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

Abstract

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.

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

“…Similar to our earlier work [14], coe cients G ( ) (y; k t) and F ( ) (y; k t) in (6) can be represented in the following matrix forms:…”

“…Introduction. We consider higher-order time interpolation functions [1][2][3]14] to obtain the time-discrete boundary integral equation. Then, we concentrate on the time-integrated kernels.…”

“…Similar to their earlier work [14], linear, quadratic and quartic time interpolation functions were utilized for a temporal discretization of the boundary integral equation. In References [1][2][3], the authors considered both one-and two-dimensional BEM formulations.…”

A fast method for solving the heat equation by layer potentials

“…While the finite element or finite difference method are highly popular for this type of problems, the method of layer potentials has distinct advantages, in particular, stability of explicit time stepping and reduction of unknowns to the boundary surface. Boundary element methods that evaluate the time convolution have been discussed in the past decades in several places, such as [4,9,11].…”

Hybrid BEM for the early stage of a 3D unsteady heat diffusion process

“…BEM applied to the C-DHT equation has been widely developed in the last decades [3,4], and a large variety of efficient formulations were established. An interesting approach for improving the accuracy at early time stage has been proposed by Grigoriev and Dargush in [5], where a high-order BEM has been established in terms of a singular flux formulation. Other approaches involve the use of time or space scaling in order to zoom into the boundary layer developed close to the interface and perform asymptotic matching with the far field solution.…”