Certified Functions for Mesh Generation

Formal methods allow for building correct-by-construction software with provable guarantees. The formal development presented here resulted in certified executable functions for mesh generation. The term certified means that their correctness is established via an artifact, or certificate, which is a statement of these functions in a formal language along with the proofs of their correctness. The term is meaningful only when qualified by a specific set of properties that are proven. This manuscript elaborates on the precise statements of the properties being proven and their role in an implementation of a version of the Isosurface Stuffing algorithm by Labelle and Shewchuk. This work makes use of the Calculus of Inductive Constructions for defining executable functions, stating their properties, and proving these properties via a direct examination of these functions (the property of liveness). The certificate is made available for inspection and execution using the Coq proof assistant

Homeomorphic Tetrahedral Tessellation for Biomedical Images

We present a novel algorithm for generating three-dimensional unstructured tetrahedral meshes for biomedical images. The method uses an octree as the background grid from which to build the final graded conforming meshes. The algorithm is fast and robust. It produces meshes with high quality since it provides dihedral angle lower bound for the output tetrahedra. Moreover, the mesh boundary is a geometrically and topologically accurate approximation of the object surface in the sense that it allows for guaranteed bounds on the two-sided Hausdorff distance and the homeomorphism between the boundaries of the mesh and the boundaries of the materials. The theory and effectiveness of our method are illustrated with the experimental evaluation on synthetic and real medical data

Homeomorphic Tetrahedralization of Multi-material Images with Quality and Fidelity Guarantees

We present a novel algorithm for generating three-dimensional unstructured tetrahedral meshes of multi-material images. The algorithm produces meshes with high quality since it provides a guaranteed dihedral angle bound of up to 19.47° for the output tetrahedra. In addition, it allows for user-specified guaranteed bounds on the two-sided Hausdorff distance between the boundaries of the mesh and the boundaries of the materials. Moreover, the mesh boundary is proved to be homeomorphic to the object surface. The algorithm is fast and robust, it produces a sufficiently small number of mesh elements that comply with these guarantees, as compared to other software. The theory and effectiveness of our method are illustrated with the experimental evaluation on synthetic and real medical data

Automatic Curvilinear Quality Mesh Generation Driven by Smooth Boundary and Guaranteed Fidelity

The development of robust high-order finite element methods requires the construction of valid high-order meshes for complex geometries without user intervention. This paper presents a novel approach for automatically generating a high-order mesh with two main features: first, the boundary of the mesh is globally smooth; second, the mesh boundary satisfies a required fidelity tolerance. Invalid elements are eliminated. Example meshes demonstrate the features of the algorithm

Multitissue Tetrahedral Image-to-Mesh Conversion with Guaranteed Quality and Fidelity

We present a novel algorithm for tetrahedral image-to-mesh conversion which allows for guaranteed bounds on the smallest dihedral angle and on the distance between the boundaries of the mesh and the boundaries of the tissues. The algorithm produces a small number of mesh elements that comply with these bounds. We also describe and evaluate our implementation of the proposed algorithm that is compatible in performance with a state-of-the art Delaunay code, but in addition solves the small dihedral angle problem. Read More: http://epubs.siam.org/doi/10.1137/10081525

Efficient Core Utilization in a Hybrid Parallel Delaunay Meshing Algorithm on Distributed-Memory Cluster

Most of the current supercomputer architectures consist of clusters of nodes that are used by many clients (users). A user wants his/her job submitted in the job queue to be scheduled promptly. However, the resource sharing and job scheduling policies that are used in the scheduling system to manage the jobs are usually beyond the control of users. Therefore, in order to reduce the waiting time of their jobs, it is becoming more and more crucial for the users to consider how to implement the algorithms that are suitable to the system scheduling policies and are able to effectively and efficiently utilize the available resources of the supercomputers. We proposed a hybrid MPI+Threads parallel mesh generation algorithm on distributed memory clusters with efficient core utilization. The algorithm takes the system scheduling information into account and is able to utilize the nodes that have been partially occupied by the jobs of other users. The experimental results demonstrated that the algorithm is effective and efficient to utilize available cores, which reduces the waiting time of the algorithm in the system job scheduling queue. It is up to 12.74 times faster than the traditional implementation without efficient core utilization when a mesh with 2.58 billion elements is created for 400 cores

Fully Generalized Two-Dimensional Constrained Delaunay Mesh Refinement

Traditional refinement algorithms insert a Steiner point from a few possible choices at each step. Our algorithm, on the contrary, defines regions from where a Steiner point can be selected and thus inserts a Steiner point among an infinite number of choices. Our algorithm significantly extends existing generalized algorithms by increasing the number and the size of these regions. The lower bound for newly created angles can be arbitrarily close to $30^{\circ}$. Both termination and good grading are guaranteed. It is the first Delaunay refinement algorithm with a $30^{\circ}$ angle bound and with grading guarantees. Experimental evaluation of our algorithm corroborates the theory