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Geometric models, such as for use in CAD/CAM or animation, are often constructed in a piece-wise fashion. Historically, these models have been made of NURBS surfaces. For various reasons it is problematic and often times mathematically impossible to combine several NURBS models into one continuous surface. The recent invention of a surface type called T-splines has made the combining of NURBS surfaces into a single continuous surface possible, but much of the mathematics has yet to be explored. This thesis explores the mathematics and algorithms necessary to merge multiple NURBS, T-spline, or T-NURCC surfaces into a single continuous surface. This thesis addresses two main problems. The first problem is merging surfaces with different parameterizations. In order to merge surfaces with different parameterizations, it is often necessary to modify the parameter values of the surface, which can change the shape of the surface. This change can be alleviated through shape control methods. The second problem is merging surfaces that meet at extraordinary points, or points with a valence other than four. Results show that the merging algorithm is able to successfully convert models composed of multiple NURBS, T-spline, or T-NURCCS surfaces into models composed of a single T-spline or T-NURCC surface. The resulting models are gap-free and contain little distortion in the parameterization.

Constructing spline models for isogeometric analysis is important in integrating design and analysis. Converting designed CAD (Computer Aided Design) models with B-reps to analysis-suitable volumetric T-spline is fundamental for the integration. In this thesis, we work on two directions to achieve this: (a) using Boolean operations and skeletons to build polycubes for feature-preserving high-genus volumetric T-spline construction; and (b) developing weighted T-splines with arbitrary degree for T-spline surface and volume modeling which can be used for analysis. In this thesis, we first develop novel algorithms to build feature-preserving polycubes for volumetric T-spline construction. Then a new type of T-spline named the weighted T-spline with arbitrary degree is defined. It is further used in converting CAD models to analysis-suitable volumetric T-splines. An algorithm is first developed to use Boolean operations in CSG (Constructive Solid Geometry) to generate polycubes robustly, then the polycubes are used to generate volumetric rational solid T-splines. By solving a harmonic field with proper boundary conditions, the input surface is automatically decomposed into regions that are classified into topologically either a cube or a torus. Two Boolean operations, union and difference, are performed with the primitives and polycubes are generated by parametric mapping. With polycubes, octree subdivision is carried out to obtain a volumetric T-mesh. The obtained T-spline surface is C2-continuous everywhere except the local region surrounding irregular nodes, where the surface continuity is elevated from C0 to G1. B´ezier elements are extracted from the constructed solid T-spline models, which are further used in isogeometric analysis. The Boolean operations preserve the topology of the models inherited from design and can generate volumetric T-spline models with better quality. Furthermore, another algorithm is developed which uses skeleton as a guidance to the polycube construction. From the skeleton of the input model, initial cubes in the interior are first constructed. By projecting corners of interior cubes onto the surface and generating a new layer of boundary cubes, the entire interior domain is split into different cubic regions. With the splitting result, octree subdivision is performed to obtain T-spline control mesh or T-mesh. Surface features are classified into three groups: open curves, closed curves and singularity features. For features without introducing new singularities like open or closed curves, we preserve them by aligning to the parametric lines during subdivision, performing volumetric parameterization from frame field, or modifying the skeleton. For features introducing new singularities, we design templates to handle them. With a valid T-mesh, we calculate rational trivariate T-splines and extract B´ezier elements for isogeometric analysis. Weighted T-spline basis functions are designed to satisfy partition of unity and linear independence. The weighted T-spline is proved to be analysis-suitable. Compared to standard T-splines, weighted T-splines have less geometrical constraint and can decrease the number of control points significantly. Trimmed NURBS surfaces of CAD models are reparameterized with weighted T-splines by a new edge interval extension algorithm, with bounded surface error introduced. With knot interval duplication, weighted T-splines are used to deal with extraordinary nodes. With B´ezier coefficient optimization, the surface continuity is elevated from C0 to G1 for the one-ring neighborhood elements. Parametric mapping and sweeping methods are developed to construct volumetric weighted T-splines for isogeometric analysis. Finally, we develop an algorithm to construct arbitrary degree T-splines. The difference between odd degree and even degree T-splines are studied in detail. The methods to extract knot intervals, calculate new weights to handle extraordinary nodes, and extract B´ezier elements for analysis are investigated with arbitrary degrees. Hybrid degree weighted Tspline is generated at designated region with basis functions of different degrees, for the purpose of performing local p-refinement. We also study the convergence rate for T-spline models of different degrees, showing that hybrid degree weighted T-splines have better performance after p-refinement. In summary, we develop novel methods to construct volumetric T-splines based on polycube and sweeping methods. Arbitrary degree weighted T-spline is proposed, with proved analysis-suitable properties. Weighted T-spline basis functions are used to reparameterize trimmed NURBS surfaces, handling extraordinary nodes, based on which surface and volumetric weighted T-spline models are constructed for isogeometric analysis.

