Simple boundary element method for three-dimensional magnetostatic problems

For numerical solution of three-dimensional magnetostatic problems the applications of finite-difference (FDM) and finite-element (FEM) methods require long calculating time and large storage capacity. In some cases these requirements can be reduced by the use of boundary element methods (BEM). A simple boundary element solution, and its application to the calculation of magnetization and stray fields of magnetic bodies, is described. The results are compared with experimental stray-field measurements performed using a vibrating pick-up loop magnetometer with high geometrical resolution [14].     

Improvement of stress singular element for crack problems in three dimensional boundary element method

In the paper, a new kind of stress singular element is introduced for crack problems. This kind of element is more simple and widely used than those presented before. In the paper, a cube with embedded circular crack and a first kind Benchmark problem are studied. The study shows that using quarter-point element and the stress singular element can obviously improve the accuracy. The influences of methods estimating stress intensity factor on accuracy are also studied.     

A new formulation of the magnetic vector potential method for three dimensional magnetostatic field problems

A novel formulation of the magnetic vector potential method for three dimensional magnetostatic field calculations is derived. Rigorously defining the interface and boundary conditions of the gauge of the vector potential, the new method gives a unique solution to the problem. The new field equation does not contain the gauge condition against the usual formulations[1], [2], [3], and takes the form of the diffusion equation. Computed results are favorably compared with the analytic solution of a test problem. This formulation is directly applicable to three dimensional eddy current problems.     

A new method for the solution of three dimensional magnetostatic fields, using a scalar potential

A new method "Equivalent Magnetized Region Solution" based on scalar potential for solving 3D magnetostatic fields is presented. The current distribution is transformed into a region of magnetic dipoles and only one scalar potential is used to calculate the field. A program "CMF3D" has been developed by finite element method, equipped with program "MESH" for subdividing field region into elements. Three examples by the solution are presented and compared with results from analytical method or experiment.     

A new boundary element method formulation for three dimensional problems in linear elasticity

Summary A new boundary element method (BEM) formulation for planar problems of linear elasticity has been proposed recently [6]. This formulation uses a kernel which has a weaker singularity relative to the corresponding kernel in the standard formulation. The most important advantage of the new formulation, relative to the standard one, is that it delivers stresses accurately at internal points that are extremely close to the boundary of a body. A corresponding BEM formulation for three dimensional problems of linear elasticity is presented in this paper. This formulation is derived through the use of Stokes' theorem and has kernels which are only 1/r singular (wherer is the distance between a source and a field point) for the displacement equation. The standard BEM formulation for three-dimensional elasticity problems has a kernel which is 1/r ² singular.With 2 Figures     

Some improvements of accuracy and efficiency in three dimensional direct boundary element method

Numerical analysis with the Boundary Element Method (BEM) has been used more and more in various engineering fields in recent years. In numerical techniques, however, there are some problems which have not been fully solved even now. The most essential one is the drop in the accuracy of results for internal points near the boundary of the structure, where the singularity of integrands in the boundary integral equation is too strong to be evaluated with the normal numerical method. For the boundary integral equation of stress, this problem became more serious, and the accuracy can be improved only partly, even though very refined boundary elements are used. In this paper, the boundary integral equation is newly formulated using a relative quantity of displacement. In this way, the singularity of boundary integrals is reduced by the order of 1/r, and the accuracy of solution is improved significantly. Furthermore, in order to integrate it more accurately, two kinds of numerical integral methods are newly developed. By using these methods, both displacement and stress can be obtained with excellent accuracy at almost any point in the structure without any numerical difficulty, although the discretization may be comparatively coarse. The generality and practicability of the present formulation and integral methods are confirmed through some examples of three dimensional elastic problems.     

A wideband fast multipole boundary element method for three dimensional acoustic shape sensitivity analysis based on direct differentiation method

This paper presents a wideband fast multipole boundary element approach for three dimensional acoustic shape sensitivity analysis. The Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem associated with the conventional boundary integral equation method in solving exterior acoustic wave problems. The sensitivity boundary integral equations are obtained by the direct differentiation method, and the concept of material derivative is used in the derivation. The iterative solver generalized minimal residual method (GMRES) and the wideband fast multipole method are employed to improve the overall computational efficiency. Several numerical examples are given to demonstrate the accuracy and efficiency of the present method.     

