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.     

A symmetric Galerkin multi-zone boundary element formulation

The recent development of the symmetric Galerkin approach to boundary element analysis (BEA) has been demonstrated to be superior to the collocation method for medium to large problems. This fact has been shown in both heat conduction and elasticity. Accounts of collocation multi-zone analysis techniques have also been prevalent in the literature, dealing with multiple boundary integral relations associated with portions of overall objects. This technique results in overall system matrices with a blocked, sparse, but unsymmetric character. It has been shown that multi-zone techniques can produce smaller solution times than a single zone analysis for large problems. These techniques are useful for multi-material problems as well. This paper presents an approach for combining the benefits of both techniques resulting in a Galerkin multi-zone method, that is overall unsymmetric but contains a significant amount of block symmetry. A condensation technique in the multi-zone solver is shown to exploit the symmetry of the Galerkin formulation as well as the blocked sparsity of the multi-zone technique. This method is compared to collocation multi-zone on two elasticity problems from the literature. It is concluded that an appropriate implementation of the symmetric Galerkin multi-zone BEA indeed has the potential to be superior to the collocation based multi-zone BEA, for medium to large-scale elasticity problems. © 1997 John Wiley & Sons, Ltd.     

A coupling of multi‐zone curved Galerkin BEM with finite elements for independently modelled sub‐domains with non‐matching nodes in elasticity

When the different parts of a structure are modelled independently by BEM or FEM methods, it is sometimes necessary to put the parts together without remeshing of the nodes along the part interfaces. Frequently the nodes do not match along the interface. In this work, the symmetric Galerkin multi‐zone curved boundary element is a fully symmetric formulation and is the method used for the boundary element part. For BEM–FEM coupling it is then necessary to interpolate the tractions in‐between the non‐matching nodes for the FEM part. Finally, the coupling is achieved by transforming the finite element domains to equivalent boundary element domains in a block symmetric formulation. This system is then coupled with a boundary element domain with non‐matching nodes in‐between. Copyright © 2004 John Wiley & Sons, Ltd.     

Convolution quadrature method‐based symmetric Galerkin boundary element method for 3‐d elastodynamics

The boundary integral equations in 3‐d elastodynamics contain convolution integrals with respect to the time. They can be performed analytically or with the convolution quadrature method. The latter time‐stepping procedure's benefit is the usage of the Laplace‐transformed fundamental solution. Therefore, it is possible to apply this method also to problems where analytical time‐dependent fundamental solutions might not be known. To obtain a symmetric formulation, the second boundary integral equation has to be used which, unfortunately, requires special care in the numerical implementation since it involves hypersingular kernel functions. Therefore, a regularization for closed surfaces of the Laplace‐transformed elastodynamic kernel functions is presented which transforms the bilinear form of the hypersingular integral operator to a weakly singular one. Supplementarily, a weakly singular formulation of the Laplace‐transformed elastodynamic double layer potential is presented. This results in a time domain boundary element formulation involving at least only weakly singular integral kernels. Finally, numerical studies validate this approach with respect to different spatial and time discretizations. Further, a comparison with the wider used collocation method is presented. It is shown numerically that the presented formulation exhibits a good convergence rate and a more stable behavior compared with collocation methods. Copyright © 2008 John Wiley & Sons, Ltd.     

A general regularization of the hypersingular integrals in the symmetric Galerkin boundary element method

The symmetric Galerkin boundary element method is used to solve boundary value problems by keeping the symmetric nature of the matrix obtained after discretization. The matrix elements are obtained from a double integral involving the double derivative of Green's operator, which is highly singular. The paper presents a regularization of the hypersingular integrals which depend only on the properties of Green's tensor. The method is presented in the case of Laplace's operator, with an example of application. The case of elasticity is finally addressed theoretically, showing an easy extension to any case of anisotropy. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical implementation of the symmetric Galerkin boundary element method in 2D elastodynamics

The symmetric Galerkin boundary element method (SGBEM) employs both the displacement integral equation and the traction integral equation which lead to a symmetric system of equations. A two‐dimensional SGBEM is implemented in this paper, with emphasis on the special treatments of singular integrals. The integrals in the time domain are carried out by an analytical method. In order to evaluate the strong singular double integrals and the hypersingular double integrals in the space domain which are associated with the fundamental solutions G ^pu and G ^pp, artificial body forces are introduced which can be used to indirectly derive the singular terms. Thus, those singular integrals which behave like 1/r and 1/r² are all avoided in the proposed SGEBM implementation. An artificial body force scheme is proposed to evaluate the body force term effectively. Two numerical examples are presented to assess the accuracy of the numerical implementation, and show similar accuracy when compared with the FEM and the analytical solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

