Shape optimization of two-dimensional anisotropic structures using the boundary element method

We review the application of the discrete wave number method to problems of scattering of seismic waves formulated in terms of boundary integral equation and boundary element methods. The approach is based on the representation of the diffracting surfaces and interfaces of the medium by surface distributions of sources or by boundary source elements, the radiation from which is equivalent to the scattered wave field produced by the diffracting boundaries. The Green's functions are evaluated by the discrete wave number method, and the boundary conditions yield a linear system of equations. The inversion of this system allows the calculation of the full wave field in the medium. We investigate the accuracy of the method and we present applications to the simulation of surface seismic surveys, to the diffraction of elastic waves by fractures, to regional crustal wave propagation and to topographic scattering.

Data-driven analysis of parametrized acoustic systems in the frequency domain

On the conventional boundary integral equation formulation for piezoelectric solids with defects or of thin shapes

Elasto-plastic plate bending analysis by a boundary element method with initial plastic moments

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.

Discrete approach for shape optimization of 2D time-harmonic acoustic radiation problems solved by BEM using the fully-analytical adjoint method

A class of challenging problems encountered in geomechanics pertains to the prediction of the transient response of elastic systems of infinite or semi-infinite extent subjected to time dependent loading and boundary conditions. The nature and complexity of such problems restrict the range of available closed form solutions to the ones of linear problems of highly simplified geometries and boundary conditions. Among the numerical techniques, the most popular and well established one is the Finite Element Method (FEM) which can routinely handle complex geometries, medium inhomogeneities, and material non-linearities, and is well suited for both frequency and time domain analysis. However, the FEM by nature encompasses the major disadvantage of modeling infinite, and semi-infinite domains by finite size meshes violating, thus, the radiation damping condition. The problem is confronted, to some extend, by the development of hybrid techniques, and transmitting boundaries, for frequency domain analysis. Yet, time domain FEM solutions involving unbounded 3-D solid media, if available at all, may increase the size of the problem to computationally prohibitive levels. An alternative to the FEM is the Boundary Element Method (BEM), which is based on integral equation formulations, developed in the beginning of the century. The first 3-D time domain BEM formulation, associated with soil-structure interaction problems, has been reported by Karabalis and Beskos [1]. In this formulation the Stokes fundamental solutions are developed on the basis of a step impulse forcing function.

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

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 face based basis function solutions to thermal conduction problems for 3-D irregular shaped bodies using BEM

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).

The influence of the magnetic configuration on the behavior of magneto-electro-elastic bimorph beams is analyzed by using a boundary element approach. The problem is formulated by using the generalized displacements and generalized tractions. The boundary integral equation formulation is obtained by extending the reciprocity theorem to magneto-electro-elastic problems; it is numerically implemented by using the boundary element method multidomain technique to address problems involving nonhomogeneous configurations. Results under different magnetic configurations are compared highlighting the characteristic features of magnetopiezoelectric behavior particularly focusing on the link between interlaminar stress and magnetic induction.

The boundary element method is basically the matricization of a boundary integral equation which can be derived several ways. The two most popular methods are (1) use of a Green’s function in Green’s theorem, and (2) use of equivalent surface sources plus the uniqueness theorem. This paper discusses the second method in the context of electromagnetic fields, and gives the analogous method for scalar fields.

SummaryThis paper presents a numerical approach for the design sensitivity analysis of modal frequencies of elastic structures submerged in an unbounded heavy fluid domain. Because the feedback of sound pressure onto the fluid‐loaded structures has to be taken into account in this case, a fully coupled scheme which combines the structural finite element method (FEM) and the acoustic boundary element method (BEM) is employed in the numerical simulation. The resulting nonlinear eigenvalue problem created by the coupled finite element and boundary element (FE‐BE) method is converted into a generalized eigenvalue problem (GEP) of reduced dimension by using a contour integral method. The design sensitivity formulation for non‐repeated eigenvalues is derived based on an adjoint method which uses both the left and right eigenvectors, and a small GEP is formed to calculate the gradients of repeated eigenvalues. The Burton‐Miller formulation is employed to shift the fictitious eigenfrequencies and their derivatives, and the parameter setting which is beneficial to the filtering of fictitious eigenfrequencies is investigated. Numerical examples are given to verify the accuracy and effectiveness of the developed method.

An electromagnetism module is under development in the commercial software LS-DYNA in order to perform coupled mechanical/thermal/electromagnetic simulations. The main part of the module is a so-called eddy current solver. This module allows us to introduce source electrical currents into solid conductors, and to compute the associated magnetic fields, electric fields, induced currents, Lorentz forces, Joule heating and so forth. The Maxwell equations are solved using a Finite Element Method (FEM) for the solid conductors coupled with a Boundary Element Method (BEM) for the surrounding air (or insulators). Both the FEM and the BEM are based on discrete differential forms (Nedelec-like elements). The use of the BEM for the air allows to handle complex 3D geometries with multiply connected conductors, very small air gaps, and motion of the conductors, due, in particular, to the electromagnetic forces. The variable of the BEM is a so-called surface current allowing the connection of the model to external current sources by simple dirichlet constraints, as well as direct computations of the self and mutual inductances of the system. In order to handle the large and dense BEM matrices, a domain decomposition is performed and low rank approximations are done on the off-diagonal blocks of the resulting block matrices. The diagonal blocks are used as an efficient preconditioner when solving the BEM part of the system. The singularities arising in the computation of the self and neighbor matrix elements of the Galerkin BEM are taken into account using a method based on Duffy transforms. The BEM method will be presented as well as benchmarks and real life application examples.