The Optimization Design of a Lightweight 2 m SiC Mirror for Ground-Based Telescopes

: The weight of the primary mirror increases as the aperture of ground-based telescopes increases, making it more challenging to maintain the positional stability and surface accuracy of the solid primary mirror. Consequently, a 2 m lightweight silicon carbide (SiC) mirror and an optimization method were proposed in this study. The relationship between the gravitational deformation of the mirror and its thickness and number of supports was derived based on force analysis of the mirror; the thickness of the mirror and the appropriate number of supports were obtained as initial parameters for optimization. The back structure of the mirror was designed in a lotus pattern to improve its rigidity. Numerous structural parameters were classified into major and non-major parameters based on the results of a sensitivity analysis. The non-major and major structural parameters were optimized using a Latin hypercube design method and a non-dominated sorting genetic algorithm, respectively. The optimized 2 m lightweight SiC mirror had a mass of 119 kg and an areal density of 38.7 kg/m 2 . The surface figure error root-mean-square ( RMS ) in the vertical state of the optical axis and the first modal resonance of the mirror assembly calculated using finite element analysis were 11.3 nm and 76.5 Hz, respectively. Modal tests of the mirror assembly were conducted using the hammering method, achieving a maximum relative frequency error of 7.4% compared with the simulation results. The optimized 2 m SiC mirror was over 50% lighter than traditional passive Zerodur mirrors of the same size.


Introduction
Ground-based telescopes serve as vital tools for space exploration.Increasing the aperture of telescopes is an effective method for improving their imaging resolution and detection ability.However, as the aperture increases, the weight of the solid primary mirror (PM) increases roughly in proportion to the cubic power of the aperture, making it more challenging to maintain the positional stability and surface accuracy of the PM [1].Moreover, an increase in the PM weight increases the overall weight of the telescope, which adversely affects the tracking of rapid targets and increases manufacturing costs.Consequently, the optimization of a lightweight PM has always been the focus of groundbased telescope research and development [2,3].
A PM with a diameter greater than 3 m such as the 4.2 m SOAR PM is usually made lightweight by means of a thin mirror and active support technology.Compared with active support technology, passive support technology has advantages such as a simple structure, high reliability, low cost, and easy maintenance.Consequently, ground-based telescopes with apertures of 2 m typically adopt passive support technology.Many studies have been conducted on lightweight mirrors that are passively supported.The 1.8 m Zerodur PM of the VLT auxiliary telescope was made lightweight by drilling hexagonal holes on the back, reducing the weight from 1384 to 605 kg.The areal density of the mirror was approximately 221 kg/m 2 [4,5].A 2 m lightweight silicon carbide (SiC) PM with an annular pattern on the back produced by CIOMP achieved a weight of 333 kg and an areal density of 105 kg/m 2 [6].Corning used abrasive waterjetting and other technologies to produce lightweight cores.These were then bonded to the front and back plates to manufacture sandwich-structure ULE mirrors.This technology was used to fabricate a 1.3 m secondary mirror weighing 185 kg for the Subaru telescope [7,8].Composite Mirror Applications, Inc. (CMA) produced a 1 m carbon fiber-reinforced polymer (CFRP) PM using composite replication technology, achieving a weight of 27 kg and an areal density of 37 kg/m 2 [9].
Zerodur, ULE, and SiC are commonly used materials for fabricating mirrors.In addition to optical performance, the mechanical and thermal properties of lightweight mirrors should be considered [10].SiC, which has a high Young's modulus and a low density, is an ideal candidate for lightweight mirrors.During the early years, SiC mirrors were expensive and used primarily for space mirrors; however, with the improvement in production technology, the cost of SiC mirrors has decreased considerably, and they have been used in ground-based telescopes [11,12].SiC mirrors are produced via a reactionbonding process akin to casting, which enables the creation of an intricate structure that is highly advantageous for lightweight mirrors.In light of the aforementioned advantages of SiC, this study focuses on the structural design of a 2 m lightweight mirror using SiC materials.
The structural design of lightweight mirrors mainly aims to eliminate mirror body components with minimal deformation resistance contribution, thus reducing the weight of the mirror without compromising its optical performance.Topology and parameter optimization have been extensively utilized in the lightweight mirror design.Liu et al. used a topology optimization technique to design a 2 m lightweight SiC primary mirror of a space telescope [13].Zhai et al. conducted a lightweight design for a 2.02 m SiC mirror using parameter optimization, resulting in a model weight of 228 kg [14].Wang et al. employed structural and parametric optimization to develop a 2 m ultralight SiC mirror weighing only 105 kg with an areal density of 34 kg/m 2 [15].Topology optimization seeks to attain the most optimal structural topology, while parameter optimization aims to determine the optimal dimensions of a given structure.This study employs a combined approach of topology and parameter optimization to achieve the lightweight design of a 2 m SiC mirror.