A boundary element method for two dimensional linear elastic fracture analysis

A boundary element method is developed for the analysis of fractures in two-dimensional solids. The solids are assumed to be linearly elastic and isotropic, and both bounded and unbounded domains are treated. The development of the boundary integral equations exploits (as usual) Somigliana's identity, but a special manipulation is carried out to regularize certain integrals associated with the crack line. The resulting integral equations consist of the conventional ordinary boundary terms and two additional terms which can be identified as a distribution of concentrated forces and a distribution of dislocations along each crack line. The strategy for establishing the integral equations is first outlined in terms of real variables, after which complex variable techniques are adopted for the detailed development. In the numerical implementation of the formulation, the ordinary boundary integrals are treated with standard boundary element techniques, while a novel numerical procedure is developed to treat the crack line integrals. The resulting numerical procedure is used to solve several sample problems for both embedded and surface-breaking cracks, and it is shown that the technique is both accurate and efficient. The utility of the method for simulating curvilinear crack propagation is also demonstrated.     

Calculation of domain integrals of two dimensional boundary element method

In this paper, a new method is applied to deal with domain integrals of boundary element method (BEM). In fact we focus to convert the domain integrals into boundary integrals for non-homogenous Laplace, Helmholtz and advection diffusion equations in two dimensional BEM. The transformation presented in this paper is based on divergence theorem. In addition, we prove the efficiency of method mathematically when the domain integrals are weakly singular. Numerical results are presented to verify the validity of this method for different geometries. Numerical implementation is done for the constant BEM, which can be implemented easily. To verify the new scheme, some test problems have been designed at end of the paper. The numerical results generally show that the new scheme has good accuracy with regards to other popular schemes.     

A multipole expansion-based boundary element method for axisymmetric potential problem

The multipole expansion is an approximation technique used to evaluate the potential field due to sources located in the far field. Based on the multipole expansion, we describe a new technique to calculate the far potential field due to ring sources which are encountered in the boundary element method (BEM) formulation of axisymmetric problems. As the sources in the near field are processed by the slower conventional BEM, it is important to maximize the amount of multipole calculations taking advantage of both interior and exterior multipole expansions. Numerical results are presented for an axisymmetric potential test problem with Neumann and Dirichlet boundary conditions. The complexity of the proposed method remains O(N²), which is equal to that of the conventional BEM. However, the proposed technique coupled with an iterative solver speeds up the solution procedure. The technique is significantly advantageous when medium and large numbers of elements are present in the domain.     

Three dimensional bi-component extrusion simulations by the boundary element method

In the present paper the boundary element method is implemented to solve fully three-dimensional bi-component flows of Newtonian fluids. The solution formulation follows that of Tran-Cong and Phan-Thien (1988a). In particular the coextrusion from square and triangular dies is studied with a view to establishing the dependence of the extrudate shape on viscosity ratio, interface shape, feed ratio and die shape.An ad hoc two-layer model is also investigated which may provide the means to predict viscoelastic extrudate shape without performing a full viscoelastic calculation.     

Boundary element method applied to three dimensional thermoelastic contact

In this paper we present a boundary element method to analyze and solve three dimensional frictionless thermoelastic contact problems. Although many problems in engineering can be solved with one-dimensional or two-dimensional models, those simplifications there are not possible in many others, such as the design of microelectronics packages. We calculate the stresses, movements, temperatures and thermal gradients on 3D solids. A thermal resistance at the contact zone depends on the local pressure is considered. The problem is solved by a double iterative method, so that in the final solution do not appear tensions in the contact zone or penetrations between the two solids. The solutions are compared with other works, where possible, to validate the method.     

A boundary face method for potential problems in three dimensions

This work presents a new implementation of the boundary node method (BNM) for numerical solution of Laplace's equation. By coupling the boundary integral equations and the moving least‐squares (MLS) approximation, the BNM is a boundary‐type meshless method. However, it still uses the standard elements for boundary integration and approximation of the geometry, thus loses the advantages of the meshless methods. In our implementation, here called the boundary face method, the boundary integration is performed on boundary faces, which are represented in parametric form exactly as the boundary representation data structure in solid modeling. The integrand quantities, such as the coordinates of Gauss integration points, Jacobian and out normal are calculated directly from the faces rather than from elements. In order to deal with thin structures, a mixed variable interpolation scheme of 1‐D MLS and Lagrange Polynomial for long and narrow faces. An adaptive integration scheme for nearly singular integrals has been developed. Numerical examples show that our implementation can provide much more accurate results than the BNM, and keep reasonable accuracy in some extreme cases, such as very irregular distribution of nodes and thin shells. Copyright © 2009 John Wiley & Sons, Ltd.     

Improving the stability of time domain dual boundary element method for three dimensional fracture problems: A time weighting approach

This paper deals with the stability of time domain dual boundary element method (DBEM). A time-weighted time domain DBEM is presented in this study and used for the first time in order to improve the stability of the standard time domain dual boundary element method. In this research a time weighting function with a prediction algorithm based on constant velocity algorithm has been utilized. The present approach was tested for three-dimensional fracture problems. The computer cost for the time of the presented approach is very close to the standard form. The results of numerical experiments carried out within this study indicate that the time weighting method, which is suggested for time domain DBEM, has more stability in comparison with the conventional method.     

