Efficiency of boundary element methods for time-dependent convective heat diffusion at high Peclet numbers

2004 ◽  
Vol 21 (4) ◽  
pp. 149-161 ◽  
Author(s):  
M. M. Grigoriev ◽  
G. F. Dargush
Keyword(s):  
Boundary Element ◽  
Convective Heat ◽  
Boundary Element Methods ◽  
Heat Diffusion ◽  
Time Dependent

A Boundary Element Method for Three-Dimensional Steady Convective Heat Diffusion

2004 ◽  
Author(s):  
M. M. Grigoriev ◽  
G. F. Dargush
Keyword(s):  
Boundary Element ◽  
Convective Heat ◽  
Sparse Matrices ◽  
Three Dimensional ◽  
Higher Order ◽  
Iterative Solvers ◽  
Boundary Element Methods ◽  
Heat Diffusion ◽  
Element Formulation ◽  
Definition Of

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.

Accurate Boundary Element Solutions for Highly Convective Unsteady Heat Flows

2005 ◽  
Vol 127 (10) ◽  
pp. 1138-1150 ◽  
Author(s):  
M. M. Grigoriev ◽  
G. F. Dargush
Keyword(s):  
Boundary Element ◽  
Peclet Number ◽  
Iterative Solvers ◽  
Heat Diffusion ◽  
Time Dependent ◽  
Time Interval ◽  
Spatial Integration ◽  
Péclet Number ◽  
Heat Flows ◽  
Analytic Integration

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 Fast Multi-Level Boundary Element Algorithm for Stokes Flows in Two-Dimensions

2004 ◽  
Author(s):  
G. F. Dargush ◽  
M. M. Grigoriev
Keyword(s):  
Boundary Element ◽  
Two Dimensions ◽  
Boundary Element Methods ◽  
Heat Diffusion ◽  
Helmholtz Equations ◽  
Stokes Flows ◽  
Singular Kernels ◽  
Multi Level ◽  
Steady State Heat ◽  
Shaped Domain

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.

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