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

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.