The boundary element-linear complementarity method for the Signorini problem

The boundary element-linear complementarity method for solving the Laplacian Signorini problem is presented in this paper. Both Green's formula and the fundamental solution of the Laplace equation have been used to solve the boundary integral equation. By imposing the Signorini constraints of the potential and its normal derivative on the boundary, the discrete integral equation can be written into a standard linear complementarity problem (LCP). In the LCP, the unique variable to be affected by the Signorini boundary constraints is the boundary potential variable. A projected successive over-relaxation (PSOR) iterative method is employed to solve the LCP, and some numerical results are presented to illustrate the efficiency of this method.     

An iterative boundary element method for Cauchy inverse problems

 We are interested in this paper in recovering an harmonic function from the knowledge of Cauchy data on some part of the boundary. A new inversion method is introduced. It reduces the Cauchy problem resolution to the determination of the resolution of a sequence of well-posed problems. The sequence of these solutions is proved to converge to the Cauchy problem solution. The algorithm is implemented in the framework of boundary elements. Displayed numerical results highlight its accuracy, as well as its robustness to noisy data. Received 6 November 2000     

A fast convergent iterative boundary element method on PVM cluster

This paper reports a fast convergent boundary element method on a Parallel Virtual Machine (PVM) (Geist et al., PVM: Parallel Virtual Machine, A Users' Guide and Tutorial for Networked Parallel Computing. MIT Press, Cambridge, 1994) cluster using the SIMD computing model (Single Instructions Multiple Data). The method uses the strategy of subdividing the domain into a number of smaller subdomains in order to reduce the size of the system matrix and to achieve overall speedup. Unlike traditional subregioning methods, where equations from all subregions are assembled into a single linear algebraic system, the present scheme is iterative and each subdomain is handled by a separate PVM node in parallel. The iterative nature of the overall solution procedure arises due to the introduction of the artificial boundaries. However, the system equations for each subdomain is now smaller and solved by direct Gaussian elimination within each iteration. Furthermore, the boundary conditions at the artificial interfaces are estimated from the result of the previous iteration by a reapplication of the boundary integral equation for internal points. This method provides a consistent mechanism for the specification of boundary conditions on artificial interfaces, both initially and during the iterative process. The method is fast convergent in comparison with other methods in the literature. The achievements of this method are therefore: (a) simplicity and consistency of methodology and implementation; (b) more flexible choice of type of boundary conditions at the artificial interfaces; (c) fast convergence; and (d) the potential to solve large problems on very affordable PVM clusters. The present parallel method is suitable where (a) one has a distributed computing environment; (b) the problem is big enough to benefit from the speedup achieved by coarse-grained parallelisation; and (c) the subregioning is such that communication overhead is only a small percentage of total computation time.     

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.     

CMRH method as iterative solver for boundary element acoustic systems

Disceretization of boundary integral equations leads to complex and fully populated linear systems. One inherent drawback of the boundary element method (BEM) is that, the dense linear system has to be constructed and solved for each frequency. For large-scale problems, BEM can be more efficient by improving the construction and solution phases of the linear system. For these problems, the application of common direct solver is inefficient. In this paper, the corresponding linear systems are solved more efficiently than common direct solvers by using the iterative technique called CMRH (Changing Minimal Residual method based on Hessenberg process). In this method, the generation of the basis vectors of the Krylov subspace is based on the Hessenberg process instead of Arnoldi's one that the most known GMRES (Generalized Minimal RESidual) solver uses. Compared to GMRES, the storage requirements are considerably reduced in CMRH.     

A fast multipole boundary element method for solving the thin plate bending problem

A fast multipole boundary element method (BEM) for solving large-scale thin plate bending problems is presented in this paper. The method is based on the Kirchhoff thin plate bending theory and the biharmonic equation governing the deflection of the plate. First, the direct boundary integral equations and the conventional BEM for thin plate bending problems are reviewed. Second, the complex notation of the kernel functions, expansions and translations in the fast multipole BEM are presented. Finally, a few numerical examples are presented to show the accuracy and efficiency of the fast multipole BEM in solving thin plate bending problems. The bending rigidity of a perforated plate is evaluated using the developed code. It is shown that the fast multipole BEM can be applied to solve plate bending problems with good accuracy. Possible improvements in the efficiency of the method are discussed.     