Constrained boundary recovery for three dimensional Delaunay triangulations

A new constrained boundary recovery method for three dimensional Delaunay triangulations is presented. It successfully resolves the difficulties related to the minimal addition of Steiner points and their good placement. Applications to full mesh generation are discussed and numerical examples are provided to illustrate the effectiveness of guaranteed recovery procedure. Copyright © 2004 John Wiley & Sons, Ltd.     

The boundary node method for three‐dimensional problems in potential theory

The boundary node method (BNM) is developed in this paper for solving potential problems in three dimensions. The BNM represents a coupling between boundary integral equations (BIE) and moving least‐squares (MLS) interpolants. The main idea here is to retain the dimensionality advantage of the former and the meshless attribute of the later. This results in decoupling of the ‘mesh’ and the interpolation procedure for the field variables. A general BNM computer code for 3‐D potential problems has been developed. Several parameters involved in the BNM need to be chosen carefully for a successful implementation of the method. An in‐depth and systematic study has been carried out in this paper in order to better understand the effects of various parameters on the performance of the method. Numerical results for spheres and cubes, subjected to different types of boundary conditions, are extremely encouraging. Copyright © 2000 John Wiley & Sons, Ltd.     

Two general algorithms for nearly singular integrals in two dimensional anisotropic boundary element method

This paper presents an extension of the previously published sinh transformation and semi-analytical method for the evaluation of nearly singular integrals. The extension involves applying the two methods to two dimensional (2D) general anisotropic boundary element method (BEM). The new feature of the present method is that the distance from the calculation point to parabolic elements is expressed as $r^{2}=(\xi -\eta )^{2}g(\xi )+b^{2}$ , where $g(\xi )$ is a well-behaved function, $\eta \hbox { and }b$ stand for the position of the projection of the nearly singular point and the shortest distance from the calculation point to the integration element, respectively. As a result, the two methods can be employed in a straightforward fashion. The accuracy and the efficiency of the proposed methods are demonstrated with four benchmark test integrals that are commonly encountered in the application of anisotropic BEM. Comparisons between the two proposed methods are also presented in the paper.     

The fast multipole boundary element method for potential problems: A tutorial

The fast multipole method (FMM) has been regarded as one of the top 10 algorithms in scientific computing that were developed in the 20th century. Combined with the FMM, the boundary element method (BEM) can now solve large-scale problems with several million degrees of freedom on a desktop computer within hours. This opened up a wide range of applications for the BEM that has been hindered for many years by the lack of efficiencies in the solution process, although it has been regarded as superb in the modeling stage. However, understanding the fast multipole BEM is even more difficult as compared with the conventional BEM, because of the added complexities and different approaches in both FMM formulations and implementations. This paper is an introduction to the fast multipole BEM for potential problems, which is aimed to overcome this hurdle for people who are familiar with the conventional BEM and want to learn and adopt the fast multipole approach. The basic concept and main procedures in the FMM for solving boundary integral equations are described in detail using the 2D potential problem as an example. The structure of a fast multipole BEM program is presented and the source code is also made available that can help the development of fast multipole BEM codes for solving other problems. Numerical examples are presented to further demonstrate the efficiency, accuracy and potentials of the fast multipole BEM for solving large-scale problems.     

Finite element analysis of three dimensional crack growth by the use of a boundary element sub model

A new automated method to model non-planar three dimensional crack growth is proposed which combines the advantages of both the boundary element method and the finite element method. The proposed method links the two methods by a submodelling strategy in which the solution of a global finite element model containing an approximation of the crack is interpolated to a much smaller boundary element model containing a fine discretization of the real crack. The method is validated through several numerical comparisons and by comparison to crack growth measured in a test specimen for an engineering structure.     

Three dimensional flow in anisotropic zoned porous media using boundary element method

Coupling the adjacent zones for seepage analysis in porous media needs compatibility and equilibrium equations (equality of potential on coinciding nodes and conservation of flowing mass between zones, respectively). When stretched coordinate transformation is applied to the anisotropic zones, the Dirichlet boundary conditions remain unchanged, but the Neumann boundary condition should also be transformed. Similarly in a zoned problem, for the interface between zones, compatibility equations remain unchanged during the transformation while the equilibrium equations should be transformed. In this paper, transformed Neumann boundary conditions and equilibrium equations for the interface of neighbor anisotropic zones for seepage problems have been developed in three dimensions. A computer program for seepage analysis of zoned anisotropic media based on the Boundary Element Method is developed. The code is used to solve several examples with isotropic and anisotropic zones. Some examples are also solved by finite element method for verification. Illustrated results show the ability and accuracy of the mathematical and the numerical model for solving different types of applied three-dimensional seepage problems that arise in engineering practice.