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Reduced basis approaches for parametrized bifurcation problems held by non-linear von Kármán equations. (English) Zbl 1427.35275

Summary: This work focuses on the computationally efficient detection of the buckling phenomena and bifurcation analysis of the parametric von Kármán plate equations based on reduced order methods and spectral analysis. The computational complexity – due to the fourth order derivative terms, the non-linearity and the parameter dependence – provides an interesting benchmark to test the importance of the reduction strategies, during the construction of the bifurcation diagram by varying the parameter(s). To this end, together the state equations, we carry out also an analysis of the linearized eigenvalue problem, that allows us to better understand the physical behaviour near the bifurcation points, where we lose the uniqueness of solution. We test this automatic methodology also in the two parameter case, understanding the evolution of the first buckling mode.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
35J60 Nonlinear elliptic equations
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
74G60 Bifurcation and buckling
65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
65K05 Numerical mathematical programming methods
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65H10 Numerical computation of solutions to systems of equations
65H17 Numerical solution of nonlinear eigenvalue and eigenvector problems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35P05 General topics in linear spectral theory for PDEs
35B32 Bifurcations in context of PDEs

Software:

rbMIT; redbKIT; RBniCS
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Full Text: DOI arXiv

References:

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