Reconstructing Loads in Nanoplates from Dynamic Data

Abstract

It was recently proved that the knowledge of the transverse displacement of a nanoplate in an open subset of its mid-plane, measured for any interval of time, allows for the unique determination of the spatial components {fm(x,y)}m=1M of the transverse load ∑m=1Mgm(t)fm(x,y), where M≥1 and {gm(t)}m=1M is a known set of linearly independent functions of the time variable. The nanoplate mechanical model is built within the strain gradient linear elasticity theory, according to the Kirchhoff–Love kinematic assumptions. In this paper, we derive a reconstruction algorithm for the above inverse source problem, and we implement a numerical procedure based on a finite element spatial discretization to approximate the loads {fm(x,y)}m=1M. The computations are developed for a uniform rectangular nanoplate clamped at the boundary. The sensitivity of the results with respect to the main parameters that influence the identification is analyzed in detail. The adoption of a regularization scheme based on the singular value decomposition turns out to be decisive for the accuracy and stability of the reconstruction.