A Galerkin symmetric and direct BIE method for Kirchhoff elastic plates: formulation and implementation

A variational Boundary Element formulation is proposed for the solution of the elastic Kirchhoff plate bending problem. The stationarity conditions of an augmented potential energy functional are first discussed. After addressing the topic of the choice of the test functions, a regularization process based on integrations by parts is developed, which allows to express the formulation in terms of double integrals, the inner being at most weakly singular and the outer regular. Standard integration procedures may then be applied for their numerical evaluation in the presence of both straight and curved boundaries. The normal slope and the vertical displacement must be C⁰ and C¹ continuous, respectively. Numerical examples show, through comparisons with analytical solutions, that a high accuracy is achieved. © 1998 John Wiley & Sons, Ltd.     

On hypersingular surface integrals in the symmetric Galerkin boundary element method: application to heat conduction in exponentially graded materials

A symmetric Galerkin formulation and implementation for heat conduction in a three‐dimensional functionally graded material is presented. The Green's function of the graded problem, in which the thermal conductivity varies exponentially in one co‐ordinate, is used to develop a boundary‐only formulation without any domain discretization. The main task is the evaluation of hypersingular and singular integrals, which is carried out using a direct ‘limit to the boundary’ approach. However, due to complexity of the Green's function for graded materials, the usual direct limit procedures have to be modified, incorporating Taylor expansions to obtain expressions that can be integrated analytically. Several test examples are provided to verify the numerical implementation. The results of test calculations are in good agreement with exact solutions and corresponding finite element method simulations. Copyright © 2004 John Wiley & Sons, Ltd.     

An efficient indirect boundary element technique for multi‐frequency acoustic analysis

An efficient indirect boundary element solution procedure for the analysis of multi‐frequency acoustic problems is developed by incorporating techniques that improve the efficiency of the integration and matrix solution phases of the computing process. The integration phase is made efficient by computing the system matrices at few predetermined key frequencies only and then evaluating the matrices at other intermediate frequencies by quadratic interpolation. The matrix solution process is made efficient by iterating the solutions using the factored form of the key frequency matrices. The effectiveness of the present development is confirmed by solving a number of example problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Interface dynamic stiffness matrix approach for three‐dimensional transient multi‐region boundary element analysis

A novel substructuring method is developed for the coupling of boundary element and finite element subdomains in order to model three‐dimensional multi‐region elastodynamic problems in the time domain. The proposed procedure is based on the interface stiffness matrix approach for static multi‐region problems using variational principles together with the concept of Duhamel integrals. Unit impulses are applied at the boundary of each region in order to evaluate the impulse response matrices of the Duhamel (convolution) integrals. Although the method is not restricted to a special discretization technique, the regions are discretized using the boundary element method combined with the convolution quadrature method. This results in a time‐domain methodology with the advantages of performing computations in the Laplace domain, which produces very accurate and stable results as verified on test examples. In addition, the assembly of the boundary element regions and the coupling to finite elements are greatly simplified and more efficient. Finally, practical applications in the area of soil–structure interaction and tunneling problems are shown. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis of symmetric domains in advanced applications with the boundary element method

The strategy in the boundary element method for the analysis of symmetric domains that does not require the modelling of contours corresponding to the axes of symmetry is investigated for a number of advanced applications. These applications include: treatment of domain loadings, two-dimensional time domain transient elastodynamics, and the analysis of probabilistic problems in elastostatics with a random geometric configuration. Both symmetric as well as arbitrary loadings acting on the symmetric objects are considered. A number of case studies are presented to provide comparisons of computer memory and CPU time requirements for the analyses of the entire object versus the analyses of only the symmetric portion of the object. The numerical data presented clearly demonstrates the advantages of developing efficient symmetric formulations for advanced applications.     

