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Recently, we have developed multi-level boundary element methods (MLBEM) for the solution of the Laplace and Helmholtz equations that involve asymptotically decaying non-oscillatory and oscillatory singular kernels, respectively. The accuracy and efficiency of the fast boundary element methods for steady-state heat diffusion and acoustics problems have been investigated for square domains. The current work extends the MLBEM methodology to the solution of Stokes equation in more complex two-dimensional domains. The performance of the fast boundary element method for the Stokes flows is first investigated for a model problem in a unit square. Then, we consider an example problem possessing an analytical solution in a rectangular domain with 5:1 aspect ratio, and finally, we study the performance of the MLBEM algorithm in a C-shaped domain.

Several recently developed boundary element formulations for time-dependent convective heat diffusion appear to provide very efficient computational tools for transient linear heat flows. More importantly, these new approaches hold much promise for the numerical solution of related nonlinear problems, e.g., Navier–Stokes flows. However, the robustness of these methods has not been examined, particularly for high Peclet number regimes. Here, we focus on these regimes for two-dimensional problems and develop the necessary temporal and spatial integration strategies. The algorithm takes advantage of the nature of the time-dependent convective kernels, and combines analytic integration over the singular portion of the time interval with numerical integration over the remaining nonsingular portion. Furthermore, the character of the kernels lets us define an influence domain and then localize the surface and volume integrations only within this domain. We show that the localization of the convective kernels becomes more prominent as the Peclet number of the flow increases. This leads to increasing sparsity and in most cases improved conditioning of the global matrix. Thus, iterative solvers become the primary choice. We consider two representative example problems of heat propagation, and perform numerical investigations of the accuracy and stability of the proposed higher-order boundary element formulations for Peclet numbers up to 105.

A higher-order boundary element method recently developed by the current authors [Numer. Heat Trans. Part B: Fundamentals 45 (2004) 109] for two-dimensional steady convective heat diffusion is generalized to three-dimensions. In order to facilitate an accurate and efficient boundary element formulation, we introduce an influence domain due to these convective kernels and then localize the surface integrations only within the domain of influence. The localization of the kernels becomes more prominent as the Peclet number of the flow increases. This, in turn, leads to increasing sparsity and improved conditioning of the global matrix. Consequently, iterative solvers for sparse matrices become the primary choice. In this paper, we consider an example problem with an exact solution and investigate the accuracy and efficiency of the boundary element formulation for Peclet numbers in the range from 10 to 1000. The bi-quartic boundary elements included in this study are shown to provide acceptable levels of resolution.

Most recently, we have developed a novel multilevel boundary element method (MLBEM) for steady Stokes flows in irregular two-dimensional domains (Grigoriev, M.M., and Dargush, G.F., Comput. Methods. Appl. Mech. Eng., 2005). The multilevel algorithm permitted boundary element solutions with slightly over 16,000 degrees of freedom, for which approximately 40-fold speedups were demonstrated for the fast MLBEM algorithm compared to a conventional Gauss elimination approach. Meanwhile, the sevenfold memory savings were attained for the fast algorithm. This paper extends the MLBEM methodology to dramatically improve the performance of the original multilevel formulation for the steady Stokes flows. For a model problem in an irregular pentagon, we demonstrate that the new MLBEM formulation reduces the CPU times by a factor of nearly 700,000. Meanwhile, the memory requirements are reduced more than 16,000 times. These superior run-time and memory reductions compared to regular boundary element methods are achieved while preserving the accuracy of the boundary element solution.

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(N 2 ) 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.

A higher-order boundary element method (BEM) recently developed by the current authors (Comput Methods Appl Mech Eng 2003; 192: 4281–4298; 4299–4312; 4313–4335) for time-dependent convective heat diffusion in two-dimensions appears to be a very attractive tool for efficient simulations of transient linear flows. However, the previous BEM formulation is restricted to relatively small time step sizes (i.e. Δt⩽4κ/V2) owing to the convergence issues of the time series for the kernel representation within a time interval. This paper extends the boundary element formulation in a way to allow time step sizes several orders of magnitude larger than in the previous approach. We consider an example problem of thermal propagation, and investigate the accuracy and efficiency of BEM formulations for Peclet numbers in the range from 103 to 105. Copyright © 2005 John Wiley & Sons, Ltd.