A multiwavelet Galerkin boundary element method for the stationary Stokes problem in 3D

In this paper, a multiwavelet Galerkin boundary element method is presented for the fast solution of the stationary Stokes problem in three dimensions. Piecewise linear discontinuous multiwavelet bases are constructed on each patch of piecewise smooth surface individually, which allow easy and efficient evaluation of the matrix entries. Because of the use of the multiwavelets, the system matrix can be compressed to O (N) (N denotes the number of unknowns) nonzero entries without compromising the order of convergence as for the conventional Galerkin boundary element method. Numerical results of two test samples are given to demonstrate the availability of the present method.     

Theoretical basis of the finite element iterative method for the eigenvalue problem in stationary cracks

In a recently published paper a finite element based iterative method was introduced for the solution of the eigenvalue problem of stationary cracks.¹ In this paper we give the theoretical basis of this iterative method and we show why it converges and how it could be extended to more complex fracture problems. The cases of cracks at interfaces are illustrated.     

Solution of the Helmholtz eigenvalue problem via the boundary element method

The numerical solution of the Helmholtz eigenvalue problem is considered. The application of the boundary element method reduces it to that of a non-linear eigenvalue problem. Through a polynomial approximation with respect to the wavenumber, the non-linear eigenvalue problem is reduced to a standard generalized eigenvalue problem. The method is applied to the test problems of a three-dimensional sphere with an axisymmetric boundary condition and a two-dimensional square.     

An algorithm based on the boundary element method for problems in engineering mechanics

The solution to a two-dimensional problem using the boundary element method requires the evaluation of a line integral over the boundary. The integrand ot this line integral is a product of a known Green's function and an unknown function. A large number of Green's functions for two-dimensional problems can be represented by a linear combination of four singular functions. By approximating the unknown function by a linear combination of known polynomials, integrals are generated whose integrand is a product of the polynomiais and one of the four singular functions. To evaluate these integrals analytically, the boundary is approximated by a sum of straight-line segments. Recursive formulae are established which reduce the generality and the complexity of the integrands to simple expressions. Analytical forms for these simple expressions are found and are used for initiating the algorithm.     

A time domain boundary element method for poroelasticity

The development of a general boundary element method (BEM) for two- and three-dimensional quasistatic poroelasticity is discussed in detail. The new formulation, for the complete Biot consolidation theory, operates directly in the time domain and requires only boundary discretization. As a result, the dimensionality of the problem is reduced by one and the method becomes quite attractive for geotechnical analyses, particularly those which involve extensive or infinite domains. The presentation includes the definition of the two key ingredients for the BEM, namely, the fundamental solutions and a reciprocal theorem. Then, once the boundary integral equations are derived, the focus shifts to an overview of the general purpose numerical implementation. This implementation includes higher-order conforming elements, self-adaptive integration and multi-region capability. Finally, several detailed examples are presented to illustrate the accuracy and suitability of this boundary element approach for consolidation analysis.     

A semi-analytic boundary element method for parabolic problems

A new semi-analytic solution method is proposed for solving linear parabolic problems using the boundary element method. This method constructs a solution as an eigenfunction expansion using separation of variables. The eigenfunctions are determined using the dual reciprocity boundary element method. This separation of variables-dual reciprocity method (SOV-DRM) allows a solution to be determined without requiring either time-stepping or domain discretisation. The accuracy and computational efficiency of the SOV-DRM is found to improve as time increases. These properties make the SOV-DRM an attractive technique for solving parabolic problems.     

Analysis of crack closure problem using the dual boundary element method

A numerical procedure of the crack closure problem solved by the dual boundary element method is developed in this paper. The dual boundary element method is used to allow for the solution to a general mixed-mode crack problem with a single regional formulation. The frictional contact problem on the crack surface is formulated with the complementary problem adapting the Coulomb's friction law. Several examples are shown to demonstrate the validity of the present procedure.     

