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The article has been retracted due to the authors’ strong personal reason about the indexing databases of Applied Mathematics.

Surface modeling and grid generation technologies have been long recognized as critical issues in practical applications of computational fluid dynamics [

Surface modeling and grid generation technologies, of course, do not produce a complete design. They are components of a complex design process. Surface grid generation is a time consuming step in the overall process. Surface models and grids are of value only so far as they allow high quality flow predictions to be made at an acceptable cost. The quality of the surface grid has a great impact on the overall quality of the final analysis product.

Researchers have shown that many methods do not ensure that the final surface grid points lie exactly on the original defined surface. It is at present very difficult to assess surface grid quality (orthogonality, curvature, stretching) by any means other than visual inspection. Inspection, of course, is not a systematic process. Further, there are very few absolute measures of quality. This approach leaves a high probability that defects will not be detected at the surface meshing stage, and they will remain in the surface grid to have a magnified impact in later steps of the process. Several methods have been constructed for the generation of good quality meshes on surfaces. The method of [^{2} objective function that depends on the coordinates at the vertices. They demonstrated their method with several examples comprising CAD and scanned meshes. A method to optimize triangular and quadrilateral meshes on parameterized surfaces was proposed by [

In this paper, a new approach to create surface meshes is developed. This approach is based on [

Method of [

Using the notation of [

where

The control functions

Given that

And since

Similarly if the dot product of Equation (1) with

Equation (4) is obtained

Since

With the assumption of orthogonality on the boundaries, it is known that

With the additional assumption of orthogonality in a thin layer of cells near the boundary, say

Since the spacing

the analogous expression for

Orthogonality is one of the important features that determine the quality of a mesh. In many cases the aim is not really to produce a completely orthogonal grid, but rather to achieve near orthogonality throughout the region and true orthogonality at the boundaries. Several procedures can be used to check for orthogonality. The ones used in this paper are orthogonality functionals and area-orthogonality functionals. In their paper, [

Recall that exact orthogonality at all grid points means that

it can be shown [

A second orthogonality functional is

two tangent vectors and not their lengths. This functional is considered to be a good grid generator. In these two integrals, if the mesh is exactly orthogonal, then

proportional to some given weight function. By taking a linear combination of area and orthogonality functionals, grid generators which simultaneously try to enforce equidistributed cell areas and orthogonality can be obtained. The name AO derives from the fact that the functional is halfway between the equal area and orthogonal-

ity functional. Two such functionals to be minimized are

Another type of functional, with a guaranteed unique minimum, was proposed by Liao. The Modified Liao functional, designed to reduce the tendency for the grids to fold, is

For our purpose, we will evaluate these functionals for the grids generated by the above methods, and look for the method which produces the minimum value.

Three surface grids are created using the present method, method created by [

In this example a mesh on a paraboloid surface with equation

Thomas and Middlecoff Method [ | Present Method | Laplace Equations | Algebraic Method | |
---|---|---|---|---|

1866 | 0.6 | 2257.8 | 15.0 | |

0.082 | 0.000 | 0.097 | 0.001 | |

1527 | 1521 | 1528 | 1521 | |

225,477 | 218,098 | 226,557 | 218,457 | |

6,064,448,516 | 5,088,636,053 | 6,222,626,312 | 5,147,728,003 | |

56.2 | 56.8 | 56.2 | 56.7 | |

Iterations | 285 | 156 | 289 | |

Error | 5.425 | 5.466 | 5.413 | 5.429 |

In this example the paraboloid given in Example 1 is modified and a mesh is generated using the polynomial

plane. The other three boundary curves are kept the same. The quality of the mesh and the number of iterations used by each method are shown in

From

In this example the paraboloid given in Example 1 is modified and a mesh is generated using the intersection of two planes with the paraboloid to replace the circle

where

kept the same. The quality of the mesh and the number of iterations used by each method are shown in

From

region near

a flow solver. To properly mesh the cusp region using these differential equations methods, multiblock approach can be used, drawing a block boundary curve from the cusp and distributing points along this curve.

A mesh can be generated using many methods. The methods used in this paper are algebraic methods, Laplace equations, Thomas and Middlecoff’s method and the extension of Barron’s method from planar regions to surfaces. To generate a rough mesh, algebraic method is the easiest to use because it creates a mesh in one iteration only. To create a smooth mesh one can use Laplace equations. Thomas and Middlecoff method can be used to obtain a good mesh in the sense of orthogonality and the number of iterations, but the best method to obtain a

Thomas and Middlecoff Method [ | Present Method | Laplace Equations | Algebraic Method | |
---|---|---|---|---|

1727 | 5526 | 3382 | 12,928 | |

0.2 | 0.3 | 0.3 | 0.7 | |

1529 | 1541 | 1538 | 1569 | |

210,096 | 210,312 | 209,307 | 214,762 | |

4,705,221,747 | 4,636,839,632 | 4,705,196,061 | 4,538,876,537 | |

81.0 | 88.4 | 75.8 | 89.6 | |

Iterations | 271 | 327 | 270 |

Thomas and Middlecoff Method [ | Present Method | Laplace Equations | Algebraic Method | |
---|---|---|---|---|

28,895 | 24,310 | 26,348 | 98,122 | |

0.9 | 0.6 | 0.8 | 2.4 | |

1608 | 1589 | 1601 | 1802 | |

291,472 | 283,370 | 284,398 | 354,063 | |

10,935,749,825 | 10,442,891,878 | 9,938,499,904 | 14,480,054,639 | |

56.6 | 57.5 | 55.1 | 64.6 | |

Iterations | 245 | 328 | 212 |

more orthogonal mesh is the present method. The number of iterations for this method to converge is almost the same as that of Thomas and Middlecoff’s method.

Bashar Zogheib,Ali El Saheli, (2015) Retraction: An Improved Approach of Surface Meshes. Applied Mathematics,06,510-519. doi: 10.4236/am.2015.63048