Isogeometric analysis of models with arbitrary topology for CAD/CAE integration

Abstract

Isogeometric Analysis (IGA) was proposed in recent years as an alternative methodology 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 the basis functions used to exactly model the geometry will also serve as the basis functions for the solution space of the numerical method. Most of the existing efforts are mainly focused on investigating different bases for IGA that should be simple for design and suitable for analysis. Basis functions of B-splines, NURBS and T-splines are popularly used for IGA over the past few years. They all work well for models with a regular control mesh. However, B-splines and NURBS have severe limitations in defining models with arbitrary topology. T-splines have excellent properties of local refinements, but have approximation problems and continuity limitations in arbitrary topology. The scenario is also true for other commonly used basis functions. To tackle the above limitations, we developed a new method for both modeling and isogeometric analysis using a control mesh with arbitrary topology. The proposed method uses mapped basis functions that guarantee the continuity at extraordinary vertices being the same as that at regular regions. Based on the input of an arbitrary quadrilateral control mesh, a global parameterization of the final surface is first defined through a Gravity Center Method (GCM). A re-parameterization method is then applied to map a basis function to others that are explicitly defined and are tailored to each of the control vertices of the given control mesh. The final surface is defined by all the input control vertices with their corresponding mapped basis functions. Depending on the continuity of the basis function used for mapping to others, the global continuity of the resulting surface, including at extraordinary vertices, can be arbitrary higher order. The proposed method using mapped basis functions can be applied with any kinds of basis functions. First, we developed a scheme using mapped cubic B-spline basis functions in both shape modeling and isogeometric analysis. We also developed a new kind of basis function, called truncated interpolatory basis function (TIBF), and an algorithm for parametric mesh regularization (PMR) for interpolatory IGA, which is useful for direct analysis of models defined using interpolatory modeling schemes. With both of the schemes using mapped cubic B-spline basis functions and mapped TIBFs, a global continuity of C2 can be achieved. Among various performance indicators, convergence rate is an important factor in evaluating a particular scheme for its suitability in isogeometric analysis. In the literature, one can find theoretical convergence rates in a regular setting for polynomial-based schemes in both isogeometric analysis and other established methods in finite element analysis (FEA). In cases of meshes of arbitrary topology or non-polynomial based schemes, however, it is rather difficult to find a theoretical evaluation of convergence rate and there is no standard numerical method for the evaluation of convergence rate in isogeometric analysis. In this research, we also propose a method for numerical evaluation of convergence rate based on L2 projection, a set of standard stencils representing the underlying geometry, and a class of scaled target functions (L2-STF) representing the underlying field solutions in the solution space. To validate the proposed method, we have applied the proposed numerical method for the evaluation of convergence rates in isogeometric analysis using B-splines for regular meshes, mapped cubic Bspline basis functions, and T-splines using unstructured meshes. The results show that the numericalmethod produces true theoretical rates for B-splines and reasonable rates for meshes of arbitrary topology. To summarize, we conducted a systematic study on isogeometric analysis and developed new schemes in isogeometric analysis using mapped basis functions for models with arbitrary topology. A numerical method is also developed for related evaluation of convergence rates. The results show that the proposed schemes using mapped basis functions have many attractive features and the proposed numerical method for convergence rate evaluation is reliable and useful in isogeometric analysis, especially for models of arbitrary topology.

Date of Award	15 Jul 2015
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Weiyin MA (Supervisor)