A Fluid–Structure Coupling Analysis of a Far-Field Flat Mirror for AliCPT-1 Telescope Calibration

Abstract

AliCPT-1 is the first cosmic microwave background (CMB) experiment in China dedicated to achieving accurate measurements of B-mode polarization. Situated in Ali of Tibet, China, this telescope is currently undergoing deployment and will operate in two frequency bands centered at 90 and 150 GHz. The far-field flat mirror (FFF) is a calibration device of the AliCPT-1 telescope for far-field beam mapping. The design of the FFF is optimized for easy assembly and adjustment. Meteorological station data reveal that the maximum wind speed near the FFF is 17.5 m/s, while the maximum wind speed on the windward side is 8 m/s. The wind pressure on the FFF was analyzed using a maximum wind speed of 17.5 m/s as the input condition, based on the fluid–structure coupling method in ANSYS. The results demonstrate that it is safe and reliable when withstanding combined gravity and wind pressure loads. The torque on the mount is within the motor rated torque. The flatness of the FFF reflective surface can be adjusted to an RMS value of 0.05 mm when taking into account the effect of gravity and assembly accuracy. The deformation caused by the maximum wind loads is approximately 0.0587 mm under the protection of the wind-proof wall. The combined deformation is 0.077 mm in RMS value combining the two influences, which is less than 1/20 of wavelength. The FFF mirror assembly is stable and precise for telescope calibration.

1. Introduction

Primordial gravitational waves refer to gravitational waves generated during the early stages of cosmic formation due to the superluminal expansion of the universe [1,2]. The aim is to detect primordial gravitational waves and explore fundamental cosmic properties in order to understand the structure and evolutionary processes of the early universe, as well as the theory of cosmic inflation [3].

The method for detecting the primordial gravitational waves is to measure the B mode polarization of the cosmic microwave background radiation (CMB) [3,4,5,6,7]. By analyzing the polarization patterns of the CMB, one can indirectly deduce the existence of primordial gravitational waves and their intensity. Currently, the construction of the Ali CMB Polarization Telescope (AliCPT-1) project is underway in China [8]. We aim to build a two-refractive-lens cosmic microwave background radiation polarization telescope, AliCPT-1, at an elevation of 5250 m in the Ali of Tibet, China. Its scientific motivation is to do precise measurements of primordial gravitational waves, thereby unraveling the origins of the universe [4,5,7].

To model the temperature-to-polarization leakage and the E-to-B leakage due to the beam mismatch, the far-field beams of the telescope need to be calibrated. To measure the far-filed beam of each detector, the calibration source must be placed at a far-field distance. The calibration source is installed ~1500 m away from the AliCPT-1 and ~7° above the horizon. Since the elevation angle of the telescope is limited from 45° to the zenith angle, a far-field flat mirror should be used to reflect the radiation from the calibration microwave source to the AliCPT-1. The support structure of the flat mirror should not obstruct the radiation from the calibration source during far-field beam calibration for accurate measurement of the beam.

Currently, only the AliCPT-1 telescope and bicep3 use the reflective mirror calibration method. However, using a reflective mirror calibration requires a stable reflective mirror structure and high surface flatness to ensure the optical performance and measurement accuracy of the reflective mirror. The AliCPT-1 FFF mirror is oriented directly toward the microwave source. When the wind blows from the source toward the FFF mirror, it is referred to as the ‘front wind’. Conversely, when the wind blows from the side of the reflector toward the source, it is termed the ‘back wind’. Therefore, when calculating wind loads, both scenarios should be taken into consideration as they represent the two extreme conditions for the FFF mirror. Utilizing these two operating conditions, the response of the FFF mirror under extreme wind loading is analyzed.

