Dirichlet Extremals for Discrete Plateau Problems in GT-Bezier Spaces via PSO

TL;DR

This paper introduces a novel method for solving discrete Plateau problems in GT-Bézier spaces using Dirichlet energy minimization and particle swarm optimization, leading to surfaces with reduced area and energy.

Contribution

It develops a new Dirichlet-energy extremal approach for GT-Bézier surfaces, combining boundary control with optimization to produce minimality-biased patches.

Findings

The method reduces Dirichlet energy compared to classical patches.

It often decreases the surface area relative to traditional constructions.

Numerical experiments confirm the effectiveness of the two-level optimization procedure.

Abstract

We study a discrete analogue of the parametric Plateau problem in a non-polynomial tensor-product surface spaces generated by the generalized trigonometric (GT)--B\'ezier basis. Boundary interpolation is imposed by prescribing the boundary rows and columns of the control net, while the interior control points are selected by a Dirichlet principle: for each admissible choice of B\'ezier basis shape parameters, we compute the unique Dirichlet-energy extremal within the corresponding GT--B\'ezier patch space, which yields a parameter-dependent symmetric linear system for the interior control net under standard nondegeneracy assumptions. The remaining design freedom is thereby reduced to a four-parameter optimization problem, which we solve by particle swarm optimization. Numerical experiments show that the resulting two-level procedure consistently decreases the Dirichlet energy and, in our tests, often reduces the realized surface area relative to classical Bernstein--B\'ezier Dirichlet patches and representative quasi-harmonic and bending-energy constructions under identical boundary control data. We further adapt the same Dirichlet-extremal methodology to a hybrid tensor-product/bilinear Coons framework, obtaining minimality-biased TB--Coons patches from sparse boundary specifications.