Adaptive hp-versions of boundary element methods for elastic contact problems

The variational formulation of elastic contact problems leads to variational inequalities on convex subsets. These variational inequalities are solved with the boundary element method (BEM) by making use of the Poincaré–Steklov operator. This operator can be represented in its discretized form by the Schur-complement of the dense Galerkin-matrices for the single layer potential operator, the double layer potential operator and the hypersingular integral operator. Due to the difficulties in discretizing the convex subsets involved, traditionally only the h-version is used for discretization. Recently, p- and hp-versions have been introduced for Signorini contact problems in Maischak and Stephan (Appl Numer Math, 2005) . In this paper we show convergence for the quasi-uniform hp-version of BEM for elastic contact problems, and derive a-posteriori error estimates together with error indicators for adaptive hp-algorithms. We present corresponding numerical experiments.     

A comparison of domain integral evaluation techniques for boundary element methods

In many cases, boundary integral equations contain a domain integral. This can be evaluated by discretization of the domain into domain elements. Historically, this was seen as going against the spirit of boundary element methods, and several methods were developed to avoid this discretization, notably dual and multiple reciprocity methods and particular solution methods. These involved the representation of the interior function with a set of basis functions, generally of the radial type. In this study, meshless methods (dual reciprocity and particular solution) are compared to the direct domain integration methods. The domain integrals are evaluated using traditional methods and also with multipole acceleration. It is found that the direct integration always results in better accuracy, as well as smaller computation times. In addition, the multipole method further improves on the computation times, in particular where multiple evaluations of the integral are required, as when iterative solvers are used. The additional error produced by the multipole acceleration is negligible. Copyright © 2001 John Wiley & Sons, Ltd.     

Adaptive boundary element for multiple subregions

The sample point error analysis and related adaptive boundary element refinement, proposed by one of the present authors, is extended to the problem with subregion partition which is often required for maintaining higher accuracy and for treatment of composite dissimilar materials. The present study is devoted to regularization of the requirement that the interface between neighboring subregions should be discretized by the unified criterion for the both, while, in general, the error influences on the point on the interface from one region differs from that from the other. Two examples concerning the two-dimensional Laplace equation are tested to verify the availability of the proposed method.     

Coupling of finite element and boundary integral methods for a capsule in a Stokes flow

We introduce a new numerical method to model the fluid–structure interaction between a microcapsule and an external flow. An explicit finite element method is used to model the large deformation of the capsule wall, which is treated as a bidimensional hyperelastic membrane. It is coupled with a boundary integral method to solve for the internal and external Stokes flows. Our results are compared with previous studies in two classical test cases: a capsule in a simple shear flow and in a planar hyperbolic flow. The method is found to be numerically stable, even when the membrane undergoes in‐plane compression, which had been shown to be a destabilizing factor for other methods. The results are in very good agreement with the literature. When the viscous forces are increased with respect to the membrane elastic forces, three regimes are found for both flow cases. Our method allows a precise characterization of the critical parameters governing the transitions. Copyright © 2010 John Wiley & Sons, Ltd.     

Degenerate scale for the analysis of circular thin plate using the boundary integral equation method and boundary element methods

In this paper, the degenerate scale for plate problem is studied. For the continuous model, we use the null-field integral equation, Fourier series and the series expansion in terms of degenerate kernel for fundamental solutions to examine the solvability of BIEM for circular thin plates. Any two of the four boundary integral equations in the plate formulation may be chosen. For the discrete model, the circulant is employed to determine the rank deficiency of the influence matrix. Both approaches, continuous and discrete models, lead to the same result of degenerate scale. We study the nonunique solution analytically for the circular plate and find degenerate scales. The similar properties of solvability condition between the membrane (Laplace) and plate (biharmonic) problems are also examined. The number of degenerate scales for the six boundary integral formulations is also determined. Tel.: 886-2-2462-2192-ext. 6140 or 6177     

A pure boundary element method approach for solving hypersingular boundary integral equations

This paper is concerned with discretization and numerical solution of a regularized version of the hypersingular boundary integral equation (HBIE) for the two-dimensional Laplace equation. This HBIE contains the primary unknown, as well as its gradient, on the boundary of a body. Traditionally, this equation has been solved by combining the boundary element method (BEM) together with tangential differentiation of the interpolated primary variable on the boundary. The present paper avoids this tangential differentiation. Instead, a “pure” BEM method is proposed for solving this class of problems. Dirichlet, Neumann and mixed problems are addressed in this paper, and some numerical examples are included in it.     

