An Application of Variational Minimization: Quasi-Harmonic Coons Patches

Abstract

For a minimal surface, the mean curvature of the surface vanishes for all possible parameterizations which results in a second-order nonlinear partial differential equation ( $pde$ ), whose solution in general is the desired surface as the unknown function of surface parameters. The solution of this partial differential equation is known only for very few cases. Instead of solving the corresponding partial differential equation, we exploit an ansatz method (Ahmad et al. (2013), Ahmad et al. (2014) Ahmad et al. (2015)), used for Coons patches spanned by finite number of boundary curves for a quasi-minimal surface as the extremal of $rms$ of mean curvature. The ansatz method targets a slightly perturbed surface (the rational blending interpolants-based Coons patch, Hermite cubic polynomials interpolants-based Coons patch, and Ferguson surface (vanishing twist vectors) in our case) that comprises initially a nonminimal surface plus the product of a real parameter with a variational function of surface parameters (vanishes at the boundary curves along the unit normal to the slightly perturbed surface). The variational function can be deliberately chosen as the product of linear functions and the mean curvature of the initial nonminimal surface such that it is zero at the boundary curves and then replace this mean curvature by the mean curvature of the resulting surface to find a surface of reduced area, and the process can be repeated for further improvement. In this article, we extend the ansatz method (1) for the rational blending functions interpolants-based Coons patch with a parameter in their form for its different values and (2) for the blending functions comprising Hermite cubic polynomial interpolants-based Coons patch (bicubically blended Coons patch (BBCP) and the Ferguson surface). The ansatz method can be extended for the variational extremal of the surfaces for fuzzy optimal control problems (Filev et al. (1992), Emamizadeh (2005), Farhadinia (2011), Mustafa et al. (2021)).