Weighted Triangulations for Geometry Processing

In this article we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion of discrete metric that defines an orthogonal dual structure for arbitrary triangle meshes and thus extends weighted Delaunay triangulations to surface meshes. We also present alternative characterizations of this primal-dual structure (through combinations of angles, areas, and lengths) and, in the process, uncover closed-form expressions of mesh energies that were previously known in implicit form only. Finally, we demonstrate how weighted triangulations provide a faster and more robust approach to a series of geometry processing applications, including the generation of well-centered meshes, self-supporting surfaces, and sphere packing

Discrete Differential Geometry of Thin Materials for Computational Mechanics

Instead of applying numerical methods directly to governing equations, another approach to computation is to discretize the geometric structure specific to the problem first, and then compute with the discrete geometry. This structure-respecting discrete-differential-geometric (DDG) approach often leads to new algorithms that more accurately track the physically behavior of the system with less computational effort. Thin objects, such as pieces of cloth, paper, sheet metal, freeform masonry, and steel-glass structures are particularly rich in geometric structure and so are well-suited for DDG. I show how understanding the geometry of time integration and contact leads to new algorithms, with strong correctness guarantees, for simulating thin elastic objects in contact; how the performance of these algorithms can be dramatically improved without harming the geometric structure, and thus the guarantees, of the original formulation; how the geometry of static equilibrium can be used to efficiently solve design problems related to masonry or glass buildings; and how discrete developable surfaces can be used to model thin sheets undergoing isometric deformation

Multiscale multiphysics simulation in composite materials

The improvements in terms of computational power provides the capability to analyze with more detail the materials behavior. On one hand, going deeper in the materials to study an increasingly smaller dimension and capture micro- or nano- changes. On the other hand, the increasing computational memory allows to perform finite elements analysis with billions of nodes, that permits to obtain more accurate results. In this sense, the focus of this work is the numerical modeling of the microscale behavior of inhomogeneous materials, with special attention to composite materials under thermo-mechanical loading conditions. This work also proposes and implements optimization tools, at a constitutive law level, as well as the level of both, macro- and micro-structural algorithms. The thesis is proposed as compendium of articles written during the last years and all published in Q1 international journals. In the first publication, a novel damage-mechanics micro-model is presented, able to represent the mechanical behaviors of masonry constituents. The proposed micro-model is based on a tension-compression continuum damage model. The adoption of appropriate failure criteria enables controlling the dilatant behavior of the material, even though this aspect is not generally associated to continuum damage models as it is for plasticity models. The study proposes a simple solution to this issue, consisting in the appropriate definition of the failure surfaces under shear stress states, together with the formulation of proper evolution laws for damage variables. The model keeps the simple and efficient format of classical damage models, where the explicit evaluation of the internal variables avoids nested iterative procedures, thus increasing computational performance and robustness. Another purpose of this research is to carry out a critical comparison of the proposed continuous micro-model with other two well-known discrete micro-modeling strategies. The second publication presents a full thermo-mechanical multiscale methodology, covering the nano-, micro-, and macroscopic scales. In such methodology, direcly deriving from the Classical First-Order Multiscale Method, fundamental material properties are determined by means of molecular dynamics simulations. Afterwards, the material properties obtained are used at the microstructural level by means of finite element analyses. Finally, the macroscale problem is solved while considering the effect of the microstructure using a thermo-mechanical homogenization on a representative volume element (RVE). The publication that close this thesis presents two computationally efficient multiscale procedures able to predict the mechanical non-linear response of composite materials. This is achieved, using an RVE Data Base (DB) calculated a-priori. Through the definitions of an equivalent damage parameter ($d_{eq}$), function of the global stress at the microscale, a series of strain controlled virtual tests of the RVE are performed storing in the DB the homogenized stress and strain state reached at certain levels of d_eq. Afterwards, the solution of the macroscale structure can be solved using the interpolation of the stored data. The first proposed procedure, named Discrete Multiscale Threshold Surface definition (DMTS), stores in the database the tenso-deformational state in which damage starts. Once reaching this state, a non-linear analysis will require the construction of the RVE to analyze the material damage evolution. On the other hand, the second method proposed, named Discrete Multiscale Constitutive Model (DMCM), is completely based on offline data and uses only the stress information stored in the DB to obtain the failure threshold and the non-linear material performance. In the article, special attention has been paid on the construction and validation of the Data Base, as well as on the study of a complete composite structure comparing the speedup obtained with both methods.En las últimas décadas, el avance en términos de poder computacional ofrece la capacidad de analizar más detalladamente el comportamiento de los materiales. Por un lado, profundizar los materiales para estudiar una dimensión cada vez más pequeña y capturar micro o nanocambios. Por otro lado, la capacidad de memoria computacional permite realizar análisis de elementos finitos con miles de millones de nodos, lo que permite obtener resultados lo más exacto posible. El objetivo de este trabajo es la modelización numérica del comportamiento microescala de materiales no homogéneos, con especial atención a los materiales compuestos, en condiciones de carga termo-mecánica, y la aplicación de herramientas de optimización de las leyes constitutivas, así como en a nivel macro y micro estructural. La tesis se propone como un compendio de artículos publicados en revistas internacionales. En la primera publicación, se presenta un micro-modelo basado en el daño mecánico, capaz de representar los comportamientos mecánicos de las estructura de mampostería. El micro-modelo propuesto se basa en un modelo de daño continuo por tensión-compresión. La adopción de criterios de daño apropiados permite al analista controlar la dilatancia del material, aunque este aspecto no está generalmente asociado a los modelos de daño continuo como lo es para los modelos de plasticidad. El estudio propone una solución simple a este problema, que consiste en la definición apropiada de las superficies de daño bajo estados de tensión de cortante junto con la formulación de leyes de evolución apropiadas para las variables de daño. El modelo mantiene el formato simple y eficiente de los modelos de daños clásicos, donde la evaluación explícita de las variables internas evita los procedimientos iterativos anidados, aumentando así el rendimiento computacional. Otro objetivo de esta investigación es realizar una comparación crítica del micro-modelo continuo propuesto con otras dos estrategias bien conocidas de micro-modelado discreto. Posteriormente, se presenta una metodología termomecánica multiescala completa, que cubre las escalas nano, micro y macroscópica. En dicha metodología, derivada directamente del Método Multiescala de Primer Orden, las propiedades fundamentales del material se determinan mediante simulaciones de dinámica molecular que se implementan en consecuencia a nivel microestructural por medio de análisis de elementos finitos. Por otro lado, el problema de macroescala se resuelve considerando el efecto de la microestructura mediante homogeneización termo-mecánica en un elemento de volumen representativo (RVE). Finalmente, se proponen dos procedimientos multiescala computacionalmente eficientes capaces de predecir la respuesta mecánica no lineal de materiales compuestos. Esto se logrará utilizando una base de datos (DB) calculada a priori. A través de las definiciones de un parámetro de daño equivalente (d_eq), funciónes de la tensión global de la microescala, se actuarán una serie de pruebas virtuales de la microescala con deformación controlada para almacenar en el DB el estrés y la tensión homogeneizadas alcanzado en ciertos niveles de d_eq. Posteriormente, la solución de la estructura de macroescala mediante el método multiescala de primer orden se reemplazará por la interpolación de los datos almacenados en el DB. El primer método propuesto, llamado Discrete Multiscale Threshold Surface (DMTS), proporcionará la generación de la RVE en la parte no lineal de la estructura, mientras que el segundo, llamado Discrete Multiscale Constitutive Model (DMCM), es completamente independiente del micromodelo porque solo se utiliza la información de estrés almacenada en el DB. En el articulo se ha prestado especial atención a la creación y validación de la base de datos y al estudio de una estructura compuesta completa comparando la aceleración en terminos de tiempo computationál obtenid