Initial Parameters of the 2 m SiC Mirror
The mirror is designed for the PM in an Alt-Az telescope capable of observing loworbit space targets.Consequently, the telescope has a high tracking speed, necessitating a lightweight mirror.The mirror, with a diameter of 2 m, has a central hole of 0.32 m in diameter and a focal ratio of 1.5.
The thickness of the mirror and the number of axial supports are traditionally determined using the empirical formulas in Equations ( 1) and ( 2), after which the parameters obtained can be adjusted based on the experience of the designer [16].
where δ denotes the maximum deformation of the mirror; t and r denote the thickness and radius of the mirror, respectively; ρ and E denote the density and Young's modulus of the material, respectively; and N denotes the number of axial supports.The parameters adjusted by different designers can vary considerably, which significantly affects the design of the mirror structure.Consequently, a theoretical analysis of the mirror thickness and the number of axial supports is necessary.
The elastic deformation of mirrors due to gravity is the primary factor influencing their surface figure error.To minimize the gravitational deformation of mirrors, two methods have been adopted: first, the stiffness of the mirror can be changed by selecting materials (such as SiC) with a high Young's modulus or by altering the geometric dimensions of the mirror (such as increasing the thickness of the mirror) to increase the area moment of inertia; second, the number of axial supports for the mirror can be increased.Reducing the distance between the supports by increasing their number can effectively reduce the gravitational deformation of the mirror (Figure 1).The parameters adjusted by different designers can vary considerably, which significantly affects the design of the mirror structure.Consequently, a theoretical analysis of the mirror thickness and the number of axial supports is necessary.
The elastic deformation of mirrors due to gravity is the primary factor influencing their surface figure error.To minimize the gravitational deformation of mirrors, two methods have been adopted: first, the stiffness of the mirror can be changed by selecting materials (such as SiC) with a high Young's modulus or by altering the geometric dimensions of the mirror (such as increasing the thickness of the mirror) to increase the area moment of inertia; second, the number of axial supports for the mirror can be increased.Reducing the distance between the supports by increasing their number can effectively reduce the gravitational deformation of the mirror (Figure 1).The section of the mirror between the two supports can be regarded as a single supported structure with a uniformly distributed gravitational load.Consequently, the mirror deformation can be expressed as follows: where I denotes the area moment of inertia, q denotes the uniformly distributed gravitational load, k denotes the distance between the two supports, M(x) denotes the bonding moment, and A and B denote integral constants.
The uniformly distributed gravitational load and the area moment of inertia can be expressed as follows: where b and h denotes the width and thickness of the structure between the two supports, respectively.The number of axial supports can be expressed as follows: where D denotes the diameter of the mirror, and d denotes the diameter of the center hole in the mirror.
From Equations ( 3)-( 6), the thickness of the mirror can be calculated as follows: For the 2 m SiC mirror, 18, 36, and 54 supports were selected, and the maximum gravitational deformations of mirrors with different thicknesses could be calculated using Equation (7).The deformation curves of the mirrors shown in Figure 2 indicate that The section of the mirror between the two supports can be regarded as a single supported structure with a uniformly distributed gravitational load.Consequently, the mirror deformation can be expressed as follows: where I denotes the area moment of inertia, q denotes the uniformly distributed gravitational load, k denotes the distance between the two supports, M(x) denotes the bonding moment, and A and B denote integral constants.
The uniformly distributed gravitational load and the area moment of inertia can be expressed as follows: q = ρbhg (4) where b and h denotes the width and thickness of the structure between the two supports, respectively.The number of axial supports can be expressed as follows: where D denotes the diameter of the mirror, and d denotes the diameter of the center hole in the mirror.
From Equations (3)-( 6), the thickness of the mirror can be calculated as follows: For the 2 m SiC mirror, 18, 36, and 54 supports were selected, and the maximum gravitational deformations of mirrors with different thicknesses could be calculated using Equation (7).The deformation curves of the mirrors shown in Figure 2 indicate that the gravitational deformation of the mirror is considerably reduced by increasing the number of supports and the mirror thickness.However, as the number of supports and mirror thickness increase, their impact on the gravitational deformation of the mirror becomes less evident.Consequently, indiscriminately increasing the number of supports or mirror thickness might not be beneficial.Excessively increasing the number of supports minimally affects the gravitational deformation of the mirror but increases the design complexity and complicates the support structure.Similarly, excessively increasing the thickness of the mirror results in a minimal change in its gravitational deformation but increases both its weight and manufacturing costs.
sign complexity and complicates the support structure.Similarly, excessively increasing the thickness of the mirror results in a minimal change in its gravitational deformation but increases both its weight and manufacturing costs.
The gravitational deformation of the 36-supported ULE mirror is shown in Figure 2.Even with 36 supports, the maximum deformation of the ULE mirror was slightly lower than that of the SiC mirror with 18 supports, owing to the high specific stiffness (elastic modulus/density) of SiC, which offers unparalleled advantages in lightweight mirrors.
The maximum gravitational deformation of the 2 m SiC mirror with 18 axial supports is limited to 39.6 nm (λ/16, λ = 632.8nm).The thickness of the solid mirror derived from Equation (7) is 122 mm, which serves as the initial parameter for lightweight optimization.The 18 supports can be arranged in inner and outer ring configurations, with 6 supports in the inner ring and 12 supports in the outer ring.

