applied sciences

Article Design of a Segmented with a Global Radius of Curvature Actuation System: Contributions of Multiple Surrogates

Songhang Wu 1,2,*, Jihong Dong 1,*, Shuyan Xu 1, Zhirong Lu 1,2 and Boqian Xu 1

1 Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China; [email protected] (S.X.); [email protected] (Z.L.); [email protected] (B.X.) 2 University of Chinese Academy of Sciences, Beijing 100049, China * Correspondence: [email protected] (S.W.); [email protected] (J.D.)

 Received: 2 November 2020; Accepted: 21 November 2020; Published: 25 November 2020 

Abstract: Due to fabrication difficulties, separately-polished segmented cannot meet the co-phasing surface shape error requirements in the segmented system. Applying the global radius of curvature (GRoC) actuation system for the individual segments has become an effective solution in space-based . In this paper, we designed a segmented mirror with a GRoC actuation system. The direct optimization by numerical simulations has low computational efficiency and is not easy to converge for optimizing the actuation point’s position on the segmented mirror. For this problem, three common surrogates, including polynomial response surface (PRS), radial basis function neural network (RBFNN), and kriging (KRG), were summed to propose the multiple surrogates (MS) which have the higher approximate ability. The surrogates were then optimized through the multi-island genetic algorithm (MIGA), and the segmented mirror met the design requirement. Compared with direct optimization through numerical simulations, the results show that the proposed multiple-surrogate-based optimization (MSBO) methodology saves computational cost significantly. Besides, it can be deployed to solve other complex optimization problems.

Keywords: multiple-surrogate-based optimization; multi-island genetic algorithm; Latin hypercube sampling; segmented mirror with a GRoC actuation system

1. Introduction There exist significant interest in using segmented mirrors instead of a monolithic to build a large-aperture telescope. The use of segmented mirrors reduces the difficulties in fabrication, transportation, and replication. Meanwhile, one challenge in the approach lies in matching the individual mirrors to yield a continuous surface. Although the six degrees of freedom adjustments in the segmented-mirror-matching process can correct some aberrations, it cannot fix the surface shape error [1]. Therefore, the existing segmented telescope system applies an active-surface shape-adjustment method. For the ground telescope, a semi-active warping harness structure was adopted in the Keck Telescope. Then, the 30 m telescope (TMT) and the European Very Large Telescope (E-ELT) also adopted this scheme [2,3]. For the space telescope, taking the James Webb Space Telescope (JWST) as an example, a global radius of curvature (GRoC) actuation system was designed for the non-insistency in the radius of curvature of segmented mirrors. This system has two main applications: (1) After rough polishing of the JWST primary mirror, use of the GRoC actuation system to adjust the GRoC to meet the requirements and record the surface shape error at this time. After removing the GRoC actuation system and supporting structures, the registered surface shape error is used as a