Higher-order boundary element methods (BEM) are presented for three-dimenisonal steady convective heat diffusion at high Peclet numbers. The boundary element formulation is facilitated by the definition of an influence domain due to convective kernels. This approach essentially localizes the surface integrations only within the domain of influence which becomes more narrowly focused as the Peclet number increases. The outcome of this phenomenon is an increased sparsity and improved conditioning of the global matrix. Therefore, iterative solvers for sparse matrices become a very efficient and robust tool for the corresponding boundary element matrices. In this paper, we consider an example problem with an exact solution and investigate the accuracy and efficiency of the higher-order BEM formulations for high Peclet numbers in the range from 1000 to 100,000. The bi-quartic boundary elements included in this study are shown to provide very efficient and extremely accurate solutions to this problem, even on a single engineering workstation.

A fast, accurate, and efficient multilevel boundary-element method (MLBEM) is developed to solve general boundary-value problems arising in computational mechanics. Here we concentrate on problems of two-dimensional steady potential flow and present a fast, direct boundary-element formulation. This novel method extends the pioneering work of Brandt and Lubrecht on multilevel multi-integration (MLMI) in several important ways to address problems with mixed boundary conditions. We utilize bi-conjugate gradient methods (BCGMs) and implement the MLMI approach for fast matrix and matrix transpose multiplication for every iteration loop. After introducing a C-cycle multigrid algorithm, we find that the number of iterations for the bi-conjugate gradient methods is independent of the boundary-element mesh discretization for a broad range of steady-state heat diffusion problems. Here, for a model problem in an L-shaped domain, we demonstrate that the computational complexity of the proposed method approaches the d...

Higher-order boundary-element methods (BEM) are presented for steady-state convective diffusion problems in two dimensions. The free-space steady convective diffusion fundamental solutions considered in this article provide an analytical upwinding for the entire Peclet number range, from zero to infinity. However, integration of the kernels over the boundary elements requires considerable attention, especially at higher Peclet numbers. We define an influence domain due to these convective kernels and then localize the surface integrations only within the domain of influence. The localization of the kernels becomes more prominent as the Peclet number of the flow increases. This, in turn, leads to increasing sparsity and improved conditioning of the global matrix. Consequently, iterative solvers become the primary choice. We consider an example problem with an exact solution, and investigate the accuracy and efficiency of the higher-order BEM formulations for Peclet numbers in the range from 200 to 200,000....

Most recently, we have developed a novel multi-level boundary element method (MLBEM) for the solution of the steady heat diffusion equation involving asymptotically decaying non-oscillatory log-singular and strongly singular kernels. This hierarchical approach generalizes the pioneering work of Brandt and Lubrecht on multi-level multi-integration (MLMI) and C-cycle multi-grid to the broader class of mixed boundary value problems. The result is a fast, accurate and efficient boundary element method. The present paper extends this new computational methodology to the solution of the Helmholtz equation involving oscillatory log-singular and strongly singular kernels for two-dimensional problems. We consider a direct boundary element formulation and, due to the nature of the fundamental solutions, split the corresponding boundary integral equation into real and imaginary parts. Then, we introduce double-noded corners to facilitate a patch-by-patch application of the MLMI algorithm for fast matrix–vector and matrix-transpose–vector multiplications within bi-conjugate gradient methods. The performance of the proposed fast MLBEM is investigated using a numerical example that possesses an exact solution. For wave numbers κ=20 and below, we demonstrate that the fast MLBEM algorithm for the Helmholtz equation is robust, accurate, and exceptionally efficient.

A boundary element method for transient convective diffusion phenomena presented in Part I of the paper is extended to two dimensional problems. We introduce a series representation for the transient convective kernel and perform a time integration for the double integrals to evaluate coefficients of the time-discrete boundary integral equation. The time-integrated kernels are evaluated for the linear, quadratic and quartic time interpolation functions utilized in the paper. Then, linear, quadratic and quartic boundary elements as well as bi-linear, bi-quadratic and bi-quartic volume cells are introduced to ensure proper resolution in space for the two-dimensional formulation. Due to the singular nature of the transient convective diffusion kernels, integration of the kernels over the boundary elements and volume cells requires a considerable effort to maintain a desired level of accuracy. We define influence domains due to time-integrated and time-delayed kernels arising for the surface and volume integrals, respectively. Note that the kernel influences are extremely localized due to the convective nature of the kernels, thus, the surface and volume integrations are performed only within these domains of influence. The localization of the kernels becomes more prominent as the Peclet number of the flow increases. Due to increasing sparsity of the global matrix, iterative solvers become the primary choice for the convective diffusion problems.

