Submitted to: Journal of Computational Physics 2
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 11/7/2005
Publication Date: 1/4/2006
Citation: Zhang, Y., Jia, Y., Wang, S.S. 2006. 2d nearly orthogonal mesh generation with controls on distortion functions. Journal of Computational Physics 2. V. 218 No. 2 pp. 549-571.
Interpretive Summary: In Computational Fluids Dynamics (CFD), practical fluids flows are often associated with complex domains, in which the generation of an orthogonal mesh becomes difficult or even impossible to attain when using classical generation systems such as the Ryskins and Leal (RL) system and conformal mapping procedures. In the RL system, the distortion function is only controlled by the local orthogonal conditions, which causes unpredictability in the distribution of the mesh density, while in the conformal mapping system, the distortion function is enforced by using a constant value. To address these shortcomings, a new method to control the distortion function was developed. In the proposed method, both the averaged scale factors and the local scale factors are used to evaluate the distortion function. It takes into account the effects of both the local orthogonal condition characterized by local scale factors and the local smoothness conditions (geometry and mesh size) characterized by the averaged scale factors. In this manner, both the strict local orthogonal condition of the RL system and the strict local smoothness condition of the conformal mapping system are relaxed, and consequently only orthogonal meshes with adequate smoothness can be obtained. Two adjustable parameters, r-xi and r-eta, are used to control the ratio of the local scale factors and the averaged scale factors in the xi and eta directions and this of the local balance of orthogonality and smoothness. These two parameters can either be user-specified with constant values through the whole domain or automatically evaluated by the indicators of the local relative smoothness conditions that are defined as the ratio of the difference over the sum of the local scale factors and the averaged scale factors. The smoother the mesh is, the smaller these two parameters are, and the smaller the effects of the averaged scale factors on the distortion function will be. Several benchmark examples were selected to demonstrate and compare the proposed method with the original RL systems of conformal mapping and the RL method with contribution factors proposed by Zhang et al. (2004). The controls on both directions significantly improved the global smoothness, but caused slightly folded meshes in some domains. With controls on only one direction, the smoothness in the corresponding direction could be improved. The automatic controls on the local balance of the orthogonality and the smoothness produced meshes with the best overall quality. For most cases, the RL method with contribution factors yielded similar results as the RL method with automatic controls of the distortion function. All methods except the RL method with contribution factors failed to generate a good mesh without distortion. Sensitivity analysis shows that the parameters r-xi and r-eta have significant effects on the mesh quality in the range of 0, 0.1, and beyond this range these two parameters have much less influence. The proposed method was also applied to two natural channels with complex geometries. It was shown that the proposed method is capable of generating nearly orthogonal meshes with a good balance between orthogonality and smoothness in geometrically complex domains where the original RL system and the conformal mapping system failed.
Technical Abstract: A method to control the distortion function of the Ryskin and Leal (RL) orthogonal mesh generation system is presented. The proposed method considers the effects from not only the local orthogonal condition but also the local smoothness condition (the geometry and the mesh size) on the distortion function. The distortion function is determined by both the scale factors and the averaged scale factors of the constant mesh lines. Two adjustable parameters are used to control the local balance of the orthogonality and the smoothness. The proposed method is successfully applied to several benchmark examples and the natural river channels with complex geometries.