Inverse Problem of Identifying a Small Defect Based on an Asymptotic Method
- PDF / 503,416 Bytes
- 7 Pages / 612 x 792 pts (letter) Page_size
- 7 Downloads / 195 Views
STIC METHODS
Inverse Problem of Identifying a Small Defect Based on an Asymptotic Method A. O. Vatul’yana, b, * and O. A. Belyakc, ** a
Southern Federal University, Rostov-on-Don, 344006 Russia
b
Southern Mathematical Institute, Vladikavkaz Scientific Center, Russian Academy of Sciences, Vladikavkaz, 362027 Russia c
Rostov State Transport University, Rostov-on-Don, 344038 Russia *e-mail: [email protected] **e-mail: [email protected] Received April 4, 2020; revised May 7, 2020; accepted June 2, 2020
Abstract—We consider the inverse problem of reconstructing a cavity of small characteristic size in an orthotropic layer based on information about the field of layer surface displacements measured within the framework of frequency sensing. Resolving equations in the inverse problem are based on the system of boundary integral equations formulated only along the cavity boundary. In the case of antiplane oscillations, based on an asymptotic approach, we derive formulas for characterizing a small defect. Results of computational experiments are presented. Keywords: orthotropic layer, defect, inverse geometric problem DOI: 10.1134/S1061830920070074
INTRODUCTION Nondestructive quality control is an important scientific and technical problem that arises when assessing the dynamic strength, e.g., in the elements of heavy-loaded vehicles, machines, and mechanisms and in aerospace products. In this case, it is important to assess the potential hazard due to both incipient defects and initially present small-scale defects such as inclusions, microcracks, and pores. A variety of methods are used to identify the presence of a defect based on measuring the components of physical fields that carry information about defect size, shape, and location. The most widely used methods are acoustic ones [1–5], which have been the subject of study in a considerable number of publications. From the mathematical point of view, identifying defects based on information about displacement fields at the boundary of an elastic body leads to poorly studied and rather complicated inverse geometric problems of elasticity theory [6, 7]. Such inverse problems can be solved based on the diffraction formulation or by considering a defect in a waveguide and determining its characteristics from the amplitudes and phases of propagating waves. For example, Karageorghis et al. [6, 7] considered the inverse geometric problem for an isotropic medium in two-dimensional and spatial formulations and determined defect parameters from the conditions of minimizing a nonquadratic residual functional. It is important to note that the body boundary has a significant effect on the formation of displacement fields in the presence of a near-surface defect; therefore, the diffraction formulation is ill-posed in this case. It should also be noted that inverse problems of reconstructing the shape of a cavity are ill-posed [8, 9]. In the present work, the inverse geometric problem of characterizing a cavity of a characteristic size small in comparison with the layer thicknes
Data Loading...
