Modelado jerárquico de objetos 3D con superficies de subdivisión

Las SSs (Superficies de Subdivisión) son un potente paradigma de modelado de objetos 3D (tridimensionales) que establece un puente entre los dos enfoques tradicionales a la aproximación de superficies, basados en mallas poligonales y de parches alabeados, que conllevan problemas uno y otro. Los esquemas de subdivisión permiten definir una superficie suave (a tramos), como las más frecuentes en la práctica, como el límite de un proceso recursivo de refinamiento de una malla de control burda, que puede ser descrita muy compactamente. Además, la recursividad inherente a las SSs establece naturalmente una relación de anidamiento piramidal entre las mallas / NDs (Niveles de Detalle) generadas/os sucesivamente, por lo que las SSs se prestan extraordinariamente al AMRO (Análisis Multiresolución mediante Ondículas) de superficies, que tiene aplicaciones prácticas inmediatas e interesantísimas, como la codificación y la edición jerárquicas de modelos 3D. Empezamos describiendo los vínculos entre las tres áreas que han servido de base a nuestro trabajo (SSs, extracción automática de NDs y AMRO) para explicar como encajan estas tres piezas del puzzle del modelado jerárquico de objetos de 3D con SSs. El AMRO consiste en descomponer una función en una versión burda suya y un conjunto de refinamientos aditivos anidados jerárquicamente llamados "coeficientes ondiculares". La teoría clásica de ondículas estudia las señales clásicas nD: las definidas sobre dominios paramétricos homeomorfos a R" o (0,1)n como el audio (n=1), las imágenes (n=2) o el vídeo (n=3). En topologías menos triviales, como las variedades 2D) (superficies en el espacio 3D), el AMRO no es tan obvio, pero sigue siendo posible si se enfoca desde la perspectiva de las SSs. Basta con partir de una malla burda que aproxime a un bajo ND la superficie considerada, subdividirla recursivamente y, al hacerlo, ir añadiendo los coeficientes ondiculares, que son los detalles 3D necesarios para obtener aproximaciones más y más finas a la superficie original. Pasamos después a las aplicaciones prácticas que constituyen nuestros principal desarrollo original y, en particular, presentamos una técnica de codificación jerárquica de modelos 3D basada en SSs, que actúa sobre los detalles 3D mencionados: los expresa en un referencial normal loscal; los organiza según una estructura jerárquica basada en facetas; los cuantifica dedicando menos bits a sus componentes tangenciales, menos energéticas, y los "escalariza"; y los codifica dinalmente gracias a una técnica similar al SPIHT (Set Partitioning In Hierarchical Tress) de Said y Pearlman. El resultado es un código completamente embebido y al menos dos veces más compacto, para superficies mayormente suaves, que los obtenidos con técnicas de codificación progresiva de mallas 3D publicadas previamente, en las que además los NDs no están anidados piramidalmente. Finalmente, describimos varios métodos auxiliares que hemos desarrollado, mejorando técnicas previas y creando otras propias, ya que una solución completa al modelado de objetos 3D con SSs requiere resolver otros dos problemas. El primero es la extracción de una malla base (triangular, en nuestro caso) de la superficie original, habitualmente dada por una malla triangular fina con conectividad arbitraria. El segundo es la generación de un remallado recursivo con conectividad de subdivisión de la malla original/objetivo mediante un refinamiento recursivo de la malla base, calculando así los detalles 3D necesarios para corregir las posiciones predichas por la subdivisión para nuevos vértices

Hexagonal global parameterization of arbitrary surfaces

In this paper we introduce hexagonal global parameterizations, a new type of parameterization in which parameter lines respect six-fold rotational symmetries (6-RoSy). Such parameterizations enable the tiling of surfaces with regular hexagonal texture and geometry patterns and can be used to generate high-quality triangular remeshing. To construct a hexagonal parameterization given a surface, we provide an automatic technique to generate a 6-RoSy field that respects directional and singularity features in the surface. This is achieved by applying the trace-and-deviator decomposition to the curvature tensor, which allows us to identify regions of appropriate directional constraints. We also introduce a technique for automatically merging and cancelling singularities. This field will then be used to generate a hexagonal global parameterization by adapting the framework of QuadCover parameterization. In particular, we formulate the energy terms needed to solve for a hexagonal parameterization. We demonstrate the usefulness of our geometry-aware global parameterization with applications such as surface tiling with regular textures and geometry patterns and triangular remeshing.Keywords: Surface parameterization, Pattern synthesis, Rotational symmetry, Triangular remeshing, Hexagonal tilin

