Identification of the material properties in nonuniform nanostructures. (4th November 2015)

• Record Type: Journal Article
• Authors: Bao, Gang; Xu, Xiang
• Is Part Of: Inverse problems. Volume 31:Number 12(2015:Dec.)
• Publication Date: 2015-11-04
• Subjects: inverse random source problem -- Euler–Bernoulli equation -- Chebyshev inequality; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/31/12/125003
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: This paper is concerned with addressing two significant challenges arising from quantifying mechanical properties of nanomaterials, namely nonuniformity of the nanomaterial and the high noise level of measurements. For nonuniformity, an explicit solution is derived for the general Euler–Bernoulli equation in terms of the Green function for the Poisson equation. Then, by examining a stochastic source, the systematic error may be removed from measurements, which leads to more accurate estimation of mechanical properties. Based on Itô integral properties, three deterministic Fredholm integral equations can be deduced to extract the stiffness and the structure of the random source from measured data. To overcome ill-posedness and high nonlinearity in solving the Fredholm equations, a Tikhonov regularization method is developed with an a priori strategy of choosing the regularization parameter. Moreover, under a regularity assumption for the stiffness coefficient and structures of the random source, the convergence rate can be obtained in the sense of probability. Numerical examples are presented to illustrate the validity and effectiveness of the novel model and regularization method.