A general 3D BEM/FEM coupling applied to elastodynamic continua/frame structures interaction analysis

This paper presents a procedure for coupling general finite element models with three‐dimensional bodies modelled by the Boundary Element Method (BEM). Shells, plates and frames are modelled by the Finite Element Method (FEM) and coupled to the BEM domain directly or by means of rigid blocks. The coupling is used for the analysis of buildings connected to half‐space by means of rigid footings, piles or plates in bending and other problems where combinations of different types of sub‐domains are required, composite domains for instance. Several numerical examples are analysed to demonstrate the robustness and accuracy of the proposed scheme. Copyright © 1999 John Wiley & Sons, Ltd.     

Static boundary element analysis of piles submitted to horizontal and vertical loads

In this paper a numerical model for the analysis of the interaction between soil and piles, with or without rigid caps, subjected to horizontal and vertical loads is presented. The piles are modelled here by the Finite Element Method (FEM) and the soil by the Boundary Element Method (BEM). In this formulation the pile is represented as one finite element and the displacements and tractions along the shaft are approximated by polynomial functions. Some examples are presented and the results obtained with this formulation are very close to those obtained with other formulations and with experimental results.     

Symmetric coupling of multi‐zone curved Galerkin boundary elements with finite elements in elasticity

The coupling of Finite Element Method (FEM) with a Boundary Element Method (BEM) is a desirable result that exploits the advantages of each. This paper examines the efficient symmetric coupling of a Symmetric Galerkin Multi‐zone Curved Boundary Element Analysis method with a Finite Element Method for 2‐D elastic problems. Existing collocation based multi‐zone boundary element methods are not symmetric. Thus, when they are coupled with FEM, it is very difficult to achieve symmetry, increasing the computational work to solve the problem. This paper uses a fully Symmetric curved Multi‐zone Galerkin Boundary Element Approach that is coupled to an FEM in a completely symmetric fashion. The symmetry is achieved by symmetrically converting the boundary zones into equivalent ‘macro finite elements’, that are symmetric, so that symmetry in the coupling is retained. This computationally efficient and fast approach can be used to solve a wide range of problems, although only 2‐D elastic problems are shown. Three elasticity problems, including one from the FEM‐BEM literature that explore the efficacy of the approach are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

p-Adaptive C k generalized finite element method for arbitrary polygonal clouds

A p-Adaptive Generalized Finite Element Method (GFEM) based on a Partition of Unity (POU) of arbitrary smoothness degree is presented. The shape functions are built from the product of a Shepard POU and enrichment functions. Shepard functions have a smoothness degree directly related to the weighting functions adopted in their definition. Here the weighting functions are obtained from boolean R-functions which allow the construction of C ^k approximations, with k arbitrarily large, defined over a polygonal patch of elements, named cloud. The Element Residual Method is used to obtain error indicators by taking into account the typical nodal enrichment scheme of the method. This procedure is enhanced by using approximations with a high degree of smoothness as it eliminates the discontinuity of the stress field in the interior of each cloud. Adaptive analysis of plane elasticity problems are presented, and the performance of the technique is investigated.     

An anisotropic linear elastic boundary element formulation for masonry wall analysis

This work presents an application of a Boundary Element Method (BEM) formulation for anisotropic body analysis using isotropic fundamental solution. The anisotropy is considered by expressing a residual elastic tensor as the difference of the anisotropic and isotropic elastic tensors. Internal variables and cell discretization of the domain are considered. Masonry is a composite material consisting of bricks (masonry units), mortar and the bond between them and it is necessary to take account of anisotropy in this type of structure. The paper presents the formulation, the elastic tensor of the anisotropic medium properties and the algebraic procedure. Two examples are shown to validate the formulation and good agreement was obtained when comparing analytical and numerical results. Two further examples in which masonry walls were simulated, are used to demonstrate that the presented formulation shows close agreement between BE numerical results and different Finite Element (FE) models.     