[pt] A Análise Isogeométrica (IGA) é um método de análise numérica de estruturas que surge com a proposta de unificação entre projeto e simulação, permitindo a criação de modelos computacionais que preservam a geometria exata do problema. Essa abordagem é possível por meio de uma classe de funções matemáticas denominadas NURBS (Non-Uniform Rational B-Splines), amplamente utilizadas em sistemas CAD (Computer-Aided Design) para modelagem de curvas e superfícies. Na análise isogeométrica, as mesmas funções que representam a geometria aproximam as variáveis de campo. Neste contexto, foi desenvolvido este trabalho que tem como objetivo fornecer uma ferramenta no âmbito da mecânica computacional para análise isogeométrica bidimensional de problemas de elasticidade linear, incluindo as etapas de modelagem, análise e visualização de resultados. O sistema é composto por dois softwares: FEMEP (Finite Element Method Educational Computer Program), desenvolvido em Python e responsável pela etapa de modelagem geométrica, e FEMOOLab (Finite Element Method Object-Oriented Laboratory), software MATLAB para análise e exibição de resultados. A ferramenta proposta apresenta uma interface gráfica de usuário (GUI) que permite a visualização e manipulação intuitiva de curvas NURBS com recursos avançados de modelagem, como interseção de curvas e recursos de reconhecimento de região que agilizam e simplificam o processo. Uma contribuição significativa deste trabalho reside na capacidade de gerar malhas isogeométricas não estruturadas, utilizando T-Splines baseadas em um algoritmo de decomposição de domínio. O sistema de código aberto permite a colaboração e o desenvolvimento contínuo pela comunidade de usuários e desenvolvedores. / [en] Isogeometric Analysis (IGA) is a numerical analysis method for structures that arises with the proposal of unification between design and simulation, allowing the creation of computational models that preserve the exact geometry of the problem. This approach is possible by a class of mathematical functions called NURBS (Non-Uniform Rational B-Splines), widely used in CAD (Computer-Aided Design) systems for modeling curves and surfaces. In isogeometric analysis, the same functions representing the geometry approximate the field variables. In this context, this work was developed to provide a tool within the scope of computational mechanics for two-dimensional isogeometric analysis of linear elasticity problems, including the steps of modeling, analysis, and visualization of results. The system consists of two software programs: FEMEP (Finite Element Method Educational Computer Program), developed in Python and responsible for the geometric modeling stage, and FEMOOLab (Finite Element Method Object-Oriented Laboratory), a MATLAB software for analysis and display of results. The proposed tool features a graphical user interface (GUI) that allows intuitive visualization and manipulation of NURBS curves with advanced modeling features such as curve intersection and region recognition features that streamline and simplify the process. A significant contribution of this work lies in the ability to generate non-structured isogeometric meshes, using T-Splines based on a domain decomposition algorithm. The open-source system allows collaboration and continuous development by the community of users and developers.

In this thesis work, a 3D Synthetic Human Face Modelling Software is implemented using C++ and OpenGL. B&eacute / zier surfaces, B-spline surfaces, Nonuniform Rational B-spline surfaces, Hierarchical B-Spline surfaces and T-spline surfaces are evaluated as options for the surface description method. T-spline surfaces are chosen since they are found to be superior considering the requirements of the work. In the modelling process, a modular approach is followed. Firstly, high detailed facial regions (i.e. nose, eyes, mouth) are modelled, then these models are unified in a complete face model employing the merging capabilities of T-splines. Local and global features of the face model are parameterized in order to have the ability to create and edit various face models. To enhance the visual quality of the model, a region-variable rendering scheme is employed. In doing this, a new file format to define T-Spline surfaces is proposed. To reduce the computational and memory cost of the software, a simplified version of the T-Spline surface description method is proposed and used.

In computer graphics we can handle unstructured triangular 3D meshes which are not too usable for processing through their irregularity. In these situations it occurs need of conversion that 3D mesh to more suitable representation. Some kind of 3D spline surface can be proper alternative because it institutes regularity in the form of control points grid and that's why it is more suitable for next processing. During conversion, which is described in this thesis, quadrilateral 3D mesh is constructed at first. This mesh has regular structure but mainly the structure corresponds to structure of control points grid of resulting 3D spline surface. Created quadrilateral 3D mesh can be saved and consequently used in specific modeling applications for T-spline surface creation.