The far-field distance from the calibration source to the reflector is calculated using the following formula:

$$\mathrm{L}=\frac{{2\mathrm{D}}^2}{\mathsf{\lambda }}$$

(1)

where λ represents the electromagnetic wave’s wavelength, and D signifies the aperture of the telescope [9]. Upon substituting D = 0.72 m and λ = 2 × 10⁻³ m, the calculated L results in 490 m. Considering these distance requirements, a pragmatic approach was taken to select a suitable installation location on the mountain for the source. This decision determined the orientation of the FFF, which is positioned at a bearing of 104° east of north. The chosen orientation, along with the prevailing wind direction, will affect the wind loading magnitude on the FFF. Consequently, the distance between the source and the reflective mirror is 1500 m, much greater than the minimum distance requirement of 490 m. The actual situation is shown in Figure 1.

The FFF reflective mirror will be installed on the top plate of the environment seal (baseplate for FFF), supported by the mount elevation (EL) stage, and exposed on the roof of the observation station, as depicted in Figure 2.

According to the calibration requirement, a 2 m by 3 m mirror reflector was needed; meanwhile, its flatness should be better than λ/20, so the FFF mirror system was designed to ensure RMS (Root Mean Square) value of flatness less than 0.1 mm under gravity and strong wind pressure. The λ/20 mirror deformation limit is a well-established criterion for proper calibration in optical systems. It is often associated with the quality assessment and calibration of optical instruments, telescopes, and other precision optical systems.

2. Structure of the Far-Field Flat Mirror

The FFF mirror, composed of a highly reflective mirror, reinforcing frame, and supporting bracket, is designed to ensure efficient high reflection and radiation transmission.

Various solutions were researched and tested. Experimental results revealed that achieving a planarity of 0.1 mm for a 1 m × 1 m honeycomb panel was extremely complex and quite difficult for the FFF mirror with 2 m width by 3 m height. Glass mirrors with metallic coating are not suitable due to both the complex manufacturing process and high costs. Considering process, performance, and cost, we chose ultra-flat aluminum panels with the reinforcing frame glued on the back as the solution. The panels are supported by 12 rods connected to six fixing points. The FFF reflector’s final dimensions are 3108.2 mm in height, 3000 mm in length, and 2791 mm in width. The back of the reflector is constructed with a frame consisting of 14 aluminum square tubes measuring 150 × 100 × 3 mm, as depicted in Figure 3.

The flatness requirement of the FFF mirror within the context of optical components should be better than 0.1 mm (λ/20). This stringent requirement is necessary because a rough reflective mirror may result in diffuse reflection instead of proper reflection, thereby rendering it ineffective for beam calibration. The signal being calibrated is at a frequency of 150 GHz, with an approximate wavelength of 2 mm. Therefore, in order to ensure its functionality, the flatness requirement for the FFF mirror should be less than 0.1 mm. There are several factors that can affect the flatness of the FFF mirror, including machining precision, weight, installation errors, and wind loading.

Achieving high flatness of the FFF mirror was challenging due to its large size. The ultra-flat aluminum panel of the FFF mirror is 0.08 mm. The panel (mirror) is bonded to a welded and reinforced frame, the surface flatness of which is within 0.05 mm. Besides 12 support rods that are adjustable, we designed adjustment points on the back crossbeam to control the central depression. A laser tracker was employed to measure the mirror and accordingly adjust the mirror flatness, mainly by loosening or tightening the stud of the back crossbeam. A flatness RMS value of 0.05 mm was achieved in the factory by comprehensive design, process, and assembly process measures. To validate the response of the mirror structure under extreme conditions, the finite element method will be used to analyze the far-field calibrated mirror structure in this paper.

3. Fluid Field Calculation

3.1. Theoretical Analysis

Establishing a fluid–structure interaction model necessitates the initial determination of the fluid flow state. The fluid domain in which the FFF mirror is located is characterized as an incompressible viscous fluid flow governed by mass conservation, energy conservation, and momentum conservation [10,11,12]. The initial step involves determining the fluid state and considering the concept of Reynolds number (Re), which serves as an indicator to distinguish between laminar and turbulent flow regimes by evaluating the relative magnitudes of inertial and viscous forces. The formula for calculating the Reynolds number is as follows:

$$\mathrm{R}\mathrm{e}=\frac{\mathsf{\rho }\mathrm{V}\mathrm{L}}{\mathsf{\mu }}$$

(2)

