An Approximation Algorithm Of Minimal Surface With Multiple Boundaries

Minimal surface,a very special and beautiful surface,has very beautiful geometric properties and some mechanical properties.It is used widely in automobile,ship,aircraft,industrial design and other fields.In order to combine with practical problems,we study the approximation problem of minimal surfaces with multiple boundaries.The minimal surface is a surface with the minimum area or mean curvature being zero in any given surfaces with multiple boundaries.Due to the flexibility and stability of triangular meshes,the initial grid structure is a triangular mesh.Thus,the area of the surface is replaced by the sum of the areas of all small triangles in order to use the special property of the minimal surface area.The mean curvature is zero in a given boundary.With the help of this special property,we use the discrete mean curvature to refine and gradually approximate the minimal surface.Actually,the problem to solve the sum of small triangle areas is a nonlinear problem which is very difficult to solve.In this thesis,we transform the nonlinear problem into a convex quadratic optimization problem,and use the alternating direction method to optimize the objective function with regarded to multiple variables.In practical applications,the problem of curved boundary continuity is widely applied.In order to meet the requirement of practical problems,this thesis proposes an approximation algorithm to solve the problem of minimal surfaces with multiple boundaries.In addition,we form a sub-boundary,which is optimized by using the smoothness of the surface splicing at the boundary.Considering the sub-boundary as a given boundary,then the problem turns to an approximation problem of minimal surfaces with new multiple boundaries.The ~1G continuous algorithm is applied to the smoothly joining problem of discrete triangular meshes.The correctness of the algorithm is proved and numerical experiments have shown that the algorithm has good applicability.We use some known minimal surfaces,such as Enneper,Scherk,spiral surface,to verify the effectiveness and accuracy of the algorithm.