Optimal Voronoi Tessellations with Hessian-based Anisotropy

International audienceThis paper presents a variational method to generate cell complexes with local anisotropy conforming to the Hessian of any given convex function and for any given local mesh density. Our formulation builds upon approximation theory to offer an anisotropic extension of Centroidal Voronoi Tessellations which can be seen as a dual form of Optimal Delaunay Triangulation. We thus refer to the resulting anisotropic polytopal meshes as Optimal Voronoi Tessel-lations. Our approach sharply contrasts with previous anisotropic versions of Voronoi diagrams as it employs first-type Bregman diagrams , a generalization of power diagrams where sites are augmented with not only a scalar-valued weight but also a vector-valued shift. As such, our OVT meshes contain only convex cells with straight edges, and admit an embedded dual triangulation that is combinatorially-regular. We show the effectiveness of our technique using off-the-shelf computational geometry libraries

Masonry shell structures with discrete equivalence classes

This paper proposes a method to model masonry shell structures where the shell elements fall into a set of discrete equivalence classes. Such shell structure can reduce the fabrication cost and simplify the physical construction due to reuse of a few template shell elements. Given a freeform surface, our goal is to generate a small set of template shell elements that can be reused to produce a seamless and buildable structure that closely resembles the surface. The major technical challenge in this process is balancing the desire for high reusability of template elements with the need for a seamless and buildable final structure. To address the challenge, we define three error metrics to measure the seamlessness and buildability of shell structures made from discrete equivalence classes and develop a hierarchical cluster-and-optimize approach to generate a small set of template elements that produce a structure closely approximating the surface with low error metrics. We demonstrate the feasibility of our approach on various freeform surfaces and geometric patterns, and validate buildability of our results with four physical prototypes. Code and data of this paper are at https://github.com/Linsanity81/TileableShell