Reproducing polynomial particle methods for boundary integral equations

Since meshless methods have been introduced to alleviate the difficulties arising in conventional finite element method, many papers on applications of meshless methods to boundary element method have been published. However, most of these papers use moving least squares approximation functions that have difficulties in prescribing essential boundary conditions. Recently, in order to strengthen the effectiveness of meshless methods, Oh et al. developed meshfree reproducing polynomial particle (RPP) shape functions, patchwise RPP and reproducing singularity particle (RSP) shape functions with use of flat-top partition of unity. All of these approximation functions satisfy the Kronecker delta property. In this paper, we report that meshfree RPP shape functions, patchwise RPP shape functions, and patchwise RSP shape functions effectively handle boundary integral equations with (or without) domain singularities.     

Maurice Jaswon and boundary element methods

In the direct boundary integral equation method, boundary-value problems are reduced to integral equations by an application of Green's theorem to the unknown function and a fundamental solution (Green's function). Discretization of the integral equation then leads to a boundary element method. This approach was pioneered by Jaswon and his students in the early 1960s. Jaswon's work is reviewed together with his influence on later workers.     

Multinode finite element based on boundary integral equations

A finite element constructed on the basis of boundary integral equations is proposed. This element has a flexible shape and arbitrary number of nodes. It also has good approximation properties. A procedure of constructing an element stiffness matrix is demonstrated first for one-dimensional case and then for two-dimensional steady-state heat conduction problem. Numerical examples demonstrate applicability and advantages of the method. © 1998 John Wiley & Sons, Ltd.     

A domain decomposition tool for boundary element methods

Domain decomposition boundary element methods have become increasingly popular over the last several years for a variety of reasons. In particular, these methods reduce the storage and CPU requirements, can result in sparse linear systems, are easy to parallelize, and, when used in conjunction with a dual reciprocity method, can significantly improve the conditioning of the associated linear system. Nevertheless, for complex geometries, determining an appropriate decomposition of the domain can be extremely difficult. A domain decomposition tool based on a graph partitioning algorithm is presented to automate the process and provide quality decompositions.     

Preconditioning for boundary element methods in domain decomposition

This paper is concerned with asymptotically almost optimal preconditioning techniques for the solution of coupled elliptic problems with piecewise continuous coefficients by domain decomposition methods. Spectrally equivalent, two- and multilevel interface preconditioners are proposed and analyzed. They are applied to two basic formulations: strongly elliptic skew-symmetric problems and symmetric, positive definite variational problems; the former involves the classical boundary potentials from the Calderon projections and the latter is based on the Steklov–Poincaré operators associated with subdomains of the decomposition. The preconditioners considered are shown to be robust with respect to both mesh-parameters and jumps in the coefficients.     

Coupling finite and boundary element methods for two-dimensional potential problems

A finite-element-boundary-element (FE-BE) coupling method based on a weighted residual variational method is presented for potential problems, governed by either the Laplace or the Poisson equations. In this method, a portion of the domain of interest is modelled by finite elements (FE) and the remainder of the region by boundary elements (BE). Because the BE fundamental solutions are valid for infinite domains, a procedure that limits the effect of the BE fundamental solution to a small region adjacent to the FE region, called the transition region (TR), is developed. This procedure involves a judicious choice of functions called the transition (T) functions that have unit values on the BE-TR interface and zero values on the FE-TR interface. The present FE-BE coupling algorithm is shown to be independent of the extent of the transition region and the choice of the transition functions. Therefore, transition regions that extend to only one layer of elements between FE and BE regions and the use of simple linear transition functions work well.     

On connections between boundary integral equations and T-matrix methods

Acoustic scattering by three-dimensional obstacles is considered, using boundary integral equations, null-field equations and the T-matrix. Connections between these techniques are explored. It is shown that solving a boundary integral equation by a particular Petrov–Galerkin method leads to the same algebraic system as obtained from the null-field equations. It is also emphasised that the T-matrix can be constructed by solving boundary integral equations rather than by solving the null-field equations.     

