Decoupling the CGAL 3D Triangulations from the Underlying Space

The {\em Computational Geometry Algorithms Library} {\sc Cgal} currently provides packages to compute triangulations in $\mathbb{R}^2$ and $\mathbb{R}^3$. In this paper we describe a new design for the 3D triangulation package that permits to easily add functionality to compute triangulations in other spaces. These design changes have been implemented, and validated on the case of the periodic space \T^3. We give a detailed description of the realized changes together with their motivation. Finally, we show benchmarks to prove that the new design does not affect the efficiency

3D mesh processing using GAMer 2 to enable reaction-diffusion simulations in realistic cellular geometries

Recent advances in electron microscopy have enabled the imaging of single cells in 3D at nanometer length scale resolutions. An uncharted frontier for in silico biology is the ability to simulate cellular processes using these observed geometries. Enabling such simulations requires watertight meshing of electron micrograph images into 3D volume meshes, which can then form the basis of computer simulations of such processes using numerical techniques such as the Finite Element Method. In this paper, we describe the use of our recently rewritten mesh processing software, GAMer 2, to bridge the gap between poorly conditioned meshes generated from segmented micrographs and boundary marked tetrahedral meshes which are compatible with simulation. We demonstrate the application of a workflow using GAMer 2 to a series of electron micrographs of neuronal dendrite morphology explored at three different length scales and show that the resulting meshes are suitable for finite element simulations. This work is an important step towards making physical simulations of biological processes in realistic geometries routine. Innovations in algorithms to reconstruct and simulate cellular length scale phenomena based on emerging structural data will enable realistic physical models and advance discovery at the interface of geometry and cellular processes. We posit that a new frontier at the intersection of computational technologies and single cell biology is now open.Comment: 39 pages, 14 figures. High resolution figures and supplemental movies available upon reques

Doctor of Philosophy

dissertationOne of the fundamental building blocks of many computational sciences is the construction and use of a discretized, geometric representation of a problem domain, often referred to as a mesh. Such a discretization enables an otherwise complex domain to be represented simply, and computation to be performed over that domain with a finite number of basis elements. As mesh generation techniques have become more sophisticated over the years, focus has largely shifted to quality mesh generation techniques that guarantee or empirically generate numerically well-behaved elements. In this dissertation, the two complementary meshing subproblems of vertex placement and element creation are analyzed, both separately and together. First, a dynamic particle system achieves adaptivity over domains by inferring feature size through a new information passing algorithm. Second, a new tetrahedral algorithm is constructed that carefully combines lattice-based stenciling and mesh warping to produce guaranteed quality meshes on multimaterial volumetric domains. Finally, the ideas of lattice cleaving and dynamic particle systems are merged into a unified framework for producing guaranteed quality, unstructured and adaptive meshing of multimaterial volumetric domains

Locally optimal Delaunay-refinement and optimisation-based mesh generation

The field of mesh generation concerns the development of efficient algorithmic techniques to construct high-quality tessellations of complex geometrical objects. In this thesis, I investigate the problem of unstructured simplicial mesh generation for problems in two- and three-dimensional spaces, in which meshes consist of collections of triangular and tetrahedral elements. I focus on the development of efficient algorithms and computer programs to produce high-quality meshes for planar, surface and volumetric objects of arbitrary complexity. I develop and implement a number of new algorithms for mesh construction based on the Frontal-Delaunay paradigm - a hybridisation of conventional Delaunay-refinement and advancing-front techniques. I show that the proposed algorithms are a significant improvement on existing approaches, typically outperforming the Delaunay-refinement technique in terms of both element shape- and size-quality, while offering significantly improved theoretical robustness compared to advancing-front techniques. I verify experimentally that the proposed methods achieve the same element shape- and size-guarantees that are typically associated with conventional Delaunay-refinement techniques. In addition to mesh construction, methods for mesh improvement are also investigated. I develop and implement a family of techniques designed to improve the element shape quality of existing simplicial meshes, using a combination of optimisation-based vertex smoothing, local topological transformation and vertex insertion techniques. These operations are interleaved according to a new priority-based schedule, and I show that the resulting algorithms are competitive with existing state-of-the-art approaches in terms of mesh quality, while offering significant improvements in computational efficiency. Optimised C++ implementations for the proposed mesh generation and mesh optimisation algorithms are provided in the JIGSAW and JITTERBUG software libraries

