Lightweight Design of Multi-Objective Topology for a Large ... - MDPI

applied sciences Article

Lightweight Design of Multi-Objective Topology for a Large-Aperture Space Mirror Yanjun Qu 1,2 , Yanru Jiang 1,2 , Liangjie Feng 1 , Xupeng Li 1,2 , Bei Liu 1,2 and Wei Wang 1, * 1

2

*

Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, China; [email protected] (Y.Q.); [email protected] (Y.J.); [email protected] (L.F.); [email protected] (X.L.); [email protected] (B.L.) University of Chinese Academy of Sciences, Beijing 100049, China Correspondence: [email protected]; Tel.: +86-29-8888-7672

Received: 9 October 2018; Accepted: 2 November 2018; Published: 15 November 2018







Abstract: For a large-aperture space telescope, one of the key techniques is the method for designing the lightweight primary mirror assembly (PMA). In order to minimize the mirror surface error under axial gravity, lateral gravity, and polishing pressure at the same time, a method for topology optimization with multi-objective function combined with parametric optimization is introduced in this paper. The weighted compliance minimum is selected as the objective function to maximum the mirror structural stiffness. Then sensitivity analysis method and size optimization are used to determine the mirror structure parameters. Compared with two types of commonly used lightweight configurations, the new configuration design shows obvious superiority. In addition, the surface figure root mean square (RMS) of the mirror mounted by given bipod flexure (BF) under 1 g lateral gravity is minimized only with a value of 3.58 nm, which proves the effectiveness of the design method proposed in this paper. Keywords: multi-objective topology optimization; sensitivity analysis; space mirror; lightweight structure

1. Introduction With the rapid development of aerospace technology, high image resolutions and large fields are in high demand for space cameras. Therefore, large aperture mirror optical systems with these advantages have been used extensively in space cameras. The space mirror must satisfy two critical requirements: (1) to ensure the image quality of the telescope system, the surface wavefront RMS aberration of the mirror is required to be better than λ/60 (λ = 632.8 nm); (2) to decrease the gravity loads and reduce the expensive launch cost, the mass of the mirror must be no less than 14 kg [1,2]. These two conflicting requirements have brought great challenges to the design of the lightweight mirror. A lot of research on the optimization of lightweight mirrors has been reported. Nelson et al. [3] and Wan et al. [4] developed the theoretical analysis based on thin plate theory to calculate mirror deformation on multiple back supports. Kihm et al. [5] presented an improved optimum design process for a lightweight 1 m diameter Zerodur mirror, and they implemented a multi-objective genetic algorithm to optimize the mirror design. Chen et al. [6] presented an integrated optomechanical design method and optimized the parameters of the mirror and bipod flexure support. In these works, the main research content is parametric optimization, and the final configuration is limited by the initial geometry of the mirror. However, configuration change in improving optical performance is more effective than changes in shape and parameters. The topology optimization method is an effective approach to acquire new configuration for preliminary lightweight designs of the mirror. Although the topological mirror has irregular shape, it will be widely used in the future with the rapid develop of the 3D printing in optical manufacturing, which is good at machining complex Appl. Sci. 2018, 8, 2259; doi:10.3390/app8112259

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shaped parts [7,8]. Park et al. [9] presented a topology optimization with the objective function of RMS surface deformation errors and subjected to different weight constraints and self-weight loading. Liu et al. [10] introduced a topology optimization model with casting constraint considered for designing the distribution and the heights of the ribs, and the compliance of the mirror was chosen as the design objective function. Although some improvements have been made in these studies, single objective function is used in topology optimization algorithm, which is difficult to ensure the optical performance under multiple working conditions. In addition, the lightweight result obtained by topology optimization cannot be used directly in practical application. Thus, a parametric optimization is needed for the detail configuration design on the topology optimization result. In this paper, a method of the topology optimization with multi-objective function is introduced for lightweight SiC primary mirror design using OptiStruct, which was launched by Altair and integrated into HyperWorks, a comprehensive computer aided engineering (CAE) simulation platform. In order to decrease the mirror surface error under axial gravity, lateral gravity, and polishing pressure at the same time, we combine the multiple objectives into a single objective function by assigning weighting factors to the objectives. In the topology optimization model, the RMS deformation error under each loading cases is replaced by the corresponding structural compliance for reducing the calculated amount, and the weighted compliance minimum is selected as the objective function. Under the constraints of the volume fraction, nodal displacement, draft direction, and cyclic symmetry condition, a new mirror configuration is proposed. To further reduce the local deformation of the mirror surface caused by gravity and polishing pressure, we add auxiliary ribs to the reestablishment model referenced to the topology optimization results. Then size optimization and sensitivity analysis method are used to determine the mirror structure parameters using ANSYS Workbench. The new configuration design mirror is compared with triangular configuration mirror and hexagonal configuration mirror in stiffness and optical performance. The results show the optimized mirror combines the advantages of these two classic lightweight mirrors. And the surface error of the mirror mounted by three given BFs under 1 g radial gravity is analyzed. We investigate the effects of non-coplanar distance L on the surface figure RMS and coefficient of astigmatism. When L is 3.5 mm, the surface error of the mirror takes the minimum value and almost has no astigmatism. The results show the mirror design method based on topology optimization with multi-objective function combined size optimization can satisfy the required optical performance and overcome the limitation of parametric studies simultaneously within the operation conditions. 2. Lightweight Design of the Primary Mirror 2.1. Fundamental Concepts Topology optimization is to find the best material distribution or force transmission path in a given design space, so that the lightweight design can be obtained under various constraints. Topology optimization of the primary mirror structure based on solid isotropic material with penalization (SIMP) was applied in this study using the software OptiStruct [11]. Its basic idea is to assume a material element that has variable relative density ρ between 0 and 1. When ρ is 1 or near 1, it means the element is required and need to be retained; when ρ is 0 or near 0, it says the element is not filled with material and needs to be removed. In order to avoid too many middle density elements, introducing penalty factor to enable continuous variables approach to 0 or 1 approximately, and the material properties of the element can be defined by Ei = (ρ0 + (ρ1 − ρ0 )ρiP ) E0 ,

