Melnik, R., Tsviliuk, O., Wang, L.
Recent Advances in Applied Mathematics (Book Series: Mathematics And Computers In Science And Engineering), Proceedings of the 14th WSEAS International Conference on Applied Mathematics, Spain, Eds, Bulucea, C.A. et al, ISBN/ISSN:1790-2769, 978-960-474-138-0, pp. 83-88, 2009
This paper focuses on the development of low dimensional approximations to coupled nonlinear systems of partial differential equations (PDE) describing phase transformations. The methodology is explained on the example ofnonlinear ferroelastic dynamics. We start from the general three-dimensional Falk-Konopka model and with the center manifold reduction obtain a Ginzburg-Landau-Devonshire one-dimensional model. The Chebyshev collocation method is applied for the numerical analysis of this latter model, followed by the application of an extended proper orthogonal decomposition. Finally, we present several numerical results where we demonstrate performance of the developed methodology in reproducing hysteresis effects occurring during phase transformations.
Keywords: coupled systems; nonlinearities; phase transformations; Proper Orthogonal Decomposition; dynamics of ferroelastic materials; Galerkin projection; shape memory alloys; thermoelasticity; model reduction
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