Abstract
Plateau’s problem is to determine the surface with minimal area that lies above an obstacle with given boundaryconditions. In this paper, a special example of this class of the problem is given and solved with the linearfinite element method. First, we triangulate the domain of definition, and transform the linear finite elementapproximation into a constrained nonlinear optimization problem. Then we introduce a simple and efficientmethod, named sequential quadratic programming, for solving the constrained nonlinear optimization problem.The sequential quadratic programming is implemented by the fmincon function in the optimization toolbox ofMATLAB. Also, we discuss the relations between the number of grids and the computing time as well as theprecision of the result.
Highlights
Plateau’s problem is to determine the surface of minimal area with a given closed curve in R3 as boundary (Elizabeth, etc., 2004, pp.39-40)
The minimal surface area depends on the number of the grid points
The volatility is weakening, i.e. it has the trend to a constant. This is in line with the theoretical result of (Shen, Shumin, 1992), which concludes that the finite element approximate solution converges to the true solution when the number of grid points tends to infinity
Summary
Plateau’s problem is to determine the surface of minimal area with a given closed curve in R3 as boundary (Elizabeth, etc., 2004, pp.). R, and the requirement is z ≥ zL for some obstacle zL The solution of this obstacle problem minimizes the function f : K → R f (z) =. The function zD : ∂D → R defines the boundary data, and zL : D → R is the obstacle. The linear finite element approximation to the minimal surface with obstacle, defined by (1) and (2), can be obtained by triangulating D and minimizing f over the space of piecewise linear functions. The linear finite element approximation for the minimal surface with obstacle is analyzed, the existence and uniqueness of the solution for the discrete problem are shown, and the error estimate of the finite element approximation is obtained
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