Universal software for solving multidimensional variational problems

E. G. Grigor'eva ; V. A. Klyachin ; A. A. Klyachin

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 2 (2017), pp. 39-55

Résumé

The article considers the problem of numerical calculation of the piecewise linear surfaces $ M $, wich is extremal for the functional type $$ F(M)=\int\limits_{M}\left(\Phi(\xi)+\alpha(x)\right)dS. $$ To solve this problem, we obtain formulas for the calculation of the gradient of the functional in the space of polygonal surfaces $M$ as a set $P$ of its vertices $$ \left(\frac{\partial \tilde{F}}{\partial h}(P)\right)^i=\frac{1}{2}\langle\sum_{j=1}^k\left(\nabla \Phi(2\xi_j|T_j|)\right)\times l_j,h\rangle+\frac{1}{2}\langle\sum_{j=1}^k\alpha(p^*_j)\xi_j\times l_j,h\rangle\,+ $$ $$ +\,\frac{1}{3}\langle\sum_{j=1}^k\nabla \alpha(p^*_j)|T_j|,h\rangle. $$ For the organization of calculations on the basis of these formulas, we construct a system of classes in the Python programming language. The system is organized as a class package wich is integrated in the simulation environment 3D Blender. This solution allows the user for the program Blender in interactive mode to perform the desired surface topology modeling, to set the boundary conditions and to visualize the results of calculation. In this paper, we present our implementation of a variational method based on approximation of piecewise-linear functions and surfaces. We proceed from the idea of creating a universal software system, most of which does not depend on the use of triangulation, integral functional, computational domain and boundary conditions. In this article we present the structure of input data, which is stored in the boundary conditions, and the vertices and the triangles of the triangulation and the description of the main program procedures. We are considering a number of illustrative examples of solving boundary value problems of mathematical physics.

Détail

Keywords: extremal surface, piecewise linear approximation, NumPy package, boundary problem.
Mots-clés : triangulation