Reduced basis approaches for parametrized bifurcation problems held by non-linear Von K\'arm\'an equations

TL;DR

This paper develops reduced basis methods for efficiently analyzing bifurcations in parametrized nonlinear Von Kármán equations, aiding in understanding buckling phenomena with reduced computational effort.

Contribution

It introduces reduced order techniques combined with spectral analysis for bifurcation detection in complex nonlinear plate equations, enhancing computational efficiency.

Findings

Effective reduction of computational complexity for bifurcation analysis.

Identification of bifurcation points through spectral analysis.

Validation of methods on nonlinear Von Kármán plate problems.

Abstract

This work focuses on the detection of the buckling phenomena and bifurcation analysis of the parametric Von K\'arm\'an 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 computational 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.