On three-dimensional boundary element methods for magnetostatics invector variables

The boundary-element solution of three-dimensional magnetostatic fields is dealt with. The formulations are based on the magnetic vector potential and the magnetic flux density. The proper boundary-conditions of the problem are discussed, and vector boundary integral equations are presented. An isoparametric boundary element method is used for the solution. Numerical examples are given for both of the formulations     

Dynamic element discretization method for solving 2D traction boundary integral equations

A sufficient condition for the existence of element singular integral of the traction boundary integral equation for elastic problems requires that the tangential derivatives of the boundary displacements are Hölder continuous at collocation points. This condition is violated if a collocation point is at the junction between two standard conforming boundary elements even if the field variables themselves are Hölder continuous there. Various methods are proposed to overcome this difficulty. Some of them are rather complicated and others are too different from the conventional boundary element method. A dynamic element discretization method to overcome this difficulty is proposed in this work. This method is novel and very simple: the form of the standard traction boundary integral equation remains the same; the standard conforming isoparametric elements are still used and all collocation points are located in the interior of elements where the continuity requirements are satisfied. For boundary elements with boundary points where the field variables themselves are singular, such as crack tips, corners and other boundary points where the stress tensors are not unique, a general procedure to construct special elements has been developed in this paper. Highly accurate numerical results for various typical examples have been obtained.     

Explicit boundary element method for nonlinear solid mechanics using domain integral reduction

Applied to solid mechanics problems with geometric nonlinearity, current finite element and boundary element methods face difficulties if the domain is highly distorted. Furthermore, current boundary element method (BEM) methods for geometrically nonlinear problems are implicit: the source term depends on the unknowns within the arguments of domain integrals. In the current study, a new BEM method is formulated which is explicit and whose stiffness matrices require no domain function evaluations. It exploits a rigorous incremental equilibrium equation. The method is also based on a Domain Integral Reduction Algorithm (DIRA), exploiting the Helmholtz decomposition to obviate domain function evaluations. The current version of DIRA introduces a major improvement compared to the initial version.     

Adaptive finite element methods for hydrodynamic lubrication with cavitation

We present an adaptive finite element method for a cavitation model based on Reynolds' equation. A posteriori error estimates and adaptive algorithms are discussed, and numerical examples illustrating the theory are supplied. Copyright © 2007 John Wiley & Sons, Ltd.     

Analytical regularization of hypersingular integral for Helmholtz equation in boundary element method

This paper presents a gradient field representation using an analytical regularization of a hypersingular boundary integral equation for a two-dimensional time harmonic wave equation called the Helmholtz equation. The regularization is based on cancelation of the hypersingularity by considering properties of hypersingular elements that are adjacent to a singular node. Advantages to this regularization include applicability to evaluate corner nodes, no limitation for element size, and reduced computational cost compared to other methods. To demonstrate capability and accuracy, regularization is estimated for a problem about plane wave propagation. As a result, it is found that even at a corner node the most significant error in the proposed method is due to truncation error of non-singular elements in discretization, and error from hypersingular elements is negligibly small.     

On the use of conjugate gradient-type methods for boundary integral equations

This paper deals with a number of iterative methods for solving matrix equations that result from boundary integral equations. The matrices are non-sparse, and in general neither positive definite nor symmetric. Traditional methods like Gauss-Seidel do not give satisfactory results, therefore the use of conjugate gradient- and Krylov-type methods is investigated based on work of Kleinman and Van den Berg, who presented a general framework for these methods. Eleven of these algorithms are given and their performance (without preconditioning) is compared in a test case involving four different integral operators arising in potential theory. For all four matrix equations the Generalized Minimal Residual method (GMRES) outperforms all other iterative methods in both computation time per iteration and total computation time. For the Fredholm equations of the first kind this method also is the fastest with respect to the number of iterations. The Bi-conjugate gradient method (Bi-CG) and the Quasi-minimal residual method (QMR) are the best alternatives. For the Fredholm equations of the second kind more methods can be used efficiently besides GMRES. The Conjugate Gradient Squared method (CGS), the Bi-conjugate gradient method (Bi-CG), its stabilized version (Bi-CGSTAB) and the Quasi Minimal Residual method (QMR) are efficient alternatives.