The existence of triangulations of non-convex polyhedra without new vertices

It is well known that a simple three-dimensional non-convex polyhedron may not be triangulated without using new vertices (so-called {\it Steiner points}). In this paper, we prove a condition that guarantees the existence of a triangulation of a non-convex polyhedron (of any dimension) without Steiner points. We briefly discuss algorithms for efficiently triangulating three-dimensional polyhedra

On Tetrahedralisations Containing Knotted and Linked Line Segments

This paper considers a set of twisted line segments in 3d such that they form a knot (a closed curve) or a link of two closed curves. Such line segments appear on the boundary of a family of 3d indecomposable polyhedra (like the Schönhardt polyhedron) whose interior cannot be tetrahedralised without additional vertices added. On the other hand, a 3d (non-convex) polyhedron whose boundary contains such line segments may still be decomposable as long as the twist is not too large. It is therefore interesting to consider the question: when there exists a tetrahedralisation contains a given set of knotted or linked line segments? In this paper, we studied a simplified question with the assumption that all vertices of the line segments are in convex position. It is straightforward to show that no tetrahedralisation of 6 vertices (the three-line-segments case) can contain a trefoil knot. Things become interesting when the number of line segments increases. Since it is necessary to create new interior edges to form a tetrahedralisation. We provided a detailed analysis for the case of a set of 4 line segments. This leads to a crucial condition on the orientation of pairs of new interior edges which determines whether this set is decomposable or not. We then prove a new theorem about the decomposability for a set of n (n ≥ 3) knotted or linked line segments. This theorem implies that the family of polyhedra generalised from the Schonhardt polyhedron by Rambau [1] are all indecomposable

pClay: A Precise Parallel Algorithm for Comparing Molecular Surfaces

Comparing binding sites as geometric solids can reveal conserved features of protein structure that bind similar molecular fragments and varying features that select different partners. Due to the subtlety of these features, algorithmic efficiency and geometric precision are essential for comparison accuracy. For these reasons, this paper presents pClay, the first structure comparison algorithm to employ fine-grained parallelism to enhance both throughput and efficiency. We evaluated the parallel performance of pClay on both multicore workstation CPUs and a 61-core Xeon Phi, observing scaleable speedup in many thread configurations. Parallelism unlocked levels of precision that were not practical with existing methods. This precision has important applications, which we demonstrate: A statistical model of steric variations in binding cavities, trained with data at the level of precision typical of existing work, can overlook 46% of authentic steric influences on specificity (p <= .02). The same model, trained with more precise data from pClay, overlooked 0% using the same standard of statistical significance. These results demonstrate how enhanced efficiency and precision can advance the detection of binding mechanisms that influence specificity

VoroCrust: Voronoi Meshing Without Clipping

Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrarily curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations. The VoroCrust algorithm is the first provably-correct algorithm for conforming polyhedral Voronoi meshing for non-convex and non-manifold domains with guarantees on the quality of both surface and volume elements. A robust refinement process estimates a suitable sizing field that enables the careful placement of Voronoi seeds across the surface circumventing the need for clipping and avoiding its many drawbacks. The algorithm has the flexibility of filling the interior by either structured or random samples, while preserving all sharp features in the output mesh. We demonstrate the capabilities of the algorithm on a variety of models and compare against state-of-the-art polyhedral meshing methods based on clipped Voronoi cells establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf. Supplemental materials available on https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd

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

On Tetrahedralisations Containing Knotted and Linked Line Segments

This paper considers a set of twisted line segments in 3d such that they form a knot (a closed curve) or a link of two closed curves. Such line segments appear on the boundary of a family of 3d indecomposable polyhedra (like the Schönhardt polyhedron) whose interior cannot be tetrahedralised without additional vertices added. On the other hand, a 3d (non-convex) polyhedron whose boundary contains such line segments may still be decomposable as long as the twist is not too large. It is therefore interesting to consider the question: when there exists a tetrahedralisation contains a given set of knotted or linked line segments? In this paper, we studied a simplified question with the assumption that all vertices of the line segments are in convex position. It is straightforward to show that no tetrahedralisation of 6 vertices (the three-line-segments case) can contain a trefoil knot. Things become interesting when the number of line segments increases. Since it is necessary to create new interior edges to form a tetrahedralisation. We provided a detailed analysis for the case of a set of 4 line segments. This leads to a crucial condition on the orientation of pairs of new interior edges which determines whether this set is decomposable or not. We then prove a new theorem about the decomposability for a set of n (n ≥ 3) knotted or linked line segments. This theorem implies that the family of polyhedra generalised from the Schonhardt polyhedron by Rambau [1] are all indecomposable

