Solution of parabolic differential equations by the boundary element method using discretisation in time

2002

SUMMARYA boundary element method (BEM) for transient heat di usion phenomena presented in Part I is extended to problems involving instantaneous rise of temperature on a portion of the boundary. The new boundary element formulation involves the use of an inÿnite ux function in order to properly capture the singular response of the ux. It is shown that the conventional ÿnite ux BEM formulation, as well as a commercial FEM code, results in a large ÿrst-time-step numerical error that cannot be reduced by mesh or time-step reÿnement. The use of the singular ux formulation for BEM demonstrates an extremely high level of accuracy for the one-dimensional case, and a signiÿcant improvement in the solutions within a two-dimensional representation. The additional errors arising due to improper time interpolation of the temperature on the boundaries adjacent to the singular ux boundary are discussed.

Section: Singular Flux Boundary Element Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Higher‐order boundary element methods for transient diffusion problems. Part II: Singular flux formulation

2002

“…Several alternative approaches for transient heat diffusion have also been proposed, including integral formulations based upon finite difference representations of the time derivative that appears in the governing equation [16,17] and methods based upon the theory of particular integrals [18]. Although these alternative approaches have significant promise, highly accurate solutions have yet to be demonstrated.…”

Section: Introductionmentioning

confidence: 98%

A fast multi‐level convolution boundary element method for transient diffusion problems

Wang

2005

SUMMARYA 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.

“…Although the Laplace (or Fourier) transformation permits solution at any time level without consideration of the temperature-time history, the applicability of the approach is limited to only linear problems. Curran et al [16] and, more recently, Tanaka et al [17] used ÿnite di erences to discretize the time derivative while utilizing steady-state kernels to recast the time-discretized equation into a boundary integral equation. Note that both implicit and explicit time operators could be constructed via time di erences, with the latter subject to time-step constraints.…”

Section: Introductionmentioning

confidence: 99%

Higher‐order boundary element methods for transient diffusion problems. Part I: Bounded flux formulation

2002

SUMMARYDespite the signiÿcant number of publications on boundary element methods (BEM) for time-dependent problems of heat di usion, 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 di usion problems with continuous temperature and bounded heat ux 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 ux 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 di usion 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 ÿnite-element code is far less than that of the boundary element methods for a given spatial and temporal discretization.