Finite Element Modeling Driven by Health Care and Aerospace Applications

This thesis concerns the development, analysis, and computer implementation of mesh generation algorithms encountered in finite element modeling in health care and aerospace. The finite element method can reduce a continuous system to a discrete idealization that can be solved in the same manner as a discrete system, provided the continuum is discretized into a finite number of simple geometric shapes (e.g., triangles in two dimensions or tetrahedrons in three dimensions). In health care, namely anatomic modeling, a discretization of the biological object is essential to compute tissue deformation for physics-based simulations. This thesis proposes an efficient procedure to convert 3-dimensional imaging data into adaptive lattice-based discretizations of well-shaped tetrahedra or mixed elements (i.e., tetrahedra, pentahedra and hexahedra). This method operates directly on segmented images, thus skipping a surface reconstruction that is required by traditional Computer-Aided Design (CAD)-based meshing techniques and is convoluted, especially in complex anatomic geometries. Our approach utilizes proper mesh gradation and tissue-specific multi-resolution, without sacrificing the fidelity and while maintaining a smooth surface to reflect a certain degree of visual reality. Image-to-mesh conversion can facilitate accurate computational modeling for biomechanical registration of Magnetic Resonance Imaging (MRI) in image-guided neurosurgery. Neuronavigation with deformable registration of preoperative MRI to intraoperative MRI allows the surgeon to view the location of surgical tools relative to the preoperative anatomical (MRI) or functional data (DT-MRI, fMRI), thereby avoiding damage to eloquent areas during tumor resection. This thesis presents a deformable registration framework that utilizes multi-tissue mesh adaptation to map preoperative MRI to intraoperative MRI of patients who have undergone a brain tumor resection. Our enhancements with mesh adaptation improve the accuracy of the registration by more than 5 times compared to rigid and traditional physics-based non-rigid registration, and by more than 4 times compared to publicly available B-Spline interpolation methods. The adaptive framework is parallelized for shared memory multiprocessor architectures. Performance analysis shows that this method could be applied, on average, in less than two minutes, achieving desirable speed for use in a clinical setting. The last part of this thesis focuses on finite element modeling of CAD data. This is an integral part of the design and optimization of components and assemblies in industry. We propose a new parallel mesh generator for efficient tetrahedralization of piecewise linear complex domains in aerospace. CAD-based meshing algorithms typically improve the shape of the elements in a post-processing step due to high complexity and cost of the operations involved. On the contrary, our method optimizes the shape of the elements throughout the generation process to obtain a maximum quality and utilizes high performance computing to reduce the overheads and improve end-user productivity. The proposed mesh generation technique is a combination of Advancing Front type point placement, direct point insertion, and parallel multi-threaded connectivity optimization schemes. The mesh optimization is based on a speculative (optimistic) approach that has been proven to perform well on hardware-shared memory. The experimental evaluation indicates that the high quality and performance attributes of this method see substantial improvement over existing state-of-the-art unstructured grid technology currently incorporated in several commercial systems. The proposed mesh generator will be part of an Extreme-Scale Anisotropic Mesh Generation Environment to meet industries expectations and NASA\u27s CFD visio

Real-Time High-Quality Image to Mesh Conversion for Finite Element Simulations

Technological Advances in Medical Imaging have enabled the acquisition of images accurately describing biological tissues. Finite Element (FE) methods on these images provide the means to simulate biological phenomena such as brain shift registration, respiratory organ motion, blood flow pressure in vessels, etc. FE methods require the domain of tissues be discretized by simpler geometric elements, such as triangles in two dimensions, tetrahedra in three, and pentatopes in four. This exact discretization is called a mesh . The accuracy and speed of FE methods depend on the quality and fidelity of the mesh used to describe the biological object. Elements with bad quality introduce numerical errors and slower solver convergence. Also, analysis based on poor fidelity meshes do not yield accurate results specially near the surface. In this dissertation, we present the theory and the implementation of both a sequential and a parallel Delaunay meshing technique for 3D and ---for the first time--- 4D space-time domains. Our method provably guarantees that the mesh is a faithful representation of the multi-tissue domain in topological and geometric sense. Moreover, we show that our method generates graded elements of bounded radius-edge and aspect ratio, which renders our technique suitable for Finite Element analysis. A notable feature of our implementation is speed and scalability. The single-threaded performance of our 3D code is faster than the state of the art open source meshing tools. Experimental evaluation shows a more than 82% weak scaling efficiency for up to 144 cores, reaching a rate of more than 14.3 million elements per second. This is the first 3D parallel Delaunay refinement method to achieve such a performance, on either distributed or shared-memory architectures. Lastly, this dissertation is the first to develop and examine the sequential and parallel high-quality and fidelity meshing of general space-time 4D multi-tissue domains

Distributed approaches for coverage missions with multiple heterogeneous UAVs for coastal areas.