Higher-order boundary element methods (BEM) for transient convective diffusion phenomena presented in Parts I and II of the paper are implemented numerically to examine their performance for a series of model problems. In order to highlight the importance of proper resolution in time and space for the transient problems, we consider both the one- and two-dimensional formulations in this paper. We utilize an implicit time-recurring formulation to advance in the time domain. The use of higher-order time interpolation functions facilitates extremely high resolution with respect to time step sizes, and thus necessitates similar levels of spatial resolution. In this paper, linear, quadratic and quartic boundary elements as well as bi-linear, bi-quadratic and bi-quartic volume cells are implemented to ensure a desired high level of accuracy both in time and in space. In order to investigate the performance of the BEM formulations, we introduce five problems of unsteady convection–diffusion that possess exact solutions. For all five numerical examples considered in this paper, the higher-order BEM demonstrate an extremely high level of accuracy even for predominantly convective flows. It is shown that the use of the convective kernels provide an analytical upwinding for any Peclet number and any mesh orientation with respect to the flow direction. The introduction of the kernel influence domains facilitates a very efficient and robust algorithm for integration over boundary elements and volume cells. Due to the nature of the convective kernels, the resulting global matrix is sparse with the non-zero elements localized around the main diagonal. The preconditioned bi-conjugate gradient method utilized in the paper allows very efficient factorization of the global sparse matrix. Moreover, the efficiency of the BEM formulations increases dramatically with the increase of the Peclet number of the flow.

A general formulation of higher-order boundary element methods (BEM) is presented for time-dependent convective diffusion problems in one- and multi-dimensions. Free-space time-dependent convective diffusion fundamental solutions originally proposed by Carslaw and Jaeger are used to obtain the boundary integral formulation. Linear, quadratic and quartic time interpolation functions are introduced in this paper for approximate representation of time-dependent boundary temperatures and normal fluxes. Closed form time integration of the kernels is mandatory to attain both accuracy and efficiency of the numerical approach. A complete set of time integrals for the one-dimensional formulation is presented here for the first time in the literature.

Despite the significant number of publications on boundary element methods (BEM) for time-dependent problems of heat diffusion, there still remain issues that need to be addressed, most importantly accuracy of the numerical modelling. Although very precise for steady-state problems, the common boundary element methods applied to transient problems do not yield highly accurate numerical solutions. This paper investigates the reasons that prohibit achievement of a high level of accuracy for transient heat diffusion problems with continuous temperature and bounded heat flux solutions. In order to greatly enhance the commonly used boundary element formulations, we propose higher-order time interpolation functions, including quadratic and quartic approximations. We show that the use of higher-order time functions greatly reduces the numerical error concentrated in the corner regions, and results in very good uniformity of the flux and temperature distributions along the boundaries for problems where uniform distributions are expected. In order to highlight the importance of proper resolution both in time and space for the transient problems, we consider one- and two-dimensional formulations in this paper. High-order boundary elements using quartic shape functions, as well as high-order bi-quartic volume cells, are used to attain mesh-independent numerical solutions. We consider four transient heat diffusion problems that possess exact solutions to investigate the convergence rate and accuracy of the higher-order boundary element formulations. A very high level of accuracy is possible for both one- and two-dimensional formulations. Additionally, we show that the accuracy of a commercially available finite-element code is far less than that of the boundary element methods for a given spatial and temporal discretization. Copyright © 2002 John Wiley & Sons, Ltd.

