The high gain, compact storage, and lightweight design of mesh reflector antennas make them suitable for application in satellite communication systems. In such antennas, metallic mesh attached to the support structures functions as the reflecting surface. When a satellite mesh antenna is deployed, the supporting structures impose a certain amount of tension on the mesh surface, forming a particularly designed shape. In the case of metallic mesh, however, this tension alters both the shape and size of mesh knitting patterns, thereby affecting its electrical properties.

Researchers have developed several insightful modeling methods based on mesh knitting patterns to investigate mesh electrical performance. For instance, Astrakhan [1] introduced formulas, in combination with a wire-grid model, to calculate the reflection and transmission coefficients of mesh. However, this model is applicable only to square or rectangular single-wire woven mesh. Later, a strip-aperture model was devised to overcome this shape restriction [2], featuring two lattices and six parameters to indicate the configuration of one mesh cell. It offered more flexibility in mesh patterns but substantially increased computational time. Further studies on the surface-patch model reduced these six modeling parameters to four, and replaced cylindrical wires with equivalent strips of the same width [3]. This strip-wire equivalence theory was employed for meshes with complex patterns, with the model diameter defined as half the strip width, and the strip used to cover the knitting wires [4]. However, this method only worked well on meshes woven using a single wire with clear pattern boundaries. Therefore, to solve the ambiguous boundary problem, a modeling method featuring an average strip width and equivalent wire diameter to replace multiple knitting wires was introduced [5]. However, the application of this approach was limited to static mesh models. Notably, the shape and size of an elastic mesh vary under different amounts of force. Consequently, its electrical properties differ from those of a static mesh model. Later, researchers proposed an angle-based method to construct models for multi-wire woven mesh with intricate knitting patterns [6]. This method accounted for the effects of force on the mesh shape, represented using model angles, but failed to offer an intuitive understanding of how the applied forces impact the mesh.

Previous research on the electrical and mechanical features of meshes has largely been conducted separately. From a mechanical standpoint, extensive research has been carried out on the structural behaviors of meshes. For instance, metallic mesh has been stretched at a controlled speed on measurement equipment, with forces applied in one or more directions [7]. Testing machines coupled with high-definition cameras have been employed to record mesh deformation, and image analysis techniques have been deployed to analyze the characteristics of knitted meshes [8–10]. In the above studies, researchers detected non-uniform tension in the mesh resulting from the sophisticated mesh structure [8], with the loading mesh exhibiting irregular strain distribution. The applied load modified the shape and size of the mesh, thereby affecting its reflectance and transmittance properties. This implies that mesh models built solely on static features are not always effective in predicting the performance of the mesh under load. However, to the best of our knowledge, no analysis has attempted to investigate the effects of applied force on the mesh by combining the dynamic structural and electrical behaviors of meshes. Overall, the above analysis indicates that a method correlating the relationship between applied force and the electrical properties of metal mesh would be of great significance.

This study provides a solution to investigating mesh performance under tension by incorporating applied forces and the electrical properties of metallic mesh. The correlation between the applied forces and the geometric parameters of the mesh offers an opportunity to develop an elastic mesh model, where the mesh adapts to tensions through shape variations. It also provides an accurate performance variation range corresponding to the tension range of the reflector’s supporting structures, thereby helping estimate the final electrical performance of the mesh reflector antenna. Notably, there are various types of deployable mesh reflector antennas, and the direction of tensions applied to the mesh surface also varies. The proposed elastic mesh model is suitable for radial-rib antennas, where tensions arise primarily from the horizontal direction for each gore surface between two ribs. For antennas with truss structures, the applicability of the proposed model depends on the specific situation.

Section II of this paper describes the representation methodology for the mesh. In Section III, a commercial mesh is used as an example to demonstrate the process of building an elastic model. Section IV presents a comparison of the measurement and simulation results. Finally, Section V concludes this work.

This section provides a detailed description of the proposed method’s application to a mesh sample. Fig. 1(a) presents a magnified picture of the complex mesh (Max mesh), along with the dimensions of its periodic knitting pattern. The mesh was woven using multiple ultra-thin molybdenum wires coated with gold, with an equivalent thread diameter of 0.08 mm. The average length l₀ and width w₀ of the knitting pattern were 2.95 mm and 2.14 mm, respectively.

