Isogeometric Collocation Methods for Analysis-suitable Parameterization and Thermal Analysis

Abstract

Isogeometric Analysis (IGA) was first introduced in 2005 as an alternative approach for finite element analysis by integrating Computer-Aided Design (CAD) and downstream analysis without the use of an intermediate mesh model. The main idea of IGA is that basis functions used for defining the geometry will also act as the basis in the process of analysis, which consequently leads to the use of the exact representation of the geometry for engineering analysis. In the literature, the basis functions of B-splines, NURBS, and T-splines are commonly used for IGA. Most of the prevailing efforts are mainly focused on isogeometric analysis based on the Galerkin formulation. Collocation methods, on the other hand, have also been investigated for producing an efficient solution with far less time to achieve the final solution compared with the Galerkin formulation. They have also been applied in various engineering applications, especially when the computation time is a key issue.

In the context of this research, we focus on collocation methods for isogeometric analysis based on non-uniform rational B-splines (NURBS). We first present isogeometric collocation (IGA-C) method for solving partial differential equations (PDEs) based on B-splines and NURBS with different boundary conditions, such as Dirichlet, Neumann, and other boundary conditions. We then investigate PDE-based methods for analysis suitable parameterization of a computational domain using the IGA-C formulation for IGA applications. The parameterization is produced based on the numerical solution of a PDE that is also solved using the IGA-C method with a smooth but simple initial parameterization of a computational domain and with boundary conditions being the known boundary representation of the desired computational domain plus other additional measures, if any, for IGA. The results show that the PDE-based IGA-C method can effectively produce satisfactory parameterization for IGA.

The above IGA-C formulation is further applied in isogeometric thermal analysis with different kinds of boundary conditions and with various forms of general internal heat sources, such as pointwise, curve-shaped, area-shaped, and mixed internal heat sources. A Gaussian distribution heat profile is used for modelling pointwise heating sources and for modelling heat distribution along curve paths and boundary curves of area-shaped heat sources. To effectively represent the solution of temperature distribution using B-splines, strategies are also developed for producing smooth adaptive knots suitable for producing solutions with different forms of internal heat sources. To validate the developed methods, some examples of thermal analysis are produced for simulating the thermal distribution of simplified printed circuit board with consideration of point, curve, and area heat sources, which demonstrates the applicability of the proposed methods in practical engineering applications.

The presented IGA-C formulation is also extended for thermal analysis under spatially varying thermal conductivity and internal heat sources. In addition to different kinds of boundary conditions, one can also easily incorporate other constraints into the final system of equations for achieving other desired properties of the final solution.

Date of Award	27 Sept 2021
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Weiyin MA (Supervisor)