Volumetric Subdivision for Efficient Integrated Modeling and Simulation
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Continuous surface representations, such as B-spline and Non-Uniform Rational B-spline (NURBS) surfaces are the de facto standard for modeling 3D objects - thin shells and solid objects alike - in the field of Computer-Aided Design (CAD). For performing physically based simulation, Finite Element Analysis (FEA) has been the industry standard for many years. In order to analyze physical properties such as stability, aerodynamics, or heat dissipation, the continuous models are discretized into finite element (FE) meshes. A tight integration of and a smooth transition between geometric design and physically based simulation are key factors for an efficient design and engineering workflow. Converting a CAD model from its continuous boundary representation (B-Rep) into a discrete volumetric representation for simulation is a time-consuming process that introduces approximation errors and often requires manual interaction by the engineer. Deriving design changes directly from the simulation results is especially difficult as the meshing process is irreversible. Isogeometric Analysis (IGA) tries to overcome this meshing hurdle by using the same representation for describing the geometry and for performing the simulation. Most commonly, IGA is performed on bivariate and trivariate spline representations (B-spline or NURBS surfaces and volumes). While existing CAD B-Rep models can be used directly for simulating thin-shell objects, simulating solid objects requires a conversion from spline surfaces to spline volumes. As spline volumes need a trivariate tensor-product topology, complex 3D objects must be represented via trimming or by connecting multiple spline volumes, limiting the continuity to C^0. As an alternative to NURBS or B-splines, subdivision models allow for representing complex topologies with as a single entity, removing the need for trimming or tiling and potentially providing higher continuity. While subdivision surfaces have shown promising results for designing and simulating shells, IGA on subdivision volumes remained mostly unexplored apart from the work of Burkhart et al. In this dissertation, I investigate how volumetric subdivision representations are beneficial for a tighter integration of geometric modeling and physically based simulation. Focusing on Catmull-Clark (CC) solids, I present novel techniques in the areas of efficient limit evaluation, volumetric modeling, numerical integration, and mesh quality analysis. I present an efficient link to FEA, as well as my IGA approach on CC solids that improves upon Burkhart et al.'s proof of concept with constant-time limit evaluation, more accurate integration, and higher mesh quality. Efficient limit evaluation is a key requirement when working with subdivision models in geometric design, visualization, simulation, and 3D printing. In this dissertation, I present the first method for constant-time volumetric limit evaluation of CC solids. It is faster than the subdivision-based approach by Burkhart et al. for every topological constellation and parameter point that would require more than two local subdivision steps. Adapting the concepts of well-known surface modeling tools, I present a volumetric modeling environment for CC-solid control meshes. Consistent volumetric modeling operations built from a set of novel volumetric Euler operators allow for creating and modifying topologically consistent volumetric meshes. Furthermore, I show how to manipulate groups of control points via parameters, how to avoid intersections with inner control points while modeling the outer surface, and how to use CC solids in the context of multi-material additive manufacturing. For coupling of volumetric subdivision models with established FE frameworks, I present an efficient and consistent tetrahedral mesh generation technique for CC solids. The technique exploits the inherent volumetric structure of CC-solid models and is at least 26 times faster than the tetrahedral meshing algorithm provided by CGAL. This allows to re-create or update the tetrahedral mesh almost instantly when changing the CC-solid model. However, the mesh quality strongly depends on the quality of the control mesh. In the context of structural analysis, I present my IGA approach on CC solids. The IGA approach yields converging stimulation results for models with fewer elements and fewer degrees of freedom than FE simulations on tetrahedral meshes with linear and higher-order basis functions. The solver also requires fewer iterations to solve the linear system due to the higher continuity throughout the simulation model provided by the subdivision basis functions. Extending Burkhart et al.'s method, my hierarchical quadrature scheme for irregular CC-solid cells increases the accuracy of the integrals for computing surface areas and element stiffnesses. Furthermore, I introduce a quality metric that quantifies the parametrization quality of the limit volume, revealing distortions, inversions, and singularities. The metric shows that cells with multiple adjacent boundary faces induce singularities in the limit, even for geometrically well-shaped control meshes. Finally, I present a set of topological operations for splitting such boundary cells - resolving the singularities. These improvements further reduce the amount of elements required to obtain converging results as well as the time required for solving the linear system.