半无限域中结构体辐射声场计算

本文从无限域Ｈｅｌｍｈｏｌｔｚ积分方程入手，用边界元法（ＢＥＭ）计算任意形状结构体辐射声场，对用ＢＥＭ计算声场出现的奇异积分，特征频率，边角点处向不连续，高频段计算误差较大等问题进行了有效，简便的处理，并用理论算例验证了这些方法有较精度。     

The green element method

This paper discusses an element-by-element approach of implementing the Boundary Element Method (BEM) which offers substantial savings in computing resource, enables handling of a wider range of problems including non-linear ones, and at the same time preserves the second-order accuracy associated with the method. Essentially, by this approach, herein called the Green Element Method (GEM), the singular integral theory of BEM is retained except that its implementation is carried out in a fashion similar to that of the Finite Element Method (FEM). Whereas the solution procedure of BEM couples the information of all nodes in the computational domain so that the global coefficient matrix is dense and full and as such difficult to invert, that of GEM, on the other hand, involves only nodes that share common elements so that the global coefficient matrix is sparse and banded and as such easy to invert. Thus, GEM has the advantage of being more computationally efficient than BEM. In addition, GEM makes the singular integral theory more flexible and versatile in the sense that GEM readily accommodates spatial variability of medium and flow parameters (e.g., flow in heterogeneous media), while other known numerical features of BEM—its second-order accuracy and ability to readily handle problems with singularities are retained by GEM. A number of schemes is incorporated into the basic Green element formulation and these schemes are examined with the goal of identifying optimum schemes of the formulation. These schemes include the use of linear and quadratic interpolation functions on triangular and rectangular elements. We found that linear elements offer acceptable accuracy and computational effort. Comparison of the modified fully implicit scheme against the generalized two-level scheme shows that the modified fully implicit scheme with weight of about 1·25 offers a marginally better approximation of the temporal derivative. The Newton–Raphson scheme is easily incoporated into GEM and provides excellent results for the time-dependent non-linear Boussinesq problem. Comparison of GEM with conventional BEM is done on various numerical examples, and it is observed that, for comparable accuracy, GEM uses less computing time. In fact, from the numerical simulations carried out, GEM uses between 15 and 45 per cent of the simulation time of BEM.     

The DRM‐MD integral equation method: an efficient approach for the numerical solution of domain dominant problems

This work presents a multi‐domain decomposition integral equation method for the numerical solution of domain dominant problems, for which it is known that the standard Boundary Element Method (BEM) is in disadvantage in comparison with classical domain schemes, such as Finite Difference (FDM) and Finite Element (FEM) methods. As in the recently developed Green Element Method (GEM), in the present approach the original domain is divided into several subdomains. In each of them the corresponding Green's integral representational formula is applied, and on the interfaces of the adjacent subregions the full matching conditions are imposed. In contrast with the GEM, where in each subregion the domain integrals are computed by the use of cell integration, here those integrals are transformed into surface integrals at the contour of each subregion via the Dual Reciprocity Method (DRM), using some of the most efficient radial basis functions known in the literature on mathematical interpolation. In the numerical examples presented in the paper, the contour elements are defined in terms of isoparametric linear elements, for which the analytical integrations of the kernels of the integral representation formula are known. As in the FEM and GEM the obtained global matrix system possesses a banded structure. However in contrast with these two methods (GEM and non‐Hermitian FEM), here one is able to solve the system for the complete internal nodal variables, i.e. the field variables and their derivatives, without any additional interpolation. Finally, some examples showing the accuracy, the efficiency, and the flexibility of the method for the solution of the linear and non‐linear convection–diffusion equation are presented. Copyright © 1999 John Wiley & Sons, Ltd.     

A numerical algorithm for simulation of the electric corona discharge in the triode system

A numerical algorithm is described to calculate the charge density, electric field and corona current distribution in the corona triode. The algorithm employs a hybrid technique based on the Boundary and Finite Element Methods (FEM). FEM is used to determine the electric field because of free space charge produced by the corona discharge. The Boundary Element Method (BEM) is applied for calculating the other component of electric filed as a result of the voltage applied to the electrodes. The Method of Characteristics (MOC) is used to update the space charge density distribution. The characteristic lines are traced backwards from points of the analysed domain to the corona wire. The current density, electric field and space charge density distributions can be controlled by changing the configuration of the system. Results of calculations in a few different cases show the influence of different parameters on the work of the corona triode.     

