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

Federico Pichi, Gianluigi Rozza|arXiv (Cornell University)|Apr 5, 2018

Model Reduction and Neural Networks30 references3 citations

TL;DR

This paper proposes a reduced basis (RB) method coupled with spectral analysis to efficiently detect buckling points and bifurcations in parametric non-linear Von Kármán plate equations. By combining RB reduction with eigenvalue analysis of the linearized problem, the approach enables rapid, high-fidelity bifurcation diagram construction across one and two parameters, achieving up to 8 solutions per parameter value and capturing complex post-buckling behavior with significant computational savings.

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.

Motivation & Objective

• To develop a computationally efficient framework for detecting buckling phenomena in parametric non-linear plate models governed by the Von Kármán equations.
• To address the high computational cost arising from fourth-order derivatives, nonlinearity, and parameter dependence in bifurcation analysis.
• To extend reduced order modeling to multi-parameter bifurcation problems, particularly for the evolution of the first buckling mode.
• To validate the accuracy and efficiency of the RB method in capturing multiple solution branches and critical points.
• To demonstrate the method’s capability on both square and rectangular plates, including 3D bifurcation plots in two-parameter settings.

Proposed method

• Application of the reduced basis (RB) method to the high-fidelity Galerkin finite element formulation of the parametric Von Kármán equations.
• Use of Newton’s method for solving the non-linear weak form, combined with RB projection to reduce online computational cost.
• Incorporation of the linearized eigenvalue problem to detect bifurcation points where solution uniqueness is lost.
• Construction of a reduced basis space from snapshots of high-fidelity solutions, enabling rapid online solution of the parametric problem.
• Implementation of a nested iterative strategy: for each parameter value, solve the non-linear problem and analyze the linearized eigenvalue spectrum.
• Extension to two-parameter problems by parametrizing both the load magnitude λ and the stress tensor σ(ψ), enabling 3D bifurcation diagrams.

Experimental results

Research questions

• RQ1 How can reduced order modeling be effectively combined with eigenvalue analysis to detect bifurcation points in parametric non-linear plate problems?
• RQ2 What is the accuracy and efficiency of the RB method in capturing multiple solution branches for the same parameter value in the Von Kármán equations?
• RQ3 How does the first buckling mode evolve when both the load magnitude λ and the load distribution ψ are varied?
• RQ4 Can the proposed methodology reliably reconstruct complex bifurcation diagrams, including post-buckling behavior, with significant computational savings?
• RQ5 What is the convergence behavior of the reduced approximation, and how does it relate to the detection of critical buckling points?

Key findings

• The proposed RB method achieved high accuracy in approximating solutions to the parametric Von Kármán equations, with significant reduction in online computational cost.
• The method successfully detected up to eight distinct solutions for a single parameter value, confirming the existence of multiple solution branches.
• The eigenvalue analysis of the linearized problem enabled precise detection of bifurcation points, where solution uniqueness is lost.
• In the two-parameter case, the method generated a 3D bifurcation diagram showing the evolution of the first buckling mode with respect to both λ and ψ, revealing complex behavior.
• The convergence test confirmed the expected order of convergence for the reduced approximation, validating the method’s reliability.
• The approach demonstrated computational efficiency in handling the nested iteration of Newton steps, parameter sweeps, and multiple initial guesses for branch tracking.