Lori A. Freitag

TETRAHEDRAL MESH IMPROVEMENT USING SWAPPING AND SMOOTHING

Automatic mesh generation and adaptive refinement methods for complex three-dimensional domains have proven to be very successful tools for the efficient solution of complex applications problems. These methods can, however, produce poorly shaped elements that cause the numerical solution to be less accurate and more difficult to compute. Fortunately, the shape of the elements can be improved through several mechanisms, including face- and edge-swapping techniques, which change local connectivity, and optimization-based mesh smoothing methods, which adjust mesh point location. We consider several criteria for each of these two methods and compare the quality of several meshes obtained by using different combinations of swapping and smoothing. Computational experiments show that swapping is critical to the improvement of general mesh quality and that optimization-based smoothing is highly effective in eliminating very small and very large angles. High-quality meshes are obtained in a computationally efficient manner by using optimization-based smoothing to improve only the worst elements and a smart variant of Laplacian smoothing on the remaining elements. Based on our experiments, we offer several recommendations for the improvement of tetrahedral meshes.

Local optimization‐based simplicial mesh untangling and improvement

We present an optimization-based approach for mesh untangling that maximizes the minimum area or volume of simplicial elements in a local submesh. These functions are linear with respect to the free vertex position; thus the problem can be formulated as a linear program that is solved by using the computationally inexpensive simplex method. We prove that the function level sets are convex regardless of the position of the free vertex, and hence the local subproblem is guaranteed to converge. Maximizing the minimum area or volume of mesh elements, although well suited for mesh untangling, is not ideal for mesh improvement, and its use often results in poor quality meshes. We therefore combine the mesh untangling technique with optimization-based mesh improvement techniques and expand previous results to show that a commonly used two-dimensional mesh quality criterion can be guaranteed to converge when starting with a valid mesh. Typical results showing the effectiveness of the combined untangling and smoothing techniques are given for both two- and three- dimensional simplicial meshes.

A Parallel Algorithm for Mesh Smoothing

Maintaining good mesh quality during the generation and refinement of unstructured meshes in finite-element applications is an important aspect in obtaining accurate discretizations and well-conditioned linear systems. In this article, we present a mesh-smoothing algorithm based on nonsmooth optimization techniques and a scalable implementation of this algorithm. We prove that the parallel algorithm has a provably fast runtime bound and executes correctly for a parallel random access machine (PRAM) computational model. We extend the PRAM algorithm to distributed memory computers and report results for two- and three-dimensional simplicial meshes that demonstrate the efficiency and scalability of this approach for a number of different test cases. We also examine the effect of different architectures on the parallel algorithm and present results for the IBM SP supercomputer and an ATM-connected network of SPARC Ultras.

Parallel components for PDEs and optimization: some issues and experiences

High-performance simulations in computational science often involve the combined software contributions of multidisciplinary teams of scientists, engineers, mathematicians, and computer scientists. One goal of component-based software engineering in large-scale scientific simulations is to help manage such complexity by enabling better interoperability among codes developed by different groups. This paper discusses recent work on building component interfaces and implementations in parallel numerical toolkits for mesh manipulations, discretization, linear algebra, and optimization. We consider several motivating applications involving partial differential equations and unconstrained minimization to demonstrate this approach and evaluate performance.

A cost/benefit analysis of simplicial mesh improvement techniques as measured by solution efficiency.

The quality of unstructured meshes has long been known to affect both the efficiency and the accuracy of the numerical solution of application problems. Mesh quality can often be improved through the use of algorithms based on local reconnection schemes, node smoothing, and adaptive refinement or coarsening. These methods typically incur a significant cost, and in this paper, we provide an analysis of the tradeoffs associated with the cost of mesh improvement in terms of solution efficiency. We first consider simple finite element applications and show the effect of increasing the number of poor quality elements in the mesh and decreasing their quality on the solution time of a number of different solvers. These simple application problems are theoretically well-understood, and we show the relationship between the quality of the mesh and the eigenvalue spectrum of the resulting linear system. We then consider realistic finite element and finite volume application problems, and show that the cost of mesh improvement is significantly less than the cost of solving the problem on a poorer quality mesh.