(1)

where ρi is the material relative density and Ei is the relative elasticity modulus of the i-th element, respectively. In the commercial software OptiStruct, the lower bound ρ0 takes the value of 0.01, that is to avoid the singularity of stiffness matrix in the process of finite element analysis, and the upper

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bound ρ1 takes the value of 1. E0 is the inherent elasticity module of a given isotropic material, and P is the penalization power. 2.2. Formulation of the Optimization Problem The purpose of this research is to obtain the optimum configuration of the mirror, which is to minimize the mirror surface distortion while reducing the weight of the mirror. The mirror surface deviation is mainly caused by the self-weight of the mirror. In addition, the polishing pressure applied in the grinding manufacturing process causes serious local deformation and has a great impact on the optical performance of the mirror. In order to reach the minimum values of mirror surface shape error caused by lateral gravity, axial gravity, and polishing pressure at the same time, the multi-objective optimization problem is selected for the design method. The practical way to deal with the multi-objective optimization problem is to combine the multiple objectives into a single objective function, which is easy to be solved by assigning weighting factors to the objectives, or choosing the most important one as the objective function and treating the other objectives as the constraint conditions [12]. The wavefront RMS aberration is a common way to evaluate the quality of the mirror surface, which is defined as the root mean square of the distances between the deformed mirror surface nodes and the fitted surface. It can be expressed as v u 2 u n 1 RMS = t ∑ ( xi − x ) n i =1

(2)

where xi is the distance from the i-th node to the fitting surface after deformation, and x is the average distance between all nodes and the fitting surface. However, the value of RMS cannot be used as the optimization objective directly in the existing commercial finite element software. Considering time consumption and computational efficiency, the RMS deformation error under each loading cases is replaced by the corresponding structural compliance. The minimum structural weighted compliance is selected as the objective function to minimize the displacements of the surface nodes, which has similar effects to decrease the RMS value [10]. The weighted structural compliance can be obtained by the formula as follows CW =

∑ Wj Cj = ∑ Wj uTj f j ,

(3)

where CW is the optimization objective weighted by Cj , which represents the compliance function of the mirror surface under the j-th load subcase, Wj is weighting factors, uj is the displacement vector, and fj is the load vector. The structural configuration can be described by the material relative density ρi , and the optimization model can be expressed as follows: Find X = (ρ1 , ρ2 , ..., ρ N ) T ,

(4)

Minimize CW = γCL + ηC A + µCP ,

(5)

optimization objective subject to n

g( X ) =

∑ Vi − αV ∗ ≤ 0,

(6)

i =1

f L = ku L , f A = ku A , f P = ku P ,

(7)

DL − D1 ≤ 0, D A − D2 ≤ 0,

(8)

ρ0 ≤ ρ i ≤ ρ1 .

(9)

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In this model, CW is the weighted compliance function, and CP is the compliance function under polishing pressure, CL and CA are the compliance functions of the mirror face under lateral gravity and axial gravity, respectively. γ, η, and µ are weighting factors. V* is the volume of the initial structure, Vi is the ith element volume, and α is the upper bound of the allowed volume fraction. fL , fA and fP are the lateral gravity, axial gravity and polishing pressure load vectors. uL , uA , and uP are the corresponding displacement vectors. DL is the nodal displacement of the mirror surface under lateral gravity, and D1 is the upper bound of lateral nodal displacement. DA is the nodal displacement of the mirror under axial gravity, and D2 is the upper bound of axial nodal displacement. The sensitivities of the objective function CW with respect ρi can be obtained by the adjoint method ∂CW ∂f ∂K ∂f ∂K ∂f ∂K = γ(u L T L − u L T u L ) + η (u A T A − u A T u ) + µ(u P T P − u P T u P ). ∂ρi ∂ρi ∂ρi ∂ρi ∂ρi A ∂ρi ∂ρi

(10)

The optimal criterion adopted in this paper is based on the Kuhn-Tuker (KT) condition, and the topology-optimized Lagrange function is constructed by introducing the KT condition to be satisfied in mathematics. The Lagrange multiplier λ0 , λ1 j , λ2 i , λ3 i is introduced to derive the Lagrange function of the continuum topology optimization problem with minimized compliance under volume constraints: L=

n 3 n n ∂CW + λ0 ( ∑ ρi Vi − αV ∗ ) + ∑ λ1 j (ku j − f j ) + ∑ λ2 i (ρ0 − ρi ) + ∑ λ3 i (ρi − ρ1 ). ∂ρi i =1 j =1 i i

(11)

The condition that the design variable takes the optimal value is ∂ku j ∂CW ∂L = + λ0 Vi + λ1 j + λ2 i + λ3 i = 0. ∂ρi ∂ρi ∂ρi

(12)

In this paper, the importance of the CL , CA , and CP are equivalent, so the weighting factors γ, η, and µ take the same value of 0.33. The initial mass of the mirror is 42.6 kg, we take a more stringent lightweight ratio of 70% compared with the demand ratio of 67.1%, so the volume fraction α takes 0.3. The design specification requires upper limits 25 nm for D1 and 160 nm for D2 [2]. The shape of the polishing plate is like pentagram and its outer diameter is 160 mm, and the static polishing load on the whole surface of the mirror is 0.2 kPa [9,10]. Design parameters in topology optimization are shown in Table 1. Table 1. Design parameters in topology optimization. γ, η, µ

α

D1

D2

P

0.33

0.3

25 nm

160 nm

0.2 kPa

γ, η, and µ are weighting factors, α is volume fraction, D1 is the upper bound of lateral nodal displacement and D2 is the upper bound of axial nodal displacement, P is the pressure of polishing load.