Re signifies the Reynolds number, ρ denotes the fluid density, V represents fluid velocity, L stands for the characteristic length, and μ represents the fluid dynamic viscosity [13,14,15]. When the Reynolds number is below a critical threshold (typically around 500,000), the flow is laminar, with viscous forces dominating and leading to a smooth and ordered flow. However, when the Reynolds number exceeds this threshold, turbulence occurs due to the dominance of inertial forces, resulting in irregular and chaotic flow patterns. This study demonstrates that with a basin length of 40 m, a fluid density of 1.225 kg/m, a viscosity of 1.81 × 10⁻⁵ [g/(m/s)], and a flow velocity of 17.5 m/s, the Reynolds number obtained from Equation (2) exceeds 500,000, confirming turbulent flow.

Turbulent flow analysis primarily employs the k-ε and k-ω turbulence models. This study utilizes the realizable k-ε turbulence model [16,17]. The turbulence model and its dissipation equations are presented as follows.

Standard model turbulence kinetic energy transport equation:

$$\frac{𝜕\left(\mathsf{\rho }\mathrm{k}\right)}{𝜕\mathrm{t}}+\frac{𝜕\left(\mathsf{\rho }\mathrm{k}{\mathrm{u}}_{\mathrm{i}}\right)}{𝜕{\mathrm{x}}_{\mathrm{i}}}=\frac{𝜕}{𝜕{\mathrm{x}}_{\mathrm{j}}}\left[\left(\mathsf{\mu }+\frac{{\mathsf{\mu }}_{\mathrm{t}}}{{\mathsf{\sigma }}_{\mathrm{k}}}\right)\frac{𝜕\mathrm{k}}{𝜕{\mathrm{x}}_{\mathrm{j}}}\right]+{\mathrm{G}}_{\mathrm{k}}+{\mathrm{G}}_{\mathrm{b}}-\mathsf{\rho }\mathsf{ϵ}-{\mathrm{Y}}_{\mathrm{M}}+{\mathrm{S}}_{\mathrm{k}}$$

(3)

Standard model turbulence dissipation rate transport equation:

$$\frac{𝜕\left(\mathsf{\rho }\mathsf{ϵ}\right)}{𝜕\mathrm{t}}+\frac{𝜕\left(\mathsf{\rho }\mathsf{ϵ}{\mathrm{u}}_{\mathrm{i}}\right)}{𝜕{\mathrm{x}}_{\mathrm{i}}}=\frac{𝜕}{𝜕{\mathrm{x}}_{\mathrm{j}}}\left[\left(\mathsf{\mu }+\frac{{\mathsf{\mu }}_{\mathrm{t}}}{{\mathsf{\sigma }}_{\mathsf{ϵ}}}\right)\frac{𝜕\mathsf{ϵ}}{𝜕{\mathrm{x}}_{\mathrm{j}}}\right]+{\mathrm{C}}_{1\mathsf{ϵ}}\frac{\mathsf{ϵ}}{\mathrm{k}}\left({\mathrm{G}}_{\mathrm{k}}+{\mathrm{C}}_{3\mathsf{ϵ}}{\mathrm{G}}_{\mathrm{b}}\right)-{\mathrm{C}}_{2\mathsf{ϵ}}\mathsf{\rho }\frac{{\mathsf{ϵ}}^2}{\mathrm{k}}+{\mathrm{S}}_{\mathsf{ϵ}}$$

(4)

G_k represents the turbulence kinetic energy generated by mean velocity gradients, G_b accounts for the turbulence kinetic energy generated by buoyancy, Y_M signifies the influence of fluctuation expansion on turbulence dissipation rate, and σ_k and σ_ϵ denote the turbulent Prandtl numbers for turbulence kinetic energy and turbulence dissipation rate, respectively [18,19].

The one-way fluid–structure coupling approach is chosen due to the rigid structure of the mirror, which has minimal influence on the fluid field. Initial fluid field calculations are conducted using the Fluent module within Workbench, providing wind pressure distribution over the mirror. These results are then used as boundary conditions for calculating mirror deformation and stress. Figure 4 presents the analysis flowchart.

