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

Parallel generalized Delaunay mesh refinement

The modeling of physical phenomena in computational fracture mechanics, computational fluid dynamics and other fields is based on solving systems of partial differential equations (PDEs). When PDEs are defined over geometrically complex domains, they often do not admit closed form solutions. In such cases, they are solved approximately using discretizations of domains into simple elements like triangles and quadrilaterals in two dimensions (2D), and tetrahedra and hexahedra in three dimensions (3D). These discretizations are called finite element meshes. Many applications, for example, real-time computer assisted surgery, or crack propagation from fracture mechanics, impose time and/or mesh size constraints that cannot be met on a single sequential machine. as a result, the development of parallel mesh generation algorithms is required.;In this dissertation, we describe a complete solution for both sequential and parallel construction of guaranteed quality Delaunay meshes for 2D and 3D geometries. First, we generalize the existing 2D and 3D Delaunay refinement algorithms along with theoretical proofs of mesh quality in terms of element shape and mesh gradation. Existing algorithms are constrained by just one or two specific positions for the insertion of a Steiner point inside a circumscribed disk of a poorly shaped element. We derive an entire 2D or 3D region for the selection of a Steiner point (i.e., infinitely many choices) inside the circumscribed disk. Second, we develop a novel theory which extends both the 2D and the 3D Generalized Delaunay Refinement methods for the concurrent and mathematically guaranteed independent insertion of Steiner points. Previous parallel algorithms are either reactive relying on implementation heuristics to resolve dependencies in parallel mesh generation computations or require the solution of a very difficult geometric optimization problem (the domain decomposition problem) which is still open for general 3D geometries. Our theory solves both of these drawbacks. Third, using our generalization of both the sequential and the parallel algorithms we implemented prototypes of practical and efficient parallel generalized guaranteed quality Delaunay refinement codes for both 2D and 3D geometries using existing state-of-the-art sequential codes for traditional Delaunay refinement methods. On a heterogeneous cluster of more than 100 processors our implementation can generate a uniform mesh with about a billion elements in less than 5 minutes. Even on a workstation with a few cores, we achieve a significant performance improvement over the corresponding state-of-the-art sequential 3D code, for graded meshes

Generalized Insertion Region Guides for Delaunay Mesh Refinement

Mesh generation by Delaunay refinement is a widely used technique for constructing guaranteed quality triangular and tetrahedral meshes. The quality guarantees are usually provided in terms of the bounds on circumradius-to-shortest-edge ratio and on the grading of the resulting mesh. Traditionally circumcenters of skinny elements and middle points of boundary faces and edges are used for the positions of inserted points. However, recently variations of the traditional algorithms are being proposed that are designed to achieve certain optimization objectives by inserting new points in neighborhoods of the center points. In this paper we propose a general approach to the selection of point positions by defining one-, two-, and three-dimensional selection regions such that any point insertion strategy based on these regions is automatically endowed with the theoretical guarantees proven here. In particular, for the input models defined by planar linear complexes under the assumption that no input angle is less than $90^\circ$, we prove the termination of the proposed generalized algorithm, as well as the fidelity and good grading of the resulting meshes

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

Extreme-Scale Parallel Mesh Generation: Telescopic Approach

In this poster we focus and present our preliminary results pertinent to the integration of multiple parallel Delaunay mesh generation methods into a coherent hierarchical framework. The goal of this project is to study our telescopic approach and to develop Delaunay-based methods to explore concurrency at all hardware layers using abstractions at (a) medium-grain level for many cores within a single chip and (b) coarse-grain level, i.e., sub-domain level using proper error metric- and application-specific continuous decomposition methods

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