Back Structure of the 2 m SiC Mirror
Many pa<erns can be used on the back of a lightweight mirror, the most commonly used being triangular, hexagonal, and annular pa<erns.
In this study, a triangular pa<ern with a spacing of k between adjacent ribs was first used for the 2 m SiC mirror (Figure 3a).The location of the support should be at the rib intersections; consequently, the inner and outer rings are not independent but are interrelated as a function of k.A finite element model of the 2 m SiC mirror was developed using the ANSYS parametric design language (APDL).The surface figure error RMS varies with distance k under a gravitational load (Figure 3b).When the distance k is 200 mm, the minimum surface figure error RMS of the mirror is 19 nm.The deformation cloud map of the mirror is shown in Figure 3c.The gravitational deformation of the 36-supported ULE mirror is shown in Figure 2.Even with 36 supports, the maximum deformation of the ULE mirror was slightly lower than that of the SiC mirror with 18 supports, owing to the high specific stiffness (elastic modulus/density) of SiC, which offers unparalleled advantages in lightweight mirrors.
The maximum gravitational deformation of the 2 m SiC mirror with 18 axial supports is limited to 39.6 nm (λ/16, λ = 632.8nm).The thickness of the solid mirror derived from Equation (7) is 122 mm, which serves as the initial parameter for lightweight optimization.The 18 supports can be arranged in inner and outer ring configurations, with 6 supports in the inner ring and 12 supports in the outer ring.

