3D boundary recovery by constrained Delaunay tetrahedralization

Three-dimensional boundary recovery is a fundamental problem in mesh generation. In this paper, we propose a practical algorithm for solving this problem. Our algorithm is based on the construction of a {\it constrained Delaunay tetrahedralization} (CDT) for a set of constraints (segments and facets). The algorithm adds additional points (so-called Steiner points) on segments only. The Steiner points are chosen in such a way that the resulting subsegments are Delaunay and their lengths are not unnecessarily short. It is theoretically guaranteed that the facets can be recovered without using Steiner points. The complexity of this algorithm is analyzed. The proposed algorithm has been implemented. Its performance is reported through various application examples

Higher-quality tetrahedral mesh generation for domains with small angles by constrained Delaunay refinement

Algorithms for generating Delaunay tetrahedral meshes have difficulty with domains whose boundary polygons meet at small angles. The requirement that all tetrahedra be Delaunay often forces mesh generators to overrefine near small domain angles---that is, to produce too many tetrahedra, making them too small. We describe a provably good algorithm that generates meshes that are constrained Delaunay triangulations, rather than purely Delaunay. Given a piecewise linear domain free of small angles, our algorithm is guaranteed to construct a mesh in which every tetrahedron has a radius-edge ratio of $2 \sqrt{2 / 3} \doteq 1.63$ or better. This is a substantial improvement over the usual bound of $2$; it is obtained by relaxing the conditions in which boundary triangles are subdivided. Given a domain with small angles, our algorithm produces a mesh in which the quality guarantee is compromised only in specific places near small domain angles. We prove that most mesh edges have lengths proportional to the domain's minimum local feature size; the exceptions span small domain angles. Our algorithm tends to generate meshes with fewer tetrahedra than purely Delaunay methods because it uses the constrained Delaunay property, rather than vertex insertions, to enforce the conformity of the mesh to the domain boundaries. An implementation demonstrates that our algorithm does not overrefine near small domain angles

Fully anisotropic split-tree adaptive refinement mesh generation using tetrahedral mesh stitching

Due to the myriad of geometric topologies that modern computational fluid dynamicists desire to mesh and run solutions on, the need for a robust Cartesian Mesh Generation algorithm is paramount. Not only do Cartesian meshes require less elements and often help resolve flow features but they also allow the grid generator to have a great deal of control in so far as element aspect ratio, size, and gradation. Fully Anisotropic Split-Tree Adaptive Refinement (FASTAR) is a code that allows the user to exert a great deal of control and ultimately generate a valid, geometry conforming mesh. Due to the split-tree nature and the use of volumetric pixels (voxels), non-unit aspect ratio meshing is easily achieved. Nodes are not generated until the end which mitigates tolerance issues. The tree is retained coherently, and viscous layers may be inserted in the space between the geometry and the Cartesian mesh before it is tetrahedralized. FASTAR uses tree traversal to determine neighbors robustly, and with the tetrahedralization of only a small amount of space around the geometry, sliver cells and inverted elements are avoided. The code uses Riemannian Metric Tensors to generate geometry-appropriate spacing and is capable of adaptive meshing from a spacing field generated either by the user or from solution data. FASTAR is a robust, general mesh generator that allows maximum flexibility with minimal post-processing

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

Lattice cleaving: a multimaterial tetrahedral meshing algorithm with guarantees

pre-printWe introduce a new algorithm for generating tetrahedral meshes that conform to physical boundaries in volumetric domains consisting of multiple materials. The proposed method allows for an arbitrary number of materials, produces high-quality tetrahedral meshes with upper and lower bounds on dihedral angles, and guarantees geometric fidelity. Moreover, the method is combinatoric so its implementation enables rapid mesh construction. These meshes are structured in a way that also allows grading, to reduce element counts in regions of homogeneity. Additionally, we provide proofs showing that both element quality and geometric fidelity are bounded using this approach