3.2. Fluid Computational Domain

The FFF mirror is shielded from mountain winds by a windshield located southwest of the AliCPT-1 observatory, as shown in Figure 5. To simulate actual working conditions, the size of the flow field model is constrained within the range of the windshield, with a final computation domain size of 30 m × 40 m × 10 m.

A regional block subdivision approach is employed to discretize the computational domain for grid partitioning. The interior domain is set at 4 m × 4 m × 3.2 m, with the reflection mirror as the central point, as depicted in Figure 6. This approach allows for adaptation to the flow field conditions near the FFF by subdividing only the grid of the interior domain, thereby reducing computational resources.

3.3. Grid Independence Verification

In the realm of computational fluid dynamic (CFD) simulations, the accuracy of calculations is heavily influenced by both the quantity and quality of the grids within the fluid model. Therefore, it is essential to ensure a well-structured grid to achieve accurate simulations [20,21,22]. Structured grids consist of regular, ordered grid cells, typically quadrilaterals or hexagons, while unstructured grids are composed of irregular, unordered grid cells, such as triangles, tetrahedra, and so on. Due to the complex structure of the reflector mirror, an unstructured grid refinement is applied to the internal domain, while a structured grid refinement is applied to the external domain.

Additionally, grid refinement is applied to the grid interface for fluid–structure coupling to maintain consistency in node quantity and position between the fluid and solid interfaces.

To balance computational accuracy and time efficiency, grid independence verification was conducted for the FFF reflector structure. Initially, six different grid configurations were generated, each with a mesh quality above 0.85. Simulations were carried out with the same boundary conditions to examine the impact of grid quantity on the maximum pressure (Pmax) and maximum velocity (Vmax), as illustrated in Figure 7. The results revealed that increasing the number of grids beyond 16,958,409 led to changes in Pmax and Vmax by less than 4.7%, indicating that further grid refinement would not significantly affect the simulation results.

Subsequently, the optimal grid quantity was determined to be 24,031,137, with a base size of 5 mm, considering a balance between computational time and accuracy. The optimized mesh of the simulation model is depicted in Figure 8.

3.4. Boundary Conditions

Ali is an area characterized by harsh natural conditions in Tibet, falling under a high-altitude frigid monsoon climate. The AliCPT-1 observatory site experiences frequent strong winds, with approximately 115 windy days per year. The extreme wind speed is 53 m/s from the southwest, so the installation of a windshield reduces the wind speed to less than 10 m/s in the direction of 220° to 315°.

An on-site meteorological station is installed on the rooftop of the observatory building to monitor weather conditions. This station is positioned near the installation site of the FFF mirror. After the installation of the windbreak, the flow field was analyzed. When the wind blows toward the windshield, the wind speed at the meteorological station is 12.1 m/s, while at the location of the FFF mirror, it is 12.8 m/s. Therefore, it can be concluded that the wind speeds at the two locations are similar and that the wind speeds at the meteorological station can be used.

Observational data were collected from November 2019 to July 2020, with a specific focus on daily instantaneous wind speeds and wind directions, as shown in Figure 9. From these data, it was determined that the max wind speed reached 17.5 m/s, with a prevailing wind direction from the northwest. The FFF mirror experienced wind from the forward direction at a bearing of 104° with a maximum wind speed of 7.5 m/s. In contrast, wind from the back direction occurs at a bearing of 290° with a maximum wind speed of 5.2 m/s. A cumulative histogram shown in Figure 9 demonstrated that wind speeds most frequently fell within the range of 6 to 14 m/s, accounting for approximately 90% of the total occurrences.

We take a maximum wind speed of 17.5 m/s as an extreme wind load to analyze the wind pressure on the reflector. Regarding boundary conditions within the computational domain, the fluid inlet velocity was set to 17.5 m/s, the fluid outlet pressure was set to 0 Pa, and we specified the no-slip boundary conditions on the surrounding walls.

