For real time monitoring of the wing state, in this paper, the inverse Finite Element Method (iFEM) is applied, which describes the displacement field of beam according to the Timoshenko theory, to sense the wing frame deformation. In order to maintain the accuracy and stability of frame deformation sensing with iFEM, an optimal placement model of strain sensors based on eigenvalue analysis is constructed. Through the model solution with the Particle Swarm Optimization (PSO) algorithm, two different optimal placement schemes of sensors are obtained. Finally, a simulation is performed on a simple cantilever beam and a static load experiment is conducted on an aluminum alloy wing frame. The results demonstrate that the iFEM is able to accurately sense the deformation of the wing frame, when the two optimal placement schemes of sensors are used.

For the inverse finite element method (iFEM), an inappropriate scheme of strain senor distribution would cause severe degradation of the deformation reconstruction accuracy. The robustness of the strain–displacement transfer relationship and the accuracy of reconstruction displacement are the two key factors of reconstruction accuracy. Previous research studies have been focused on single-objective optimization for the robustness of the strain–displacement transfer relationship. However, researchers found that it was difficult to reach a mutual balance between robustness and accuracy using single-objective optimization. In order to solve this problem, a bi-objective optimal model for the scheme of sensor distribution was proposed for this paper, where multi-objective particle swarm optimization (MOPSO) was employed to optimize the robustness and the accuracy. Initially, a hollow circular beam subjected to various loads was used as a case to perform the static analysis. Next, the optimization model was established and two different schemes of strain sensor were obtained correspondingly. Finally, the proposed schemes were successfully implemented in both the simulation calculation and the experiment test. It was found that the results from the proposed optimization model in this paper proved to be a promising tool for the selection of the scheme of strain sensor distribution.

The inverse finite element method (iFEM) is one of the most effective deformation reconstruction techniques for shape sensing, which is widely applied in structural health monitoring. The distribution of strain sensors affects the reconstruction accuracy of the structure in iFEM. This paper proposes a method to optimize the layout of sensors rationally. Firstly, this paper constructs a dual-objective model based on the accuracy and robustness indexes. Then, an improved adaptive multi-objective particle swarm optimization (IAMOPSO) algorithm is developed for this model, which introduces initialization strategy, the adaptive inertia weight strategy, the guided particle selection strategy and the external candidate solution (ECS) set maintenance strategy to multi-objective particle swarm optimization (MOPSO). Afterwards, the performance of IAMOPSO is verified by comparing with MOPSO applied on the existing inverse beam model. Finally, the IAMOPSO is employed to the deformation reconstruction of complex plate-beam model. The numerical and experimental results demonstrate that the IAMOPSO is an excellent tool for sensor layout in iFEM.

The inverse finite element method (iFEM) for the 3D framework deformation reconstruction was introduced. As the process of iFEM did not require a priori knowledge, such as the modal shape, the loading, and the elastic-inertial material information of the structure, it presented high potential in the framework deformation reconstruction. With the current research, it was observed that the key step in the deformation reconstruction of the frame structure with iFEM was the section strains computing of the beam element from the surface strain measurements. The corresponding stability was severely affected by the placement of strain sensors. Therefore, it was necessary to discover a suitable sensor placement to maintain the stability of section strains computing. For this problem, one optimal model of sensor placement was proposed in this paper. Firstly, the well-separated eigenvalues were applied as the optimization target to construct the optimal model. Following, an optimal sensor placement was obtained through the optimal placement model solution, with the particle swarm optimization (PSO) method. Finally, the effectiveness of optimal placement was verified though the accuracy comparison of iFEM deformation reconstruction of a wing-like frame subjected to various loads for different schemes of sensor placement.

Reconstruction of structural deformation using discrete strain sensors, i.e., shape sensing, is the core of structural health monitoring (SHM). However, the isoparametric transformation utilizing the Jacobian matrix in the reconstruction has the obvious drawback that the numerical integration can be necessarily used in the inverse of the Jacobian matrix, reducing the computational accuracy. To address this problem, the inverse-plate quadrilateral area coordinates method (iQAC) is proposed in this article, combining the original inverse finite-element method (iFEM) as the fundamental framework and the quadrilateral area coordinates method as an improvement. In this method, the introduction of the area coordinates technique ensures that isoparametric transformation can be expressed as higher order polynomials, and accurate solutions can be calculated by integrating explicitly whether the elements are distorted. Finally, the numerical and experimental results for the rectangular plate and the wing-shaped plate demonstrate that the iQAC method has high accuracy and generality.

For the problem of Timoshenko beams with multiple singularities in presence of discontinuities in the stiffness or curvature, the traditional inverse finite element method (iFEM) adopts the piecewise method to measure the real-time deformed displacements. However, as the number of singularity increases, the dimension of the strain-displacement matrix and the required number of strain sensors dramatically increases. In order to solve this problem, a novel deformation measurement approach is presented. Initially, distribution theory is employed to describe the rigidities of beams with multiple singularities involving axial rigidity, shear rigidity, torsional rigidity, and bending rigidity, and the corresponding constitutive formulations are presented. Then, considering displacement function presenting peak at these singularities, the isogeometric analysis approach is proposed to be used as basis functions to describe the displacement fields. Next, the strain-tensor transformation relations suitable for discontinuous beam structure are presented. Finally, a straight beam with circular cross sections is used as a case to verify the accuracy and effectiveness of the presented measurement method. The numerical results from simulation analysis and experiment tests demonstrate the superior accuracy capability and potential applicability of presented measurement approach for performing accurate shape prediction of the beam with multiple singularities.