3. Application Examples of the Topology Optimization Method 3.1. Initial Design of the Primary Mirror For this study, we considered a large aperture primary mirror with a center hole from a Cassegrain space telescope. The material of the mirror is SiC with the mechanical properties shown in Table 2. E0 is Young’s modulus; µ is Poisson’s ratio; and ρ is density. The mirror is mounted by three BFs, as indicated in Figure 1. Table 2. Parameters and material properties of the primary mirror. Outside Diameter 610 mm

Inside Diameter 86 mm

SiC

Outer Edge Thickness 70 mm

E0

µ

ρ

280 GPa

0.17

2800 kg/m3

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Figure 1. Primary mirror mounted on on bipod flexures. Figure 1. Primary mirror mounted bipod flexures.

The finite finite element elementmodel modelisisestablished establishedinin the commercial software HyperMesh. To improve the commercial software HyperMesh. To improve the the computational efficiency of topology optimization and ensure the rotation characteristics, 1/3 computational efficiency of topology optimization and ensure the rotation characteristics, a 1/3 afinite finite element method (FEM) model cyclic symmetry conditionisisproposed. proposed. Considering Considering the element method (FEM) model withwith cyclic symmetry condition manufacturability, manufacturability, the the material material is is removed along the height direction of the mirror to obtain an open back configuration. This is achieved by setting the draw direction configuration. This direction in optimization-topology optimization-topology panel in OptiStruct. The vector vector from from an an anchor anchor node node on on the mirror surface surface to its projection node on the back OptiStruct. The surface determinesthe thedirection directionofofthe thedraft, draft,and and elements also established in every along surface determines elements areare also established in every rowrow along the the direction. The distribution best distribution the material is described through calculating the relative draftdraft direction. The best of theof material is described through calculating the relative density density of the elements. 2 shows 2/3model FEM model the primary mirror. The FEM model of of the elements. Figure Figure 2 shows the 2/3the FEM of theof primary mirror. The FEM model of the the primary mirror is divided design non-design area. non-design area distinguished primary mirror is divided intointo design areaarea andand non-design area. TheThe non-design area distinguished in in color includes front mirror surface, outer edge, inner edge, wherein initial thickness of redred color includes front mirror surface, outer edge, andand inner edge, wherein the the initial thickness of the the front mirror surface 4 mm, outer edge inner edge thickness 5 mm. remaining part front mirror surface is 4 ismm, the the outer edge andand inner edge thickness areare 5 mm. TheThe remaining part of of mirror distinguished in green is design The nodes thearea fixed area are constrained thethe mirror distinguished in green colorcolor is design area. area. The nodes of the of fixed are constrained during during the optimization There are elements 116,520 elements andnodes. 129,912 nodes. the optimization process.process. There are 116,520 and 129,912

Figure 2. The FEM model of the primary mirror. 2/32/3 FEM model of the primary mirror. Figure 2. The

3.2. Topology Optimization 3.2. Topology Optimization Results Results The topology optimization optimization results results tend tend to to converge converge through through 53 53 iterations, iterations, as as shown shown in in Figure Figure 3. 3. The topology Figure 4 shows the optimal material distribution of the primary mirror. The material in the red area Figure 4 shows the optimal material distribution of the primary mirror. The material in the red area makes to to structural stiffness and and needsneeds to be retained or enhanced, and the material makes aagreat greatcontribution contribution structural stiffness to be retained or enhanced, and the in the blue area has little effect on structural stiffness and should be removed. From the topology material in the blue area has little effect on structural stiffness and should be removed. From the optimization results, we can infer thatinfer the elements on the lines between the threethe fixed surfaces topology optimization results, we can that the elements on the lines between three fixed should be preserved, and the inner ring and outer ring should be connected with the elements between surfaces should be preserved, and the inner ring and outer ring should be connected with the the three fixed surfaces. Thefixed elements withThe highelements relative with density form a triangular configuration, elements between the three surfaces. high relative density net form a triangular which is generally considered to possess a high stiffness. net configuration, which is generally considered to possess a high stiffness.

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Figure Iteration history oftopology topology optimization. Figure 3.3.Iteration history of optimization. Figure 3. Iteration history of topology optimization. Figure 3. Iteration history of topology optimization.

Figure Optimal material distribution ofthe the primary mirror. Figure 4.4.Optimal material distribution of primary mirror. Figure 4. Optimal material distribution of the primary mirror. Figure 4. Optimal material distribution of the primary mirror.

tothe Accordingto therelative relativedensities, densities,the theelements elementsretained retainedin inthe thetopology topologyoptimization optimizationresult resultare are According According to the relative densities, the elements retained in the topology optimization result are divided into three types of ribs, as shown in Figure 5. The initial design based on topology optimization into three three types types of of ribs, ribs, as as shown shown in in Figure Figure 5.5. The The initial initial design design based based on on topology topology divided into divided into three types of ribs, as shown in Figure 5. The initial design based on topology is shown in Figure 6a.in has led ushas to led obtain a obtain conceptual optimal optimization shown inAlthough Figure6a. 6a.topology Althoughoptimization topologyoptimization optimization has ledus usto to obtainaaconceptual conceptual optimization isisshown Figure Although topology optimization is shown in Figure 6a. Although topology optimization has led us to obtain a conceptual design, design, itdesign, coulditit not be implemented in engineering directlydirectly because of the unequal quilting effects optimal could notbe beimplemented implemented inengineering engineering directly because ofthe theunequal unequal quilting optimal could not in because of quilting optimal design, it could not be implemented in engineering directly because of the unequal quilting caused by the unevenly distributed ribs during mirror fabrication [13]. effectscaused causedby bythe theunevenly unevenlydistributed distributedribs ribsduring duringmirror mirrorfabrication fabrication[13]. [13]. effects effects caused by the unevenly distributed ribs during mirror fabrication [13].