A boundary element method (BEM) for steady viscous fluid flow at high Reynolds numbers is presented. The new integral formulation with a poly-region approach involves the use of the convective kernel with slight compressibility that was previously employed by Grigoriev and Fafurin [1] for driven cavity flows with Reynolds numbers up to 1000. In order to avoid the overdeterminancy of the global set of equations when using eight-noded rectangular volume cells from that previous work, 12-noded hexagonal volume regions are introduced. As a result, the number of linearly independent integral equations for each node becomes equal to the degrees of freedom of the node. The numerical results for square-driven cavity flow having Reynolds numbers up to 5000 are compared to those obtained by Ghia et al. [2] and demonstrate a high level of accuracy even in resolving the secondary vortices at the corners of the cavity. Next, a comprehensive study is done for backward-facing step flows at Re=500 and 800 using the BEM, along with a standard Galerkin-based finite element methods (FEM). The numerical methods are in excellent agreement with the benchmark solution published by Gartling [3]. However, several additional aspects of the problem are also considered, including the effect of the inflow boundary location and the traction singularity at the step corner. Furthermore, a preliminary comparative study of the poly-region BEM versus the standard FEM indicates that the new method is more than competitive in terms of accuracy and efficiency. Copyright © 1999 John Wiley & Sons, Ltd.

A new boundary element method is presented for steady incompressible flow at moderate and high Reynolds numbers. The whole domain is discretized into a number of eight-noded cells, for each of which the governing boundary integral equation is formulated exclusively in terms of velocities and tractions. The kernels used in this paper are the fundamental solutions of the linearized Navier-Stokes equations with artificial compressibility. Significant attention is given to the numerical evaluation of the integrals over quadratic boundary elements as well as over quadratic quadrilateral volume cells in order to ensure a high accuracy level at high Reynolds numbers. As an illustration, square driven cavity flows are considered for Reynolds numbers up to 1000. Numerical results demonstrate both the high convergence rate, even when using simple (direct) iterations, and the appropriate level of accuracy of the proposed method. Although the method yields a high level of accuracy in the primary vortex region, the secondary vortices are not properly resolved.

We have recently developed a novel multi-level boundary element method (MLBEM) for steady heat diffusion in irregular two-dimensional domains (Numerical Heat Transfer Part B: Fundamentals, 46 : 329–356, 2004). This presentation extends the MLBEM methodology to three-dimensional problems. First, we outline a 3-D MLBEM formulation for steady heat diffusion and discuss the differences between multi-level algorithms for two and three dimensions. Then, we consider an example problem that involves heat conduction in a semi-infinite three-dimensional domain. We investigate the performance of the MLBEM formulation using a single-patch approach. The MLBEM algorithms are shown facilitate fast and accurate numerical solutions with no loss of the solution accuracy. More dramatic speed-ups can be achieved provided that patch-edge corrections are also evaluated using multi-level technique.Copyright © 2005 by ASME

Previous work on the development of multi-level boundary element methods (MLBEM) has produced dramatic computational efficiencies compared to traditional methods. However, for problems involving the oscillatory kernels associated with the Helmholtz equation, the performance improvements demonstrated to date have been limited due to the presence of coarse mesh Nyquist constraints. In the present paper, we examine an extension of the original Helmholtz MLBEM to remove these limitations, based on a polar decomposition of the kernels and field variables originally proposed by Brandt. In this initial numerical implementation of these ideas, the algorithm is applied to a pair of two-dimensional half-space problems and, therefore, is limited to the fast evaluation of Rayleigh integrals. In general, excellent performance is achieved. For example, in the second problem with a non-dimensional wave number equal to 105 , we obtain speed-up factors of approximately 300,000 versus a traditional BEM approach, while maintaining comparable levels of accuracy.Copyright © 2005 by ASME

Higher-order boundary element methods (BEM) are presented for three-dimensional steady convective heat diffusion at high Peclet numbers. An accurate and efficient boundary element formulation is facilitated by the definition of an influence domain due to convective kernels. This approach essentially localizes the surface integrations only within the domain of influence which becomes more narrowly focused as the Peclet number increases. The outcome of this phenomenon is an increased sparsity and improved conditioning of the global matrix. Therefore, iterative solvers for sparse matrices become a very efficient and robust tool for the corresponding boundary element matrices. In this paper, we consider an example problem with an exact solution and investigate the accuracy and efficiency of the higher-order BEM formulations for high Peclet numbers in the range from 1,000 to 100,000. The bi-quartic boundary elements included in this study are shown to provide very efficient and extremely accurate solutions, even on a single engineering workstation.Copyright © 2004 by ASME

