{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/112966"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/112966","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"A multilevel method for meshless solution of the poisson equation in heat transfer and fluid flow","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Anand Radhakrishnan, -"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"M.S.","degree_level":"Thesis","degree_discipline":"Mechanical Engineering","degree_department":null,"school":null,"contributors":["Vanka, Surya Pratap"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-01-12T21:45:20Z","date_published":"2022-01-12T21:45:20Z","updated_at":"2026-07-22T22:24:52Z","subjects":["Meshless","Multigrid","Radial Basis Function based Finite Difference","Polyharmonic Spline","Poisson Equation","Incompressible Navier--Stokes Equation"],"languages":["en"],"rights":["Copyright 2021 - Anand Radhakrishnan"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/112966","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Vanka, Surya Pratap"]},{"key":"dc:creator","label":"Author","values":["Anand Radhakrishnan, -"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-01-12T21:45:20Z","2021-06-23","2021-08"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mechanical Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.S."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Meshless","Multigrid","Radial Basis Function based Finite Difference","Polyharmonic Spline","Poisson Equation","Incompressible Navier--Stokes Equation"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2021 - Anand Radhakrishnan"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/112966"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-01-12 without embargo terms","The student, - Anand Radhakrishnan, accepted the attached license on 2021-06-18 at 20:07.","The student, - Anand Radhakrishnan, submitted this Thesis for approval on 2021-06-18 at 20:13.","This Thesis was approved for publication on 2021-06-23 at 13:08.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16698 on 2022-01-12 at 12:43:25","Made available in DSpace on 2022-01-12T21:45:20Z (GMT). No. of bitstreams: 2 ANANDRADHAKRISHNAN-THESIS-2021.pdf: 6326171 bytes, checksum: 0912e791e7323d70f4de68c1b5c9ecfc (MD5) LICENSE.txt: 4218 bytes, checksum: 0ee5681545c65db8dbcf6696a526f7b8 (MD5) Previous issue date: 2021-06-23"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["A multilevel method for meshless solution of the poisson equation in heat transfer and fluid flow"]}]}],"canonical_facts":{"dc:contributor":["Vanka, Surya Pratap"],"dc:creator":["Anand Radhakrishnan, -"],"dc:date":["2022-01-12T21:45:20Z","2021-06-23","2021-08"],"dc:description":["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.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-01-12 without embargo terms","The student, - Anand Radhakrishnan, accepted the attached license on 2021-06-18 at 20:07.","The student, - Anand Radhakrishnan, submitted this Thesis for approval on 2021-06-18 at 20:13.","This Thesis was approved for publication on 2021-06-23 at 13:08.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16698 on 2022-01-12 at 12:43:25","Made available in DSpace on 2022-01-12T21:45:20Z (GMT). No. of bitstreams: 2 ANANDRADHAKRISHNAN-THESIS-2021.pdf: 6326171 bytes, checksum: 0912e791e7323d70f4de68c1b5c9ecfc (MD5) LICENSE.txt: 4218 bytes, checksum: 0ee5681545c65db8dbcf6696a526f7b8 (MD5) Previous issue date: 2021-06-23"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/112966"],"dc:language":["en"],"dc:rights":["Copyright 2021 - Anand Radhakrishnan"],"dc:subject":["Meshless","Multigrid","Radial Basis Function based Finite Difference","Polyharmonic Spline","Poisson Equation","Incompressible Navier--Stokes Equation"],"dc:title":["A multilevel method for meshless solution of the poisson equation in heat transfer and fluid flow"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mechanical Engineering"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["M.S."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:52Z"}