A computational study of the effect of unstructured mesh quality on solution efficiency

It is well known that mesh quality affects both efficiency and accuracy of CFD solutions. Meshes with distorted elements make solutions both more difficult to compute and less accurate. We review a recently proposed technique for improving mesh quality as measured by element angle (dihedral angle in three dimensions) using a combination of optimization-based smoothing techniques and local reconnection schemes. Typical results that quantify mesh improvement for a number of application meshes are presented. We then examine effects of mesh quality as measured by the maximum angle in the mesh on the convergence rates of two commonly used CFD solution techniques. Numerical experiments are performed that quantify the cost and benefit of using mesh optimization schemes for incompressible flow over a cylinder and weakly compressible flow over a cylinder.

The Scalability of Mesh Improvement Algorithms

In this paper we develop a common framework to explore the scalability of three improvement strategies for unstructured meshes: adaptive refinement, vertex smoothing, and edge flipping. We give a general parallel algorithm for these strategies based on defining, for each algorithm, an elemental operation and a task graph. By choosing the correct task graph, we can ensure the correct parallel execution of the algorithms independent of implementation. Finally, we present experimental results obtained on an IBM SP system and use these results to investigate, in practice, the scaling and relative costs of these algorithms.

Adaptive, Multiresolution Visualization of Large Data Sets using a Distributed Memory Octree

The interactive visualization and exploration of large scientific data sets is a challenging and difficult task; their size often far exceeds the performance and memory capacity of even the most powerful graphics workstations. To address this problem, we have created a technique that combines hierarchical data reduction methods with parallel computing to allow interactive exploration of large data sets while retaining full-resolution capability. The user may interactively change the resolution of the reduced data set either globally or by specifying a region of interest. In this way, high resolution can be obtained in local subregions without sacrificing graphics performance. We describe the software architecture of the system, give details pertaining to the use of a distributed memory octree used to create the reduced data set, and present performance results for the visualization of Rayleigh-Taylor instability and x-ray burst simulation data sets.

Comparison of remote visualization strategies for interactive exploration of large data sets

We compare three remote visualization strategies used for interactive exploration of large data sets: image-based rendering, parallel visualization servers, and subsampling. We review each strategy and provide details for an adaptive multiresolution subsampling technique that we have developed. To determine the problem regimes for which each approach is most cost effective, we develop performance models to analyze the costs of computation and communication associated with the common visualization task of isosurface generation. Using these models, we investigate a number of hardware system configurations and task complexity scenarios when parameters such as problem size, visualization demands, and network bandwidth change. For one particular strategy, subsampling, we further investigate the tradeoffs between multiresolution and uniform grid methods in terms of performance and approximation errors.

Users manual for Opt-MS : local methods for simplicial mesh smoothing and untangling.

Creating meshes containing good-quality elements is a challenging, yet critical, problem facing computational scientists today. Several researchers have shown that the size of the mesh, the shape of the elements within that mesh, and their relationship to the physical application of interest can profoundly affect the efficiency and accuracy of many numerical approximation techniques. If the application contains anisotropic physics, the mesh can be improved by considering both local characteristics of the approximate application solution and the geometry of the computational domain. If the application is isotropic, regularly shaped elements in the mesh reduce the discretization error, and the mesh can be improved a priori by considering geometric criteria only. The Opt-MS package provides several local node point smoothing techniques that improve elements in the mesh by adjusting grid point location using geometric, criteria. The package is easy to use; only three subroutine calls are required for the user to begin using the software. The package is also flexible; the user may change the technique, function, or dimension of the problem at any time during the mesh smoothing process. Opt-MS is designed to interface with C and C++ codes, ad examples for both two-and three-dimensional meshes are provided.