Back Structure of the 2 m SiC Mirror
Many patterns can be used on the back of a lightweight mirror, the most commonly used being triangular, hexagonal, and annular patterns.
In this study, a triangular pattern with a spacing of k between adjacent ribs was first used for the 2 m SiC mirror (Figure 3a).The location of the support should be at the rib intersections; consequently, the inner and outer rings are not independent but are interrelated as a function of k.A finite element model of the 2 m SiC mirror was developed using the ANSYS parametric design language (APDL).The surface figure error RMS varies with distance k under a gravitational load (Figure 3b).When the distance k is 200 mm, the minimum surface figure error RMS of the mirror is 19 nm.The deformation cloud map of the mirror is shown in Figure 3c.
The 2 m SiC mirror was also tested using an annular pattern where the inner and outer rings were independent of each other (Figure 4a).A finite element model of the mirror was developed using the APDL.Different outer ring radii (r out ) were set and the inner ring radius (r in ) was varied from 330 to 400 mm.The surface figure error RMS of the mirror was computed under a gravitational load, with the variation curve of surface figure error RMS indicated in Figure 4b.The minimum surface figure error RMS of the mirror is 19.5 nm when r in and r out are 380 and 840 mm, respectively.A deformation cloud map of the mirror is shown in Figure 4c.The 2 m SiC mirror was also tested using an annular pa<ern where the inner and outer rings were independent of each other (Figure 4a).A finite element model of the mirror was developed using the APDL.Different outer ring radii (rout) were set and the inner ring radius (rin) was varied from 330 to 400 mm The essence of the pa<ern on the back of the mirror is to enhance the area moment of inertia of the mirror to improve its bending resistance.To achieve this, a lotus pa<ern for the 2 m SiC mirror was implemented (Figure 5a).A finite element model of the mirror was developed using the APDL.Different outer radii (rout) were set, and the inner radius (rin) was varied from 330 to 400 mm.The surface figure error RMS of the mirror was calculated under a gravitational load, with the variation curve of the surface figure error RMS indicated in Figure 5b.The minimum surface figure error RMS of the mirror is 17.7 nm when rin and rout are 370 and 830 mm, respectively.A deformation cloud map of the mirror is shown in Figure 5c.The 2 m SiC mirror was also tested using an annular pa<ern where the inner and outer rings were independent of each other (Figure 4a).A finite element model of the mirror was developed using the APDL.The essence of the pa<ern on the back of the mirror is to enhance the area moment of inertia of the mirror to improve its bending resistance.To achieve this, a lotus pa<ern for the 2 m SiC mirror was implemented (Figure 5a).A finite element model of the mirror was developed using the APDL.Different outer radii (rout) were set, and the inner radius (rin) was varied from 330 to 400 mm.The surface figure error RMS of the mirror was calculated under a gravitational load, with the variation curve of the surface figure error RMS indicated in Figure 5b.The minimum surface figure error RMS of the mirror is 17.7 nm when rin and rout are 370 and 830 mm, respectively.A deformation cloud map of the mirror is shown in Figure 5c.The essence of the pattern on the back of the mirror is to enhance the area moment of inertia of the mirror to improve its bending resistance.To achieve this, a lotus pattern for the 2 m SiC mirror was implemented (Figure 5a).A finite element model of the mirror was developed using the APDL.Different outer radii (r out ) were set, and the inner radius (r in ) was varied from 330 to 400 mm.The surface figure error RMS of the mirror was calculated under a gravitational load, with the variation curve of the surface figure error RMS indicated in Figure 5b.The minimum surface figure error RMS of the mirror is 17.7 nm when r in and r out are 370 and 830 mm, respectively.A deformation cloud map of the mirror is shown in Figure 5c.
Comparing three different patterns for the 2 m SiC mirror, the lotus pattern mirror has surface figure error RMS of 17.7 nm, which is superior to those of the triangular and annular patterns.
A topology optimization method was employed to improve the lightweight of the 2 m SiC mirror.The surface plate and support locations were designated as non-optimized regions, whereas the remaining parts of the mirror were designated as optimized regions.The result of the topology optimization is shown in Figure 6a, with red indicating the parts to be retained and blue indicating those that can be removed.Based on the optimization results, the height of the outer edge of the mirror was reduced.Considering the grinding process, auxiliary reinforcing ribs were added to the back of the surface plate to enhance its rigidity.To enhance the natural frequency of the mirror and position the center of gravity behind the surface plate, semi-open back plates were designed on the back of the mirror.The final structure of the 2 m SiC mirror is shown in Figure 6b.Comparing three different pa<erns for the 2 m SiC mirror, the lotus pa<ern mirror has surface figure error RMS of 17.7 nm, which is superior to those of the triangular and annular pa<erns.
A topology optimization method was employed to improve the lightweight of the 2 m SiC mirror.The surface plate and support locations were designated as non-optimized regions, whereas the remaining parts of the mirror were designated as optimized regions.The result of the topology optimization is shown in Figure 6a, with red indicating the parts to be retained and blue indicating those that can be removed.Based on the optimization results, the height of the outer edge of the mirror was reduced.Considering the grinding process, auxiliary reinforcing ribs were added to the back of the surface plate to enhance its rigidity.To enhance the natural frequency of the mirror and position the center of gravity behind the surface plate, semi-open back plates were designed on the back of the mirror.The final structure of the 2 m SiC mirror is shown in Figure 6b.

Optimization of Mirror Structure Parameters
Optimization of the structural parameters could be performed after determining the structure of the 2 m SiC mirror.A 1/6 mirror was selected for optimization to enhance computational efficiency.The mirror is an axisymmetric geometric body, evenly divided into six sections at 60° intervals.Each section includes three axial supports, with one on the inner ring and two on the outer ring.The radial stiffness of the mirror significantly exceeds its axial stiffness, and the radial support positions are related to the mirror's center of gravity, remaining insensitive to variations in specific structural parameters.Meanwhile, the axial support positions significantly impact the optimization of structur-  Comparing three different pa<erns for the 2 m SiC mirror, the lotus pa<ern mirror has surface figure error RMS of 17.7 nm, which is superior to those of the triangular and annular pa<erns.
A topology optimization method was employed to improve the lightweight of the 2 m SiC mirror.The surface plate and support locations were designated as non-optimized regions, whereas the remaining parts of the mirror were designated as optimized regions.The result of the topology optimization is shown in Figure 6a, with red indicating the parts to be retained and blue indicating those that can be removed.Based on the optimization results, the height of the outer edge of the mirror was reduced.Considering the grinding process, auxiliary reinforcing ribs were added to the back of the surface plate to enhance its rigidity.To enhance the natural frequency of the mirror and position the center of gravity behind the surface plate, semi-open back plates were designed on the back of the mirror.The final structure of the 2 m SiC mirror is shown in Figure 6b.

