Estimation of higher-order damping terms in linear plate models
type: Item
instance of: scholarly article
title: Estimation of higher-order damping terms in linear plate models
zbMATH Open document ID: 0677.73020
DOI: 10.1016/0096-3003(89)90085-4
published in: Applied Mathematics and Computation
publication date: 1989-01-01T00:00:00Z
review text: Timoshenko's model for computation of plates with internal damping involves a fifth order differential equation that can be written in the form (1) \(u_{tt}+Bu_ t+Au=f\) in \(\Omega\times [0,T]\), where A and B are fourth-order elliptic operators, familiar to researchers in linear elasticity. For example A can be the Kirchhoff operator \[ A\phi (x,y)=\nu \Delta (a(x,y)\Delta \phi)+(1- \nu)[(a\phi_{xx})_{xx}+2(a\phi_{xy})_{xy}+(a\phi_{yy})_{yy}], \] \[ and\quad B\phi =\Delta (b(x,y)\Delta \phi) \] (unfortunately letters a, b are abused in the article). The author sets the problem in the \(L^ 2\times H^ 2_ 0\) setting and introduces appropriate bilinear forms. Existence of solutions to (1) with zero initial conditions is easily established. Then, using a sequence of bilinear forms, convergence to a unique limit is proved in weak star \(L^{\infty}([0,T],H^ 2_ 0)\) for the Kevin-Voigt model with coefficients in the equation (1) not known, but prescribed within certain limits. Displacements at specifically assigned points are regarded as the (measured) data. Compact embedding of \(H^ 2(\Omega)\) in \(L^{\infty}(\Omega)\) is used to prove uniform convergence of the bilinear forms. The author determines optimality conditions and establishes criteria for positivity of Lagrangian multipliers. This leads to results concerning stability with respect to perturbations of the data. The article concludes with specific examples of computations. This is an excellent, carefully written article combining functional analysis, numerical methods and estimating techniques to derive results that should be of interest to engineers working on plates and shells. It is clear that authors' results should be easily extended to several types of shell models and problems in other areas of engineering.
Mathematics Subject Classification ID: 74S30
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 74H45
Mathematics Subject Classification ID: 74K20
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 4110231
zbMATH Keywords: optimal regularized output least squares estimators
zbMATH Keywords: regularity properties of optimal estimators
zbMATH Keywords: Timoshenko's model
zbMATH Keywords: fifth order differential equation
zbMATH Keywords: fourth-order elliptic operators
zbMATH Keywords: sequence of bilinear forms, convergence to a unique limit
zbMATH Keywords: weak star
zbMATH Keywords: Kevin-Voigt model
zbMATH Keywords: Compact embedding
zbMATH Keywords: uniform convergence
zbMATH Keywords: optimality conditions
zbMATH Keywords: positivity of Lagrangian multipliers
zbMATH Keywords: perturbations of the data
author: Luther White
reviewed by: Vadim Komkov
MaRDI profile type: Publication
full work available at URL: 0096-3003(89)90085-4
OpenAlex ID: W2032789711
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description: scientific article; zbMATH DE number 4110231