Currently, the wall approach primarily utilizes the function approach and the viscous sublayer resolving approach. Wall functions are based on the law of the wall, valid for attached flow with favorable or zero pressure gradient and for relatively simple flow geometries such as flat plate or channel flows. The viscous sublayer resolving approach uses a highly precise grid to resolve steep profiles within the viscous sublayer. This approach is more accurate for heat transfer simulations, natural convection, cases with flow impingement, and cases requiring accurate prediction of skin friction. Since the FFF mainly involves natural convection and flow impingement, this study employs the viscous sublayer resolving approach within the wall approach. In Fluent, the k-ε model is selected, and the near-wall treatment is set to enhanced wall treatment. Fluent automatically selects the viscous sublayer resolving approach. Mesh resolution requirements are expressed in terms of y+. By reviewing the analysis results of the reflective mirror, it is found that the y+ value is 19.9, which falls within the range of 10–110 as required. Upon reviewing the analysis results of the reflector, the y+ value of 19.9 falls within the range of 10–110, which is considered acceptable.

A realizable k-ε turbulence model was employed for turbulent simulation, with model constants set to C₂ϵ = 1.9, σ_k = 1, and σ_ϵ = 1.2. For the inlet boundary conditions for the turbulence model, the specification method employs the intensity and viscosity ratio method. Turbulent intensity measures the strength of turbulent fluctuations, while the turbulent viscosity ratio is defined as the ratio of turbulent viscosity to fluid viscosity. Normal turbulent intensities typically range from 1% to 5%. For external flows, a turbulent intensity of 1% or lower and a turbulent viscosity ratio of 1–10 are generally recommended. In consideration of the flow characteristics of this study, a turbulent intensity of 5% and a turbulent viscosity ratio of 10 are applied.

The flow surface adopts a pressure outlet boundary condition with a pressure of 0 Pa. The reflector surface adopts a no-slip wall boundary condition, while the top and both side surfaces adopt free-slip wall boundary conditions.

The coupled algorithm was utilized for pressure–velocity coupling, treating pressure and velocity as two unknowns and imposing simultaneous constraints on their relationship to derive analytical solutions. Concurrently, a coupled analysis was conducted on fluid parameters such as velocity, pressure, density, and temperature to depict fluid behavior accurately. A mixed initialization approach was adopted to solve the Laplace equation, using distinct interpolation methods for different boundary conditions to initialize the pressure and velocity fields of the computational domain.

3.5. Convergence Assessment

The assessment of convergence is crucial for ensuring the accuracy and reliability of computational results. It serves as an evaluation method for validating the correctness of various factors, including mesh quality, physical model selection, and boundary condition configuration.

In Fluent simulations, convergence is typically determined by monitoring the trend of residuals or customizing one of the physical parameters. Fluent’s default convergence criteria state that convergence is achieved when all variable residuals fall below 1 × 10⁻³, except for energy residuals, which have even more stringent requirements, demanding that they be less than 1 × 10⁻⁶ [23]. Residuals refer to the sum of fluxes on each face of each element, predominantly monitoring physical quantities like mass, energy, velocity, turbulence, and more.

In the current computational process, when the number of iterations reaches 56, the mass residual is below 1 × 10⁻³, thus meeting the convergence criteria.

3.6. Results Analysis

After achieving computational convergence, the relevant parameters related to the reflector are obtained. A symmetrical plane perpendicular to the ground and passing through the mirror was established, and pressure contour plots and streamline plots were examined on this plane, as shown in Figure 10.

When the mirror withstands the front wind, the pressure distribution map reveals that the front side of the reflector experiences the maximum pressure at 211 Pa. Conversely, the mirror’s backside exhibits lower pressure, which is easily attributed to the formation of small vortices on the windward side. The pressure distribution on the reflector surface, as depicted in Figure 11, emphasizes the substantial pressure on the front side of the reflector and the windward face of the support structure during forward wind loading.

In contrast, when the wind is loaded in the back direction, the distribution map reveals that the maximum pressure on the backside of the reflector reaches 263 Pa. Additionally, the low-pressure region during backloading is more significant compared to forward loading, which should be attributed to a deflection effect of the upward tile angle. Figure 11 illustrates that the pressure primarily concentrates on the reflector’s back side and the support structure’s windward face. Since the support structure has a smaller surface area, it is less conducive to vortex formation, resulting in the lowest pressure on the back side.

