A new construction of smooth surfaces from triangle meshes using parametric pseudo-manifolds

We introduce a new manifold-based construction for fitting a smooth surface to a triangle mesh of arbitrary topology. Our construction combines in novel ways most of the best features of previous constructions and, thus, it fills the gap left by them. We also introduce a theoretical framework that provides a sound justification for the correctness of our construction. Finally, we demonstrate the effectiveness of our manifold-based construction with a few concrete examples

A new construction of smooth surfaces from triangle meshes using parametric pseudo-manifolds

We introduce a new manifold-based construction for fitting a smooth surface to a triangle mesh of arbitrary topology. Our construction combines in novel ways most of the best features of previous constructions and, thus, it fills the gap left by them. We also introduce a theoretical framework that provides a sound justification for the correctness of our construction. Finally, we demonstrate the effectiveness of our manifold-based construction with a few concrete examples

Construction of C\u3csup\u3e∞\u3c/sup\u3e Surfaces From Triangular Meshes Using Parametric Pseudo-Manifolds

We present a new constructive solution for the problem of fitting a smooth surface to a given triangle mesh. Our construction is based on the manifold-based approach pioneered by Grimm and Hughes. The key idea behind this approach is to define a surface by overlapping surface patches via a gluing process, as opposed to stitching them together along their common boundary curves. The manifold based approach has proved to be well-suited to fit with relative ease, Ck-continuous parametric surfaces to triangle and quadrilateral meshes, for any arbitrary finite k or even k = ∞. Smooth surfaces generated by the manifold-based approach share some of the most important properties of splines surfaces, such as local shape control and fixed-sized local support for basis functions. In addition, the differential structure of a manifold provides us with a natural setting for solving equations on the surface boundary of 3D shapes. Our new manifold-based solution possesses most of the best features of previous constructions. In particular, our construction is simple, compact, powerful, and flexible in ways of defining the geometry of the resulting surface. Unlike some of the most recent manifold-based solutions, ours has been devised to work with triangle meshes. These meshes are far more popular than any other kind of mesh encountered in computer graphics and geometry processing applications. We also provide a mathematically sound theoretical framework to undergird our solution. This theoretical framework slightly improves upon the one given by Grimm and Hughes, which was used by most manifold-based constructions introduced before

Parametric pseudo-manifolds

We introduce a novel and constructive definition of gluing data, and give the first rigorous proof that a universal manifold satisfying the Hausdorff condition can always be constructed from any set of gluing data. We also present a class of spaces called parametric pseudo-manifolds, which under certain conditions, are manifolds embedded in Rn and defined from sets of gluing data. We give a construction for building a set of gluing data from any simplicial surface in R3. This construction is an improvement of the construction given in Siqueira et al. (2009), where the results were stated without proof. We also give a complete proof of the correctness of this construction making use of the crucial “property A.” The above results enable us to develop a methodology that explicitly yields manifolds in Rn arising in several graphics and engineering applications

Parametric Pseudo-Manifolds

We introduce a novel and constructive definition of gluing data, and prove that a universal manifold can always be constructed from any set of gluing data. We also present a class of spaces called parametric pseudo-manifolds, which under certain conditions are manifolds embedded in Rn and defined from sets of gluing data. The combination of both definitions is equivalent to a constructive definition of manifolds. They also enable us to develop constructions that explicitly yield manifolds in Rn arising in several graphics and engineering applications

Unstructured spline spaces for isogeometric analysis based on spline manifolds

Based on spline manifolds we introduce and study a mathematical framework for analysis-suitable unstructured B-spline spaces. In this setting the parameter domain has a manifold structure, which allows for the definition of function spaces that have a tensor-product structure locally, but not globally. This includes configurations such as B-splines over multi-patch domains with extraordinary points, analysis-suitable unstructured T-splines, or more general constructions. Within this framework, we generalize the concept of dual-compatible B-splines, which was originally developed for structured T-splines. This allows us to prove the key properties that are needed for isogeometric analysis, such as linear independence and optimal approximation properties for $h$-refined meshes

Large-scale Geometric Data Decomposition, Processing and Structured Mesh Generation

Mesh generation is a fundamental and critical problem in geometric data modeling and processing. In most scientific and engineering tasks that involve numerical computations and simulations on 2D/3D regions or on curved geometric objects, discretizing or approximating the geometric data using a polygonal or polyhedral meshes is always the first step of the procedure. The quality of this tessellation often dictates the subsequent computation accuracy, efficiency, and numerical stability. When compared with unstructured meshes, the structured meshes are favored in many scientific/engineering tasks due to their good properties. However, generating high-quality structured mesh remains challenging, especially for complex or large-scale geometric data. In industrial Computer-aided Design/Engineering (CAD/CAE) pipelines, the geometry processing to create a desirable structural mesh of the complex model is the most costly step. This step is semi-manual, and often takes up to several weeks to finish. Several technical challenges remains unsolved in existing structured mesh generation techniques. This dissertation studies the effective generation of structural mesh on large and complex geometric data. We study a general geometric computation paradigm to solve this problem via model partitioning and divide-and-conquer. To apply effective divide-and-conquer, we study two key technical components: the shape decomposition in the divide stage, and the structured meshing in the conquer stage. We test our algorithm on vairous data set, the results demonstrate the efficiency and effectiveness of our framework. The comparisons also show our algorithm outperforms existing partitioning methods in final meshing quality. We also show our pipeline scales up efficiently on HPC environment

Deformable Simplicial Complexes

In this dissertation we present a novel method for deformable interface tracking in 2D and 3D|deformable simplicial complexes (DSC). Deformable interfaces are used in several applications, such as fluid simulation, image analysis, reconstruction or structural optimization. In the DSC method, the interface (curve in 2D; surface in 3D) is represented explicitly as a piecewise linear curve or surface. However, the domain is also subject to discretization: triangulation in 2D; tetrahedralization in 3D. This way, the interface can be alternatively represented as a set of edges/triangles separating triangles/tetrahedra marked as outside from those marked as inside. Such an approach allows for robust topological adaptivity. Among other advantages of the deformable simplicial complexes there are: space adaptivity, ability to handle and preserve sharp features, possibility for topology control. We demonstrate those strengths in several applications. In particular, a novel, DSC-based fluid dynamics solver has been developed during the PhD project. A special feature of this solver is that due to the fact that DSC maintains an explicit interface representation, surface tension is more easily dealt with. One particular advantage of DSC is the fact that as an alternative to topology adaptivity, topology control is also possible. This is exploited in the construction of cut loci on tori where a front expands from a single point on a torus and stops when it self-intersects