface integral transformation associated with the term for the thermal effects in the BIE, a new kernel function is constructed. All formulations are implemented in an existing BEM code for 3D elastostatic analysis. Some numerical examples are presented to demonstrate the veracity of the formulations and the implementation, where the numerical results are compared with those obtained using the finite element method (FEM).

P 211. Efficient method for computing the TMS-induced electric field in spherically symmetric head models, the triangle Construction

This paper presents a perturbation analysis of the excitation problem for an open domain. As a fundamental study, it addresses wave scattering by a crack in a two-dimensional domain to avoid fictitious eigenfrequency issues, which are encountered in a boundary integral equation. The system comprises a homogeneous, isotropic, and linearly elastic half plane containing interior and surface-breaking cracks. The modal amplitude is approximated using the perturbation method with the Taylor expansion around the complex-valued eigenfrequency for the open domain. The eigenfrequency problem for an open domain is solved using boundary element method and Sakurai-Sugiura method, which is one of the nonlinear eigensolvers. Several numerical results demonstrate that the proposed perturbation method approximately evaluates the modal amplitude due to real-valued frequency excitation.

This paper deals with seismic response evaluation of a finite multilayered geological structure with non-parallel boundaries rested on semi-infinite elastic half-plane subjected to transient seismic wave. The hybrid modelling approach is applied. It is based on the decomposition of the whole domain under consideration into two subregions: a finite-sized near-field zone and the open semi-infinite far-field region. The near-field range is presented by a finite geological profile, which is heterogeneous due to non-parallel layering. The far-field region is semi-infinite elastic isotropic homogeneous medium where the finite layered structure is located. An efficient 2D hybrid computational tool based on the boundary element method (BEM) for the far-field zone and the finite element method (FEM) for the finite layered domain is developed and verified in frequency domain. The BEM model of the seismically active far-field zone is inserted as a macro-finite element (MFE) in the FEM commercial program ANSYS. The hybrid model is of the FEM hosted type, where the boundary element domain (BED) is converted into macro-finite element, the nodal forces along the interface between two zones are expressed through the BED nodal tractions and the entire BEM formulation is converted to FEM-like approach. This hybrid model is applied for modelling of a real geological profile of the main wall of the “Liulyakovitsa” tailing pond, near Panagyurishte town, Bulgaria. Simulation results are given in the form of free surface synthetic seismograms stemming from the real seismic signals recorded at bedrock so as to establish the tailing pond’s seismic response.

Finite element method (FEM) is widely used in the modeling of electromagnetic systems and as a reliable method to verify analytical modeling of electromagnetic devices in design step. As the method is based on discretization of domains, the electromagnetic continuity conditions at the boundaries of elements should be confirmed. Electromagnetic continuity conditions in FEM modeling were investigated in this article. It was shown that the continuity of tangential component of magnetic field intensity is not guaranteed at the boundaries of elements. The inability of FEM to establish the continuity condition leads to some virtual surface current densities at the boundaries of adjacent elements which does not exist in reality. The existence of virtual surface currents is proved in an example. Then, with the help of a complicated example, it is shown that large number of elements and, consequently, the large number of boundaries between elements may cause great errors in FEM postprocess results, especially in calculating magnetic stored energy and electromagnetic forces acting on moving parts. It has been shown that it is very important to take into account the impact of these virtual current densities in energy conversion analysis by FEM.

A finite element–boundary element method for advection–diffusion problems with variable advective fields and infinite domains

An isogeometric Burton-Miller method for the transmission loss optimization with application to mufflers with internal extended tubes

The three-dimensional neutron diffusion equation has been solved using the boundary element method employing quadratic isoparametric boundary elements. Direct computation of the singular integral which arises in the present formulation can be avoided using standard mapping techniques, i.e., polar coordinates transformation is introduced after the quadrilateral element is subdivided into four triangles. The domain integral in the boundary integral equation corresponding to the neutron diffusion equation is converted into an equivalent boundary one when the integral is related to a uniform source or a slowing-down source. No boundary elements are required on the outer surface of an infinite reflector. The boundary elements for a symmetry plane can be also omitted when the method of images is adopted. Test calculations show that the present techniques provide accurate solutions for problems of irregular geometries.

Numerical calculation of regular and singular integrals in boundary integral equations using Clenshaw–Curtis quadrature rules

In engineering industries, composite laminates have been widely applied for various applications. This work presents an efficient analysis of the interlaminar stresses in three-dimensional thin layered anisotropic composites by the boundary element method (BEM). Due to the nearly singular integrals in the boundary integral equation, the conventional BEM approach cannot be applied to analyze the composite layers that are very thin. The present work employs the self-regularization scheme to analyze the interlaminar stresses in thin anisotropic composites. In the end, a few benchmark examples are presented to show the applicability of the present approach.

When a vibrating thin structure moves in a heavy flow field, the effect of sound waves on the structure cannot be ignored. This effect is especially pronounced when the relative velocity of the structure and the flow field increases. The hybrid numerical method of finite element method (FEM) and convective boundary element method (BEM) is proposed to solve acoustic-structural interaction problems in uniform relative motion between structure source and fluid. Besides, an acoustic-analogy Lorentz transform is proposed to handle the direct convective boundary integral equation (BIE), which makes the convective BIE has the same form as traditional BIE so that there are no more convective terms and extra singular integrals. The continuity condition at the fluid-structure interface is also proposed to generate the coupling equation. Compared with the traditional FEM, this hybrid method has the advantages of greatly reducing the number of finite elements outside the structure as well as automatically satisfying the radiation condition at far field. And compared with the direct convective BIE, this method reduces the computational complexity and time. It is potential to deliver a solver with higher accuracy and less memory consumption. Some numerical examples are designed to study the influence of the convective effect. Keywords: Acoustic-structural interaction, Finite element method, Boundary element method, FEM-BEM coupling

Boundary Element Analysis of Cracks in Shear Deformable Plates and Shells. Topics in Engineering, Vol 43

The limiting process that leads to the formulation of hypersingular boundary integral equations is first discussed in detail. It is shown that boundary integral equations with hypersingular kernels are perfectly meaningful even at non-smooth boundary points, and that special interpretations of the integrals involved are not necessary. Careful analysis of the limiting process has also strong relevance for the development of an appropriate numerical algorithm. In the second part, a new general method for the evaluation of hypersingular surface integrals in the boundary element method (BEM) is presented. The proposed method can be systematically applied in any BEM analysis, either with open or closed surfaces, and with curved boundary elements of any kind and order (of course, provided the density function meets necessary regularity requirements at each collocation point). The algorithm operates in the parameter plane of intrinsic coordinates and allows any hypersingular integral in the BEM to be directly transformed into a sum of a double and a one-dimensional regular integrals. Since all singular integrations are performed analytically, standard quadrature formulae can be used. For the first time, numerical results are presented for hypersingular integrals on curved (distorted) elements for three-dimensional problems.