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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.

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.

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....

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 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.

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

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.