Tetrahedralization of a Hexahedral Mesh

Two important classes of three-dimensional elements in computational meshes are hexahedra and tetrahedra. While several efficient methods exist that convert a hexahedral element to a tetrahedral elements, the existing algorithm for tetrahedralization of a hexahedral complex is the marching tetrahedron algorithm which limits pre-selection of face divisions. We generalize a procedure for tetrahedralizing triangular prisms to tetrahedralizing cubes, and combine it with certain heuristics to design an algorithm that can triangulate any hexahedra.Comment: The previous version had an error in the proof of Observation 2.1, which has since been rectified in this version. Formatting and title change

Développement d'un modèle numérique simplifié du tronc pour simuler l'effet d'une chirurgie de la scoliose sur l'apparence externe d'un patient

RÉSUMÉ La scoliose est une maladie du système musculo-squelettique caractérisée par une déformation tridimensionnelle complexe du tronc. Pour les cas les plus sévères, un traitement chirurgical est nécessaire. Celui-ci consiste à redresser la colonne vertébrale à l’aide de tiges métalliques ancrées aux vertèbres moyennant des vis et des crochets. La prédiction du résultat chirurgical s’avère un élément fondamental à toute planification opératoire. Actuellement, la méthode clinique pour élaborer une stratégie opératoire et estimer le résultat de la correction de la scoliose réside principalement dans l’analyse radiographique de la flexibilité du rachis jumelée à l’expérience du chirurgien. Afin d’assister davantage ce dernier lors de la planification, un simulateur biomécanique permettant d’identifier la configuration optimale des implants qui corrigera le mieux les déformations de la colonne est en cours de développement au Centre Hospitalier Universitaire Sainte-Justine à Montréal. Toutefois, ni ce simulateur ni l’analyse du chirurgien ne tient compte des tissus mous du tronc et ne fournit d’information sur l’apparence externe après l’intervention. Pour le chirurgien, le résultat de la chirurgie sur l’apparence externe s’avère hautement subjectif et son expérience demeure son seul atout. Tout ceci reste donc fort problématique, considérant que la principale raison pour prescrire une opération provient d’abord du mécontentement du patient vis-à-vis son apparence esthétique. Ne possédant aucune idée du niveau de correction esthétique qu’une chirurgie quelconque peut lui offrir, ce dernier est parfois déçu de l’imperfection de son apparence après l’intervention. L’objectif principal du projet consiste donc à définir une modélisation physique simplifiée des tissus déformables entre l’épiderme (surface de la peau) et les structures osseuses du tronc, afin de visualiser en 3D et d’évaluer l’effet d’une chirurgie de la scoliose sur l’apparence externe du patient. Cette étude s’attarde uniquement aux tissus mous car la modélisation biomécanique des structures osseuses fait l’objet d’un autre projet. Ainsi, pour prédire les résultats des corrections à l’externe, on utilise une configuration postopératoire déjà connue des structures osseuses.----------ABSTRACT Scoliosis is a musculoskeletal disorder characterized by a complex three-dimensional deformation of the trunk. For severe cases, surgical treatment is necessary. This procedure consists in rectifying the spine shape using metal rods anchored to the vertebrae by means of screws and hooks. The prediction of surgical outcome is a fundamental element of any preoperative evaluation. Currently, the clinical method to define a surgical strategy and estimate the result of curve correction relies primarily on radiographic analysis of spinal flexibility and on the surgeon's own experience. To further assist the clinician during surgical planning, a biomechanical simulator is currently being developed at Montreal’s Sainte-Justine University Hospital Center to identify the optimal configuration of the implants to best correct the spinal deformities. However, neither this simulator nor the spinal flexibility analysis consider the soft tissues of the trunk in order to provide information on the patient’s external appearance after the intervention. For the surgeon, the residual trunk asymmetry proves highly subjective and his experience remains his only asset. This is problematic considering that the main reason to prescribe an operation comes initially from the patient’s dissatisfaction towards their apparent deformity. Having no prior knowledge of the level of aesthetic improvement a surgery can offer him, the patient is sometimes disappointed by the imperfection of his appearance after the intervention. Therefore, the goal of this project is to define a simplified physical model of the deformable tissues between the skin surface (epidermis) and bone structures of the trunk in order to visualize in 3D and assess the effect of scoliosis surgery on the patient’s external appearance. This research focuses only on the soft tissues since biomechanical modeling of the bone structures is the subject of another ongoing project. Consequently, a known postoperative configuration of the bone structures serves as our basis to predict the external appearance after scoliosis surgery. To achieve our goal, we first propose a methodology to build a simplified system to model the different deformable structures of the trunk. Initially, 3D pre and postoperative reconstructions of the bone structures are obtained from standard radiographs while non-invasive 3D optical digitizers acquire the external surface of the trunk using white non-ionizing structure light. Following certain mesh preprocessing, we develop a generic method to generate three different tetrahedral layers starting from the external surface of the trunk to represent the skin, fat and muscles. From these new layers a generalized particle system based on elastic potential energy is defined. Forces preserving distance, area and volume constraints are calculated to describe the physical behavior of the various soft tissues. Finally, a rigid articulated model of the bone structures is created in order to transform the internal preoperative configuration to the postoperative state. By solving a set of dynamic equations, the displacements of this rigid model deform the simplified soft tissue layers of the trunk in order to predict the external appearance after scoliosis surgery. A suitable numerical integration scheme to compute the dynami

The adaptive thin shell tetrahedral mesh

Tetrahedral meshes are often used for simulating deformable objects. Unlike engineering disciplines that often focuses on accuracy, computer graphics is biased towards stable, robust, and fast methods. In that spirit we present an approach for building an adaptive inward shell of the surface of an object. The goal is to device a simple and fast algorithm capable of building a topologically consistent tetrahedral mesh. The tetrahedral mesh can be used with several different simulation method, such as the finite element method (FEM), and the main contribution of this paper is a novel tetrahedral mesh generation method based on adaptive surface extrusion