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

TL;DR

This work develops a discrete Plateau construction within GT-Bézier tensor-product spaces by fixing boundary rows/columns and selecting interior points as a Dirichlet-energy extremal, yielding a parameter-dependent linear system for the interior controls. An outer PSO loop optimizes the four GT-Bézier shape parameters to minimize the reduced Dirichlet energy, producing surfaces with consistently lower energy and often smaller area than classical Bernstein–Dirichlet or quasi-harmonic variants under the same boundary data. The approach is extended to a hybrid TB–Coons framework, producing minimality-biased surfaces from sparse boundary information, and is connected to harmonicity by measuring and reducing Laplacian defects within both GT-Bézier and Coons-type constructions. Overall, the method demonstrates how integrating shape-parameterized non-polynomial bases with Dirichlet relaxation yields numerically efficient, energy-efficient discrete minimal surfaces suitable for geometric design and analysis.

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)--Bézier 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 Bézier basis shape parameters, we compute the unique Dirichlet-energy extremal within the corresponding GT--Bézier 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--Bézier 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.

Figures (7)

• Figure 1: Cubic GT-Bézier curves and corresponding curvature plots under varying shape parameters.
• Figure 2: Bicubic surfaces constructed from the same boundary control points by different approaches.
• Figure 3: Bicubic surfaces constructed from the same boundary control points by different approaches.
• Figure 4: PSO convergence histories for minimizing $\mathcal{J}(\boldsymbol{\alpha})$ (10 runs): (a) Example \ref{['ex:quaddirichlet']}; (b) Example \ref{['ex:dirichletcubic']}.
• Figure 5: Approximately harmonic GT--Bézier surfaces from partial boundary control data.

Theorems & Definitions (19)

• Definition 1: Bernstein polynomials farouki2012bernsteinPhillips2003
• Definition 2: Bézier curve farin2014curves
• Definition 3: Quadratic GT-Bézier (base case)
• Definition 4: Recursive degree elevation (GT-Bézier basis of degree $n\ge 3$)
• Proposition 2.1: Partition of unity and endpoint interpolation
• proof
• Remark 2.1: Link to the surface parameters of Section \ref{['sec:intro']}
• Definition 5: Univariate GT-Bézier curve
• Theorem 1: Dirichlet extremals for GT-Bézier surfaces
• proof