Optimization of Mirror Structure Parameters
Optimization of the structural parameters could be performed after determining the structure of the 2 m SiC mirror.A 1/6 mirror was selected for optimization to enhance computational efficiency.The mirror is an axisymmetric geometric body, evenly divided into six sections at 60° intervals.Each section includes three axial supports, with one on the inner ring and two on the outer ring.The radial stiffness of the mirror significantly exceeds its axial stiffness, and the radial support positions are related to the mirror's center of gravity, remaining insensitive to variations in specific structural parameters.Meanwhile, the axial support positions significantly impact the optimization of structur-

Optimization of Mirror Structure Parameters
Optimization of the structural parameters could be performed after determining the structure of the 2 m SiC mirror.A 1/6 mirror was selected for optimization to enhance computational efficiency.The mirror is an axisymmetric geometric body, evenly divided into six sections at 60 • intervals.Each section includes three axial supports, with one on the inner ring and two on the outer ring.The radial stiffness of the mirror significantly exceeds its axial stiffness, and the radial support positions are related to the mirror's center of gravity, remaining insensitive to variations in specific structural parameters.Meanwhile, the axial support positions significantly impact the optimization of structural parameters.Consequently, the lightweight optimization process included axial support positions and excluded radial support positions.Symmetrical constraints were applied to the connecting surfaces.A flowchart for the structural parameter optimization of the 2 m SiC mirror is shown in Figure 7.
Numerous structural parameters of the 2 m SiC mirror could be further optimized, and changes to certain parameters may have minimal effects on the surface figure error RMS and mass of the mirror.Simultaneous optimization of these parameters would significantly reduce computational speed, particularly for structural parameters with minimal impacts on the surface figure error RMS and mass, resulting in a waste of computational resources.Therefore, it was imperative to differentiate these parameters to discern the sensitivity of the surface figure error RMS and mass to parameter variations.A parametric study method was employed to conduct a sensitivity analysis on the eleven structural parameters.Each factor (parameter) independently varied from low to high across 30 specific levels (parameter values) within the experimental space, while the other factors were constant at their baseline values.A total of 331 analytical calculations were conducted to identify the impact of each factor on the outcomes.The surface figure error RMS and mass variations with respect to the structural parameters are shown in Figure 8a and Figure 8b, respectively.The greater the surface figure error RMS and mass variations, the more sensitive they are to changes in the structural parameters.Numerous structural parameters of the 2 m SiC mirror could be further optimized, and changes to certain parameters may have minimal effects on the surface figure error RMS and mass of the mirror.Simultaneous optimization of these parameters would significantly reduce computational speed, particularly for structural parameters with minimal impacts on the surface figure error RMS and mass, resulting in a waste of computational resources.Therefore, it was imperative to differentiate these parameters to discern the sensitivity of the surface figure error RMS and mass to parameter variations.A parametric study method was employed to conduct a sensitivity analysis on the eleven structural parameters.Each factor (parameter) independently varied from low to high across 30 specific levels (parameter values) within the experimental space, while the other factors were constant at their baseline values.A total of 331 analytical calculations were conducted to identify the impact of each factor on the outcomes.The surface figure error RMS and mass variations with respect to the structural parameters are shown in Figure 8a and 8b, respectively.The greater the surface figure error RMS and mass variations, the more sensitive they are to changes in the structural parameters.Figure 8a shows that the outer ring radius (r 2 ) has the greatest impact on the surface figure error RMS, followed by the main rib thickness (t r ), surface plate thickness (t f ), and inner ring radius (r 1 ). Figure 8b shows that the main rib thickness (t r ) has the greatest impact on the mirror mass, followed by the surface plate thickness (t f ), back plate width (w b ), and mirror thickness (h).The remaining structural parameters, including the outer edge height (h o ), auxiliary rib height (h s ), outer edge thickness (t o ), auxiliary rib thickness (t s ), and back plate thickness (t b ), were defined as the non-major structural parameters because they cause less than 10% variation in the surface figure error RMS and mass within the experimental space.The Latin hypercube design (LHD) method was employed for non-major structural parameters, and the function can be expressed as follows: The optimization objective was to minimize the surface figure error RMS and mass of the mirror, using a sample size of 100.The level distribution of each factor is shown in Figure 9, demonstrating that the LHD method uniformly samples the factors within the design matrix.The responses of the surface figure error RMS and mass are shown in Figure 10.The preferred values of non-major structural parameters are summarized in Table 1. Figure 8a shows that the outer ring radius (r2) has the greatest impact on the surface figure error RMS, followed by the main rib thickness (tr), surface plate thickness (tf), and inner ring radius (r1).Figure 8b shows that the main rib thickness (tr) has the greatest impact on the mirror mass, followed by the surface plate thickness (tf), back plate width (wb), and mirror thickness (h).The remaining structural parameters, including the outer edge height (ho), auxiliary rib height (hs), outer edge thickness (to), auxiliary rib thickness (ts), and back plate thickness (tb), were defined as the non-major structural parameters because they cause less than 10% variation in the surface figure error RMS and mass within the experimental space.The Latin hypercube design (LHD) method was employed for non-major structural parameters, and the function can be expressed as follows: The optimization objective was to minimize the surface figure error RMS and mass of the mirror, using a sample size of 100.The level distribution of each factor is shown in Figure 9, demonstrating that the LHD method uniformly samples the factors within the design matrix.The responses of the surface figure error RMS and mass are shown in Figure 10.The preferred values of non-major structural parameters are summarized in Table 1.After completing the design of non-major structural parameters, further optimization was required for the six major structural parameters, including the outer ring radius (r 2 ), main rib thickness (t r ), surface plate thickness (t f ), inner ring radius (r 1 ), mirror thickness (h), and back plate width (w b ).The mass of the 1/6 mirror was limited to 21 kg.The optimization function can be expressed as follows: The optimization objective was to minimize the surface figure error RMS of the mirror and reduce its mass.Various parameters are interrelated and jointly affect the optimization result.The Non-dominated Sorting Genetic Algorithm-II(NSGA-II) was adopted for its excellent search performance.After completing the design of non-major structural parameters, further optimization was required for the six major structural parameters, including the outer ring radius The parameters for the NSGA-II are listed in Table 2.After 241 iterations, an optimal solution with a surface figure error RMS value of 11.1 nm and a mass of 19.8 kg was obtained.The optimization processes for the surface figure error RMS and mass are shown in Figure 11.Histograms of the major structural parameters are shown in Figure 12.The horizontal axis represents the range of parameter values, while the vertical axis indicates the number of times each value was selected during the optimization process.The convergence of the optimization objective is more likely the more a parameter value was selected.The optimization values of the major structural parameters are summarized in Table 3.
Table 3. lected.The optimization values of the major structural parameters are summarized in Table 3.

