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Algebraic grid generation on trimmed parametric surface using non-self-overlapping planar Coons patch. (English) Zbl 1059.65113

Summary: Using a Coons patch mapping to generate a structured grid in the parametric region of a trimmed surface can avoid the singularity of elliptic partial differential equation methods when only ${C}^{1}$ continuous boundary is given; the error of converting generic parametric ${C}^{1}$ boundary curves into a specified representation form is also avoided. However, overlap may happen on some portions of the algebraically generated grid when a linear or naïve cubic blending function is used in the mapping; this severely limits its usage in most of engineering and scientific applications where a grid system of non-self-overlapping is strictly required.

To solve the problem, non-trivial blending functions in a Coons patch mapping should be determined adaptively by the given boundary so that self-overlapping can be averted. We address the adaptive determination problem by a functional optimization method. The governing equation of the optimization is derived by adding a virtual dimension in the parametric space of the given trimmed surface. Both one- and two-parameter blending functions are studied. To resolve the difficulty of guessing good initial blending functions for the conjugate gradient method used, a progressive optimization algorithm is then proposed which has been shown to be very effective in a variety of practical examples. Also, an extension is added to the objective function to control the element shape. Finally, experiment results are shown to illustrate the usefulness and effectiveness of the presented method.

##### MSC:
 65N50 Mesh generation and refinement (BVP of PDE) 65D18 Computer graphics, image analysis, and computational geometry