A generalized boundary integral method for transient heat diffusion in isotropic heterogeneous media

In this dissertation, a Laplace transform method has been developed to solve the transient heat diffusion equation in isotropic heterogeneous media. It has been necessary to extend the boundary integral method developed by Divo and Kassab[0] for steady heat conduction to account for this new Laplace transform formulation. Using the Laplace transform, an integral equation is derived in the Laplace transform space, and it involves both boundary and domain integrals. A new domain integral arises, in addition to the sifting deviation integral encountered in the steady state case. The domain integral is manipulated analytically and written as the sum of two integrals, both of which are approximated using radial basis functions (RBF's). This operation effectively and significantly reduces the error in approximating the single original sifting deviation integral by RBF's. RBF's are radially symmetric expansion functions, which in turn are chosen to be the divergence of certain (vector-) functions appropriate to the problem under consideration. This expansion subsequently allows for the transformation of domain integrals into boundary integrals. It is noted that this formulation can also be used in the case of constant conductivity. That is, a general method has been proposed, herein, which solves the Laplace transform heat conduction problem by relying on the steady-state fundamental solution, whether it is the usual potential fundamental solution (−(1/2π)lnr in 2-D or 1/4πr in 3-D) in the case of constant conductivity, or the Divo and Kassab generalized fundamental solution, in the case of spatially varying thermal conductivity.; Following a brief review of the generalized BEM developed by Divo and Kassab[0], the proposed Laplace transform solution to the transient problem in non-homogeneous media is presented in detail. The Stehfest transform was used to invert the Laplace transformed boundary element solution for the temperature back to the time domain. Due to the ill-conditioned nature of the Laplace transform inversion process, very accurate spatial resolution of the temperature is required to produce accurate temperatures in the time domain. As such, considerable attention was given to addressing issues critical for the successful implementation of the Laplace transform method for transient heat conduction in heterogeneous media.; First, higher order discretization of boundary integrals were developed. Discontinuous bilinear and biquadratic elements were derived and incorporated into the 3-D code. Higher order elements produced much improved boundary fluxes and temperatures. Furthermore, accurate evaluation of the sifting deviation area integral is shown to be critical to further improvement of the heat flux. It is shown that clustering of the expansion points used in the radial-basis-function approximation of the domain integral is necessary to obtain accurate values of the sifting deviation integral. A modified RBF algorithm is proposed to implement the clustering method. Here the temperature difference in the integrand itself is modeled using RBF's, and the integrand as a whole is subsequently approximated using RBFs with clustering, without introducing any new unknowns. Effectively, the original set of poles at which the temperature is evaluated at the interior is retained, while refining the actual pole distribution used to approximate the sifting deviation integral. Results from numerical experiments validate this viable implementation of the clustering scheme. Finally, a series of 3-D numerical examples demonstrate that the proposed approach yields accurate predictions in the time domain.