The FFF is fixed on a three-axis rotating mount. The mount will stand straight (without elevation tilt) when the telescope is calibrated. The elevation (EL) axis was locked up by two motors. The torque caused by wind pressure on the mirror should not be larger than the maximum load torque of the motors.

With a maximum pressure of 263 Pa applied to a mirror with an area of 8.33 m², multiplied by the distance of 2.925 m between the center of gravity and the EL axis, and considering the FFF mirror inclined at an angle of 45 degrees, the maximum torque caused by wind pressure was calculated as 4531 N·m. The EL motor reducer output torque is up to 18,945 N·m; in addition, the motor is even more robust in a locked state than a rotating one, so the mount can support the FFF in the maximum wind speed loading condition.

Overall, the results analysis provides a comprehensive understanding of the pressure distribution and torque effects on the FFF mirror, confirming the suitability of the mount to withstand the wind loading conditions.

4. Structural Analysis

4.1. FEA Model

The finite element method was chosen for FFF structural analysis [24]. In finite element analysis, the matrix expression for the control equations by

$$\left[\mathrm{K}\right]\left\{\mathrm{x}\right\}=\left\{\mathrm{F}\right\}$$

(5)

where {K} is the stiffness coefficient matrix, {x} is the displacement vector, and {F} is the force vector. In static structural analysis, assuming {F} to be a constant matrix and continuous, materials should adhere to elastic, small deformation theory. Nonlinear boundary conditions may be applied, and {F} represents the static loads applied to the model, which is independent of time variation and inertial forces [24].

We employed a static strength analysis in ANSYS Workbench to analyze the deformation of the reflection mirror under maximum wind pressure and simultaneously evaluate the stability of the support rod structure.

Firstly, simplify the model. Reduce the number of mesh subdivisions by removing bolts, small holes, and threads from the model to minimize the amount of numerical calculations and save time. The structural outline was effectively captured through mesh refinement in finite element analysis, and additional nodes are available for displacement and stress calculation. Localized refinement to the connection between the support structure of the FFF mirror and the base plate to ensure the accuracy of critical structures and to mitigate potential stress concentration points. For optimal computational performance and practical conditions, tetrahedral elements with a size of 10 mm were chosen, with denser meshing at complex connection points. The final model encompasses approximately 9 million elements and 12 million nodes, as depicted in Figure 12.

Almost all structures within FFF are composed of aluminum alloy 6061 (6061 for short), which exhibits fine corrosion resistance, mechanical properties, formability, weldability, and machinability; additionally, the superior specific strength is the most crucial factor for the lightweight design of FFF. The key parameter of which is detailed in

Table 1. AL6061 material properties.

Material Properties	Material Parameters
Density (t/mm3)	2.77 × 10−9
Young’s Modulus (MPa)	71,000
Poisson’s Ratio	0.33
Tensile Yield Strength (MPa)	280
Tensile Ultimate Strength	310

.

Analyze the FFF deformation and stress with forward and backward wind load, assuming the baseplate is fixed, as depicted in Figure 13. The figure shows that the surface of the FFF mirror experiences the highest pressure with the supporting bracket. This uneven pressure distribution can cause a decrease in the accuracy of the mirror’s shape. Additionally, stress concentration may occur in the supporting bracket.

4.2. Results Analysis

The analytical results show that the maximum FFF deformation of 0.059 mm occurs in the mirror surface under 17.5 m/s wind speed applied in the front wind direction. The maximum stress of 4.5 MPa is concentrated at the interface between the support rod and the base plate, as illustrated in Figure 14 and Figure 15.

The surface of the FFF mirror, measured with a three-coordinate measuring machine, indicates deformations of 0.05 mm resulting from processing and self-weight. The final Root Mean Square (RMS) flatness is obtained by combining the 0.05 mm with 0.0587 mm, which is 0.077 mm. This value is less than the required flatness of 0.1 mm. The material at this location is AL6061, and its yield strength is 280 MPa, which shows that the FFF can be used normally without damage in this extreme working condition. Maximum deformation and stress in backward wind pressure have an even lesser influence on the FFF. The result showed that the telescope can be calibrated in this extreme operating condition.

