A multilevel method for meshless solution of the poisson equation in heat transfer and fluid flow

Meshless methods using radial basis functions (RBF) are an attractive alternative to grid based methods for solving partial differential equations in complex geometries. Gaussian, Multiquadratics and inverse Multiquadratics are some of the more popular RBF's, but the require a shape paramter for a stable and accurate solution and also face stagnation issues. Recently, Polyharmonic splines (PHS) with appended polynomials have overcome the aforementioned issues and offer spectral convergence of the discretization errors with the degree of appended polynomials. In this thesis, we present a non-nested multilevel algorithm using the PHS-RBF meshless method for the soluution of the Poisson equation, which commonly arises in numerous heat transfer and fluid flow applications. The PHS-RBF discretization of the Poisson equation leads to a sparse set of equations with unknown variables at each of the scattered point. The non-nested multilevel algorithm solves this set of equations by restricting and prolongating the values and corrections between multiple independently generated coarse set of points by making use of RBF interpolation. The performance of this algorithm is tested for the Poisson equation in three model geometries, using manufactured solutions. Rapid convergence of the residual is observed with Dirichlet boundary conditions using Successive Over-Relaxation(SOR) as the relaxation scheme . However, convergence is seen to be quite modest for the all-Neumann boundary condition, but this poor convergence is ameliorated by using the multilevel algorithm as a preconditioner to the GMRES, which is a Krylov Subspace Projection (KSP) method. This rapid convergence of the all-Neumann equation is then applied to the pressure Poisson equation arising in the fractional step method, with explicit convection and explicit diffusion. We demonstrate fast convergence, both with refinement of number of points and degree of appended polynomials, for various fluid flow problems in complex domains with high accuracy using the meshless fractional step algorithm.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mechanical Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Anand Radhakrishnan, -

Contributors dc:contributor

• Vanka, Surya Pratap

• Meshless
• Multigrid
• Radial Basis Function based Finite Difference
• Polyharmonic Spline
• Poisson Equation
• Incompressible Navier--Stokes Equation

Statement dc:rights

• Copyright 2021 - Anand Radhakrishnan

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/112966

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/112966