Homogenized yarn-level cloth

We present a method for animating yarn-level cloth effects using a thin-shell solver. We accomplish this through numerical homogenization: we first use a large number of yarn-level simulations to build a model of the potential energy density of the cloth, and then use this energy density function to compute forces in a thin shell simulator. We model several yarn-based materials, including both woven and knitted fabrics. Our model faithfully reproduces expected effects like the stiffness of woven fabrics, and the highly deformable nature and anisotropy of knitted fabrics. Our approach does not require any real-world experiments nor measurements; because the method is based entirely on simulations, it can generate entirely new material models quickly, without the need for testing apparatuses or human intervention. We provide data-driven models of several woven and knitted fabrics, which can be used for efficient simulation with an off-the-shelf cloth solver

Restructuring surface tessellation with irregular boundary conditions

In this paper, the surface tessellation problem is explored, in particular, the task of meshing a surface with the added consideration of incorporating constructible building components. When a surface is tessellated into discrete counterparts, certain unexpected conditions usually occur at the boundary of the surface, in particular, when the surface is being trimmed. For example, irregularly shaped panels form at the trimmed edges. To reduce the number of irregular panels that may form during the tessellation process, this paper presents an algorithmic approach to restructuring the surface tessellation by investigating irregular boundary conditions. The objective of this approach is to provide an alternative way for freeform surface manifestation from a well-structured discrete model of the given surface

Particle Methods for Geophysical Flow on the Sphere.

We present a Lagrangian Particle-Panel Method (LPPM) for geophysical fluid flow on a rotating sphere motivated by problems in atmosphere and ocean dynamics. We focus here on the barotropic vorticity equation and 2D passive scalar advection, as a step towards the development of a new dynamical core for global circulation models. The LPPM method employs the Lagrangian form of the equations of motion. The flow map is discretized as a set of Lagrangian particles and panels. Particle velocity is computed by applying a midpoint rule/point vortex approximation to the Biot-Savart integral with quadrature weights determined by the panel areas. We consider several discretizations of the sphere including the cubed sphere mesh, icosahedral triangles, and spherical Voronoi tesselations. The ordinary differential equations for particle motion are integrated by the fourth order Runge-Kutta method. Mesh distortion is addressed using a combination of adaptive mesh refinement (AMR) and a new Lagrangian remeshing procedure. In contrast with Eulerian schemes, the LPPM method avoids explicit discretization of the advective derivative. In the case of passive scalar advection, LPPM preserves tracer ranges and both linear and nonlinear tracer correlations exactly. We present results for the barotropic vorticity equation applied to several test cases including solid body rotation, Rossby-Haurwitz waves, Gaussian vortices, jet streams, and a model for the breakdown of the polar vortex during sudden stratospheric warming events. The combination of AMR and remeshing enables the LPPM scheme to efficiently resolve thin fronts and filaments that develop in the vorticity distribution. We validate the accuracy of LPPM by comparing with results obtained using the Eulerian based Lin-Rood advection scheme. We examine how energy and enstrophy conservation in the LPPM scheme are affected by the time step and spatial discretization. We conclude with a discussion of how the method may be extended to the shallow water equations.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/99936/1/pbosler_1.pd

