Achieving Finite Element Mesh Quality via Optimization of the Jacobian Matrix Norm and Associated Quantities, Part II - A Framework for Volume Mesh Optimization and the Condition Number of the Jacobian Matrix

Three-dimensional unstructured tetrahedral and hexahedral finite element mesh optimization is studied from a theoretical perspective and by computer experiments to determine what objective functions are most effective in attaining valid, high quality meshes. The approach uses matrices and matrix norms to extend the work in Part I to build suitable 3D objective functions. Because certain matrix norm identities which hold for 2 x 2 matrices do not hold for 3 x 3 matrices. significant differences arise between surface and volume mesh optimization objective functions. It is shown, for example, that the equivalence in two-dimensions of the Smoothness and Condition Number of the Jacobian matrix objective functions does not extend to three dimensions and further. that the equivalence of the Oddy and Condition Number of the Metric Tensor objective functions in two-dimensions also fails to extend to three-dimensions. Matrix norm identities are used to systematically construct dimensionally homogeneous groups of objective functions. The concept of an ideal minimizing matrix is introduced for both hexahedral and tetrahedral elements. Non-dimensional objective functions having barriers are emphasized as the most logical choice for mesh optimization. The performance of a number of objective functions in improving mesh quality was assessed on a suite of realistic test problems, focusing particularly on all-hexahedral ''whisker-weaved'' meshes. Performance is investigated on both structured and unstructured meshes and on both hexahedral and tetrahedral meshes. Although several objective functions are competitive, the condition number objective function is particularly attractive. The objective functions are closely related to mesh quality measures. To illustrate, it is shown that the condition number metric can be viewed as a new tetrahedral element quality measure

Achieving Finite Element Mesh Quality via Optimization of the Jacobian Matrix Norm and Associated Quantities, Part 1 - A Framework for Surface Mesh Optimization

Structured mesh quality optimization methods are extended to optimization of unstructured triangular, quadrilateral, and mixed finite element meshes. N"ew interpretations of well-known nodally-bssed objective functions are made possible using matrices and matrix norms. The matrix perspective also suggests several new objective functions. Particularly significant is the interpretation of the Oddy metric and the Smoothness objective functions in terms of the condition number of the metric tensor and Jacobian matrix, respectively. Objective functions are grouped according to dimensionality to form weighted combinations. A simple unconstrained local optimum is computed using a modiiied N-ewton iteration. The optimization approach was implemented in the CUBIT mesh generation code and tested on several problems. Results were compared against several standard element-based quaIity measures to demonstrate that good mesh quality can be achieved with nodally-based objective functions

The role of regional groundwater flow in the hydrogeology of the Culebra member of the Rustler formation at the Waste Isolation Pilot Plant (WIPP), southeastern New Mexico

Numerical simulation has been used to enhance conceptual understanding, of the hydrogeology of the Culebra Dolomite in the context of regional groundwater flow. The hydrogeology is of interest because this unit is a possible pathway for offsite migration of radionuclides from a proposed repository for defense-generated transuranic wastes (the Waste Isolation Pilot Plant). The numerical model used is three-dimensional, extends laterally to topographic features that form the actual boundaries of a regional groundwater system, and uses a free-surface upper boundary condition to simulate the effect of change in the rate of recharge on groundwater flow. Steady-state simulations were performed to examine the sensitivity of simulation results to assumed values for hydraulic conductivity and recharge rate. Transient simulations, covering the time period from 14,000 years in the past to 10,000 years in the future, provided insight into how patterns of groundwater flow respond to changes in climate. Simulation results suggest that rates and directions of Groundwater flow in the Culebra change with time due to interaction between recharge, movement of the water table, and the topography of the land surface. The gentle east-to-west slope of the land surface in the vicinity of the WIPP caused groundwater in the Culebra to flow toward and discharge into Nash Draw, a topographic depression. Modern-day flow directions in the Culebra reflect regional rather than local features of the topography. Changes in Groundwater flow, however, lagged behind changes in the rate of recharge. The present-day position of the water table is still adjusting to the decrease in recharge that ended 8,000 years ago. Contaminants introduced into the Culebra will travel toward the accessible environment along the Culebra rather than by leaking upward or downward into other units. Natural changes in flow over the next 10,000 years will be small and will mainly reflect future short-term wet periods