In this presentation, we re-visit the poly-region boundary element methods (BEM) proposed earlier for the steady Navier-Stokes [1] and Boussinesq [2] flows, and develop a novel higher-order BEM formulation for the thermoviscous fluid flows that involves the definition of the domains of kernel influences due to steady Oseenlets. We introduce region-by-region implementation of the steady-state Oseenlets within the poly-region boundary element fequatramework, and perform integration only over the (parts of) higher-order boundary elements and volume cells that are influenced by the kernels. No integration outside the domains of the kernel influences are needed. Owing to the properties of the convective Oseenlets, the kernel influences are very local and propagate upstream. The localization becomes more prominent as the Reynolds number of the flow increases. This improves the conditioning of the global matrix, which in turn, facilitates an efficient use of the iterative solvers for the sparse matrices [3]. Here, we consider quartic boundary elements and bi-quartic volume cells to ensure a high level resolution in space. Similar to the previous developments [4–6], coefficients of the discrete boundary integral equations are evaluated with the sufficient precision using semi-analytic approach to ensure exceptional accuracy of the boundary element formulation. To demonstrate the attractiveness of the poly-region BEM formulation, we consider a numerical example of the well-known Rayleigh-Benard problem governed by the Boussinesq equations.Copyright © 2004 by ASME

A new space-time multi-level boundary element method (MLBEM) is developed for transient heat diffusion problems in two-dimensions. This approach extends the MLBEM approach for steady heat diffusion [1] to accommodate fast time convolution algorithm [2]. The space-time MLBEM algorithm developed in this presentation provides fast, accurate and memory efficient numerical solutions for time-dependent heat diffusion problems. Conventional BEM approaches using M boundary elements result in operation counts of order O(M2N2) 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(M log MN3/2) for a two-dimensional model problem using the multi-level boundary element algorithm. Memory requirements are also significantly reduced, while maintaining the same level of accuracy as the conventional time domain BEM approach.

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(N3/2 ) for a couple of 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 © 2004 by ASME

Recently, we have developed multi-level boundary element methods (MLBEM) for the solution of the Laplace and Helmholtz equations that involve asymptotically decaying non-oscillatory and oscillatory singular kernels, respectively. The accuracy and efficiency of the fast boundary element methods for steady-state heat diffusion and accoustics problems have been investigated for square domains. The current work extends the MLBEM methodology to the solution of Stokes equation in more complex two dimensional domains. The performance of the fast boundary element method for the Stokes flows is first investigated for a model problem in a unit square. Then, we study the performance of the MLBEM algorithm in a C-shaped domain.Copyright © 2004 by ASME

An accurate and efficient multi-grid boundary element method is developed to solve steady heat diffusion problems. This novel method extends the pioneering work of Brandt and Lubrecht on multi-level multi-integration in several important ways to address problems with mixed boundary conditions. We utilize bi-conjugate gradient methods and implement the multi-level multi-integration approach for fast matrix and matrix transpose multiplication for e very iteration loop. Furthermore, by introducing a C-c ycle multigrid algorithm, we find that the number of iterations for the bi-conjugate gradient methods is independent of the boundary element mesh discretization.

A fast, accurate and efficient multi-level boundary element method is developed to solve general boundary value problems. Here we concentrate on problems of two-dimensional steady potential flow and present a fast direct boundary element formulation. This novel method extends the pioneering work of Brandt and Lubrecht on multi-level multi-integration (MLMI) in several important ways to address problems with mixed boundary conditions. We utilize bi-conjugate gradient methods (BCGM) and implement the MLMI approach for fast matrix and matrix transpose multiplication for every iteration loop. Furthermore, by introducing a C-cycle multigrid algorithm, we find that the number of iterations for the bi-conjugate gradient methods is independent of the boundary element mesh discretization for problems of steady-state heat diffusion considered in this paper. As a result, the computational complexity of the proposed method is proportional to only N · log(N), where N is the number of degrees of freedom.Copyright © 2003 by ASME

