A Variational Formulation of The Boundary Element Method in Transient Poroelasticity

In linear poroelasticity so far only collocation boundary element methods have been available. However, in some applications, e.g., when coupling with finite elements is desired, a symmetric formulation is preferable. Choosing a Galerkin approach which involves the second boundary integral equation, such a formulation is possible. Here, a previously presented integration by part technique for the regularization of the first boundary integral equation is extended to the second boundary integral equation as well. While the weakly singular representation of the double layer operator has been presented before, the emphasis lies here on the so called hyper‐singular boundary integral operator. Due to the regularization, this operator can be evaluated numerically and, hence, be used within a numerical scheme for the first time. Different numerical studies will be presented to show the behavior of the established symmetric Galerkin boundary element method, also comparing it with collocation boundary element methods. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The boundary element method for elasticity problems with concentrated loads based on displacement singular elements

We present a divide-and-conquer algorithm to improve the three-dimensional (3-D) boundary element method (BEM) for capacitance extraction. We divide large interconnect structures into small sections, set new boundary conditions using the border for each section, solve each section, and then combine the results to derive the capacitance. The target application is critical nets, clock trees, or packages where 3-D accuracy is required. Our algorithm is a significant improvement over the traditional BEMs and their enhancements, such as the "window" method, where conductors far away are dropped, and the "shield" method where conductors hidden behind other conductors are dropped. Experimental results show that our algorithm is a magnitude faster than the traditional BEM and the window+shield method, for medium to large structures. The error of the capacitance computed by the new algorithm is within 2% for self capacitance and 7% for coupling capacitance, compared with the results obtained by solving the entire system using BEM. Furthermore, our algorithms gives accurate distributed RC, where none of the previous 3-D BEM algorithms and their enhancements can.

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

To solve boundary integral equations for potential problems using collocation Boundary Element Method (BEM) on smooth curved 3D geometries, an analytical singularity extraction technique is employed. By adopting the isoparametric approach, curved geometries that are represented by mapped rectangles or triangles from the parametric domain are considered. The singularity extraction on the governing singular integrals can be performed either as an operation of subtraction or division, each having some advantage.A particular series expansion of a singular kernel about a source point is investigated. The series in the parametric coordinates consists of functions of a type Rpxqyr, where R is a square root of a quadratic bivariate homogeneous polynomial with coefficients corresponding to the first fundamental form of a smooth surface, and p,q,r are integers, satisfying p≤−1 and q,r≥0. By extracting more terms from the series expansion of the singular kernel, the smoothness of the regularized kernel at the source point can be increased. Analytical formulae for integrals of such terms are obtained from antiderivatives of Rpxqyr, using recurrence formulae, and by evaluating them at the edges of rectangular or triangular parametric domains.Numerical tests demonstrate that the singularity extraction technique can be a useful prerequisite for a numerical quadrature scheme to obtain high order of convergence for the governing singular integrals in 3D collocation BEM with respect to the number of quadrature nodes or with respect to size of the discretization mesh cells.

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.

The boundary element–Rayleigh integral method (BERIM) is developed for the solution of acoustic problems consisting of a cavity with a single opening. The method is a hybrid of the interior boundary element method (BEM) and the Rayleigh integral method for the solution of the Helmholtz equation. FORTRAN codes for expressing the method for axisymmetric problems and general three-dimensional problems are developed. Both methods are applied to the problem of simulating the acoustic field produced by horn loudspeakers. The results are compared with measured data and the traditional exterior BEM. It is shown that the method can have significant computational advantages over the traditional BEM for problems consisting of an open cavity.

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.

Finite Element Method (FEM1) is pervasively used in most of 3D product design analysis, in which Computer Aided Design (CAD) models need to be converted in to mesh models first and then enriched with some material features and boundary conditions data, etc. The interaction between CAD models and FEM models is intensive. Boundary Element Method (BEM) has been expected to be advantageous in large-scale problems in recent years owing to its reduction of the dimensionality and its reduced complexity in mesh generation. However, the BEM application has so far been limited to relatively small problems due to the memory and computational complexity for matrix buildup are O(N2). The fast multipole BEM (FMBEM) combined with BEM and fast multipole method (FMM) can overcome the defect of the traditional BEM, and provides an effective method to solve the large-scale problem. Combining GPU parallel computing with FMBEM can further improve its efficiency significantly. Based on the three-dimensional elastic mechanics problems, the parallelisms of the multipole moment (ME), multipole moment to multipole moment (M2M) translation, multipole moment to local expansion (M2L) translation, local expansion to local expansion (L2L) translation and near-field direct calculation were analyzed respectively according to the characteristics of the FMM, and the parallel strategies under CUDA were presented in this paper. Three main major parts are included herein: (1) FMBEM theory in 3D elastostatics, (2) the parallel FMBEM algorithm using CUDA, and (3) comparison the GPU parallel FMBEM with BEM, FEM and FMBEM respectively by engineering examples. Numerical example results show the 3D elastostatics GPU FMBEM not only can speed up the boundary element calculation process, but also save memory which can be effective to solve the large-scale engineering problems.

A Monte Carlo solution method for linear elasticity

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

The Modified Local Green’s Function Method (MLGFM) is an integral method hybrid of the Finite Element Method (FEM) and the Boundary Element Method (BEM). The method uses the FEM to create discrete projections of the Green’s functions that will be used as fundamental solutions in BEM formulation. The MLGFM has the advantage of presenting high convergence for the displacements in the domain, inherited from the FEM, and for the normal stress on the boundary, inherited from the BEM. Despite these advantages, the accuracy of the recovered stresses in the domain has not been studied in previous works. As the MLGFM is a hybrid of the FEM and BEM, techniques used in both methods will be explored in this paper to study the advantages and disadvantages of each one. The techniques that will be used here are the Least Squares procedure, the Zienkiewicz and Zhu (ZZ) recovery, both widely used in FEM, and the integral form derived from the fundamental solution used in the BEM. The techniques will be analyzed in terms of errors and computational cost of each one.

The consistent boundary element method for potential and elasticity: Part II — machine-precision numerical evaluations for 2D problems

Evaluation of surface electric fields, which is well described by Poisson equation, is of great importance on many occasions in power system. This paper proposes a boundary element method for long slender conductors. Examples show the method is convergent and shows great potential for the evaluation of electric field generated by slender conductors.