An Approximation Algorithm Of Discrete Minimal Surface

The minimal surface is a very important kind of surfaces.The mean curvature of the minimal surface is everywhere zero in three-dimensional space.Given a closed space curve,the area of the minimal surface is the smallest one in all surfaces bounded by this curve.Many scholars have put forward a lot of researches about the minimal surface in the last 200 years,such as the stability of the minimal surface,the Plateau problem,the solution of the minimal surface equation.The minimal surface has been applied widely in the architectural design field and the industrial geometric design field,because of its unique geometric properties.In this thesis,we propose an approximation algorithm to solve the Plateau problem.That is to find a discrete minimal surface bounded by the given closed space curve.The mathematical model is an optimization problem with the area of the surface as the objective function.When the meshes are small enough,the area of the surface can be approximated by the sum of the areas of all small meshes.We introduce two kinds of initial meshes construction methods based on two kinds of boundaries.The nonlinear optimization model can be converted into an easily solved quadratic programming model,and thus we can use alternating direction method to solve this problem.The algorithm is analyzed in theory in this thesis.In addition,considering of the boundary continuity conditions,we present another algorithm with1 G and2G continuity conditions.Similarly it can be converted into another quadratic programming model with equality constraints.Furthermore,this method can be used to solve hole filling problem.We analyze the errors under different initial conditions.Numerical experiment results show that this algorithm works well in different cases.