Numerical study of the stability of equilibrium surfaces using NumPy package

V. A. Klyachin ; E. G. Grigoreva

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 2 (2015), pp. 17-30

Résumé

The article is devoted to numerical investigation of stability for equilibrium surfaces. These surfaces are models for surfaces between two media. Moreover, these surfaces are extremal surfaces for the functional of the follwing type $$ W({\mathcal M})=A({\mathcal M})+G({\mathcal M}), $$ where $$ A({\mathcal M})=\int\limits_{{\mathcal M}}\alpha(x)d\mathcal M, \quad G({\mathcal M})=\int\limits_{\Omega_1}\varphi(x)dx, $$ and domains $\Omega\subset R^{n+1}$, $\Omega_1\subset \Omega$ such that $\partial \Omega_1\cap\partial \Omega = {\mathcal M}$. The problem of study a stability of equilibrium surfaces is reduced to investigate the value of kind $$ \inf_{h}\frac{\int\limits_{\mathcal M}|\nabla h|^2d\mathcal M}{\int\limits_{\mathcal M}||A||^2h^2d\mathcal M}, $$ where $||A||$ is norm of second fudamental form for surface $\mathcal M\subset R^n$, and gradient $\nabla h$ is calculated in Riemann metric of $\mathcal M$. Using piecewise linear interpolation this value can be approximated by the value $$ \min_{\bar{h}}\frac{\langle A\bar{h},\bar{h}\rangle}{\langle B\bar{h},\bar{h}\rangle}, $$ where $A,B$ are symmetric positive definite matrixes. The article describes Python package NDimVar implemented on the basis package NumPy for solution of the above pointed problem. In addition, the study of stability for minimal surface of catenoid $$ \left\{ \begin{array}{lcl} x_1= \cosh\frac{t}{a}\cos\varphi\\ x_2= \cosh\frac{t}{a}\sin\varphi\\ x_3=, \; |t|\end{array} \right. $$ is considered. It is calculated maximal value of $T$ under which catenoid is stable minimal surface.

Détail

Keywords: extremal surface, piecewise linear approximation, main frequency, package NumPy.
Mots-clés : triangulation