Results and Modal Test of the 2 m SiC Mirror Assembly
A finite element model of the 2 m SiC mirror was developed based on the optimization results.The mirror was supported by an axial whiffletree structure and a radial circulararch flexible structure.The 18 supports of the whiffletree were connected to the mirror using flexible rods, limiting the z-axis degrees of freedom.The circular-arc flexible structure located in the hole of the mirror was a special flexible hinge with radial degrees of freedom, comprising 12 identical flexible curved beams symmetrical in the direction of rotation [17].Each flexible curved beam exhibited tangential stiffness and radial flexibility, with all 12 curved beams collaboratively limiting the displacement of the mirror in both the x-and y-directions.A gravitational load was applied to the model.The surface deformation of the mirror when the optical axis of the mirror was vertical is shown in Figure 13a, with a surface peak-valley (PV) value of 56 nm and a surface figure error RMS value of 11.3 nm.The surface deformation of the mirror when the optical axis of the mirror was horizontal is shown in Figure 13b, with an overall tilt of 0.6".The tilt of the mirror represents a rigid body displacement, which does not impact its surface figure error RMS.Consequently, the surface deformation, free of tilt error, is shown in Figure 13c, with a PV value of 115 nm and a surface figure error RMS value of 28.8 nm.The tilt of the mirror impacts the alignment of the optical path, which is compensated by the Stewart platform of a secondary mirror.Modal tests were conducted on the mirror assembly using the hammering method [18].The mirror assembly was positioned on three identical support blocks, and two high-precision tri-axial accelerometers were fixed perpendicular one to another on the outer edge of the mirror.The test only achieved the eigen frequency of the mirror assembly, without its mode shape, owing to the limited number of accelerometers.The finite element model and modal test of the mirror assembly are shown in Figure 14b and Figure 14c, respectively.The simulation and test results are shown in Figure 15, indicating substantial agreement between the simulation and experimental results.The largest Modal tests were conducted on the mirror assembly using the hammering method [18].The mirror assembly was positioned on three identical support blocks, and two highprecision tri-axial accelerometers were fixed perpendicular one to another on the outer edge of the mirror.The test only achieved the eigen frequency of the mirror assembly, without its mode shape, owing to the limited number of accelerometers.The finite element model and modal test of the mirror assembly are shown in Figure 14b and Figure 14c, respectively.The simulation and test results are shown in Figure 15, indicating substantial agreement between the simulation and experimental results.The largest discrepancy is observed between the test and simulation results for the first-order resonant frequency.The test result for the first-order resonant frequency was 71.22 Hz, with a damping ratio of 1.12%.In contrast, the simulation result for the first-order resonant frequency was 76.5 Hz, exhibiting a relative frequency error of 7.4%.The modal test results indirectly confirm the validity of the optimization method.Modal tests were conducted on the mirror assembly using the hammering method [18].The mirror assembly was positioned on three identical support blocks, and two high-precision tri-axial accelerometers were fixed perpendicular one to another on the outer edge of the mirror.The test only achieved the eigen frequency of the mirror assembly, without its mode shape, owing to the limited number of accelerometers.The finite element model and modal test of the mirror assembly are shown in Figure 14b and Figure 14c, respectively.The simulation and test results are shown in Figure 15, indicating substantial agreement between the simulation and experimental results.The largest discrepancy is observed between the test and simulation results for the first-order resonant frequency.The test result for the first-order resonant frequency was 71.22 Hz, with a damping ratio of 1.12%.In contrast, the simulation result for the first-order resonant frequency was 76.5 Hz, exhibiting a relative frequency error of 7.4%.The modal test results indirectly confirm the validity of the optimization method.

