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

Abstract

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.

A general algorithm for the numerical evaluation of domain integrals in 3D boundary element method for transient heat conduction

“…We may call these integrals as pseudo-singular integrals. The difficulty of numerically integrating a function with such behavior can introduce numerical unstable problems into the solution, as reported in [11][12][13][14]. Thus accurate calculation of the domain integrals is important to the successful implementation of the pseudo-initial condition method.…”

A new method for the numerical evaluation of domain integrals in a 3D boundary element method for transient heat conduction

“…This is because as the time step approaches zero, the integrand in the domain integral (the time-dependent fundamental solution) is close to singular as the source point is on the integration element. The difficulty of numerically integrating a function with such behavior can introduce numerical unstable problems into the solution, as reported in [11][12][13][14]. Thus accurate calculation of the domain integrals is important for the successful implementation of the pseudo-initial condition method.…”