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arXiv · 2601.11677

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

Muhammad Ammad·Md Yushalify Misro·Samia Bibi·Ahmad Ramli
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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.

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Muhammad Ammad, Md Yushalify Misro, Samia Bibi, Ahmad Ramli. 2026-01-16. Dirichlet Extremals for Discrete Plateau Problems in GT-Bezier Spaces via PSO. https://arxiv.org/abs/2601.11677

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