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The paper gives the description of boundary element method (BEM) with subdomains for the solution of convection—diffusion equations with variable coefficients and Burgers’ equations. At first, the whole domain is discretized into K subdomains, in which linearization of equations by representing convective velocity by the sum of constant and variable parts is carried out. Then using fundamental solutions for convection—diffusion linear equations for each subdomain the boundary integral equation (in which the part of the convective term with the constant convective velocity is not included into the pseudo‐body force) is formulated. Only part of the convective term with the variable velocity, which is, as a rule, more than one order less than convective velocity constant part contribution, is left as the pseudo‐source. On the one hand, this does not disturb the numerical BEM—algorithm stability and, on the other hand, this leads to significant improvement in the accuracy of solution. The global matrix, similar to the case of finite element method, has block band structure whereas its width depends only on the numeration order of nodes and subdomains. It is noted, that in comparison with the direct boundary element method the number of global matrix non‐zero elements is not proportional to the square of the number of nodes, but only to the total number of nodal points. This allows us to use the BEM for the solution of problems with very fine space discretization. The proposed BEM with subdomains technique has been used for the numerical solution of one‐dimensional linear steady‐state convective—diffusion problem with variable coefficients and one‐dimensional non‐linear Burgers’ equation for which exact analytical solutions are available. It made it possible to find out the BEM correctness according to both time and space. High precision of the numerical method is noted. The good point of the BEM is the high iteration convergence, which is disturbed neither by high Reynolds numbers nor by the presence of negative velocity zones.

The purpose of this paper is to develop an easy-to-implement and accurate fast boundary element method (BEM) for solving large-scale elastodynamic problems in frequency and time domains.

A newly developed kernel-independent fast multipole method (KIFMM) is applied to accelerating the evaluation of displacements, strains and stresses in frequency domain elastodynamic BEM analysis, in which the far-field interactions are evaluated efficiently utilizing equivalent densities and check potentials. Although there are six boundary integrals with unique kernel functions, by using the elastic theory, the authors managed to accelerate these six boundary integrals by KIFMM with the same kind of equivalent densities and check potentials. The boundary integral equations are discretized by Nyström method with curved quadratic elements. The method is further used to conduct the time-domain analysis by using the frequency-domain approach.

Numerical results show that by the fast BEM, high accuracy can be achieved and the computational complexity is brought down to linear. The performance of the present method is further demonstrated by large-scale simulations with more than two millions of unknowns in the frequency domain and one million of unknowns in the time domain. Besides, the method is applied to the topological derivatives for solving elastodynamic inverse problems.

An efficient KIFMM is implemented in the acceleration of the elastodynamic BEM. Combining with the Nyström discretization based on quadratic elements and the frequency-domain approach, an accurate and highly efficient fast BEM is achieved for large-scale elastodynamic frequency domain analysis and time-domain analysis.

The state space method (SSM) is good at analyzing the interfacial physical quantities in laminated materials, while it has difficulty in calculating the mechanical quantities of interior points, which can be easily evaluated by the boundary element method (BEM). However, the material has to be divided into many subdomains when the traditional BEM is applied to analyze the functionally graded material (FGM), so that the computational amount will be increased enormously. This study aims to couple these two methods to strengthen their advantages and overcome their disadvantages.

Herein, a state space BEM in which the SSM is coupled by the BEM is proposed to analyze the elasticity of FGMs, where one BEM domain is set and the others belong to SSM domains. The discretized elements occur only on the boundary of the BEM domain and at the interfaces between different SSM domains. In SSM domains, the horizontal interfaces of FGMs are discretized by linear elements and the variables along the vertical direction are yielded by the precise integration method.

The accuracy of the proposed method is verified by comparing the present results with the ones from the finite element method (FEM). It is found that the present method can provide accurate displacements and stresses in the FGMs by fewer freedom degrees in comparison with the FEM. In addition, the present method can provide continuous interfacial stresses at the interfaces between different material domains, while the interfacial stresses by the FEM are discontinuous.

The system equations of the state space BEM are built by combining the boundary integral equation with the state equations according to the continuity conditions at the interfaces. The mechanical parameters of any inner point can be evaluated by the boundary integral equation after the unknowns on the boundaries and interfaces are determined by the system equation.

Domain integrals, known as volume potentials in 3D elasticity problems, exist in many boundary-type methods, such as the boundary element method (BEM) for inhomogeneous partial differential equations. The purpose of this paper is to develop an accurate and reliable technique to effectively evaluate the volume potentials in 3D elasticity problems.

An adaptive background cell-based domain integration method is proposed for treatment of volume potentials in 3D elasticity problems. The background cells are constructed from the information of the boundary elements based on an oct-tree structure, and the domain integrals are evaluated over the cells rather than volume elements. The cells that contain the boundary elements can be subdivided into smaller sub-cells adaptively according to the sizes and levels of the boundary elements. The fast multipole method (FMM) is further applied in the proposed method to reduce the time complexity of large-scale computation.

