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Application of an RBF blending interpolation method to problems with shocks

Harris M., Kassab A., Divo E.

Computer Assisted Methods in Engineering and Science

2015

Vol. 22, no. 3

229--241

Radial basis functions (RBF) have become an area of research in recent years, especially in the use of solving partial differential equations (PDE). Radial basis functions have an impressive capability in interpolating scattered data, even for data with discontinuities. Although, for infinitely smooth radial basis functions such as the multi-quadrics and inverse multi-quadrics, the shape parameter must be chosen properly to obtain accurate approximations while avoiding ill-conditioning of the interpolating matrices. The optimum shape parameter can vary depending on the field, such as in locations of sharp gradients or shocks. Typically, the shape parameter is chosen to maintain a high conditioning number for the interpolation matrix, rendering the RBF smooth [1–10]. However, this strategy fails for a problem with a shock or sharp discontinuity. Instead, in such cases the conditioning number must be kept small. The focus of this work is then to demonstrate the use of RBF interpolation in the approximation of sharp gradients or shocks by use of a RBF blending interpolation approach. This RBF blending interpolation approach is used to maintain the optimum shape parameter depending on the field. The approach is able to sense gradients or shocks in the field and adjust the shape parameter accordingly to keep excellent accuracy. Presented in this work, is an explanation of the RBF blending interpolation methodology and testing of the RBF blending interpolation approach by solving the Burger’s equation using the virtual finite difference method.

A model-integrated localized collocation meshless method (MIMS)

Gerace S., Erhart K., Kassab A., Divo E.

Computer Assisted Methods in Engineering and Science

2013

Vol. 20, no. 3

207--225

A model integrated meshless solver (MIMS) tailored to solve practical large-scale industrial problems is based on robust meshless methods strategies that integrate a native model-based point generation procedures. The MIMS approach fully exploits strengths of meshless methods to achieve automation, stability, and accuracy by blending meshless solution strategies based on a variety of shape functions to achieve stable and accurate iteration process that is integrated with a newly developed, highly adaptive model generation employing quaternary triangular surface discretization for the boundary, a binary-subdivision discretization for the interior, and a unique shadow layer discretization for near-boundary regions. Together, these discretization techniques provide directionally independent, automatic refinement of the underlying native problem model to generate accurate adaptive solutions without need for intermediate user intervention. By coupling the model generation with the solution process, MIMS addresses issues posed by ill-constructed geometric input and pathologies often generated from solid models in the course of CAD design.