A coupled complex boundary method for the Cauchy problem

Considered in this paper is a Cauchy problem governed by an elliptic partial differential equation. In the Cauchy problem, one wants to recover the unknown Neumann and Dirichlet data on a part of the boundary from the measured Neumann and Dirichlet data, usually contaminated with noise, on the remaining part of the boundary. The Cauchy problem is an inverse problem with severe ill-posedness. In this paper, a coupled complex boundary method (CCBM), originally proposed in [Cheng XL, Gong RF, Han W, et al. A novel coupled complex boundary method for solving inverse source problems. Inverse Prob. 2014;30:055002], is applied to solve the Cauchy problem stably. With the CCBM, all the data, including the known and unknown ones on the boundary are used in a complex Robin boundary on the whole boundary. As a result, the Cauchy problem is transferred into a complex Robin boundary problem of finding the unknown data such that the imaginary part of the solution equals zero in the domain. Then the Tikhonov regularization is applied to the resulting new formulation. Some theoretical analysis is performed on the CCBM-based Tikhonov regularization framework. Moreover, through the adjoint technique, a simple solver is proposed to compute the regularized solution. The finite-element method is used for the discretization. Numerical results are given to show the feasibility and effectiveness of the proposed method.     

A boundary element solution to the soap bubble problem

The boundary element method (BEM) is applied to the soap bubble problem, that is to the problem of determining the surface that a soap bubble constrained by bounding contours assumes under the action of molecular forces. This is also the shape of a uniformly stretched membrane bounded by one or more non-intersecting curves. As the slopes of the membrane surface are finite, their square can not be neglected and the resulting governing equation is non-linear. The problem is solved using the analogue equation method (AEM). According to this method the non-linear membrane is substituted by a linear one subjected to a fictitious transverse load. The fictitious load is established using the BEM. Numerical examples are presented which illustrate the method and demonstrate its accuracy. This application of the BEM to non-linear problems shows that BEM is a versatile computational method for all-purpose use in engineering analysis. The solution of the problem at hand is very important in engineering, since the soap bubble surface can be used as the best initial form for membrane roofs.     

A boundary element solution of the edge crack problem

The equations of the boundary element method and of a stressed semi-infinite crack in an infinite plane are combined to formulate the solution to the finite edge cracked plate. The coupled integral equations are solved numerically by the most elementary form of the boundary element method and by Gaussian quadrature. Results for the stress intensity factor and crack opening displacements are presented for several fundamental problems.

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Résumé On combine les équations de la méthode des contours élémentaires et d'une fissure semi-infinie soumise à contrainte dans une plaque infinie en vue de formuler la solution applicable à une plaque finie fissurée sur ses bords. Les équations intégrales couplées sont résolues de manière numérique par la forme la plus simple de la méthode des contours élémentaires ainsi que par une quadrature de Gauss. On présente les résultats pour le facteur d'intensité de contrainte et pour le déplacement d'ouverture de la fissure dans le cas de divers problèmes fondamentaux.

     

An iterative solution scheme for systems of boundary element equations

In this paper an iterative scheme of first order is developed for the purpose of solving linear systems of equations. In particular, systems that are derived from boundary integral equations are investigated. The iterative schemes to be considered are of the form Ex^(k+1) = Dx^(k) + d, where E and D are square matrices. It will be assumed that E is a lower matrix, i.e. the coefficients above the central diagonal are zero. It will be shown that by considering matrix D embedded in a vector space and reducing its size with respect to a chosen metric, that convergence rates can be substantially improved. Equation ordering and parameter matrices are used to reduce the magnitude of D. A number of examples are tested to illustrate the importance of the choice of metric, equation ordering and the parameter matrix. Computation times are determined for both the iterative procedure and Gauss elimination indicating the usefulness of iteration which can be orders of magnitude faster.     

Grid optimization for the boundary element method

The quality of solution obtained using the boundary element method (BEM) is dependent on how the boundary is discretized. This is particularly true in domains of complex geometry. A rule for grid optimization for the BEM is derived on the bases of an asymptotic measure of the boundary element error that preserves the number of elements (degrees of freedom). Three example problems are provided to show the advantages of grid optimization in terms of accuracy and cost.