The black semicircles in Fig. 1(a) show that the knitting pattern is circular when no force is applied. The static mesh model, as depicted in Fig. 1(b), was built in Computer Simulation Technology (CST) Studio Suite—an electromagnetic analysis software. The diameter of the static mesh model was determined based on [6]. A model diameter of d = 1.17d_t was directly employed, since the mesh material employed in this study is the same as that in [6]. Notably, the periodic mesh knitting pattern provided the opportunity to perform simulations considering periodic boundary conditions, thus reducing the simulation time. The mesh model was simulated as a unit cell in CST and analyzed based on Floquet boundaries using the frequency-domain solver.

Upon the application of a certain amount of force, the mesh pattern became elliptical. Consequently, an elliptical mesh model was adopted to precisely represent the knitting pattern under tension. The elliptical mesh model was formulated as follows:

where a = (l − s)/2 and b = w/2 are the major and minor semi-axes of the ellipse, depending on whether the force is applied along the x-axis or the y-axis, respectively. For static mesh models with fixed width and length parameters, a = b = w₀/2. However, when subjected to tension, a and b take different values. Here, s denotes the distance between two mesh ellipses along the x-axis, calculated as s = l₀ − w₀ . The change in length l and width w of the mesh model in response of the applied forces are expressed in the following sections.

Next, the applied forces and the corresponding displacement of the mesh were measured using an INSTRON 5583 universal testing machine, as shown in Fig. 2. A tensile test was performed on a square mesh specimen of size 50 mm at a tensile speed of 100 mm/min. In addition, a uniaxial tension test was conducted considering two orthogonal directions: the weave direction (WD) and perpendicular to the weave direction (PWD). The recorded data were then analyzed using linear curve fitting to determine the linear elastic deformation range. The results are shown in Fig. 3, where the red dots and black squares denote the measured data pertaining to the WD and PWD, respectively. Notably, the mesh underwent the three stages of deformation—linear elastic deformation, nonlinear plastic deformation, and fracture. The magnified figure in the inset presents the averaged results of three independent experiments, where the red and black solid lines correspond to the linear curve fitting results.

Furthermore, Young’s modulus for the two directions (WD and PWD) was obtained by conducting the least squares curve fitting. In this context, the force–displacement relationship, considering force F applied in the WD and PWD, can be expressed as F_WD = 0.987Δl and F_PWD = 0.132Δw, respectively, where Δl and Δw refer to displacements of the mesh in the WD and PWD, respectively. Moreover, Poisson’s ratio v at WD and PWD was calculated using the measurement results, as follows:

Subsequently, the WD and PWD along the x- and y-axes were defined, respectively. The constitutive relation between the orthotropic mesh and the forces applied to it along the x-axis can be expressed as follows:

Notably, the engineering constants were calculated from tensile tests conducted on the specimens, which were 50 mm in length (l_sp) and width (w_sp). In this context, it must be noted that the mesh model represents one unit of the mesh specimen. Therefore, specimen displacement can be calculated as the linear accumulation of the displacements of the mesh model. The displacement of the mesh specimen in the WD can then be formulated as follows:

where l_sp/l₀ refer to the number of model units of the mesh specimen in the WD. After obtaining all concerned coefficients, the constitutive relation of the mesh was combined with the geometrical mesh model. The length and width of the elastic mesh-model in terms of the applied forces were derived as follows:

where w_sp/w₀ denote the amount of model units of the mesh specimen in the PWD. Similarly, the constitutive relation between the orthotropic mesh and the forces applied to it along the y-axis can be expressed as follows:

Combining these constitutive relations with the equations for the length and width of the geometrical mesh model, we get the following:

Table 1 summarizes the relationships between the ellipse parameters and the forces applied to the elastic mesh model. The final force-based elastic mesh model was formulated by substituting the corresponding length and width equations into elliptical mesh model equations. Subsequently, variations in the electrical performance of the mesh under tensions were examined by constructing the elliptical mesh model in electromagnetic analysis software, with forces applied toward the x-axis (WD) and force F_WD set as the variable. Fig. 4 demonstrates the deformation of the elastic mesh model, considering the Max mesh built in CST, in response to different forces applied in the direction of the x-axis, depicting that the circular model stretches into an elliptical model.

The force–displacement relationship derived from the measured data implies that linear elastic deformation occurs only in the initial stage under a small amount of applied force. However, controlling the applied force precisely during the measurement process is difficult. Consequently, displacement was employed to examine mesh deformation under varying force conditions. Furthermore, the simulation results were compared with the measured coefficients of the Max mesh at different deformation states along the WD to verify the applicability of the elastic mesh model.

