Application of a High Order Accurate Meshless Method to Solution of Heat Conduction in Complex Geometries

Abstract

In recent years, a variety of meshless methods have been developed to solve partial differential equations in complex domains. Meshless methods discretize the partial differential equations over scattered points instead of grids. Radial basis functions (RBFs) have been popularly used as high accuracy interpolants of function values at scattered locations. In this paper, we apply the polyharmonic splines (PHS) as the RBF together with appended polynomial and solve the heat conduction equation in several geometries using a collocation procedure. We demonstrate the expected exponential convergence of the numerical solution as the degree of the appended polynomial is increased. The method holds promise to solve several different governing equations in thermal sciences.

Consistency and Convergence of a High Order Accurate Meshless Method for Solution of Incompressible Fluid Flows

“…There are several types of meshless methods used for the solutions of heat transfer and fluid flow problems [14][15][16][17][18][19][20][21][22][23][24][25][26]. The intent of this work is to study the consistency and convergence characteristics of a recently developed high order meshless technique [25] that solves the incompressible Navier-Stokes equations using Polyharmonic Spline Radial Basis Function (PHS-RBF) interpolations of scattered data.…”

Smaller stencil preconditioners for polyharmonic spline RBF-FD discretizations

“…For example, a (restarted) generalized minimal residual (GMRES) method is used in [34,25,6] while a stabilized biconjugate gradient (BiCGstab) method solves the systems in [11,27,43,44,5,17,2]. Krylov solvers with ILU preconditioners (possibly after some prior reordering) are used in [11,27,34,43,44,25,5,17,2,6,5,6,11]. Iterative solvers with ILU preconditioners typically suffer from an increase in the problem size and become less efficient if the number of nonzeros in the matrix increases [6,5].…”

“…While earlier papers on RBF-FD often assumed direct solvers for the linear systems, more recent literature includes discussions on iterative solvers (and preconditioners) but mostly uses well-known methods in a blackbox manner, i.e., without any adaptations taking into account the RBF-FD origin of the linear systems. For example, a (restarted) generalized minimal residual (GMRES) method is used in [34,25,6] while a stabilized biconjugate gradient (BiCGstab) method solves the systems in [11,27,43,44,5,17,2]. Krylov solvers with ILU preconditioners (possibly after some prior reordering) are used in [11,27,34,43,44,25,5,17,2,6,5,6,11].…”

Shear-Driven Flow in an Elliptical Enclosure Generated by an Inner Rotating Circular Cylinder

“…Collocating the governing equation at a given point gives an implicit equation connecting value at the local point with values at the points in the cloud. Further details of the derivation and properties of the derivative and Laplacian coefficients are given in ourprevious publications[52,54,57,58].…”