Control functions and grid qualities measurements in the elliptic grid generation around arbitrary surfaces

Kim, Sungcho

doi:10.1002/(sici)1097-0363(20000515)33:1<81::aid-fld3>3.0.co;2-0

Cited by 6 publications

(7 citation statements)

References 9 publications

(8 reference statements)

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“…This section describes the necessary formulas and procedures for solving the constrained optimization problem as defined in Eq. (13). A penalty function based method is adopted to convert this constrained optimization problem into an unconstrained optimization problem, where our numerical implementation minimizes the objective function by means of the conjugate gradient method.…”

Section: Numerical Implementationmentioning

confidence: 99%

“…Automatic 2D mesh generation has been studied for decades [1][2][3]. Among various approaches, the three most popular ones are Delaunay-triangulation method [4][5][6][7][8], the partial differential equation (PDE) method [9][10][11][12][13], and the boundary-conforming mapping (BCM) method [14][15][16][17]. The Delaunay-triangulation approach meshes a region R by incrementally inserting points into the interior of R while maintaining certain special geometric properties on the resulting triangles.…”

Section: Introductionmentioning

confidence: 99%

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Non-self-overlapping Hermite interpolation mapping: a practical solution for structured quadrilateral meshing

Wang¹,

Tang²

2005

Computer-Aided Design

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This paper addresses the problem of constructing a structured quadrilateral grid inside a given four-sided 2D region by a particular boundary-conforming mapping scheme-Hermite Interpolation Mapping (HIM). When the four given boundary curves are concave and convoluted, all boundary-conform mapping methods suffer from potential self-overlapping problem. Under HIM, the geometry of the grid depends on both the four boundary curves and the tangent vector functions associated with the curves. While the four boundary curves are fixed, the tangent functions in HIM can be varied to suit the need of controlling the characteristics of the mesh inside the given region so to prevent self-overlapping. Besides tangent functions, the four twist vectors at the corners of the region can also be adjusted to influence the distribution of the inner grid elements. In our approach, a constrained functional optimization scheme is adopted to adjust the tangent functions and the twist vectors, adaptive to the geometry of the boundary curves, so that the resulting HIM will be free of selfoverlapping. The optimization is carried out on the shape control energy that measures the overall mesh quality of the underlying HIM while the self-overlapping is strongly prevented in the form of constraints to the optimization. Experimental results show the promise of the proposed method as a practical and effective solution for structured grid generation.

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Section: Numerical Implementationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Non-self-overlapping Hermite interpolation mapping: a practical solution for structured quadrilateral meshing

Wang¹,

Tang²

2005

Computer-Aided Design

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“…An example of a trimmed n-sided patch and its parametric area is given in Figure 1. The task of approximating a trimmed surface by a complex of simple planar elements (either triangular or quadrilateral) plays an important role in engineering computing; this is referred to as the surface meshing operation, which has been studied for many years [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The two most powerful analysis tools in engineering are the ÿnite element methods and the ÿnite di erence methods.…”

Section: Deÿnition 11mentioning

confidence: 99%

“…Also, error may be generated when converting generic boundary curves into curves with a speciÿed representation. Grid generation is actually a boundary-value problem, so grids can be generated from point distribution on boundaries by solving elliptic PDEs in the ÿeld [6][7][8][9][10]. The smoothness properties and extremum principles of some such PDE systems can serve to produce smooth grids without boundary overlapping.…”

Section: Deÿnition 11mentioning

confidence: 99%

Non‐self‐overlapping structured grid generation on an n‐sided surface

Wang

Tang

2004

Numerical Methods in Fluids

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SUMMARYMost existing meshing algorithms for a 2D or shell ÿgure requires the ÿgure to have exactly four sides. Generating structured grids in the n-sided parametric region of a trimmed surface thus usually requires to ÿrst partition the region into four-sided sub-regions. We address the automatic structured grid generation problem in an n-sided region by ÿtting a planar Gregory patch so that the partition requirement is naturally avoided. However, self-overlapping may occur in some portions of the algebraically generated grid; this severely limits its usage in most of engineering and scientiÿc applications where a grid system with no self-intersecting is strictly required. To solve the problem, we use a functional optimization approach to move grid nodes in the u−v domain of the trimmed surface to eliminate the self-overlapping. The derivatives of a Gregory patch, which are extremely di cult to compute analytically, are not required in our method. Thus, our optimization algorithm compares favourably at least in terms of speed with some other mesh optimization algorithms, such as the elliptic PDE method. In addition, to overcome the di culty of guessing a good initial position of every grid node for the conjugate gradient method, a progressive optimization algorithm is incorporated in our optimization. Experiment results are given to illustrate the usefulness and e ectiveness of the presented method.

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“…Currently, there are mainly two classes of methods for generating structured quadrilateral grids in a planar four-sided region, the partial differential equation (PDE) method [1][2][3][4][5], and the boundary-conforming mapping (BCM) method [6][7][8][9]. The PDE method generates a grid by first distributing points on the boundary curves and then solving elliptic PDEs in the field.…”

Section: Introductionmentioning

confidence: 99%

Generating strictly non-self-overlapping structured quadrilateral grids

Lin

Tang

Joneja

et al. 2007

Computer-Aided Design

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In this paper, we present a BPM (Bézier Patch Mapping) algorithm which generates a strictly non-self-overlapping structured quadrilateral grid in a given four-sided planar region. Given four pieces of polynomial curves which enclose a simple region in the plane, the algorithm first constructs a Bézier patch which interpolates the four curves (as its four boundary curves), while the inner control points of its control grid remain unknown. In this paper, we show that, for the bijective condition to be satisfied, it is sufficient that the interior points satisfy a set of quadratic inequality equations. Exploiting this key result, we formulate the mapping algorithm as an optimization problem where the constraints are the bijective condition of the Bézier Patch Mapping, and the objective is to find out the best from all of the non-self-overlapping grids. Thus, commercial optimization solvers can be used to find the bijective mapping. If a solution to the optimization problems exists, then so does a solution to the mapping problem, and vice-versa. The BPM method is simple and intuitive, and some examples presented in this paper demonstrate its effectiveness

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Control functions and grid qualities measurements in the elliptic grid generation around arbitrary surfaces

Cited by 6 publications

References 9 publications

Non-self-overlapping Hermite interpolation mapping: a practical solution for structured quadrilateral meshing

Non-self-overlapping Hermite interpolation mapping: a practical solution for structured quadrilateral meshing

Non‐self‐overlapping structured grid generation on an n‐sided surface

Generating strictly non-self-overlapping structured quadrilateral grids

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