Identification Problem for Nonlinear Gao Beam

Jana Burkotová,Jitka Machalová,Jana Radová

doi:10.3390/math8111916

Jana Burkotová, Jitka Machalová + Show 1 more

Open Access

https://doi.org/10.3390/math8111916

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Journal: Mathematics	Publication Date: Nov 1, 2020
Citations: 4	License type: CC BY 4.0

Affiliation: Palacký University, Olomouc

Abstract

This paper deals with the identification of coefficients in the nonlinear beam model which was first introduced by D. Y. Gao in 1996. For the identification of coefficients, an optimal control approach is used. The unknown coefficients are material parameters of the beam and play the role of the control variables. The existence of at least one solution of the optimal control problem is proved. For the studied problem the finite element approximation is provided. Finally some illustrative examples are introduced.

Highlights

Beams are important components in many engineering applications
E I w IV − E α (w0 )2 w00 + P μ w00 = f where w = w( x ) is a deflection of the beam, E stands for Young’s modulus, I is the constant area moment of inertia while functions α = 3tb(1 − ν2 ), μ = (1 + ν)(1 − ν2 ) and f = (1 − ν2 )q depend on the Poisson ratio ν, t stands for half-thickness, b is a width of the beam and q is the distributed transverse load
We focus on an identification of material parameters for the static Gao beam model

Summary

Introduction

Beams are important components in many engineering applications. The Euler- Bernoulli beam is the most popular model but it is linear and its validity is limited only to small deflections. Mathematics 2020, 8, 1916 there is only one paper [7] dealing with the inverse problem for a dynamic Gao beam using a collage-based method, where data from [8] were considered for a numerical example. The material parameters are coefficients of a nonlinear differential equation and their calculation can be based on the solution of the corresponding state problem for given data. For this purpose, an optimal control approach can be used. The contribution of the presented paper can be seen in proof of the existence of at least one solution of the studied problem, the convergence analysis of its discretization and the sensitivity analysis that provides the gradient for the efficient numerical realization

Gao Beam Model

Parameter Identification

Discretization and Convergence Analysis

Sensitivity Analysis

Numerical Examples

Conclusions