This study employed a boundary condition of an instantaneous wind speed of 17.5 m/s, which occurred only at specific times in the last three years: 5:53 a.m. on 13 January 2020 and 2:19 a.m. on 14 January 2020; normally, the chosen weather had relatively low winds for calibration work for equipment safety. Furthermore, the maximum wind speeds are from wind directions of 40 degrees southeast, not in the vertical direction of the FFF mirror, which will have a lesser influence on the mirror. So, actually, the deformation of the FFF mirror is smaller compared to the analyzed conditions.

5. Structural Analysis of the Connection

In the overall analysis, the connection structure of the FFF mirror has been simplified. This structure has a connecting pipe threaded to rod 1 and rod 2, as shown in Figure 16. The tilt angle of the FFF mirror is controlled by adjusting the coordination between the connecting pipe and the two rods. A detailed analysis and calculation of the connection structure are required.

The nut acts to lock the thread, so it is simplified in the calculation of the connection structure. Instead, the threads on each rod were simplified by using contact types. The material chosen for the connection structure is 6061.

Mesh the model primarily using hexahedral grid elements with a size of 3 mm. A total of 27,471 mesh elements and 112,932 nodes were yielded, and the mesh quality exceeded 0.85, as illustrated in Figure 17.

Concerning boundary conditions, it was chosen to fix the inner wall of the hole on Rod 1, subjecting it to full constraint.

To investigate the structural strength of the connecting structure under wind load, as previously studied, the maximum displacement of the support rod under wind load is 0.05 mm, set as displacement load in Y-direction on rod2.

The analysis results are shown in Figure 18. The maximum stress within the connecting structure is 31.27 MPa, located in the relief notch of Rod 1 and presented as the concentration of stresses in the root corner. A yield strength of 6061 (280 MPa) is eight times the maximum stress, so it can be concluded that this connecting structure remains safe under actual working conditions according to the principles of the fourth strength theory.

A failure analysis was conducted to investigate the ultimate strength of the connection structure by increased displacement load. When the displacement reached 0.45 mm, the stress exceeded the material’s ultimate strength (281.5 MPa), which means the structure deformation cannot be restored. Figure 19 illustrates the stress distribution contour plot at a displacement of 0.44 mm. The maximum stress still appears at the relief notch, and the region of elevated stress in rod 1 extends from the relief notch to the threaded area. Therefore, the thickness of the pipe should be increased if it is to be used in these extreme conditions.

6. Conclusions

The far-field flat mirror in a simulated environment of AliCPT-1 observatory was analyzed using fluid–structure coupling FEA. The following conclusions can be drawn:

(1) The FFF structure was designed to ensure safety and stability at a low cost. Furthermore, the installation and precise adjusting plan are straightforward and convenient. When in a static state, the mirror demonstrates a high level of flatness. Even under extreme wind loads, the mirror’s deformation is small. The overall impact on flatness is below λ/20, which satisfies the requirement of flatness.

(2) Under the maximum wind load, the safety factors of the mirror, mirror support frame structure, and support rods exceed 8. The torque impact on the mount EL axis caused by the wind load remains within the motor reducers’ capacity. The analysis results demonstrate the safety and reliability of the FFF system under the influence of both gravity and wind pressure loads.

(3) The convergence of the grid was verified, and validation was performed for actual operating conditions. Grid convergence conducted on the internal and external flow domains demonstrated a good consistency between grid quality and the number of grids in the flow field.

(4) The application of the fluid–structure interaction (FSI) method proves to be an effective approach for simulating the wind-induced deformation of the FFF. The results demonstrate that, under a wind speed of 17.5 m/s, the maximum deformation of the FFF reflective mirror, occurring at the center, is 0.0587 mm. The overall analysis suggests that the FFF reflective mirror can maintain normal functionality under wind loading conditions.

(5) Finite element analysis has shown that the FFF reflective mirror maintains structural stability under extreme wind loading conditions. This structural design approach can be applied to the design of large-scale, slender-supported reflective mirror structures in the future.