This Thesis focuses on a high-level framework proposal for heterogeneous aerial, fixed wing teams of robots, which operate in complex coastal areas. Recent advances in the computational capabilities of modern processors along with the decrement of small scale aerial platform manufacturing costs, have given researchers the opportunity to propose efficient and low-cost solutions to a wide variety of problems. Regarding marine sciences and more generally coastal or sea operations, the use of aerial robots brings forth a number of advantages, including information redundancy and operator safety. This Thesis initially deals with complex coastal decomposition in relation with a vehicles’ on-board sensor. This decomposition decreases the computational complexity of planning a flight path, while respecting various aerial or ground restrictions. The sensor-based area decomposition also facilitates a team-wide heterogeneous solution for any team of aerial vehicles. Then, it proposes a novel algorithmic approach of partitioning any given complex area, for an arbitrary number of Unmanned Aerial Vehicles (UAV). This partitioning schema, respects the relative flight autonomy capabilities of the robots, providing them a corresponding region of interest. In addition, a set of algorithms is proposed for obtaining coverage waypoint plans for those areas. These algorithms are designed to afford the non-holonomic nature of fixed-wing vehicles and the restrictions their dynamics impose. Moreover, this Thesis also proposes a variation of a well-known path tracking algorithm, in order to further reduce the flight error of waypoint following, by introducing intermediate waypoints and providing an autopilot parametrisation. Finally, a marine studies test case of buoy information extraction is presented, demonstrating in that manner the flexibility and modular nature of the proposed framework.Esta tesis se centra en la propuesta de un marco de alto nivel para equipos heterogéneos de robots de ala fija que operan en áreas costeras complejas. Los avances recientes en las capacidades computacionales de los procesadores modernos, junto con la disminución de los costes de fabricación de plataformas aéreas a pequeña escala, han brindado a los investigadores la oportunidad de proponer soluciones eficientes y de bajo coste para enfrentar un amplio abanico de cuestiones. Con respecto a las ciencias marinas y, en términos más generales, a las operaciones costeras o marítimas, el uso de robots aéreos conlleva una serie de ventajas, incluidas la redundancia de la información y la seguridad del operador. Esta tesis trata inicialmente con la descomposición de áreas costeras complejas en relación con el sensor a bordo de un vehículo. Esta descomposición disminuye la complejidad computacional de la planificación de una trayectoria de vuelo, al tiempo que respeta varias restricciones aéreas o terrestres. La descomposición del área basada en sensores también facilita una solución heterogénea para todo el equipo para cualquier equipo de vehículos aéreos. Luego, propone un novedoso enfoque algorítmico de partición de cualquier área compleja dada, para un número arbitrario de vehículos aéreos no tripulados (UAV). Este esquema de partición respeta las capacidades relativas de autonomía de vuelo de los robots, proporcionándoles una región de interés correspondiente. Además, se propone un conjunto de algoritmos para obtener planes de puntos de cobertura para esas áreas. Estos algoritmos están diseñados teniendo en cuenta la naturaleza no holonómica de los vehículos de ala fija y las restricciones que impone su dinámica. En ese sentido, esta Tesis también ofrece una variación de un algoritmo de seguimiento de rutas bien conocido, con el fin de reducir aún más el error de vuelo del siguiente punto de recorrido, introduciendo puntos intermedios y proporcionando una parametrización del piloto automático. Finalmente, se presenta un caso de prueba de estudios marinos de extracción de información de boyas, que demuestra de esa manera la flexibilidad y el carácter modular del marco propuesto

Collective dynamics of flexible active particles on substrates : from cells to tissues

We study the effects of disorder in epithelial confluent tissues through the Voronoi model for dense tissues. The modeling of epithelial tissues relies on three different mechanisms: cell-cell and cell-medium interactions, and propulsion or activity. First, we focus on the role of cell-cell interaction in this model by exploring, in the athermal limit, its anomalous jamming behavior. We introduce a new metric that allows us to find a hierarchical structure in its energy landscape similar to colloidal particle systems. We then introduce a cell-medium interaction by explicitly considering an interaction between the cells and their underlying substrate. We consider that the targeted geometry of the cells changes according to their spatial position and in turn affects the cells motility. We show that when the characteristic length scale of the disorder is smaller than the cell size, the cell motility increases when compared to its homogeneous counterpart. This result is in sharp contrast to what has been reported for tissues with heterogeneity in the mechanical properties of the individual cells, where the disorder favors rigidity. Due to the internal biological complexity of the cells, changes to the cell-substrate interaction should trigger a hierarchy of biochemical responses in the cell that lead to its adaptation to the new substrate region. As such, the process of cell adaptation to its underlying structure is not instantaneous but requires a finite time that in many cases competes with other relevant timescales for the dynamics such as, for example, the diffusion timescale. With this in mind, we then introduce a characteristic adaptation time of the cells to the cell-substrate interaction changes. We study how the competition between the adaptation of the cells and their mobility can compromise the fidelity of the substrate and by relating this with the previous disordered substrate propose a typical time scale for the adaptation of cells that is relevant for experiments. Lastly, we consider non-confluent tissues by allowing the cells to break from one another and create empty spaces. This change opens the door to the study of the surface properties of cell colonies and it is a first step towards the study of the transition from a single cell to confluent tissue. Implications of our findings in the field of Soft Condensed Matter Physics are discussed