Tensor Representation of a Numerical Method for Solving the Plateau Problem

Source abstract

This paper proposes an operator approach for solving computational problems on irregular meshes in the form of triangulations. The proposed method is based on organizing the computational process as a sequence of operations on multidimensional tensors that represent all the processed data, including a description of the triangulation structure, triangulation vertices, triangle sides and normals to them, and the values of mesh functions at the triangulation vertices. To demonstrate the effectiveness of the proposed method, the author conducted a series of numerical experiments on calculating the gradient of a piecewise linear function on triangulations. A comparison of the time was given with the direct calculation method, which involves performing calculations for each individual triangle. As a nontrivial example of the application of this approach, this paper considers the numerical solution of the Plateau problem of finding a polygonal surface of minimum area in R3 with a fixed boundary contour in the form of a spatial polygonal line. The solution to this problem is sought using the gradient descent method. The computational scheme is presented in the form of tensor operations, a list of which is given in the paper. The software implementation is implemented using the Python programming language and its NumPy library, designed for working with multidimensional arrays. Examples of calculating approximate solutions to the area minimum problem are provided, modeling well-known minimal surfaces: the helicoid, the catenoid, and the Scherk surface.