Grid-quality measures for structured and unstructured meshes

This paper presents and evaluates four measures that provide posteriori estimates of the error in computed CFD solutions. These measures, which can be applied to both structured and unstructured meshes, link information about the geometry of the cells in a mesh to the computed solution. Three of the measures gauge how well a mesh resolves the flow. One is based on a length scale representing the curvature of scalar field in threedimensions (3D) and an effective cell dimension in the direction of change. The second is based on the directional change in the velocity vector. The third is based on the spectral radius of the deformation tensor and a cell dimension along its eigenvector. The fourth measure gauges the smoothness of cell dimensions along the flow direction. The four measures were evaluated by establishing and interrogating a database, made up of a series of CFD solutions for a three-dimensional test problem in which each solution is generated by using a different mesh that range from "poor" to "good." The interrogation shows the four measures to correlate with the relative error, defined with respect to the grid-independent solution. Graduate Student. Professor. # Professor. Associate Fellow AIAA. Supervisor § Product Development Engineer Engineering Supervisor Copyright © 2001 by T. I-P. Shih. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. INTRODUCTION The accuracy of computational fluid dynamics (CFD) solutions depends not just on the physical model and the numerical scheme, but also on the mesh. The mesh must represent the geometry and enable the physical model and the numerical scheme to resolve the relevant flow physics to the required accuracy. This requirement on the mesh can be difficult to achieve for three-dimensional (3D) problems with complicated geometries and flow features. One reason for this difficulty is that CFD is constantly being challenged to address the most complicated problem possible based on the latest computing capability. As a result, the number of grid points or cells that can be used in a mesh is often restricted by either the available computer memory or a need to have a practical turn-around time in computing a solution. With a limit on the number of grid points or cells that can be used, accuracy considerations demand grid points or cells to be placed in regions where they are most needed to resolve the geometry and flow physics and sparsely distributing them elsewhere. Unfortunately, this non-uniform distribution can create what are referred to as poor quality cells, which can induce considerable errors in the computed solutions. Thus, it is important to understand the relationship between mesh and solution accuracy. For both structured and unstructured meshes, this relationship is not well understood. In an effort to obtain better understanding, many indicators have been developed to evaluate meshes. These indicators, referred to as grid-quality measures, can be classified into two broad groups those that involve only information about the mesh geometry and those that involve information about the mesh geometry and the solution obtained on that mesh. Each of these two groups can further be classified as local (involving only one cell) and macro (involving the cell and adjacent neighboring (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. cells). Grid-quality measures, that involve only geometry and are local, include edge lengths, volume size, aspect ratio, and orthogonality or skewness. Grid-quality measures that involve only geometry but are macro include smoothness, which relate change in grid spacing about a cell. Grid-quality measures that involve geometry plus solution can also be local or macro. Local ones include those that measure resolution such as the y value of the first few grid points normal to the wall. Macro ones include smoothness in conjunction with a solution variable. Most grid-quality measures developed to date focus on the geometry of the cells in a mesh (see, e.g., Liu & Joe (1994)). Relatively few investigators studied grid-quality measures that account for both geometry and solution. The exceptions are studies on solution-adaptive methods in conjunction with r-, h-, and p-grid-refinement techniques. For these methods, the error indicators used to refine the mesh may be used as grid-quality measures. Comprehensive reviews on error indicators used to refine meshes can be found in Carey (1996) and Roache (1998). Research on error indicators that account for the solution has so far considered only gradients of scalar fields (see, e.g., Peraire, et al. (1976), Carey (1996), Diaz (1982), Zienkiewicz & Zhu (1987), Rippa (1992), Celik (1996), Berzins (1998), and references cited there). In CFD, the solution has vectors and tensors, and the different components of the vector and tensor fields should be considered collectively instead of individually. For example, a highly anisotropic mesh can be optimal if all cells are aligned with the flow direction, which requires a link between the cell geometry and a vector quantity. A more serious deficiency of existing error indicators for use as grid-quality measures is that they do not directly link solutions errors to grid quality. They only show where error at one location may be higher than another by using either Taylor series expansions or higher-order polynomial approximations, which is of course adequate for mesh refinement. Finally, error indicators developed so far cannot readily be normalized so that their magnitudes are problem dependent. To be useful as a grid-quality measure, it would be desirable for error indicators to be nearly problem-independent so that certain magnitudes imply certain errors or confidence levels, at least for classes of flows. Thus, the ideal grid-quality measures should have the following characteristics: (1) Account for the geometry of the cells in a mesh and the solution obtained on that mesh. (2) Account for the vector and tensor nature of the computed solution. (3) Are normalized and problem independent. (4) Correlate well with die solution error. The objective of this study is to present and evaluate gridquality measures for structured and unstructured meshes that may have these characteristics. The organization of the rest of this paper is as follows. First, the approach used to develop and evaluate gridquality measure is outlined. Afterwards, the grid-quality measures developed are described. Next, a description of the database used to evaluate the grid-quality measures is given. This is then followed by an evaluation of the gridquality measures developed.