Optimum interpolation functions for boundary element method

In the Boundary Element Method (BEM) the density functions are approximated by interpolation functions which are chosen to satisfy appropriate continuity requirements. The error of approximation inside an element depends upon the location of the collocation points that are used in constructing the interpolation functions. The location of collocation points also affects the nodal values of the density function and, hence, the total error in the analysis if boundary conditions are satisfied in a collocation sense. In this paper, we minimize the error inside the element using the L₁ norm to obtain the optimum location of collocation points. Results show that irrespective of the continuity requirement at the element end, the location of collocation points computed by the algorithm presented in this paper results in an error that is less than the error corresponding to uniformly spaced collocation points. Results for optimum location of collocation points and the average error are presented for Lagrange polynomials up to order fifteen and for Hermite polynomials that ensure continuity up to the seventh order of derivative at the element end. The information of the optimum location of interpolation points for Lagrange and Hermite polynomials should be useful to other researchers in BEM who could incorporate it into their current programs without making significant changes that would be needed for incorporating the algorithm. The algorithm presented is independent of the BEM application in two-dimensions, provided that the density functions are approximated by polynomials and is applicable to direct and indirect formulations. Two numerical examples show the application of the algorithm to an elastostatic problem in which one boundary is represented by integrals of the Direct BEM while the other boundary by the Indirect BEM and a fracture mechanics problem by Direct method in which the crack is represented by displacement discontinuity density function.     

Viscoplastic plane‐strain analysis of infinite solids using elastic supports

A highly efficient novel Finite Element Boundary Element Method (FEBEM) is proposed for the elasto‐viscoplastic plane‐strain analysis of displacements and stresses in infinite solids. The proposed method takes advantage of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) to achieve higher efficiency and accuracy by using the concept of elastic supports to simulate the effects of unbounded solid mass surrounding the region of interest. The BEM is used to compute the stiffnesses of elastic supports and to estimate the location of the truncation boundary for the finite element model. As compared to the conventional coupled FEBEM, the proposed method has three main computational advantages. Firstly, the symmetrical and highly banded form of the standard finite element stiffness matrix is not disturbed. Secondly, the proposed technique may be implemented simply by using standard codes for elasto‐viscoplastic finite element analysis and elastic boundary element analysis. Thirdly, the yielded zone is approximately located in advance by using the BEM and hence, an unnecessarily large extent of the domain does not have to be discretized for the finite element modelling. The efficiency and accuracy of the proposed method are demonstrated by computing elastic and elasto‐plastic displacements and stresses around ‘deep’ underground openings in rock mass subject to hydrostatic and non‐hydrostatic in situ stresses. Results obtained by the proposed method are compared with ‘exact’ solutions and with those obtained by using a BEM and a coupled FEBEM. Copyright © 1999 John Wiley & Sons, Ltd.     

An immersed boundary element method for level‐set based topology optimization

In this paper, we propose a new BEM for level‐set based topology optimization. In the proposed BEM, the nodal coordinates of the boundary element are replaced with the nodal level‐set function and the nodal coordinates of the Eulerian mesh that maintains the level‐set function. Because this replacement causes the nodal coordinates of the boundary element to disappear, the boundary element mesh appears to be immersed in the Eulerian mesh. Therefore, we call the proposed BEM an immersed BEM. The relationship between the nodal coordinates of the boundary element and the nodal level‐set function of the Eulerian mesh is clearly represented, and therefore, the sensitivities with respect to the nodal level‐set function are strictly derived in the immersed BEM. Furthermore, the immersed BEM completely eliminates grayscale elements that are known to cause numerical difficulties in topology optimization. By using the immersed BEM, we construct a concrete topology optimization method for solving the minimum compliance problem. We provide some numerical examples and discuss the usefulness of the constructed optimization method on the basis of the obtained results. Copyright © 2012 John Wiley & Sons, Ltd.     

DOMAIN DECOMPOSITION WITH BEM AND FEM

The Finite Element Method and the Boundary Element Method are two different structure analysis methods with a totally different numerical character. Therefore, it makes no sense to couple these two methods pointwise at the interface. In contrast to a lot of coupling strategies in the past, in this paper a method is constructed where we have coupling of the two different methods in a weak form. As a result we can analyse the given structure with two different grids independent of each other. On this account, we see that the big advantage of the proposed method is in its ablity to couple BEM and FEM. The construction of a robust and reliable numerical algorithm depends on the adaptive control of symmetry and definiteness of the coupling matrix. Therefore, we use an iterative method for solving the boundary integral equation by expanding the Calderon projector in a Neumann series. Numerical results show the preciseness and efficiency of the method. © 1997 John Willey & Sons Ltd.     

Alternative time-marching schemes for elastodynamic analysis with the domain boundary element method formulation

This work presents alternative time-marching schemes for performing elastodynamic analysis by the Boundary Element Method. The use of the static fundamental solution and the maintenance of the domain integral associated to the accelerations characterize the formulation employed in this work. It is called D-BEM, D meaning domain. Time response is obtained by employing step-by-step time-marching procedures similar to those adopted in the Finite Element Method. Among all integration procedures, Houbolt scheme became the most popular used to march in time with D-BEM formulation, in spite of the presence of a high numerical damping. In order to improve the integration, this work presents alternative schemes that can be used to perform elastodynamic analysis by the BEM with a better damping control. In order to verify the accuracy of the proposed scheme, three examples are presented and discussed at the end of this work.     

