Klyachin V.A., Grigorеva E.G. Numerical Study of the Stability of Equilibrium Surfaces Using NumPY Package
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Klyachin Vladimir Alеksandrovich
Doctor of Physical and Mathematical Sciences, Head of Department of Computer Science and Experimental Mathematics, Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Grigorеva Elеna Gеnnadiеvna
Doctor of Physical and Mathematical Sciences, Associate Professor, Department of Computer Science and Experimental Mathematics, Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Abstract. 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(M) =A(M) + G(M),
where A(M) =M∫︁α(x)dM, G(M)=Ω1∫︁φ(x)dx and domains Ω ⊂ Rn+1, Ω1 ⊂ Ω such that ∂Ω1 ∩ ∂Ω =M. The problem of study a stability of equilibrium surfaces is reduced to investigate the value of kind
infh (M∫|∇h|2dM)/(M∫||A||2h2dM)
where ||A|| is norm of second fudamental form for surfaceM⊂ Rn, and gradient ∇ℎ is calculated in Riemann metric of M. Using piecewise linear interpolation this value can be approximated by the value
minh <Ah,h>/<Bh,h>
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
x1=a cosh(t/a) cosφ,
x2=a cosh(t/a) sinφ,
x3=t,t<|T|
is considered. It is calculated maximal value of T under which catenoid is stable minimal surface.
Key words: extremal surface, triangulation, piecewise linear approximation,
main frequency, package NumPy.
Numerical Study of the Stability of Equilibrium Surfaces Using NumPY Package by Klyachin V.A., Grigorеva E.G. is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №2 (27) 2015 pp. 17-30
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