Higher-order boundary element methods (BEM) are presented for time-dependent convective diffusion in two dimensions. The time-dependent convective diffusion free-space fundamental solutions originally proposed by Carslaw and Jaeger are used to obtain the boundary integral formulation. Boundary element method solutions up to the Peclet number 106 are obtained for an example problem of unsteady convection-diffusion that possesses an exact solution. We investigate the convergence rate and accuracy of the higher-order boundary element formulations. An extremely high accuracy of the BEM solutions for highly convective flows is demonstrated. Moreover, it is shown that the use of time-dependent convective kernels provides an automatic upwinding for the entire range of Peclet numbers and also leads to very efficient algorithms as the Peclet number increases.Copyright © 2003 by ASME

Publisher Summary Higher-order boundary element methods (BEM) are presented for time-dependent convective diffusion in two dimensions. To facilitate accurate evaluation of the discrete integral equation coefficients over both singular and nonsingular boundary elements because of complex behavior of the time-dependent convective kernels, the chapter develops and implements a semianalytical algorithm. The time-dependent convective-diffusion-free-space fundamental solutions originally proposed by Carslaw and Jaeger are used to obtain the boundary integral formulation. In this chapter, BEM solutions up to the Peclet number 106 are obtained for a problem of unsteady convection-diffusion that possesses an exact solution. An extremely high accuracy of the BEM solutions for highly convective flows is demonstrated. Moreover, it is shown that the use of time-dependent convective kernels provide an automatic upwinding for the entire range of Peclet numbers and also lead to very efficient algorithms as the Peclet number increases. Higher-order boundary element methods for time-dependent convective diffusion problems are also presented. Linear, quadratic, and quartic interpolation functions are utilized for temporal and spatial discretizations of the boundary integral equation.

Publisher Summary This chapter develops a fast, accurate, and efficient multilevel iterative method to solve general boundary value problems arising in computational mechanics. It concentrates on problems of two-dimensional steady potential flow and presents a fast direct boundary element method. This novel method extends the work on multilevel multiintegration (MLMI) in several important ways to address problems with arbitrary geometry and mixed boundary conditions. Researchers use biconjugate gradient methods and implement the MLMI approach for fast matrix and matrix transpose multiplication for every iteration loop. A multigrid algorithm is introduced to find that the number of iterations for the bi-conjugate gradient methods is independent of the boundary-element mesh discretization for problems of steady-state heat diffusion discussed in this chapter. The method can be extended in a straightforward manner to the solution of many problems in science and engineering that result in very large sets of matrix equations when the associated integral equations are discretized.

In order to better understand the performance of clutches, brakes and seals, there is a need to study the multiscale thermomechanical interactions that occur across rough interfacial sliding contacts. While the full three-dimensional problem is at present somewhat beyond reach from a computational standpoint, we can gain some insight into the problem by examining the more idealized axisymmetric case. Here we develop a transient coupled thermomechanical boundary element method to investigate the behaviour of sliding rings with axisymmetric roughness. By incorporating our recently developed fast convolution algorithm, we are able to consider the spatially and temporally localized response at asperity-level contacts, along with the evolution toward a global component-level steady state. In particular, we investigate behaviour for the sliding ring problem by considering surface profiles that incorporate axisymmetric roughness over several orders-of-magnitude. One of the most interesting findings that emerge from these simulations is the development of roughness-dependent steady-state shapes. Details of the short-time transient interactions are found to significantly affect the long-term component level response. This suggests that one may be able to control overall component deformations through the careful specification of surface profiles.

Higher-order boundary element methods (BEM) are presented for time-dependent convective diffusion problems. The time-dependent convective diffusion free-space fundamental solutions originally proposed by Carslaw and Jaeger are used to obtain the boundary integral formulation. For the linear, quadratic and quartic time interpolation functions considered in this paper, a complete set of closed form time integrals for the one-dimensional formulation are developed. Boundary element method solutions are obtained for four problems of unsteady convection-diffusion, including shock wave propagation. It is shown that the BEM solutions are extremely accurate in contrast to finite-difference and finite-element methods. Moreover, no upwinding is needed for the boundary element methods, even for high Peclet number flows. Finally, the conventional BEM formulation is extended to a problem involving singular flux arising due to a sudden rise of temperature on the boundary. This infinite flux BEM formulation provides significantly more accurate numerical results than the conventional approach.