Duality and radial basis functions (RBFs) are compelling principles which are currently underrepresented in computational engineering. Following an introduction to the mathematics, these methodologies are applied to a series of toy problems.
To begin, both methodologies are integrated into a predictor-corrector algorithm, which is then used to model transient thermal diffusion in a one-dimensional domain. Following discretization of the heat equation with the finite element method (FEM) and RBFs, Euclidean temperature errors of 9.613E-2 (RBF) and 1.442E-1 (FEM) are observed at five seconds and 7.635E-2 (RBF) and 1.136E-1 (FEM) at ten seconds. In addition, duality is successfully applied as an error bound on the numerical solutions; the duality gap is cheaply computed by the algorithm and converges to zero as the thermal flux and temperature fields iteratively approach their exact solutions.
The RBF methodology is next applied to the interpolation of two complicated data sets. The first is a two-dimensional velocity field associated with a scattered collection of particles. Using a subset of these particles and their velocities as input, the RBFs accurately interpolate the scattered velocity data back on to the original collection of particles. The addition of a high-order polynomial to the RBFs is seen to improve accuracy further with an increase in algorithm complexity. The second test interpolates two-dimensional stress and stress gradient tensor fields on scattered particles. In supplement to the standard RBF technique of the first test, a Hermite RBF technique is devised which uses given stress and stress gradient data as input. The Hermite technique achieves an improvement of one order of magnitude in stress interpolation and one to two orders of magnitude in stress gradient interpolation over the standard technique.
In the final tests, RBFs are tested alongside finite differences in the discretization of two-phase heat exchange problems. In the first two cases, the RBFs more accurately compute the position of the planar phase interface with time, as well as the temperature fields in the vapour and liquid-phases. In the third case, it is found that the RBFs track the instantaneous position and velocity of a spherical bubble with higher accuracy.