A hybrid nodal-integral/finite-element approach for solution of PDEs in arbitrary geometries

Nodal-integral methods (NIM) are a class of efficient coarse mesh methods. The nodal integral method can achieve the same accuracy as many conventional numerical methods, like the finite-element and finite-volume method using a coarser mesh and lower CPU time. The transverse integration procedure employed in NIM limits its application to regular domains that can be accurately discretized using either rectangular (2D) or cuboid (3D) cells. To take advantage of the efficient NIM method and extend its applicability to arbitrary domains, a hybrid nodal-integral/finite-element method (NI-FEM) is developed in this dissertation. The bulk of the domain is discretized using coarse NIM cells and the rest of the domain, which cannot be accurately captured by coarse NIM cells, is discretized using FEM elements. This allows hybrid NI-FEM to combine the advantages of the NIM with the versatility of the FEM. Since the NIM is used in the bulk of the domain, this results in significant reduction in computation time compared to discretizing the domain entirely using FEM elements. The interfacial conditions at the common interfaces between the NIM cells and the FEM elements relate the transverse-averaged discrete unknowns of the NIM with FEM's point discrete unknowns. The interfacial conditions are developed based on the continuity across the common interface. In this dissertation, the hybrid NI-FEM is developed to solve several partial differential equations (PDEs) relevant to nuclear engineering; such as, 1) Convection-diffusion equation (CDE), 2) Burgers equation, and finally 3) the incompressible Navier-Stokes equation. Extensive benchmarking is carried out for these PDEs by comparing the numerical solutions to the exact (or often manufactured) solutions. The order of accuracy in both space and time are reported here for the hybrid method. The hybrid method is also compared with standalone FEM approach and is proven to be efficient for similar level of accuracy. A parallel implementation of the hybrid NI-FEM is developed. The effect of the ratio of the NIM to FEM subdomain sizes on the efficiency is carried out to determine when the hybrid approach is more attractive than the conventional FEM.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Nuclear, Plasma, Radiolgc Engr

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Namala, Sundar

Contributors dc:contributor

• Uddin, Rizwan
• Kozlowski, Tomasz
• Jewett, Brian
• Munk, Madicken

• Hybrid Method
• Nodal Integral Method (NIM)
• Finite Element Method (FEM)

Statement dc:rights

• Copyright 2023 Sundar Namala

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/120290

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F120290