Fig. 5 presents photographs of the measurement setup. The Agilent Network Analyzer E8361A and an X-band WR90 waveguide are presented in Fig. 5(a). In Fig. 5(b), the mesh sample attached to a vernier caliper shows an initial length of 41.54 mm—the same length as that of the calibration kit. Two sides of the mesh are fixed to the vernier caliper, while the other two sides are free, indicating the same condition as in the force– displacement measurement experiment. After calibration of the experimental setup, the mesh sample was placed between two waveguides to measure its electrical performance. The reflectance and transmittance of the mesh under different forces in the WD were acquired by precisely controlling the deformation length of the mesh sample, as shown in the bottom panel of Fig. 5(b).

To obtain the reflectance and transmittance, the elastic mesh model was simulated in CST considering a range of forces. The simulation results are represented using solid lines in Fig. 6, and the measured results are displayed as dots. The x-axis at the bottom denotes the force, on top is the displacement, the y-axis to the left signifies the reflectance, and the right side indicates the transmittance. The characteristics of the mesh sample upon the application of force in the WD can be summarized as follows: first, Fig. 6 shows that mesh reflectance negatively correlates with the applied force and displacement. On the contrary, the transmittance increases with increasing force. In particular, Fig. 6 demonstrates the reflectance and transmittance at four frequencies, indicating that the measured reflectance and transmittance align well with the simulated results, especially when the frequency is 9.6 GHz. Furthermore, it is observed that the difference between the measurement and simulation results increases as the frequency differs from 9.6 GHz, such as 11.6 GHz. This is because the mesh model was built considering a center frequency of 9.6 GHz and a measured reflectance within the frequency range of 9.3–9.9 GHz. Based on these findings, we can conclude that the proposed elastic mesh model accurately presents the electrical characteristics of the mesh under different tensions within a specific frequency range.

The force-based elastic mesh model, by relating mesh electrical properties to applied forces, also enables an analysis of the impact of tension on the performance of mesh reflector antennas. For instance, as illustrated by the red line in Fig. 6, the transmission loss caused by the metallic mesh is 0.04 dB under a force of 0.3 N at 9.6 GHz. However, when the force increases to 4 N, the transmittance increases and reflectance reduces, resulting in a mesh loss of 0.05 dB. In this context, it must be noted that the influence of the mesh on reflector performance is also affected by other factors, such as feed illumination angle. By employing the elastic mesh model, the effects of various illumination angles on mesh or mesh reflector antennas can be systematically analyzed through simulations. Overall, the proposed elastic mesh model serves as a reliable method for the precise evaluation of the impacts of metallic mesh on the performance of mesh reflector antennas.

Given that the reflectance value ranges from 0 to −0.1 dB, discrepancies between the simulated and measurement results are minimal. Therefore, Table 2 compares the measured transmittance of the Max mesh with that of two dynamic mesh models— the elastic mesh model and the angle-based model [6]— under identical simulation settings. Notably, since deformation in the angle-based mesh model depends on the model’s angles, whereas the elastic mesh model responds to applied forces, length displacement (Δl) was adopted as the common variable to ensure a consistent basis for comparison. The results are reported in Table 2, where the root mean square error (RMSE) is employed to quantify the differences precisely. RMSEs of 0.3410 and 0.7270 indicate that the elastic mesh model achieved higher accuracy than the angle-based model.

This paper presents an elastic mesh modeling method to detect variations in the electrical properties of a mesh in response to varying forces. The elastic deformation of the mesh is studied, and correlations between the applied forces and the geometric parameters of the mesh are derived. Furthermore, a commercial mesh is used as an example to detail the establishment of the elastic mesh. The force–displacement relationship of the sample mesh is acquired and applied to build the elastic mesh model. It is observed that the simulation results of the elastic elliptical mesh model match well with the mesh measurement results, thereby proving the feasibility of the proposed model.

In contrast to existing mesh modeling approaches, the proposed model exhibits the ability to predict the dynamic electrical properties of the mesh under varying tensions by integrating applied forces with the geometric parameters of the mesh. This is particularly helpful for accurately calculating mesh losses in mesh reflector antennas, which reveals the presence of an uneven tension distribution across the mesh surface. The proposed modeling method can be applied to single-wire woven mesh, multi-wire woven mesh, simple knitting pattern mesh, complex knitting pattern mesh, and even a combination of the above types of mesh.