A Time-Space Multi-Level Boundary Element Approach for Time-Dependent Heat Diffusion

2004 ◽  
Author(s):  
M. M. Grigoriev ◽  
G. F. Dargush
Keyword(s):  
Boundary Element ◽  
Space Time ◽  
Two Dimensions ◽  
Heat Diffusion ◽  
Time Dependent ◽  
Fast Time ◽  
Convolution Algorithm ◽  
Multi Level ◽  
Transient Heat Diffusion ◽  
Diffusion Problems

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 Fast Time Convolution Algorithm for Unsteady Heat Diffusion Problems

2004 ◽  
Author(s):  
C. H. Wang ◽  
M. M. Grigoriev ◽  
G. F. Dargush
Keyword(s):  
Time Domain ◽  
Boundary Element Methods ◽  
Heat Diffusion ◽  
Fast Time ◽  
Convolution Integrals ◽  
Convolution Algorithm ◽  
Multi Level ◽  
Linear Problems ◽  
Transient Heat Diffusion ◽  
Memory Efficient

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.

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.

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.

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