A Framework for Modeling the Growth and Development of Neurons and Networks

The development of neural tissue is a complex organizing process, in which it is difficult to grasp how the various localized interactions between dividing cells leads relentlessly to global network organization. Simulation is a useful tool for exploring such complex processes because it permits rigorous analysis of observed global behavior in terms of the mechanistic axioms declared in the simulated model. We describe a novel simulation tool, CX3D, for modeling the development of large realistic neural networks such as the neocortex, in a physical 3D space. In CX3D, as in biology, neurons arise by the replication and migration of precursors, which mature into cells able to extend axons and dendrites. Individual neurons are discretized into spherical (for the soma) and cylindrical (for neurites) elements that have appropriate mechanical properties. The growth functions of each neuron are encapsulated in set of pre-defined modules that are automatically distributed across its segments during growth. The extracellular space is also discretized, and allows for the diffusion of extracellular signaling molecules, as well as the physical interactions of the many developing neurons. We demonstrate the utility of CX3D by simulating three interesting developmental processes: neocortical lamination based on mechanical properties of tissues; a growth model of a neocortical pyramidal cell based on layer-specific guidance cues; and the formation of a neural network in vitro by employing neurite fasciculation. We also provide some examples in which previous models from the literature are re-implemented in CX3D. Our results suggest that CX3D is a powerful tool for understanding neural development

Non-invasive localization of atrial ectopic beats by using simulated body surface P-wave integral maps

Non-invasive localization of continuous atrial ectopic beats remains a cornerstone for the treatment of atrial arrhythmias. The lack of accurate tools to guide electrophysiologists leads to an increase in the recurrence rate of ablation procedures. Existing approaches are based on the analysis of the P-waves main characteristics and the forward body surface potential maps (BSPMs) or on the inverse estimation of the electric activity of the heart from those BSPMs. These methods have not provided an efficient and systematic tool to localize ectopic triggers. In this work, we propose the use of machine learning techniques to spatially cluster and classify ectopic atrial foci into clearly differentiated atrial regions by using the body surface P-wave integral map (BSPiM) as a biomarker. Our simulated results show that ectopic foci with similar BSPiM naturally cluster into differentiated non-intersected atrial regions and that new patterns could be correctly classified with an accuracy of 97% when considering 2 clusters and 96% for 4 clusters. Our results also suggest that an increase in the number of clusters is feasible at the cost of decreasing accuracy.This work was partially supported by The "Programa Prometeu" from Conselleria d'Educacio Formacio I Ocupacio, Generalitat Valenciana (www.edu.gva.es/fio/index_es.asp) Award Number: PROMETEU/2016/088 to JS; The "Plan Estatal de Investigacion Cientifica y Tecnica y de Innovacion 2013-2016" from the Ministerio de Economia, Industria y Competitividad of Spain, Agencia Estatal de Investigacion (www.mineco.gob.es) and the European Commission (European Regional Development Funds - ERDF -FEDER) (ec.europa.eu/regional_policy/es/funding/erdf/) Award Number: DPI2016-75799-R to JS and The "Programa Estatal de Investigacion, Desarrollo e Innovacion Orientado a los Retos de la Sociedad" from the Ministerio de Economia y Competitividad of Spain, Agencia Estatal de Investigacion (www.mineco.gob.es) and the European Commission (European Regional Development Funds - ERDF -FEDER) (ec.europa.eu/regional_policy/es/funding/erdf/) Award Number: TIN2014-59932-JIN to AFA and RS. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Ferrer Albero, A.; Godoy, EJ.; Lozano, M.; Martínez Mateu, L.; Alonso Atienza, F.; Saiz Rodríguez, FJ.; Sebastián Aguilar, R. (2017). Non-invasive localization of atrial ectopic beats by using simulated body surface P-wave integral maps. PLoS ONE. 12(7):1-23. https://doi.org/10.1371/journal.pone.0181263S12312

Mesh generation for voxel -based objects

A new physically-based approach to unstructured mesh generation via Monte-Carlo simulation is proposed. Geometrical objects to be meshed are represented by systems of interacting particles with a given interaction potential. A new way of distributing nodes in complex domains is proposed based on a concept of dynamic equilibrium ensemble, which represents a liquid state of matter. The algorithm is simple, numerically stable and produces uniform node distributions in domains of complex geometries and different dimensions. Well-shaped triangles or tetrahedra can be created by connecting a set of uniformly-spaced nodes. The proposed method has many advantages and potential applications.;The new method is applied to the problem of meshing of voxel-based objects. By customizing system potential energy function to reflect surface features, particles can be distributed into desired locations, such as sharp corners and edges. Feature-preserved surface mesh can then be constructed by connecting the node set.;A heuristic algorithm using an advancing front approach is proposed to generate triangulated surface meshes on voxel-based objects. The resultant surface meshes do not inherit the anisotropy of the underlying hexagonal grid. However, the important surface features, such as edges and corners may not be preserved in the mesh.;To overcome this problem, surface features such as edges, corners need to be detected. A new approach of edge capturing is proposed and demonstrated. The approach is based on a Laplace solver with incomplete Jacobi iterations, and as such is very simple and efficient. This edge capturing approach combined with the mesh generation methods above forms a simple and robust technique of unstructured mesh generation on voxel-based objects.;A graphical user interface (GUI) capable of complex geometric design and remote simulation control was implemented. The GUI was used in simulations of large fuel-cell stacks. It enables one to setup, run and monitor simulations remotely through secure shell (SSH2) connections. A voxel-based 3D geometrical modeling module is built along with the GUI. The flexibility of voxel-based geometry representation enables one to use this technique for both geometric design and visualization of volume data