# A Refined Shape Sensing Method for Skin Antenna Structure Based on Inverse Finite Element Method
## 1 Introduction  
Today, with the rapid development of integrated antenna of radar technology, morphing-wing-embedded antenna arrays have an extensive application, such as military and civilian aerospace [1]. However, these structures are subjected to complex loads, such as wind and atmospheric pressure, which will produce various types of deformation leading to decrease the pointing accuracy of array antenna dramatically [2]. Therefore, real-time measurement of the deformation of antenna unit, which provides feedback to actuation and control systems of structure, is significant for human and environment safety [3; 4]. The method on dynamically monitoring the deformation of structure, known as shape sensing, and the process of precisely approximating the integrated displacements based on strain measurement belongs to strictly the inverse problem [5]. To settle the inverse problem, [PERSON] et al. proposed a method included a regularization term to enhance the smoothness degree for approximating solution [6], and the technique was extensively employed for the inverse procedure. During the past few decades, the utilization of shape sensing algorithms for practical engineering has been demonstrated extensively. However, these approaches are mainly classified into the inverse finite element method (iFEM), modal method, and [PERSON]'s displacement method. [PERSON] and [PERSON] proposed a modal method derived from the idea of polynomial basis function or spline function interpolation [7], which establishes the relationship between strain and deformation mode coordinates explicitly. Reference [8] use the modal method to reconstruct the displacement field of the beam structure precisely. However, the reconstruction accuracy is strictly limited on the number of sensors, which are at least higher than the mode orders extracted from structure. Thus, the method inevitably needs a high fidelity physical model, which is hard to do outside the laboratory environment. Then, [PERSON] et al. develop a strategy, referred to as [PERSON]'s displacement method [9], which is based on the Euler-Bernoullibeam theory, and are extensively applicable to structural reconstruction such as beams, wing-boxes, and plates. The technique included the displacement superposition principle, which builds the quantitative relation about strain and element degrees of freedom definitely, resulting in the true shape sensing of slender structures. References [10; 11] employ the numerical and experimental means to examine its feasibility. Although the method has a sufficiently accurate reconstruction of a slender structure, it is not suitable for the regeneration of three-dimensional structures. Hence, a well-suited shape-sensing algorithm, in addition to satisfying high accuracy, fast calculation, and strong robustness, also needs to consider the complexity of structural boundary conditions and geometric topology. Due to the deficiencies of the modal and the KO method, they are not suited for broad use in the field of structural health monitoring (SHM). [PERSON] and [PERSON] proposed an innovative methodology [12], named the inverse finite element method (iFEM), embodying the aforementioned characteristics required for a powerful SHM algorithm. The technology is based on the minimization of weighted-least-squares functional, which generally is applicable to complex structures subjected to complicated boundary conditions at any time [13]. The framework is sufficiently precise, powerful, and fast for applications in an engineering structure loaded statically and dynamically. Furthermore, the strain-displacement relationship was merely considered in the formulation, and materials of structure can be strictly ignored [14]. Benefiting from the potentiality of iFEM, the complex structure can be equivalent to simple models topologically, such as beam, plate, or shell. Moreover, based on the kinematic assumption of the Timoshenko beam theory, an inverse frame finite element is developed by [PERSON] and [PERSON] [15].