Figure 5. Material distribution under different relative densities. Figure5.5.Material Materialdistribution distributionunder underdifferent differentrelative relativedensities. densities. Figure Figure 5. Material distribution under different relative densities.

In order to reduce the local deformation, we add auxiliary ribs into the topology optimization. In order toreduce reducelightening thelocal localdeformation, deformation, we addauxiliary auxiliary ribsinto into thefor topology optimization. order to the add ribs the topology optimization. AndIn we make some holes in thewe material concentration area further weight loss. In order to reduce the local deformation, we add auxiliary ribs into the topology optimization. Andspecific wemake make somecriteria lightening holesthe inthe thematerial materialconcentration concentrationarea areafor forfurther furtherweight weightloss. loss.The The And we some lightening holes in The design can refer following: And we make some lightening holes in the material concentration area for further weight loss. The specificdesign designcriteria criteriacan canrefer referthe thefollowing: following: specific specific design criteria ribs can pass refer through the following: (1) The additional the center of the areas without material filling to reduce the (1) The The additional ribs pass through the center of of the the areas areas without without material material filling fillingto to reduce reduce the the (1) additional ribs pass through the center local deformation; (1) The additional ribs pass through the center of the areas without material filling to reduce the localdeformation; deformation; local local deformation;

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(2) The additional ribs pass through the intersection of the main ribs to shorten the transmission (2) The additional ribs pass through the intersection of the main ribs to shorten the transmission path of the force; path of the force; (3) The Thearrangement arrangementof ofthe theadditional additionalribs ribsmust mustnot notaffect affectthe thecyclic cyclicsymmetry symmetryof of the the mirror; mirror; (3) (4) Increasing Increasinglightening lighteningholes holesin inthe theareas areaswhere wherethe thematerial material is is excessively excessively concentrated. concentrated. (4) Then we we construct model using computer-aided design (CAD) software, as shown Then constructthe theverification verification model using computer-aided design (CAD) software, as in Figure 6b. shown in Figure 6b.

Figure 6. 6. (a) (a) Initial Initial design design based based on on topology topology optimization; optimization; (b) (b) Modified Modified rib rib design design considering considering Figure quilting effects. quilting effects.

3.3. Sensitive Analysis and Size Optimization 3.3. Sensitive Analysis and Size Optimization Topology optimization is the design method in the structural conceptual design stage, so the Topology optimization is the design method in the structural conceptual design stage, so the reestablishment of the three-dimensional model has large subjective factors in the dimensions of each reestablishment of the three-dimensional model has large subjective factors in the dimensions of each part. We perform design sensitivity analysis and size optimization on the final model of the primary part. We perform design sensitivity analysis and size optimization on the final model of the primary mirror to approach the optimal thicknesses of each part. From Figure 3, it can be found that the mirror to approach the optimal thicknesses of each part. From Figure 3, it can be found that the compliance function C under the axial gravity is the main impact factor, so only RMS under axial compliance function CAAunder the axial gravity is the main impact factor, so only RMS under axial gravity (RMS ) is selected as the objective function in sensitivity analysis and size optimization. gravity (RMSAA) is selected as the objective function in sensitivity analysis and size optimization. Initially, eight dimensional variables (DVs) are selected, and the initial specific design parameters Initially, eight dimensional variables (DVs) are selected, and the initial specific design of the mirror are shown in Table 3. To reduce the number of DVs and computation, sensitivities of parameters of the mirror are shown in Table 3. To reduce the number of DVs and computation, the RMSA with respect to the unit change of dimensional parameters of the mirror are analyzed with sensitivities of the RMSA with respect to the unit change of dimensional parameters of the mirror are FEM and the results are shown in Figure 7. According to the results of the sensitivity analysis shown analyzed with FEM and the results are shown in Figure 7. According to the results of the sensitivity in Figure 7, five DVs are selected for further optimization, namely, DV1, DV2, DV4, DV5, and DV6. analysis shown in Figure 7, five DVs are selected for further optimization, namely, DV1, DV2, DV4, Table 4 summarizes the lower bound, upper bound, initial values, and optimum values of the DVs for DV5, and DV6. Table 4 summarizes the lower bound, upper bound, initial values, and optimum primary mirror size optimization. For further lightweight, the dimension parameters of DV3 and DV8, values of the DVs for primary mirror size optimization. For further lightweight, the dimension which have low sensitivities, are reduced to 4 and 3 mm, respectively, and we delete the cylindrical parameters of DV3 and DV8, which have low sensitivities, are reduced to 4 and 3 mm, respectively, structure for that DV7 which has the lowest sensitivity. and we delete the cylindrical structure for that DV7 which has the lowest sensitivity. Table 3. DVs of the Φ650 mm SiC mirror. Table 3. DVs of the Φ650 mm SiC mirror. Design Variable