Discussion
Lightweight mirrors not only reduce their own weight but also reduce the weight and cost of the entire system.Space-based mirrors thus offer a higher lightweight rate owing to launch weight constraints.For instance, the areal density of the 3.5 m Herschel PM is 25 kg/m² [11] and that of the 2 m ultralight space mirror is 34 kg/m² [15].By contrast, the areal densities of the 2 m and 4 m SiC PM produced by CIOMP for ground-based telescopes are 105 kg/m² and 120 kg/m², respectively [6,12].Compared with space telescope mirrors, the lightweight rate of mirrors for ground-based telescopes needs further enhancement.This study focused on the capability of using SiC in intricate structures and proposed a 2 m SiC lightweight mirror with a lotus back structure, ideal for ground-based telescopes.The areal density of the optimized 2 m SiC mirror is 38.7 kg/m 2 , which is significantly lower than that of the SiC mirrors in the aforementioned ground-based telescopes.

Discussion
Lightweight mirrors not only reduce their own weight but also reduce the weight and cost of the entire system.Space-based mirrors thus offer a higher lightweight rate owing to launch weight constraints.For instance, the areal density of the 3.5 m Herschel PM is 25 kg/m² [11] and that of the 2 m ultralight space mirror is 34 kg/m² [15].By contrast, the areal densities of the 2 m and 4 m SiC PM produced by CIOMP for ground-based telescopes are 105 kg/m² and 120 kg/m², respectively [6,12].Compared with space telescope mirrors, the lightweight rate of mirrors for ground-based telescopes needs further enhancement.
This study focused on the capability of using SiC in intricate structures and proposed a 2 m SiC lightweight mirror with a lotus back structure, ideal for ground-based telescopes.The areal density of the optimized 2 m SiC mirror is 38.7 kg/m 2 , which is significantly lower than that of the SiC mirrors in the aforementioned ground-based telescopes.
The methods for determining the number of axial supports and mirror thickness can be applied to mirrors of various materials in ground-based telescopes.The optimization method can be applied to SiC mirror designs for ground-based telescopes and can also serve as a reference for designing space-based SiC mirrors.However, the proposed design and optimization methods for lightweight structures are subject to certain limitations for mirrors made with other materials.

Conclusions
This study focused primarily on the lightweight optimization of a 2 m SiC mirror for ground-based telescopes.The mirror thickness and number of axial supports were preliminarily selected by analyzing the gravitational deformation of the mirror.The back structure of the mirror was designed as a lotus pattern owing to its improved rigidity.Numerous structural parameters were categorized into major and non-major structural parameters via a sensitivity analysis of the surface figure error RMS and mass.The nonmajor structural parameters were optimized using the LHD method, whereas the major structural parameters were optimized using the NSGA-II model.The optimized 2 m SiC lightweight mirror weighed 119 kg, with an areal density of 38.7 kg/m 2 , which was more than 50% lighter than traditional passive Zerodur mirrors of the same size.The proposed structure and optimization method can serve as a reference for the design of SiC mirrors.

Figure 1 .
Figure 1.Effect of supports on gravitational deformation of the mirror.

Figure 1 .
Figure 1.Effect of supports on gravitational deformation of the mirror.

Figure 2 .
Figure 2. Effect of the number of axial supports and mirror thickness on deformation of 2 m mirrors.

Figure 2 .
Figure 2. Effect of the number of axial supports and mirror thickness on deformation of 2 m mirrors.

Figure 3 .
Figure 3. (a) The triangular pa<ern on the back of the 2 m SiC mirror.(b) The curve of surface figure error RMS versus distance k.(c) The deformation cloud map of the mirror when distance k is 200 mm.

Figure 4 .
Figure 4. (a) The annular pa<ern on the back of the 2 m SiC mirror.(b) The curve of the surface figure error RMS versus the inner ring radius Rin for different outer ring radii Rout.(c) The deformation cloud map of the mirror when Rin and Rout are 380 and 840 mm, respectively.

Figure 3 .
Figure 3. (a) The triangular pattern on the back of the 2 m SiC mirror.(b) The curve of surface figure error RMS versus distance k.(c) The deformation cloud map of the mirror when distance k is 200 mm.

