Geodesic
A numerical solution of the membrane eigenproblem by the model order reduction
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1088-1099.

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In this paper the Model Order Reduction technique to solve the problem of free oscillations of a heterogeneous rectangular elastic membrane is applied. Instead of solving 2D problem for the membrane in the exact formulation, we substitute it by a special network of 1D elastic strings. We present the characteristic equations for the spectrum of free oscillations of this network and develop the numerilal algorithm to solve the problem. We investigate the behavior of eigenvalues of a rectangular network and show that the eigenvalues and eigenvectors of rectangular networks of elastic strings and the rectangular membrane are close. The problem solution for the network of elastic strings has significantly less computational cost compared with the solution of free oscillations of a heterogeneous rectangular elastic membrane.
Keywords: networks of elastic strings, eigenvalue, eigenvector, model order reduction, finite-difference method. 
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B. K. Kaldybekova; G. V. Reshetova. A numerical solution of the membrane eigenproblem by the model order reduction. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1088-1099. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a110/

[1] Schilders W., van der Vorst H., Rommes J. (eds.), Model Order Reduction: Theory, Research Aspects and Applications, The European Consortium for Mathematics in Industry, XIII, 2008 | DOI | MR | Zbl

[2] Komarov A. V., Penkin O. M., Pokornyi Yu. V., “On the spectrum of a uniform network of strings”, Izv. Vuzov, Mat., 2000, no. 4, 23–27 | MR | Zbl

[3] Nicaise S., Penkin O. M., “Relationship between the lower frequency spectrum of plates and networks of beams”, Math. Meth. Appl. Sci., 23 (2000), 1389–1399 | 3.0.CO;2-K class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[4] Pokorny Yu. V., Penkin O. M., Pryadiev V. L. and others, Differential equations on geometric graphs, Fiziko-Matematicheskaya Literatura, M., 2005 | MR

[5] Friedman A., Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York–Montreal, Que.–London, 1969 | MR | Zbl

[6] Kaldybekova B. K., Penkin O. M., “On the low frequencies of natural oscillations of a special grid of strings”, Scientific journal of Abai KazNPU, Almaty, 2016, 28 pp.

[7] Lisitsa V., Reshetova G., Tcheverda V., “Finite-difference algorithm with local time-space grid refinement for simulation of waves”, Computational Geosciences, 16:1 (2012), 39–54 | DOI | Zbl

[8] Mikhaylenko B. G., Reshetova G. V., “Mathematical modeling of the distribution of seismic and acoustic gravitational waves for the inhomogeneous Earth-Atmosphere model”, Geologiya i Geofizika, 47:5 (2006), 547–556

[9] Quarteroni A., Valli A., Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23, Springer-Verlag, Berlin, 1994 | MR | Zbl

[10] PDE Toolbox, http://www.math.ntu.edu.tw/s̃hyue/myclass/pde00/misc/pde.pdf