Parallel mesh adaptive techniques for complex flow simulation

Dynamic mesh adaptation on unstructured grids, by localised refinement and derefinement, is a very efficient tool for enhancing solution accuracy and optimise computational time. One of the major drawbacks however resides in the projection of the new nodes created, during the refinement process, onto the boundary surfaces. This can be addressed by the introduction of a library capable of handling geometric properties given by a CAD (Computer Aided Design) description. This is of particular interest also to enhance the adaptation module when the mesh is being smoothed, and hence moved, to then re-project it onto the surface of the exact geometry. However, the above procedure is not always possibly due to either faulty or too complex designs, which require a higher level of complexity in the CAD library. It is therefore paramount to have a built-in algorithm able to place the new nodes, belonging to the boundary, closer to the geometric definition of it. Such a procedure is proposed in this work, based on the idea of interpolating subdivision. In order to efficiently and effectively adapt a mesh to a solution field, the criteria used for the adaptation process needs to be as accurate as possible. Due to the nature of the solution, which is obtained by discretisation of a continuum model, numerical error is intrinsic in the calculation. A posteriori error estimation allows us to somewhat assess the accuracy by using the computed solution itself. In particular, an a posteriori error estimator based on the Zienkievicz Zhu model is introduced. This can be used in the adaptation procedure to refine the mesh in those areas where the local error exceeds a set tolerance, hence further increasing the accuracy of the solution in those regions during the next computational step. Variants of this error estimator have also been studied and implemented. One of the important aspects of this project is the fact that the algorithmic concepts are developed thinking parallel, i.e. the algorithms take into account the possibility of multiprocessor implementation. Indeed these concepts require complex programming if one tries to parallelise them, once they have been devised serially. Another important and innovative aspect of this work is the consistency of the algorithms with parallel processor execution

A new graphical user interface for a 3D topological mesh modeler

In this thesis, I present a new platform-independent, open source, intuitive graphical user interface for TopMod, an application designed for interacting with 3-dimensional manifold meshes represented by a Doubly Linked Face List (DLFL). This new interface, created using the Trolltech Qt user interface library, enables users to construct and interact with complex manifold meshes much faster and more easily than was previously possible. I also present a method for the rapid creation of a successful online community of users and developers, by integrating a variety of open source web-based software packages. The new website, which includes a discussion forum, a news blog, a collaborative user and developer wiki, and a source code repository and release manager, received an average of 250 unique visits per day during the first two months of its existence, and it continues to be utilized by a variety of users and developers worldwide

Analysis and Parameterization of Triangulated Surfaces

This dissertation deals with the analysis and parameterization of surfaces represented by triangle meshes, that is, piecewise linear surfaces which enable a simple representation of 3D models commonly used in mathematics and computer science. Providing equivalent and high-level representations of a 3D triangle mesh M is of basic importance for approaching different computational problems and applications in the research fields of Computational Geometry, Computer Graphics, Geometry Processing, and Shape Modeling. The aim of the thesis is to show how high-level representations of a given surface M can be used to find other high-level or equivalent descriptions of M and vice versa. Furthermore, this analysis is related to the study of local and global properties of triangle meshes depending on the information that we want to capture and needed by the application context. The local analysis of an arbitrary triangle mesh M is based on a multi-scale segmentation of M together with the induced local parameterization, where we replace the common hypothesis of decomposing M into a family of disc-like patches (i.e., 0-genus and one boundary component) with a feature-based segmentation of M into regions of 0-genus without constraining the number of boundary components of each patch. This choice and extension is motivated by the necessity of identifying surface patches with features, of reducing the parameterization distortion, and of better supporting standard applications of the parameterization such as remeshing or more generally surface approximation, texture mapping, and compression. The global analysis, characterization, and abstraction of M take into account its topological and geometric aspects represented by the combinatorial structure of M (i.e., the mesh connectivity) with the associated embedding in R^3. Duality and dual Laplacian smoothing are the first characterizations of M presented with the final aim of a better understanding of the relations between mesh connectivity and geometry, as discussed by several works in this research area, and extended in the thesis to the case of 3D parameterization. The global analysis of M has been also approached by defining a real function on M which induces a Reeb graph invariant with respect to affine transformations and best suited for applications such as shape matching and comparison. Morse theory and the Reeb graph were also used for supporting a new and simple method for solving the global parameterization problem, that is, the search of a cut graph of an arbitrary triangle mesh M. The main characteristics of the proposed approach with respect to previous work are its capability of defining a family of cut graphs, instead of just one cut, of bordered and closed surfaces which are treated with a unique approach. Furthermore, each cut graph is smooth and the way it is built is based on the cutting procedure of 0-genus surfaces that was used for the local parameterization of M. As discussed in the thesis, defining a family of cut graphs provides a great flexibility and effective simplifications of the analysis, modeling, and visualization of (time-depending) scalar and vector fields; in fact, the global parameterization of M enables to reduce th

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells