Shape sensing as a crucial component of structural health monitoring plays a vital role in real-time actuation and control of smart structures, and monitoring of structural integrity. As a model-based method, the inverse finite element method (iFEM) has been proved to be a valuable shape sensing tool that is suitable for complex structures. In this paper, we propose a novel approach for the shape sensing of thin shell structures with iFEM. Considering the structural form and stress characteristics of thin-walled structure, the error function consists of membrane and bending section strains only which is consistent with the Kirchhoff–Love shell theory. For numerical implementation, a new four-node quadrilateral inverse-shell element, iDKQ4, is developed by utilizing the kinematics of the classical shell theory. This new element includes hierarchical drilling rotation degrees-of-freedom (DOF) which enhance applicability to complex structures. Firstly, the reconstruction performance is examined numerically using a cantilever plate model. Following the validation cases, the applicability of the iDKQ4 element to more complex structures is demonstrated by the analysis of a thin wallpanel. Finally, the deformation of a typical aerospace thin-wall structure (the composite tank) is reconstructed with sparse strain data with the help of iDKQ4 element.

In the last decades, curved thin-shell structure such as composite tank of spacecraft has been widely used in aerospace because of its excellent bearing capacity and weight-saving [

The dynamic reconstruction of the three-dimensional displacement field of a structure known as shape sensing is a crucial component of structural health monitoring which provides data support for subsequent calculation of stress and strain and failure prediction. Furthermore, the real-time evaluation of the deformed shape is also a vital technology for the development of smart structures such as morphed capability and embedded conformal antennas that require real-time shape sensing to provide feedback for their actuation and control systems.

Numerous studies on shape sensing found in the open literature can be divided into the following categories: (1) the Modal Method (MM) [

In order to reconstruct the three-dimensional displacement field in real-time with strain data obtained from the structure surface, Tessler et al. [

The main focus of this work is to redefine the weighted-least-square error functional based on classical plate theory. Subsequently, a new four-node quadrilateral inverse-shell element, iDKQ4, is developed for numerical implementation. The new element includes hierarchical drilling rotation degrees-of-freedom (DOF) to enhance applicability in modeling complex structures. This study is organized as follows: the iDKQ4 element is presented in brief in

Consistent with obtaining a flat element formulation, the inverse shell element can also be regarded as a superposition of a plate-bending element and a membrane element. In this paper, the four node plane stress (as shown in

The 4-node membrane element with drilling DOFs is derived by combining the in-plane displacements using Allman-type interpolation functions [

where

At the four corner nodes, two bending rotations

Therefore, these kinematic variables are related using the shape functions developed by Batoz for DKQ element [

From the strain-displacement relationship of linear elastic theory, we can know that

It should be noted that the plane stress assumption

The generalized strains vector consisting of membrane strain

where

where the node displacement vector of iDKQ4 element can be expressed as

In order to decouple plane strain and curvatures, the strain rosettes need to be attached to the top and bottom surfaces of the element as shown in

The counterpart of membrane strains and bending curvatures calculated from

where

For an individual inverse element, the error functional with respect to DOFs of the entire discretization can be expressed as:

where

The squared norms expressed in

where

If all the values in

Minimizing the error function with respect to the unknown nodal displacement DOF gives rise to

where

Firstly, the cantilever plate model is used to verify the accuracy of iDKQ4 element. As shown in ^{3}). A concentrated force of 25.728 n is applied along the negative direction of the z-axis near the tip. Bogert et al. [

In order to validate the bending capability of iDKQ4 element, the cantilever plate is discretized with 28 inverse elements to ensure that the position of the strain-rosette is coincident with the selection in the work by Tessler and Kefal. As depicted in

High-fidelity FEM analysis is performed with ABAQUS, a commercially available finite element software, to generate the strain data as input of iFEM calculation. Moreover, The displacement field calculated by FEM analysis can also be used as a benchmark to examine the reconstruction accuracy of iFEM. Contour plots for the transverse displacement are compared between the iFEM and high fidelity FEM analyses as shown in

In

The finite element convergence is studied to establish an exact reference solution of the problem. In FEM calculation, 930 rectangular S4R elements are used to discretize the cylindrical shell uniformly. In order to facilitate the transmission of strain data, iFEM calculation adopts the same discretization with FEM analysis. As shown in

The displacement field calculated by direct FEM analysis and the reconstructed result by iFEM are shown in

In

Although having numerous advantages of high specific strength, high specific modulus, corrosion resistance and designable performance, the composite is prone to damage such as delamination and debonding, which often is invisible and has a fatal impact on the bearing capacity of the tank. The robustness and adaptability of the improved inverse finite element method/iDKQ4 element are verified by cantilever plate and cylindrical shell structures. In this section, the composite tank is investigated with the improved iFEM algorithm.

The geometric dimensions of the tank are shown in

Young’s modulus [GPa] |
Shear modulus [GPa] |
Poisson’s ratio |
Density [ |
---|---|---|---|

135/7.579/7.579 | 4.49/4.49/3.2 | 0.32/0.32/0.49 | 1620 |

Firstly, the linear static analysis of the tank is carried out in ABAQUS using a high fidelity grid composed of 10886 S4R shear deformation shell elements. The same grids is used for iFEM calculation and FEM analysis. The strain calculated by FEM is used as the calculation input of iFEM, and the displacement field obtained by FEM analysis is used to evaluate the prediction ability of iFEM. In order to avoid introducing errors in calculating local strain field, when the input strain field is not fully defined, elements should have a rectangular shape aligned with the input strain field direction. It is obvious that the elements of the covers do not meet this requirement, but fortunately we do not care about the results of the metal covers. In the first case study, the strain of all elements except covers can be obtain as presented in

To assess the global displacement, it is convenient to compute the axial displacement

This study proposes an improved iFEM the mathematical foundation of which is based on a least-squares functional error embracing membrane strain and curvature to solve the shape sensing problem of thin-shell structure. Subsequently, a new four-node quadrilateral inverse-shell element, named iDKQ4, is developed for numerical implementation. The robustness and adaptability of the element are verified by a cantilever plate model and a cylindrical shell model. Then, a composite tank is employed to evaluate the iFEM/iQS4 technology for application to engineering structures. Finally, the improved iFEM/iDKQ4 technology can be easily implemented and ready to applied for real-time structural health monitoring of general thin plate and shell structures.