The method is a boundary-only discretization method, and it can be applied in the BEM easily. Much computational time is saved by coupling with the FMM. Numerical examples demonstrate the accuracy and efficiency of the proposed method..

Boundary elements are used to create adaptive background cells, and domain integrals are evaluated over the cells rather than volume elements. Large-scale computation is made possible by coupling with the FMM.

This paper is concerned with the application of radial basis function networks (RBFNs) as interpolation functions for all boundary values in the boundary element method (BEM) for the numerical solution of heat transfer problems. The quality of the estimate of boundary integrals is greatly affected by the type of functions used to interpolate the temperature, its normal derivative and the geometry along the boundary from the nodal values. In this paper, instead of conventional Lagrange polynomials, interpolation functions representing these variables are based on the “universal approximator” RBFNs, resulting in much better estimates. The proposed method is verified on problems with different variations of temperature on the boundary from linear level to higher orders. Numerical results obtained show that the BEM with indirect RBFN (IRBFN) interpolation performs much better than the one with linear or quadratic elements in terms of accuracy and convergence rate. For example, for the solution of Laplace's equation in 2D, the BEM can achieve the norm of error of the boundary solution of O(10⁻⁵) by using IRBFN interpolation while quadratic BEM can achieve a norm only of O(10⁻²) with the same boundary points employed. The IRBFN‐BEM also appears to have achieved a higher efficiency. Furthermore, the convergence rates are of O(h^1.38) and O(h^4.78) for the quadratic BEM and the IRBFN‐based BEM, respectively, where h is the nodal spacing.

This paper gives a bibliographical review of the finite element and boundary element parallel processing techniques from the theoretical and application points of view. Topics include: theory – domain decomposition/partitioning, load balancing, parallel solvers/algorithms, parallel mesh generation, adaptive methods, and visualization/graphics; applications – structural mechanics problems, dynamic problems, material/geometrical non‐linear problems, contact problems, fracture mechanics, field problems, coupled problems, sensitivity and optimization, and other problems; hardware and software environments – hardware environments, programming techniques, and software development and presentations. The bibliography at the end of this paper contains 850 references to papers, conference proceedings and theses/dissertations dealing with presented subjects that were published between 1996 and 2002.

A boundary element method (BEM) formulation for the solution of transient conduction‐convection problems is developed in this paper. A time‐dependent fundamental solution for moving heat source problems is utilized for this purpose. This reduces the governing parabolic partial differential equations to a boundary‐only form and obviates the need for any internal discretization. Such a formulation is also expected to be stable at high Peclet numbers. Numerical examples are included to establish the validity of the approach and to demonstrate the salient features of the BEM algorithm.

The boundary element method (BEM) and the finite element method (FEM) may be computationally expensive if complex problems are to be solved; thus there is the need of implementing them on fast computer architectures, especially parallel computers. Because these methods are complementary to each other, the coupling of FEM and BEM is widely used. In this paper, the coupling of displacement‐based FEM and collocation BEM and its implementation on a distributed memory system (Parsytec MultiCluster2) is described. The parallelization is performed by data partitioning which leads to a very high efficiency. As model problems, we assume linear elasticity for the boundary element method and elastoplasticity for the finite element method. The efficiency of our implementation is shown by various test examples. By numerical examples we show that a multiplicative Schwarz method for coupling BEM with FEM is very well suited for parallel implementation.

We report on the progress in the development and application of a coupled boundary element/finite volume method temperature‐forward/flux‐back algorithm developed to solve conjugate heat transfer arising in 3D film‐cooled turbine blades. We adopt a loosely coupled strategy where each set of field equations is solved to provide boundary conditions for the other. Iteration is carried out until interfacial continuity of temperature and heat flux is enforced. The NASA‐Glenn explicit finite volume Navier‐Stokes code Glenn‐HT is coupled to a 3D BEM steady‐state heat conduction solver. Results from a CHT simulation of a 3D film‐cooled blade section are compared with those obtained from the standard two temperature model, revealing that a significant difference in the level and distribution of metal temperatures is found between the two. Finally, current developments of an iterative strategy accommodating large numbers of unknowns by a domain decomposition approach is presented. An iterative scheme is developed along with a physically‐based initial guess and a coarse grid solution to provide a good starting point for the iteration. Results from a 3D simulation show the process that converges efficiently and offers substantial computational and storage savings.

This paper presents a hybrid finite element — boundary element method for the steady state thermal analysis of energy installations. The coupling of the two techniques is presented: finite elements are used in a bounded region containing thermal sources while the complementary domain is treated with boundary elements. With such a combination the number of unknowns is reduced and an accurate prediction of temperature is obtained. As an example, the temperature rise is computed for the case of three power cables laid in a thermal backfill: the finite element method (FEM) is used for the cables and the backfill while the homogeneous soil is taken into account with the boundary element method (BEM).