论文信息 - A trust region method for constructing triangle-mesh approximations of parametric minimal surfaces

A trust region method for constructing triangle-mesh approximations of parametric minimal surfaces

Given a function f"0 defined on the unit square @W with values in R^3, we construct a piecewise linear function f on a triangulation of @W such that f agrees with f"0 on the boundary nodes, and the image of f has minimal surface area. The problem is formulated as that of minimizing a discretization of a least squares functional whose critical points are uniformly parameterized minimal surfaces. The nonlinear least squares problem is treated by a trust region method in which the trust region radius is defined by a stepwise-variable Sobolev metric. Test results demonstrate the effectiveness of the method.

Robert J. Renka

[1] Stephen J. Wright,et al. Numerical Optimization , 2018, Fundamental Statistical Inference.

[2] Ulrich Pinkall,et al. Computing Discrete Minimal Surfaces and Their Conjugates , 1993, Exp. Math..

[3] Caiming Zhang,et al. Robust modeling of constant mean curvature surfaces , 2012, ACM Trans. Graph..

[4] Robert J. Renka,et al. A Levenberg–Marquardt method based on Sobolev gradients , 2012 .

[5] Robert J. Renka,et al. Minimal Surfaces and Sobolev Gradients , 1995, SIAM J. Sci. Comput..

[6] R. Renka. A Sobolev gradient method for treating the steady-state incompressible Navier-Stokes equations , 2013 .

[7] Konrad Polthier,et al. Discrete Constant Mean Curvature Surfaces and Their Index (離散可積分系の研究の進展--超離散化・量子化) , 2001 .

[8] John William Neuberger,et al. Sobolev gradients and differential equations , 1997 .

[9] Kenneth A. Brakke,et al. The Surface Evolver , 1992, Exp. Math..

[10] Jianmin Zheng,et al. Constructing Triangular Meshes of Minimal Area , 2008 .

[11] Robert J. Renka,et al. Nonlinear least squares and Sobolev gradients , 2013 .

[12] Robert J. Renka,et al. Minimization of the Ginzburg-Landau energy functional by a Sobolev gradient trust-region method , 2013, Appl. Math. Comput..