Figure 3 .
Figure 3. (a) The triangular pa<ern on the back of the 2 m SiC mirror.(b) The curve of surface figure error RMS versus distance k.(c) The deformation cloud map of the mirror when distance k is 200 mm.

Figure 4 .
Figure 4. (a) The annular pa<ern on the back of the 2 m SiC mirror.(b) The curve of the surface figure error RMS versus the inner ring radius Rin for different outer ring radii Rout.(c) The deformation cloud map of the mirror when Rin and Rout are 380 and 840 mm, respectively.

Figure 4 .
Figure 4. (a) The annular pattern on the back of the 2 m SiC mirror.(b) The curve of the surface figure error RMS versus the inner ring radius R in for different outer ring radii R out .(c) The deformation cloud map of the mirror when R in and R out are 380 and 840 mm, respectively.

Figure 6 .
Figure 6.(a) The topology optimization result of the 2 m SiC mirror.(b) The final structure of the 2 m SiC mirror with a lotus pa<ern.

Figure 5 .
Figure 5. (a) A lotus pattern on the back of the 2 m SiC mirror.(b) The curve of the surface figure error RMS versus the inner ring radius R in for different outer ring radii R out .(c) The deformation cloud map of the mirror when R in and R out are 370 and 830 mm, respectively.

Figure 5 .
Figure 5. (a) A lotus pa<ern on the back of the 2 m SiC mirror.(b) The curve of the surface figure error RMS versus the inner ring radius Rin for different outer ring radii Rout.(c) The deformation cloud map of the mirror when Rin and Rout are 370 and 830 mm, respectively.

Figure 6 .
Figure 6.(a) The topology optimization result of the 2 m SiC mirror.(b) The final structure of the 2 m SiC mirror with a lotus pa<ern.

Figure 6 .
Figure 6.(a) The topology optimization result of the 2 m SiC mirror.(b) The final structure of the 2 m SiC mirror with a lotus pattern.

Figure 7 .
Figure 7. Flowchart for the structural parameter optimization of the 2 m SiC mirror.

Figure 7 .
Figure 7. Flowchart for the structural parameter optimization of the 2 m SiC mirror.

Figure 8 .
Figure 8.(a) Curves of surface figure error RMS versus the structural parameters of the 2 m SiC mirror.(b) Curves of mass versus the structural parameters of the 2 m SiC mirror.

Figure 10 .
Figure 10.(a) Response of the surface figure error RMS.(b) Response of the mass.

Figure 10 .
Figure 10.(a) Response of the surface figure error RMS.(b) Response of the mass.

Table 2 .Figure 11 .Figure 11 .
Figure 11.Optimization process of objectives.Red indicated infeasible, black indicated feasible, blue indicated feasible-tie, and green indicated feasible be<er.(a) Optimization process of the surface figure error RMS.(b) Optimization process of the mass.

Table 2 .Figure 11 .Figure 12 .
Figure 11.Optimization process of objectives.Red indicated infeasible, black indicated feasible, blue indicated feasible-tie, and green indicated feasible be<er.(a) Optimization process of the surface figure error RMS.(b) Optimization process of the mass.

Figure 13 .
Figure 13.Surface deformation of the 2 m SiC mirror (a) The optical axis is in a vertical state.(b) The optical axis is in a horizontal state.(c) After compensating for the tilt of 0.6".

Figure 13 .
Figure 13.Surface deformation of the 2 m SiC mirror (a) The optical axis is in a vertical state.(b) The optical axis is in a horizontal state.(c) After compensating for the tilt of 0.6".

Figure 14 .
Figure 14.(a) Axial and radial support structures for the 2 m SiC mirror.(b) Finite element model of the 2 m SiC mirror assembly.(c) The modal test of the mirror assembly.Figure 14.(a) Axial and radial support structures for the 2 m SiC mirror.(b) Finite element model of the 2 m SiC mirror assembly.(c) The modal test of the mirror assembly.

Figure 14 .
Figure 14.(a) Axial and radial support structures for the 2 m SiC mirror.(b) Finite element model of the 2 m SiC mirror assembly.(c) The modal test of the mirror assembly.Figure 14.(a) Axial and radial support structures for the 2 m SiC mirror.(b) Finite element model of the 2 m SiC mirror assembly.(c) The modal test of the mirror assembly.Photonics 2024, 11, x FOR PEER REVIEW 13 of 14

Table 1 .
Values of non-major structural parameters of the 2 m SiC mirror.

Table 1 .
Values of non-major structural parameters of the 2 m SiC mirror.

Table 1 .
Values of non-major structural parameters of the 2 m SiC mirror.

Table 3 .
Values of major structural parameters of the 2 m SiC mirror.

Table 3 .
Values of major structural parameters of the 2 m SiC mirror.