Modelling and structural analysis of historical masonry systems including vaulted structure

The conservation of historic structures has been given special attention due to their cultural, social and economic importance. However they often show considerable structural vulnerability and have been seriously damaged by natural disasters including earthquakes. An excessive loss of architectural heritage has occurred because of earthquakes. A safety assessment and restoration practice on historical structures has been tackled extensively by professionals including architects and engineers. However, structural assessment of historical buildings is a complex task. Complexity comes from insufficient understanding of the characteristic of historical materials, limited knowledge of the seismic response of historical structures and yet-unknown structural deterioration due to the past natural disasters. Today it is perceived that nonlinear FEM analysis permits detailed study of historical masonry structures. However, in some cases, its application poses difficulties. The difficulties derive from the definition of material properties, the definition of a complex geometry and the analysis procedures. The results depend on the material properties considerably. However, it is not easy to describe appropriately the behaviour of historical materials including masonry in the FEM analysis. The definition of a complex geometry is challenging although the discretisation of accurate geometry is crucial. As for the analysis procedure, one of the difficulties is observed in seismic assessment. FEM-based nonlinear dynamic analysis permits close observation of seismic response of a historical masonry structure but it requires excessive computational effort, for a large-scale structure in particular. On the other hand, pushover can be adopted more efficiently than nonlinear dynamic analysis but the obtained result can be less reliable. All these considerations indicate that the understanding of FEM approaches still needs to be deepened to adopt more accurately and at the same time efficiently for the analysis of historical structures. The present research discusses the applicability of existing nonlinear FEM approaches to the study of masonry historical structures. The FEM analysis is adopted to the analysis of real and complex structures including mixed steel and masonry vaulted systems belonging to the Hospital de Sant Pau in Barcelona and a large single-nave church damaged by the 2009 Abruzzo earthquake. As a final outcome of the research, the conclusions provided criteria and guidelines for the analysis of these types of structures under vertical loading and seismic forces. The achievement of the research will contribute to both engineers and researchers who are involved in the conservation of historical masonry structures especially by means of FEM analysis.La conservación de las estructuras históricas llama la atención debido a su importancia cultural, social y económica. Sin embargo, muestran considerablemente vulnerabilidad estructural y se han dañado seriamente por desastres naturales como terremotos. La excesiva pérdida de patrimonio arquitectónico ha ocurrido a causa de los terremotos. Se ha llevado a cabo la evaluación de la seguridad y la práctica de restauración de estructuras históricas ampliamente por los profesionales incluso arquitectos e ingenieros. No obstante, la evaluación estructural de los edificios históricos es una tarea compleja. La complejidad viene de la comprensión insuficiente de las características de los materiales históricos, conocimiento limitado de la respuesta sísmica de estructuras históricas y deterioro estructural todavía desconocido debido a los desastres naturales pasados. Hoy en día se percibe que el análisis de elementos finitos (FEA) no lineal permite el estudio detallado de las estructuras de mampostería históricos. Con todo, en algunos casos, no es sencilla la aplicación de ello. Las dificultades vienen de la definición de las propiedades del material, la definición de una geometría compleja y los procedimientos de análisis. Los resultados dependen de las propiedades del material considerablemente. Sin embargo, no es fácil describir adecuadamente en el FEA el comportamiento de materiales históricos como mampostería. Es difícil definir la geometría compleja es crucial aunque la discretización de la geometría exacta. En cuanto al procedimiento de análisis, se observa una de las dificultades en la evaluación sísmica. Análisis dinámico no lineal del FEA permite la observación precisa de la respuesta sísmica de las estructuras de mampostería histórica pero requiere el esfuerzo computacional excesivo, especialmente por una estructura a gran escala. Por otro lado, pushover puede ser más eficiente que el análisis dinámico no lineal pero el resultado obtenido por ello puede ser menos fiable. Estas consideraciones indican que la compresión del FEA necesita profundizarse para que se adopte FEA más precisamente y más eficientemente para el análisis de estructuras históricas. La presente investigación analiza la aplicabilidad del FEA no lineal acerca del estudio de las estructuras históricas de mampostería. El FEA se adopta para el análisis de las estructuras reales y complejas incluso los sistemas abovedados de la combinación del acero y mampostería pertenecientes al Hospital de Sant Pau de Barcelona y una gran iglesia de una sola nave dañada por el terremoto de Abruzzo 2009. Como resultado final de la investigación, las conclusiones presentan criterios y directrices para el análisis de estés tipos de estructuras bajo cargas verticales y sísmicas. El fruto de la investigación contribuirá a ambos ingenieros e investigadores que participan en la conservación de las estructuras de mampostería históricos sobre todo por medio del FEA