Design Variable DV1 DV1 DV2 DV2 DV3 DV3 DV4 DV4 DV5 DV5 DV6 DV6 DV7 DV8 DV7 DV8

Term

Term Front platethickness thickness Front plate Outer ring thickness Outer ring thickness Inner ring thickness InnerRib1 ringthickness thickness Rib1 thickness Rib2 thickness Rib3 thickness Rib2 thickness Cylindrical structure radius Rib3 thickness Auxiliary rib thickness Cylindrical structure radius Auxiliary rib thickness

Value

Value

5 mm 5 mm 5 mm 5 mm 5 mm 5 mm 15 mm 15 mm 8 mm 6 mm 8 mm 162 mm 6 mm 4 mm

162 mm 4 mm

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Figure 7. Sensitivity analysis using the FEM on primary mirror. Table 4. Lower bound, bound,analysis and optimum value of the DVs for mirror. size optimization. Figure upper 7. Sensitivity using the FEM on primary Figure 7. Sensitivity analysis using the FEM on primary mirror. Design Lower Table 4. Lower bound, upper bound, and optimumUpper value of the DVsOptimum for size optimization. Variable Bound Value Table 4. Lower bound, upper bound, and optimumBound value of the DVs for size optimization. Design Variable Lower Bound Upper Bound Optimum Value

DV1 4 mm 8 mm Design Lower Upper DV1 44 mm 88 mm DV2 mm mm Variable Bound Bound DV2 4 mm 8 mm DV4 12 mm 18 mm DV4 12 mm 18 mm DV1 4 mm 8 mm DV5 4 mm mm 10 mm mm DV5 4 10 DV2 4 mm 8 mm DV6 mm 10 mm mm DV6 44 mm 10 DV4 12 mm 18 mm DV5 4 mm 10 mm 4. of the New Configuration 4. Performance Performance Evaluation Evaluation of the New Configuration DV6 4 mm 10 mm

4 mm 4Optimum mm 5.6 mm 5.6 Value mm 16.4 mm 16.44mm mm 5.2 mm 5.25.6 mm mm 4 mm mm 16.4 mm 5.2 mm 4 mm

4.1. Comparison of Optical Performance of Three Naked Mirrors 4. Performance Evaluation of the New Configuration In order to confirm the optical performance of the optimized mirror, we construct two classic lightweight formsof of mirrors with hexagonal lightening lightening holes and triangular triangular lightening lightening holes, holes, 4.1. Comparison Optical Performance of Three Naked Mirrorsholes respectively, whichareare commonly and shown Figure 8a,b The triangular respectively, which commonly used used and shown in Figurein8a,b [14–17]. The[14–17]. triangular configuration In order to confirm the optical performance of the optimized mirror, we construct two classic configuration lightweight mirrorconsidered is generallytoconsidered to have a high lightweight ratio, and the lightweight mirror is generally have a high lightweight ratio, and the hexagonal lightweight forms of mirrors with hexagonal lightening holes and triangular lightening holes, hexagonal configuration lightweight mirror is generally considered to possess a high stiffness configuration lightweight mirror is generally considered to possess a high stiffness [18,19]. To[18,19]. prove respectively, which are commonly used and shown in Figure 8a,b [14–17]. The triangular To the effectiveness the new lightweight configuration, two classic designs the prove effectiveness of the new of lightweight configuration, we make the we twomake classicthe designs have the same configuration lightweight mirror is generally considered to have a high lightweight ratio, and the have the same thickness in front plate, outer ring, and inner ring with the optimized mirror. thickness in front plate, outer ring, and inner ring with the optimized mirror. The diameter ofThe the hexagonal configuration lightweight mirror is generally considered to possess a high stiffness [18,19]. diameter the of inscribed circle of theislightening hole is 60 mm, which is 15 times thefront thickness of[20]. the inscribed of circle the lightening hole 60 mm, which 15 times the thickness of the surface To prove the effectiveness of the new lightweight configuration, we make the two classic designs frontthe surface [20].thickness And the of optimal thickness of the obtained by size optimization. And optimal the ribs is obtained byribs sizeisoptimization. have the same thickness in front plate, outer ring, and inner ring with the optimized mirror. The diameter of the inscribed circle of the lightening hole is 60 mm, which is 15 times the thickness of the front surface [20]. And the optimal thickness of the ribs is obtained by size optimization.

Figure 8. 8. Two Twoclassic classiclightweight lightweightmirrors. mirrors.(a) (a) Triangular configuration lightweight mirror; Figure Triangular configuration lightweight mirror; (b) (b) Hexagonal configuration lightweight mirror. Hexagonal configuration lightweight mirror.

The influences ofclassic the on under gravity (RMS ), RMS RMS under lateral lateral gravity Figure 8. Twoof lightweight mirrors. Triangular configuration lightweight mirror; (b) The influences the ribs ribs on the the RMS RMS under(a)lateral lateral gravity (RMSLL), under gravity (RMS ), and total mass (m) of the mirror are equivalent. Each of the influence is normalized by their Hexagonal configuration lightweight mirror. A (RMSA), and total mass (m) of the mirror are equivalent. Each of the influence is normalized by their The influences of the ribs on the RMS under lateral gravity (RMSL), RMS under lateral gravity (RMSA), and total mass (m) of the mirror are equivalent. Each of the influence is normalized by their

