By using the FTIM and Lie-group methods to identify unknown force in the Euler-Bernoulli beam

Abstract

The present paper mainly discusses the direct problem and the inverse problem of the Euler-Bernoulli beam dynamic equation. For the direct problem, we use the Euler method, the fourth-order Runge-Kutta method (RK4) and the group preserving scheme (GPS). The group preserving scheme (GPS) is a new form for solving the non-linear dynamical system, and it can preserve the internal symmetry group of the considered ordinary differential equations (ODEs) system. For the inverse problem, we use the fictitious time integration method (FTIM), the Lie-group shooting method (LGSM) and the Lie-group adaptive method (LGAM). Particularly, the group preserving scheme (GPS) also is a very important basic theory for the Lie-group methods. We have applied the fictitious time integration method (FTIM) and the Lie-group shooting method (LGSM) to deriving the algebraic equations and solved them in a closed-form. In contrast to other estimation methods, the advantages of the Lie-group shooting method (LGSM) are that it does not need any prior information on the functional form of the external force, no iterations are required and the closed-form solution is available. The other Lie-group method we used is the Lie-group adaptive method (LGAM) which is using the layer-stripping technique which can be used to find the unknown force layer by layer via iterations. The layer-stripping technique together with the Lie-group adaptive method (LGAM) leads to that solving the inverse Euler-Bernoulli beam equation does not require the extra measurement of data, in addition to the usual boundary conditions and initial conditions for the direct problem. In this paper, we introduce all of the methods which are used, deeply and thoroughly, and via using them to calculate the Euler-Bernoulli beam dynamic equation. Moreover, we use the programming language, FORTRAN, to analyze the numerical identifications. For the ill-posed behavior of inverse problems, we have tested and verified that the Lie-group methods are very useful to be directed against them, namely using the Lie-group methods we can obtain good results even for the ill-posed problems.