Optimization of unstructured grid

Abstract

This work explores constrained optimization techniques with various basic objective functions to move nodal points around, and to improve the quality of unstructured grids. The negative volumes can be avoided with appropriate volume constraints. The objective function of distance and aspect ratio type combined with volume constraints are useful equilateral smoothers. The objective function of volume type combined with Delaunay triangulation can change the connectivity rapidly and equalize the volume distribution. The constrained optimization technique combined with a grid generation procedure can become a useful tool for obtaining a user-desired grid. I. I N T R O D U C T I O N The grid generation is an important element in flow simulation. For extremely complicated configurations, an unstructured grid generator is a more automatic approach than a patched or overlapped multi-zonal grid generator. The spent on zoning the grid can be totally eliminated. Popular procedures for generating unstructured grid are the advancing front technique and Delaunay triangularization. A few examples of these grid generation procedures can be found in Refs. [l-71. A sophisticated unstructured flow solver can tolera te the grid quality more than a regular unstructured flow solver. However, such a sophisticated flow solver may become quite computationally intensive. Therefore, the quality control of the grid remains an important topic for practical application. If the original grid generator provides reasonable connectivity, for many cases, the quality of grid can be improved by carefully redistributing the nodal point coordinates without changing the connectivity, adding and deleting nodal points. Contrarily, casually moving the grids points not only destroys the grid quality, but can also cause the cell volumes to become negative. Carefully moving the grid points around can adjust the shape of cells and i t can also enlarge the undesired small cells. The too small cells may not cause serious is* Member Technical Staff; Member, AIAA t MetacomD Technoloeies Inc.: Member. AIAA 4 sues in steady-state computation, but these cells are preferably avoided in time wave propagation computation, because an unnecessarily small cell can cause a very small step for satisfying the CFL condition and can cost more computing time. Also using such a small step for most other large cells could induce very small CFL. For those schemes preferring CFL close to stability bound, the small CFL could cause inefficiency as well as large numerical errors in the long-time ~ o l u t i o n . ~ ~ . Furthermore, the dynamic grid can be useful for problems in which the boundary of computational domain changes arbitrarily with time. The dynamic deformation of aircraft is one of these applications. Apparently, automatically moving the grid points to the appropriate locations is critical to predicting accurate solutions with least expense. In present effort, we mainly explore use of the constrained optimization technique to redistributing unstructured grid points. The connectivity may or may not change. The effects of various basic objectives functions are examined. First, we will define the problem and describe the fundamental solution procedure in section 11. Next, the results for various cases (2-D, 3-D, local and global) will be discussed in section 111. The potential of constrained optimization approach will be shown. 11. N U M E R I C A L A P P R O A C H The grid point coordinates may be updated based on nodal point, element or group of nodes. With the nodal point approach, nodal coordinates x(z, y, z ) are updated point by point. Only three variables are to be solved. With an element based approach, nodal coordinates are updated element by element such that four points, twelve variables, are updated simultaneously for a tetrahedral element. The number of variables are further increased based on a group of nodes. Here we will illustrate only the procedure with the nodal based approach since the basic concept behind these variations is the same. Suppose we decide t o move around a node A in Fig. 1. For various reasons, the cell S is too small ., Copyright 01995 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved