Reduced Basis Approaches for Parametrized Bifurcation Problems held by Non-linear Von Kármán Equations

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.

[1]  F. Brezzi,et al.  Finite element approximations of the von Kármán equations , 1978 .

[2]  Yvon Maday,et al.  RB (Reduced basis) for RB (Rayleigh–Bénard) , 2013 .

[3]  Gianluigi Rozza,et al.  On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics , 2017, J. Sci. Comput..

[4]  Claudio Canuto,et al.  A Posteriori Error Analysis of the Reduced Basis Method for Nonaffine Parametrized Nonlinear PDEs , 2009, SIAM J. Numer. Anal..

[5]  Karen Paula L. Veroy,et al.  Reduced-basis methods applied to problems in elasticity : analysis and applications , 2003 .

[6]  A. Quarteroni,et al.  Reduced Basis Methods for Partial Differential Equations: An Introduction , 2015 .

[7]  A. Ambrosetti,et al.  A primer of nonlinear analysis , 1993 .

[8]  Volker Mehrmann,et al.  Numerical methods for parametric model reduction in the simulation of disk brake squeal , 2016 .

[9]  E. Allgower,et al.  Introduction to Numerical Continuation Methods , 1987 .

[10]  Philippe G. Ciarlet,et al.  Mathematical elasticity. volume II, Theory of plates , 1997 .

[11]  A. Patera,et al.  Certified real‐time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced‐basis a posteriori error bounds , 2005 .

[12]  Peter Monk,et al.  A mixed finite element method for the biharmonic equation , 1987 .

[13]  A. Quarteroni.  Numerical Models for Differential Problems , 2009 .

[14]  Sheng Zhang,et al.  Invalidity of decoupling a biharmonic equation of two Poisson equations of non-convex polygons , 2008 .

[15]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[16]  J. Rappaz,et al.  Numerical analysis for nonlinear and bifurcation problems , 1997 .

[17]  N. Nguyen,et al.  EFFICIENT REDUCED-BASIS TREATMENT OF NONAFFINE AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 2007 .

[18]  Jacques Rappaz,et al.  Finite Dimensional Approximation of Non-Linear Problems .1. Branches of Nonsingular Solutions , 1980 .

[19]  Edward L. Reiss,et al.  Nonlinear Buckling of Rectangular Plates , 2011 .

[20]  M. S. Chen,et al.  Multiple Bifurcation in the von Kármán Equations , 1997, SIAM J. Sci. Comput..

[21]  P. G. Ciarlet,et al.  Linear and Nonlinear Functional Analysis with Applications , 2013 .

[22]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[23]  Annalisa Quaini,et al.  Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology , 2017, J. Comput. Phys..

[24]  I. Babuska,et al.  Regularity and numerical solution of eigenvalue problems with piecewise analytic data , 1989 .

[25]  Paul C. Fife,et al.  On von Karman's equations and the buckling of a thin elastic plate , 1966 .

[26]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[27]  J. Hesthaven,et al.  Certified Reduced Basis Methods for Parametrized Partial Differential Equations , 2015 .

[28]  G. Rozza,et al.  Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings , 2018, 1801.06553.

[29]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[30]  J. Hesthaven,et al.  Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations , 2007 .

[31]  Melvyn S. Berger,et al.  On von kármán's equations and the buckling of a thin elastic plate, I the clamped plate , 1967 .

[32]  Ahmed K. Noor,et al.  Reduced Basis Technique for Nonlinear Analysis of Structures , 1980 .

[33]  L. Reinhart,et al.  On the numerical analysis of the Von Karman equations: Mixed finite element approximation and continuation techniques , 1982 .

[34]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[35]  Felipe Millar,et al.  A finite element method for the buckling problem of simply supported Kirchhoff plates , 2015, J. Comput. Appl. Math..

[36]  T. Kármán.  Festigkeitsprobleme im Maschinenbau , 1907 .

[37]  Ahmed K. Noor,et al.  Multiple‐parameter reduced basis technique for bifurcation and post‐buckling analyses of composite plates , 1983 .

[38]  Anders Logg,et al.  Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book , 2012 .

[39]  Ahmed K. Noor,et al.  On making large nonlinear problems small , 1982 .