A quadtree data structure for the coupled finite‐element/element‐free Galerkin method

In the present paper, we derive an efficient data structure for the organization of the nodes in the coupled finite element/element‐free Galerkin method. With respect to its implementation, we compare various approaches of recursive spatial discretizations that facilitate most flexible handling of the nodes. The goal of the paper is to refine the implementation issues of the data structure which is fundamental to the element‐free Galerkin method and thus to speed‐up this otherwise computationally rather expensive meshfree method. Copyright © 2001 John Wiley & Sons, Ltd.     

Improved multi‐level Newton solvers for fully coupled multi‐physics problems

An algorithm is suggested to improve the efficiency of the multi‐level Newton method that is used to solve multi‐physics problems. It accounts for full coupling between the subsystems by using the direct differentiation method rather than error prone finite difference calculations and retains the advantage of greater flexibility over the tightly coupled approaches. Performance of the algorithm is demonstrated by solving a fluid–structure interaction problem. Copyright © 2003 John Wiley & Sons, Ltd.     

A high‐order immersed boundary discontinuous‐Galerkin method for Poisson's equation with discontinuous coefficients and singular sources

We adopt a numerical method to solve Poisson's equation on a fixed grid with embedded boundary conditions, where we put a special focus on the accurate representation of the normal gradient on the boundary. The lack of accuracy in the gradient evaluation on the boundary is a common issue with low‐order embedded boundary methods. Whereas a direct evaluation of the gradient is preferable, one typically uses post‐processing techniques to improve the quality of the gradient. Here, we adopt a new method based on the discontinuous‐Galerkin (DG) finite element method, inspired by the recent work of [A.J. Lew and G.C. Buscaglia. A discontinuous‐Galerkin‐based immersed boundary method. International Journal for Numerical Methods in Engineering, 76:427‐454, 2008]. The method has been enhanced in two aspects: firstly, we approximate the boundary shape locally by higher‐order geometric primitives. Secondly, we employ higher‐order shape functions within intersected elements. These are derived for the various geometric features of the boundary based on analytical solutions of the underlying partial differential equation. The development includes three basic geometric features in two dimensions for the solution of Poisson's equation: a straight boundary, a circular boundary, and a boundary with a discontinuity. We demonstrate the performance of the method via analytical benchmark examples with a smooth circular boundary as well as in the presence of a singularity due to a re‐entrant corner. Results are compared to a low‐order extended finite element method as well as the DG method of [1]. We report improved accuracy of the gradient on the boundary by one order of magnitude, as well as improved convergence rates in the presence of a singular source. In principle, the method can be extended to three dimensions, more complicated boundary shapes, and other partial differential equations. Copyright © 2014 John Wiley & Sons, Ltd.     

A discontinuous‐Galerkin‐based immersed boundary method

A numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced. The proposed method is conceptually simple and free of user‐defined parameters. Starting with a conforming finite element mesh, the key ingredient is to switch those elements intersected by the Dirichlet boundary to a discontinuous‐Galerkin approximation and impose the Dirichlet boundary conditions strongly. By virtue of relaxing the continuity constraint at those elements, boundary locking is avoided and optimal‐order convergence is achieved. This is shown through numerical experiments in reaction–diffusion problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Hat interpolation wavelet‐based multi‐scale Galerkin method for thin‐walled box beam analysis

The objective of the present work is to propose a new adaptive wavelet‐Galerkin method based on the lowest‐order hat interpolation wavelets. The specific application of the present method is made on the one‐dimensional analysis of thin‐walled box beam problems exhibiting rapidly varying local end effects. Higher‐order interpolation wavelets have been used in the wavelet‐collocation setting, but the lowest‐order hat interpolation is applied here first and a hat interpolation wavelet‐based Galerkin method is newly formulated. Unlike existing orthogonal or biorthogonal wavelet‐based Galerkin methods, the present method does not require special treatment in dealing with general boundary conditions. Furthermore, the present method directly works with nodal values and does not require special formula for the evaluation of system matrices. Though interpolation wavelets do not have any vanishing moment, an adaptive scheme based on multi‐resolution approximations is possible and a preconditioned conjugate gradient method can be used to enhance numerical efficiency. Copyright © 2001 John Wiley & Sons, Ltd.     