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maximumvalue valueand andthe theobjective objectivefunction functionisisobtained obtainedby byusing usingunification-object unification-objectmethod method[21], [21],ititcan can maximum beexpressed expressedasasfollows follows be

maximum value and the objective function is obtained using method [21], it can RMSAby RMSL unification-object m RMS A RMS L m FF(t()t )   . . be expressed as follows (13) (13) RMS RMS RMS MM RMS Minimize RMS A1 1 RMS L2 2 m Minimize • Minimize F (t) = • . (13) RMS1 RMS2 M where RMS RMS1,1,RMS RMS2,2,and and M Mare arethe themaximum maximum values valuesofofRMS RMSA,A,RMS RMSL,L,and andm, m,respectively. respectively.The The where where RMS1between , between RMS2 , objective and M are the maximum values of RMS m, respectively. A , isRMS L , and relationship objective function andthe thethickness thickness therib rib isshown shown Figure andthe the relationship function and ofofthe ininFigure 9,9,and The relationship between objective function and the thickness of the rib is shown in Figure 9, and the optimalthickness thicknessofofrib ribofofthe thetwo twoclassic classiclightweight lightweightmirrors mirrorsboth bothare are44mm. mm. optimal optimal thickness of rib of the two classic lightweight mirrors both are 4 mm.

Figure Figure9.9.9.The Theobjective objectivefunctions functionsconsidered consideredRMS RMS ,RMS RMS andmmof thetriangular triangularand andhexagonal hexagonal Figure The objective functions considered RMS ofofthe the triangular and hexagonal A,A,and LL,,LRMS A configurationlightweight lightweight mirror with respect the thickness the ribs configuration lightweight mirror with respect toto the thickness the ribs configuration mirror with respect to the thickness ofofof the ribs t.t.t.

Then FEM is is used to analyze the deformation of theofof mirror under under lateral gravity, axial gravity, Thenthe the FEM isused used analyze thedeformation deformation themirror mirror underlateral lateralgravity, gravity, axial Then the FEM totoanalyze the the axial and polishing pressure, Figure 10a–c illustrates the axial gravity induced deformation gravity, and polishing polishingrespectively. pressure, respectively. respectively. Figure 10a–c illustrates the axial axial gravity gravity induced gravity, and pressure, Figure 10a–c illustrates the induced of these threeof mirrors. The simulated deformation the mirror of surface is transferred into Zernike deformation ofthese thesethree three mirrors.The The simulatedof deformation ofthe themirror mirror surfaceis istransferred transferred deformation mirrors. simulated deformation surface polynomials optomechanical codes written in MATLAB, peak valley (PV),valley and RMS intoZernike Zernikethrough polynomials throughoptomechanical optomechanical codes writtenininthe MATLAB, thepeak peak valley (PV), into polynomials through codes written MATLAB, the (PV), deviation be calculated after removing bias, tilt, and defocus [22,23]. andRMS RMScan deviation canbe becalculated calculated afterthe removing the bias, tilt,and and defocus[22,23]. [22,23]. and deviation can after removing the bias, tilt, defocus

Figure Figure10. 10.Deformation Deformationof themirror mirrorsurface surfaceunder underaxial axialgravity. gravity.(a) (a)Triangular Triangularconfiguration configurationdesign; design; Figure 10. Deformation ofofthe the mirror surface under axial gravity. (a) Triangular configuration design; (b) Hexagonal configuration design; (c) New configuration design. (b)Hexagonal Hexagonalconfiguration configurationdesign; design;(c) (c)New Newconfiguration configurationdesign. design. (b)

Then we compare the new configuration design mirror with two classic mirrors. The PV values, RMS values, fundamental frequencies, mass and lightweight ratios of these three mirrors

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Then we compare the new configuration design mirror with two classic mirrors. The PV values, Then we compare the new configuration design mirror with two classic mirrors. The PV values, RMS values, fundamental frequencies, mass and lightweight ratios of these three mirrors are listed RMSSci. values, frequencies, mass and lightweight ratios of these three mirrors are10listed Appl. 2018, 8, fundamental 2259 of 13 in Table 5. The RMS values and PV values under axial gravity are much larger than that under the in Table 5. The RMS values and PV values under axial gravity are much larger than that under the lateral gravity, which is consistent with the iterative curve characteristics of topology optimization in lateral gravity, which is consistent with the iterative curve characteristics of topology optimization in Figure 3. in The hexagonal configuration has under a highaxial lightweight ratio, andlarger the triangular are listed Table 5. The RMS values andmirror PV values gravity are much than that Figure 3. The hexagonal configuration mirror has a high lightweight ratio, and the triangular configuration mirror has which superiority in optical The new design configuration under the lateral gravity, is consistent with performance. the iterative curve characteristics of topology configuration mirror has superiority in optical performance. The new design configuration lightweight mirror combines advantages of the two mirror classicalhas lightweight mirrors, and hasand a high optimization in Figure 3. Thethe hexagonal configuration a high lightweight ratio, the lightweight mirror combines the advantages of the two classical lightweight mirrors, and has a high fundamental frequency.mirror It alsohas shows the superiority the new configuration mirrorconfiguration obtained by triangular configuration superiority in opticalofperformance. The new design fundamental frequency. It also shows the superiority of the new configuration mirror obtained by the optimization proposed in this of paper compared the mirror optimized size lightweight mirrormethod combines the advantages the two classicalwith lightweight mirrors, and hasby a high the optimization method proposed in this paper compared with the mirror optimized by size optimization or single-objective topology optimization [6,9]. fundamental frequency. It also shows the superiority of the new configuration mirror obtained by the optimization or single-objective topology optimization [6,9]. optimization method proposed in this paper compared with the mirror optimized by size optimization Table 5. Comparisons of the classic and new configuration lightweight mirror design. or single-objective topology optimization Table 5. Comparisons of the classic [6,9]. and new configuration lightweight mirror design. Tri-

Hex-

New

Terms Tri- and new configuration Hex-lightweight mirror design. New Table 5. Comparisons ofConfiguration the classic Configuration Configuration

