#### Date of Award

Fall 11-2017

#### Embargo Period

12-13-2017

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mechanical Engineering

#### Advisor(s)

Yongjie Jessica Zhang

#### Abstract

To tightly integrate Computer Aided Design (CAD) and Finite Element Analysis (FEA), isogeometric analysis was proposed by employing the same basis functions for geometric representation in analysis, aiming to eliminate the time-consuming mesh generation in traditional FEA. Non-Uniform Rational B-Spline (NURBS), as the current industry standard in CAD community, was the first candidate utilized in isogeometric analysis. However, due to the global tensor-product nature of NURBS, it does not support local refinement and arbitrary topology. Local refinement is essential in both local editing (from perspective of design) and adaptive analysis (from perspective of analysis), which are known for high efficiency in terms of computation cost. Arbitrary topology, on the other hand, is unavoidable in complex geometries. In this dissertation, we develop various methods to address highly localized refinement that supports arbitrary topology in the context of isogeometric analysis. The proposed methods are suitable for both geometric design and engineering analysis, with necessary properties carefully investigated. We present Truncated Hierarchical Catmull-Clark Subdivision (THCCS), which generalizes truncated hierarchical B-splines to control grids of arbitrary topology. By coupling truncated hierarchical B-splines and Catmull-Clark subdivision, THCCS takes advantages of both methods and hence it possesses several nice properties suitable for geometric design and adaptive analysis. THCCS basis functions satisfy partition of unity, are linearly independent, and are locally refinable. THCCS also preserves geometry during adaptive h-refinement and thus inherits the surface continuity of Catmull-Clark subdivision, namely C2-continuous everywhere except at the local region surrounding extraordinary nodes, where the surface continuity is C1. However, the local refinement of THCCS imposes a strong restraint that in each refinement step, at least the two-ring neighborhood of a control point needs to be refined. To improve its refinement locality, an extended version of THCCS, namely extended Truncated Hierarchical Catmull-Clark Subdivision, is proposed. Based on the Stam’s method, we first build a set of basis functions over arbitrary quadrilateral meshes and apply them to THCCS. Then, a new basis-function-insertion scheme is developed with the aid of the truncation mechanism, which refines one-ring neighborhood rather than two-ring neighborhood. Therefore, eTHCCS significantly improves the efficiency of local refinement compared with THCCS. Despite the flexibility for Catmull-Clark subdivision in handling extraordinary points, it is not compatible with the current CAD industry standard, i.e., NURBS. T-splines, on the other hand, as a superior alternative to NURBS, have no such issue. We present Truncated T-splines as a new type of T-splines suitable for both geometric design and analysis, supporting highly localized refinement. By introducing truncation mechanism into T-spline basis functions, truncated T-splines release the strong T-mesh constraints imposed in analysis-suitable T-splines. Thus, truncated T-splines induce less refinement propagation than analysis-suitable T-splines. Truncated T-spline basis functions are piecewise polynomials that are linearly independent and form a partition of unity. Refinement of truncated T-splines preserves geometry and produces nested spline spaces. Furthermore, we study truncated T-splines and local refinement on the general domain (2-manifold) with extraordinary points in the T-mesh. G1 continuity is attained around extraordinary points by properly capping quartic B´ezier patches, where a constrained optimization problem is solved. The above-mentioned three methods, namely THCCS, eTHCCS and truncated T-splines, work only for quad meshes. To study local refinement on complex volumetric geometries, we propose a novel method termed truncated hierarchical tricubic splines (TH-spline3D) that take unstructured hexahedral meshes as input control meshes. Initially on the input mesh, the blending functions are defined as linear combinations of tricubic Bernstein polynomials, which is an extension of B´ezier extraction on unstructured quad meshes. We next build the hierarchical structure and apply the truncation mechanism to such blending functions for highly localized refinement. During the refinement, high-level blending functions are added in the solution space, whereas certain low-level ones are discarded or truncated depending on the region that is marked for refinement. The hierarchical blending functions are polynomials that form a partition of unity, and their support is reduced due to the truncation mechanism, yielding sparser stiness matrices than the classical hierarchical refinement. TH-spline3D support B´ezier extraction such that it can be easily incorporated into the existing finite element frameworks. However, the optimal convergence rate is lost in analysis when we use Catmull-Clark subdivision functions or express basis functions as linear combinations of Bernstein polynomials around extraordinary points. We present a novel blended B-spline parameterization method to restore optimal convergence rates when unstructured quadrilateral/hexahedral meshes are taken as input control meshes. C0 and C1 B-spline functions are adopted in boundary elements and irregular elements around extraordinary edges/vertices, whereas C1 and C2 B-spline functions are used in regular elements away from extraordinary edges/vertices. The truncation mechanism is employed for a smooth transition from irregular to regular elements. The resulting geometry of such blended construction is C0-continuous around extraordinary edges/vertices, C1-continuous across the interface between irregular and regular elements, and C2-continuous elsewhere. The blended B-spline construction yields consistent parameterization over refinement. Moreover, it exhibits optimal convergence rates. Spline functions in the blended construction form a non-negative partition of unity, are linearly independent, capture desired sharp features on the boundary surface, and naturally support B´ezier extraction such that the construction can be used in existing finite element frameworks. In summary, we propose several methods to support local refinement on complex geometries, aiming to achieve a high refinement locality. Several properties are particularly investigated, such as partition of unity, geometry preservation, nestedness of spline space, as well as (global) linear independence. With these nice properties, the proposed methods can serve as promising candidates for isogeometric analysis.

#### Recommended Citation

Wei, Xiaodong, "Highly Localized Refinement Using Truncation Mechanism on Unstructured Quadrilateral and Hexahedral Meshes" (2017). *Dissertations*. 1097.

http://repository.cmu.edu/dissertations/1097