A fast multi‐level convolution boundary element method for transient diffusion problems

A new algorithm is developed to evaluate the time convolution integrals that are associated with boundary element methods (BEM) for transient diffusion. This approach, which is based upon the multi‐level multi‐integration concepts of Brandt and Lubrecht, provides a fast, accurate and memory efficient time domain method for this entire class of problems. Conventional BEM approaches result in operation counts of order O(N²) for the discrete time convolution over N time steps. Here we focus on the formulation for linear problems of transient heat diffusion and demonstrate reduced computational complexity to order O(N^3/2) for three two‐dimensional model problems using the multi‐level convolution BEM. Memory requirements are also significantly reduced, while maintaining the same level of accuracy as the conventional time domain BEM approach. Copyright © 2005 John Wiley & Sons, Ltd.     

Trefftz boundary elements—multi‐region formulations

This paper is concerned with an effective numerical implementation of the Trefftz boundary element method, for the analysis of two‐dimensional potential problems, defined in arbitrarily shaped domains. The domain is first discretized into multiple subdomains or regions. Each region is treated as a single domain, either finite or infinite, for which a complete set of solutions of the problem is known in the form of an expansion with unknown coefficients. Through the use of weighted residuals, this solution expansion is then forced to satisfy the boundary conditions of the actual domain of the problem, leading thus to a system of equations, from which the unknowns can be readily determined. When this basic procedure is adopted, in the analysis of multiple‐region problems, proper boundary integral equations must be used, along common region interfaces, in order to couple to each other the unknowns of the solution expansions relative to the neighbouring regions. These boundary integrals are obtained from weighted residuals of the coupling conditions which allow the implementation of any order of continuity of the potential field, across the interface boundary, between neighbouring regions. The technique used in the formulation of the region‐coupling conditions drives the performance of the Trefftz boundary element method. While both of the collocation and Galerkin techniques do not generate new unknowns in the problem, the technique of Galerkin presents an additional and unique feature: the size of the matrix of the final algebraic system of equations which is always square and symmetric, does not depend on the number of boundary elements used in the discretization of both the actual and region‐interface boundaries. This feature which is not shared by other numerical methods, allows the Galerkin technique of the Trefftz boundary element method to be effectively applied to problems with multiple regions, as a simple, economic and accurate solution technique. A very difficult example is analysed with this procedure. The accuracy and efficiency of the implementations described herein make the Trefftz boundary element method ideal for the study of potential problems in general arbitrarily‐shaped two‐dimensional domains. Copyright © 1999 John Wiley & Sons, Ltd.     

A simple boundary element method for problems of potential in non‐homogeneous media

A simple boundary element method for solving potential problems in non‐homogeneous media is presented. A physical parameter (e.g. heat conductivity, permeability, permittivity, resistivity, magnetic permeability) has a spatial distribution that varies with one or more co‐ordinates. For certain classes of material variations the non‐homogeneous problem can be transformed to known homogeneous problems such as those governed by the Laplace, Helmholtz and modified Helmholtz equations. A three‐dimensional Galerkin boundary element method implementation is presented for these cases. However, the present development is not restricted to Galerkin schemes and can be readily extended to other boundary integral methods such as standard collocation. A few test examples are given to verify the proposed formulation. The paper is supplemented by an Appendix, which presents an ABAQUS user‐subroutine for graded finite elements. The results from the finite element simulations are used for comparison with the present boundary element solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

The 3‐D elastodynamic boundary element method: semi‐analytical integration for linear isoparametric triangular elements

A semi‐analytical integration scheme is described in this paper which is designed to reduce the errors incurred when integrals with singular integrands are evaluated numerically. This new scheme can be applied to linear triangular elements for use in steady‐state elastodynamic BEM problems and is particularly useful for predicting displacement to high accuracy, close to surfaces for a spectrum of frequencies. The scheme involves the application of Taylor expansions to formulate the integrals into two parts. One part is regular and is evaluated numerically and the other part is singular but sufficiently simple to enable its transforma tion into a line integral. The line integral is solved numerically using Gauss–Legendre quadrature. This approach caters for all the integral types that appear in steady‐state elastodynamic boundary elements but, in particular, no special treatment is required for the evaluation of the Cauchy principal value singular integrals. Numerical tests are performed on a simple test‐problem for which a known analytical solution exists. The results obtained using the semi‐analytical approach are shown to be considerably more accurate than those obtained using standard quadrature methods. Copyright © 1999 John Wiley & Sons, Ltd.