Most of the modern engineering analysis methods (Finite Element, Finite Difference, Finite Volume, etc.) rely on space discretizations of the underlying geometric model. Such spatial meshes have to conform to the geometric model in order to approximate boundary conditions, construct basis functions with good local properties as well as perform numerical integration and visualization of the modeling results. Despite recent advances in automatic mesh generation, spatial meshing still remains difficult problem which often requires geometry simplification and feature removal. Conforming spatial mesh also restricts motions and variations of the geometry and breaks design-analysis cycle. In order to overcome difficulties and restrictions of the mesh-based methods, the alternative analysis methods have been proposed. We present a numerical technique for solving engineering analysis problems that combines meshfree method with distance fields, radial basis functions and collocation technique. The proposed approach enhances the collocation method with exact treatment of boundary conditions at all boundary points. It makes it possible to exclude boundary conditions from the collocation equations. This reduces the size of the algebraic system which results in faster solutions. On another hand, the boundary collocation points can be used to enforce the governing equation of the problem which enhances the solutions accuracy. Ability to use unstructured grids empowers the meshfree method with distance fields with higher level of geometric flexibility. In our presentation we demonstrate comparisons of the numerical results given by the combined approach with results delivered by the traditional collocation technique and meshfree method with distance fields.
Highly Accurate Numerical Technique for Population Models via Rational Chebyshev Collocation Method