Terms PVA Terms & RMSA PVA & RMSA PVL & RMSL PVLA&&RMS RMSLA PV PV P &&RMS PV RMSPL L PVP & RMS P PV & RMS (+2 P°C) PVP & RMS PV & RMS (+2 ◦ C) PVFrequency & RMS (+2 °C) Frequency Frequency Mass Mass Mass Lightweight Lightweightratio ratio Lightweight ratio

Configuration Configuration Configuration 125.73 & 28.65 nm 150.57 & 34.43 nm 119.53 & 27.58 nm 8.08 & 1.68 nm 10.38 & 1.92 nm 7.03 & 1.28 nm 125.73 && 28.65 & 34.43 nmnm 119.53 & 27.58 nmnm 8.08 1.68nm nm 150.57 10.38 & 1.92 7.03 & 1.28 55.80 & 13.56 nm 60.74 & 14.05 nm 45.65 & 10.10 8.08 & 1.68 nm 10.38 & 1.92 nm 7.03 & 1.28 nm nm 55.80 & 13.56 nm 60.74 & 14.05 nm 45.65 & 10.10 nm 61.6&&13.56 10.48 100.5 & 17.5 & 8.78 55.80 nmnm 60.74 & 14.05 nmnm 45.6550.1 & 10.10 nmnm 61.6 & 10.48 nm 100.5 & 17.5 nm 50.1 & 8.78 nm 61.6 & 10.48 nm 100.5 & 17.5 nm 50.1 & 8.78 nm 1400.8 Hz 1344.5 Hz 1555.3 Hz 1400.8 1344.5 1555.3 1400.8 HzHz 1344.5 Hz Hz 1555.3 Hz Hz 13.48 kg 11.54 kg 11.08 kg 13.48 kg kg 11.54 kg kg 11.08 kg kg 13.48 11.54 11.08 68.4% 72.9% 74.0% 68.4% 72.9% 74.0% 68.4% 72.9% 74.0%

125.73 & 28.65 nm Hex-Configuration 150.57 & 34.43 nm New119.53 & 27.58 nm Tri-Configuration Configuration

4.2. Optical Performance of the PMA 4.2. 4.2. Optical Optical Performance Performance of of the the PMA PMA We can find that the mirror surface distortion under lateral gravity is far better than that under We thatthe themirror mirrorsurface surface distortion under lateral gravity far better thanunder that We can can find that distortion under lateral gravity is farisbetter than that axial gravity through our previous work, so the primary mirror is measured with an optical under axial gravity through our previous work, so the primary mirrorisismeasured measured with with an optical axial gravity through our previous work, so the primary mirror optical interferometer horizontally, as shown in Figure 11. The adhesive position of the mirror and bipod interferometer interferometer horizontally, horizontally, as as shown shown in in Figure Figure11. 11. The The adhesive adhesive position position of of the the mirror mirror and and bipod bipod flexures affects the surface error due to the varied distance between the apex of BF and plane of the flexures flexures affects affects the the surface surface error error due due to tothe thevaried varied distance distance between between the the apex apex of of BF BF and and plane plane of of the the center of gravity (CG), as illustrated in Figure 12. The misalignment generates torque and the coupled center centerof ofgravity gravity(CG), (CG),as as illustrated illustratedin in Figure Figure12. 12. The The misalignment misalignment generates generates torque torqueand and the the coupled coupled reaction of bending moment and gravity will cause mirror distortion substantially. reaction reactionof ofbending bendingmoment momentand andgravity gravitywill willcause causemirror mirrordistortion distortionsubstantially. substantially.

Figure 11.11. Horizontal setup forfor measuring thethe mirror surface with anan optical interferometer. Figure Horizontal setup measuring mirror surface with optical interferometer. Figure 11. Horizontal setup for measuring the mirror surface with an optical interferometer.

Figure 12. The non-coplanar distance L of the PMA.

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Figure 12. The non-coplanar distance L of the PMA.

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We investigate the effects of non-coplanar distance L on the surface error of the PMA under 1 g Wegravity. investigate thethe effects distance L on surface or error PMA under 1g lateral When apexofisnon-coplanar above the plane of CG, L the is positive elseofLthe is negative. Fringe lateral gravity. When the apex is above the surface plane ofdeviation CG, L is positive or else L is negative. Fringe Zernike Zernike polynomials are used to fit the [24]. The primary aberration terms of the polynomials to fit surface deviation [24]. The primary aberration terms of polynomials, the polynomials polynomialsare is used shown in the Table 6. Through comparing coefficients of the Zernike we isfind shown 6. Through comparing coefficients of the Zernikecomponent. polynomials, find that the that in theTable primary astigmatism-A (Pri-Astig-A) is the dominant Thewe analysis results primary astigmatism-A (Pri-Astig-A) the dominant Theinanalysis resultsFrom of surface of surface figure RMS and Pri-Astig-Ais coefficient due component. to L are shown Figure 13a,b. Figure figure RMS coefficient due are shown in Figure 13a,b. From Figure we can 13a, we canand findPri-Astig-A there is a concave curve in to theL plot of the surface figure RMS. When L is13a, 3.5 mm, the find is aerror concave inisthe plot of the surface RMS. When L is 3.5 mm, RMSthere surface of thecurve mirror minimized to 3.58 nm. figure As shown in Figure 13b, when L isthe lessRMS than surface of the mirror is minimized to 3.58 nm. As shown in Figure 13b, whenwhen L is L less than 3.5 mm,error the Pri-Astig-A is positive, it means the mirror will have lateral astigmatism; is larger 3.5 mm, Pri-Astig-A is positive, it means theitmirror have lateral astigmatism; when L is larger than 3.5the mm, the Pri-Astig-A is negative, and meanswill the mirror will have lengthways astigmatism. than 3.5 mm, the Pri-Astig-A negative, andisitillustrated means theinmirror have lengthways For visualization, the residualissurface error Figurewill 14. When L is 3.5 mm,astigmatism. there is only For visualization, the residual surface error is illustrated in Figure When L is 3.5 mm, is only a small scale deformation on the mirror surface and almost has no14. astigmatism. Based onthere the analysis aabove, small scale deformation mirror surface and almost has no astigmatism. on performance the analysis we can infer that on thethe new configuration mirror could exhibit excellentBased optical above, wegiven can infer based on BFs. that the new configuration mirror could exhibit excellent optical performance based on given BFs. Table 6. Fringe Zernike polynomials. Table 6. Fringe Zernike polynomials.

