Non-Uniform Rational B-Splines and Rational Bezier Triangles for Isogeometric Analysis of Structural Applications

Abstract

Isogeometric Analysis (IGA) is a major advancement in computational analysis that bridges the gap between a computer-aided design (CAD) model, which is typically constructed using Non-Uniform Rational B-splines (NURBS), and a computational model that traditionally uses Lagrange polynomials to represent the geometry and solution variables. In IGA, the same shape functions that are used in CAD are employed for analysis. The direct manipulation of CAD data eliminates approximation errors that emanate from the process of converting the geometry from CAD to Finite Element Analysis (FEA). As a result, IGA allows the exact geometry to be represented at the coarsest level and maintained throughout the analysis process. While IGA was initially introduced to streamline the design and analysis process, this dissertation shows that IGA can also provide improved computational results for complex and highly nonlinear problems in structural mechanics. This dissertation addresses various problems in structural mechanics in the context of IGA, with the use of NURBS and rational Bézier triangles for the description of the parametric and physical spaces. The approaches considered here show that a number of important properties (e.g., high-order smoothness, geometric exactness, reduced number of degrees of freedom, and increased flexibility in discretization) can be achieved, leading to improved numerical solutions. Specifically, using B-splines and a layer-based discretization, a distributed plasticity isogeometric frame model is formulated to capture the spread of plasticity in large-deformation frames. The modeling approach includes an adaptive analysis where the structure of interest is initially modeled with coarse mesh and knots are inserted based on the yielding information at the quadrature points. It is demonstrated that improvement on efficiency and convergence rates is attained. With NURBS, an isogeometric rotation-free multi-layered plate formulation is developed based on a layerwise deformation theory. The derivation assumes a separate displacement field expansion within each layer, and considers transverse displacement component as C0-continuous at dissimilar material interfaces, which is enforced via knot repetition. The separate integration of the in-plane and through-thickness directions allows to capture the complete 3D stresses in a 2D setting. The proposed method is used to predict the behavior of advanced materials such as laminated composites, and the results show advantages in efficiency and accuracy. To increase the flexibility in discretizing complex geometries, rational Bézier triangles for domain triangulation is studied. They are further coupled with a Delaunay-based feature-preserving discretization algorithm for static bending and free vibration analysis of Kirchhoff plates. Lagrange multipliers are employed to explicitly impose high-order continuity constraints and the augmented system is solved iteratively without increasing the matrix size. The resulting discretization is geometrically exact, admits small geometric features, and constitutes C1-continuity. The feature-preserving rational Bézier triangles are further applied to smeared damage modeling of quasi-brittle materials. Due to the ability of Lagrange multipliers to raise global continuity to any desired order, the implicit fourth- and sixth-order gradient damage models are analyzed. The inclusion of higher-order terms in the nonlocal Taylor expansion improves solution accuracy. A local refinement algorithm that resolves marked regions with high resolution while keeping the resulting mesh conforming and well-conditioned is also utilized to improve efficiency. The outcome is a unified modeling framework where the feature-preserving discretization is able to capture the damage initiation and early-stage propagation, and the local refinement technique can then be applied to adaptively refine the mesh in the direction of damage propagation.PHDCivil EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147668/1/ningliu_1.pd