A new inverse analysis approach for multi-region heat conduction BEM using complex-variable-differentiation method

This paper presents a new inverse analysis approach for identifying material properties and unknown geometries for multi-region problems using the Boundary Element Method (BEM). In this approach, the material properties and coordinates of an unknown region boundary are taken as the optimization variables, and the sensitivity coefficients are computed by the Complex-Variable-Differentiation Method (CVDM). Due to the use of CVDM, the sensitivity coefficients can be accurately determined in a way that is as simple to use as the Finite Difference Method (FDM) and an inverse analysis for a complex composite structure can be easily performed through a similar procedure to the direct computation. Although basic integral equations are presented for heat conduction problems, the application of the proposed algorithm to other problems, such as elastic problems, is straightforward. Two numerical examples are given to demonstrate the potential of the proposed approach.     

Use of constant,linear and quadratic boundary elements in 3D wave diffraction analysis

The performance of the Boundary Element Method (BEM) depends on the size of the elements and the interpolation function used. However, improvements in accuracy and efficiency obtained with both expansion and grid refinement increases demand on the computational effort. This paper evaluates the performance of constant, linear and quadratic elements in the analysis of the three-dimensional scattering caused by a cylindrical cavity buried in an infinite homogeneous elastic medium subjected to a point load. A circular cylindrical cavity for which analytical solutions are known is used in the simulation analysis. First, the dominant BEM errors are identified in the frequency domain and related to the natural vibration modes of the inclusion. Comparisons of BEM errors are then made for different types of boundary elements, maintaining similar computational costs. Finally, the accuracy of the BEM solution is evaluated when the nodal points are moved inside linear and quadratic discontinuous elements.     

拆除爆破研究中数值分析方法的比较与选择

概述了数值分析法的分类。介绍了平面杆系有限元法、离散元法、数值流形法和不连续变形分析等几种数值分析方法。简单地讨论了平面杆系有限元法的分析步骤以及在拆除爆破中适于解决的问题 ;同时叙述了流形分析中采用的有限覆盖技术。通过分析和比较这几种方法在拆除爆破研究中的应用 ,作者认为 ,当前应用传统的有限元法进行爆破理论研究或拆除爆破模拟存在一些困难 ;离散元法用于拆除爆破理论的研究是可行的 ;不连续变形分析法对于拆除爆破模拟研究是一种具有良好前景的数值方法     

拆除爆破研究中数值分析方法的比较与选择

概述了数值分析法的分类。介绍了平面杆系有限元法、离散元法、数值流形法和不连续变形分析等几种数值分析方法。简单地讨论了平面杆系有限元法的分析步骤以及在拆除爆破中适于解决的问题 ;同时叙述了流形分析中采用的有限覆盖技术。通过分析和比较这几种方法在拆除爆破研究中的应用 ,作者认为 ,当前应用传统的有限元法进行爆破理论研究或拆除爆破模拟存在一些困难 ;离散元法用于拆除爆破理论的研究是可行的 ;不连续变形分析法对于拆除爆破模拟研究是一种具有良好前景的数值方法     

Boundary element method homogenization of the periodic linear elastic fiber composites

The paper presented is devoted to the Boundary Element Method based homogenization of the periodic transversely isotropic linear elastic fiber-reinforced composites. The composite material under consideration has deterministically defined elastic properties while its components are perfectly bonded. To have a good comparison with the FEM-based computational techniques used previously, the additional Finite Element discretization is presented and compared numerically against BEM homogenization implementation on the example of engineering glass–epoxy composite. The homogenization method proposed has rather general characteristics and, as it is shown, can be easily extended on n-component composites. On the contrary, we can consider and homogenize the heterogeneous media with randomly defined material properties using Monte-Carlo simulation technique or second order perturbation second probabilistic moment approach.     

Probabilistic boundary element method

In this paper, attention is focused on obtaining probabilistic density function (PDF) random variables in complex structures by the use of a Boundary Element Method (BEM). The method of incorporating random variables into convention BEM and the post-process treatment of the PDF of random variables are presented in this paper. As a numerical example, a plane crack problem is analysed, in which the length of the crack and the applied loads are taken as random variables.