Name Name

Term Term

Focus 2r2−1 2 Pri Astig-A r cos(2θ) Focus 2r2 − 1 2 Pri Astig-A r cos(2θ) Pri Astig-B r2sin(2θ) 2 sin(2θ) Pri Astig-B 2 Pri Coma-A (3r −2r2r)cos(θ) Pri Coma-A (3r − 2r)cos(θ) Pri Coma-B (3r2 −2 2r)sin(θ) Pri Coma-B (3r − 2r)sin(θ) 2− Pri Spherical 6r6r4 4− − 6r6r 21 Pri Spherical −1 3 3 Pri Trefoil-A r cos(3θ) Pri Trefoil-A r cos(3θ) Pri Trefoil-B r3 sin(3θ) Pri Trefoil-B r3sin(3θ)

L = 1 mm, Coefficient/nm L = 1 mm, Coefficient/nm 3.73 × 10−2 3.739.06 × 10−2 9.06 −5.19 × 10−2 −1.37 5.19 ××10 10−2−2 1.37 × 10−−12 2.77 × 10 2.77 × 10−1 −2 −2.15 × 10 −2.15 × 10−2 −1 − 9.62×× 10 10 1 9.62 −33 − 6.53 × −6.53 × 10−

L = 3.5 mm,

L = 6 mm,

LCoefficient/nm = 3.5 mm, L =Coefficient/nm 6 mm, −2 Coefficient/nm Coefficient/nm 2.19 × 10 9.78 × 10−3 2.192.31 × 10×−210−1 2.31−2.67 × 10−×110−2 −2 −2.672.15 × 10 ×−210−2 2.15 × 10 2.97 × 10−1 2.97 × 10−1 −9.32 3 −3 −9.32 × 10×−10 − 2 1.041.04 × 10× 10−2 3 −3 −7.83 × 10×−10 −7.83

3 −8.72 9.78 × 10− −8.72 −5.66 × 10−2 −5.66 ×1.65 10−2× 10−2 1.65 × 10−2 3.17 × 10−1 3.17 × 10−1 −4 −3.46 −3.46 × 10−4× 10 − 2 −1.02 −1.02 × 10 × 10−2 −8.05 × 10−3× 10−3 −8.05

Figure L.L. (a)(a) Surface figure RMS due to to L under 1 g1 Figure13. 13.The Theoptical opticalperformance performanceofofPM PMwith withrespect respecttoto Surface figure RMS due L under Appl. 2018, 8, x FOR PEER REVIEW radial gravity; (b)(b) The coefficient of Pri Astig-A duedue to L. g Sci. radial gravity; The coefficient of Pri Astig-A to L.

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Figure14. 14.The Theresidual residual surface error due to varied L under g lateral gravity. Figure surface error due to varied L under 1g1 lateral gravity.

5. Conclusions Using a topology optimization with multi-objective function combined with parametric optimization method, we designed a new configuration lightweight mirror considering axial gravity, radial gravity, and polishing pressure simultaneously. In the application example, the RMS value

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5. Conclusions Using a topology optimization with multi-objective function combined with parametric optimization method, we designed a new configuration lightweight mirror considering axial gravity, radial gravity, and polishing pressure simultaneously. In the application example, the RMS value under each loading case was replaced by the corresponding structural compliance, and the minimum structural weighted compliance was selected as the objective function. Under the constraints of the volume fraction, nodal displacement, draft direction, and cyclic symmetry condition, a new mirror configuration was proposed. Then sensitivity analysis and parametric optimization methods were used to determine the size of the mirror structure. Compared with two classic lightweight mirrors, we found that the optimized mirror combines the high lightweight ratio of the hexagonal configuration lightweight mirror and the excellent optical performance of the triangular configuration lightweight mirror. In addition, the surface error of the mirror mounted by given BF under 1 g radial gravity was analyzed. We investigated the effects of non-coplanar distance L on the surface figure RMS and coefficient of Pri Astig-A. When L is 3.5 mm, the surface error of the mirror is minimized only with a RMS value of 3.58 nm. It shows an extraordinary merit that the surface figure of the PMA is far better than the optical requirement (λ/60 RMS, λ = 632.8 nm). The results show superiority of the new configuration mirror and illustrate the effectiveness of the proposed optimization design method. Author Contributions: W.W. and Y.Q. conceived and designed the experiments; Y.Q., Y.J., L.F., X.L., and B.L. performed the experiments; W.W. and Y.Q. analyzed the data; Y.Q., and Y.J. contributed reagents/materials/analysis tools; Y.Q. wrote the paper. Funding: This work was supported by the National Natural Science Foundation of China (Grant No. 51402351). Conflicts of Interest: The authors declare no conflict of interest.

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