Appl. Sci. 2020, 10, 8375 ; doi:10.3390/app10238375 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 8375 2 of 14 target map for the finely-polished. The surface shape error root mean square (RMS) after the second polishing can reach 20 nm [4,5]. (2) In the co-phasing process, the system performs micro-control on the GRoC. The first application has a broad adjustment range ( 5 mm), and the second application has ± a small adjustment range ( 0.15 mm) [6]. Since the actuation point of the GRoC actuation system on ± the segmented mirror’s back directly affects the curvature adjustment accuracy, the actuation point’s position needs to be optimized. Complicated structural optimization through numerical simulations is unavailable since searching for the optimum typically requires thousands or even millions of simulations. To alleviate the computational burden, the surrogate-based optimization methodology (SBOM) becomes the right choice. It simplifies the relationship between output responses and design variables through surrogates, then directly optimizes the surrogates. Wang et al. optimized the 2.8 m diameter circular mirror’s main structural parameters based on the back propagation (BP) neural network surrogate. The error between the surrogate and the simulation was less than 7.09% [7]. Hong et al. and Xiang et al. used the radial basis function neural network (RBFNN) for the design of the sealing structure and shelter location problem under uncertainty of road networks, respectively, which significantly improved optimization efficiency [8,9]. Ribeiro et al. optimized the functionally-graded plates using an RBFNN surrogate [10]. Wang et al. carried out a many-objective optimization for a deep-sea aquaculture vessel by RBFNN surrogate [11]. Based on the response surface method (RSM) surrogate, J.C. Hsiao et al. carried out a multi-objective optimization design on the robot arm’s size parameters, which improved the computational efficiency by 90% compared with optimization by numerical simulation [12]. Ahmed et al. used an RSM surrogate to replace the 3-D finite element method models in the design of a transverse-flux permanent magnet linear synchronous motor [13]. With the help of an RSM surrogate, Rafiee et al. and Chau et al. achieved good results in designing an outer rotor permanent magnet motor and designing a new compliant planar spring, respectively [14,15]. Hakjin et al. used a Kriging surrogate to replace the complicated optimization problem of multiple wing sails considering the flow interaction effect, which effectively improved the computational efficiency [16]. Zhenxu et al. built a Kriging surrogate based on the aerodynamic noise’s influence on a high-speed trains’s shape design and performed multi-objective optimization [17]. Guo et al. and Das et al. optimized the stiffness of composite cylinders and the nonlinear energy sink by Kriging surrogate, respectively, and the optimization efficiency was remarkably improved [18,19]. However, the approximate ability of the surrogate is insufficient for some complicated optimization, mainly in: (1) the approximate ability of different surrogates is different, and it is not easy to find the best surrogate for one optimization; and (2) a surrogate is easily affected by changes in sample points, and it is difficult to maintain the approximation ability as the number of sample points increases. To solve this problem, we proposed multiple surrogates to optimize the activation point’s position on the segmented mirror’s back. The multiple surrogates aimed to minimize the generalized mean square error (GMSE) of the output response and select the combination that was closest to the output response among surrogates (RBFNN [9], PRS [20], and KRG [21]). After the MIGA optimized them, the segmented mirror had met the design requirement. This work attempts to provide an efficient methodology to solve the complicated optimization problem in the design of a segmented mirror with a GRoC actuation system. The rest of this paper is organized as follows: In Section2, the multiple surrogates and the multiple-surrogate-based optimization methodology are described and a method to solve the superposition coefficients is introduced. The design process of a segmented mirror with a GRoC actuation system, including the support design and the actuation point’s position design, is described in Section3. Finally, conclusions are drawn in Section4. Appl. Sci. 2020, 10, 8375 3 of 14

2. Proposed Multiple-Surrogate-Based Optimization Methodology

2.1. Multiple Surrogates The surrogate is a mathematical method to predict the unknown point’s responses by building an approximate relationship between design variables and responses relying on limited sample point data information. Several expressions of popular surrogate models, including PRS, RBFNN, and KRG, are as follows: XN XN NX1 XN 2 − PRS : yp = a0 + bixi + ciixi + dijxixj, (1) i=1 i=1 i=1 j=i+1 where yp is the output of PRS, xi and xj are the input variables, N is the number of input variables, and a0, bi, cii, and dij are regression coefficients.

XM RBFNN : yr = α ϕ( x x , c), (2) i k − ik i=1 where yr is the output of RBFNN, x is the input variable, M is the number of input variables, αi is coefficient, is the Euler norm, ϕ() is basis function, and spread constant, c, is 0.5. k k XK KRG : yk = βiφi(x) + z(x), (3) i=1 where yk is the output of KRG, x is the input variable, K is the number of input variables, βi is coefficient, φi() is basis function, and z() is the function of random error. The approximation accuracy of a surrogate is affected by sampling points. In other words, the best surrogate may be the worst one in the next iteration. And multiple surrogates are linearly superimposed by surrogate models. The best combination of surrogates for approximation can be gained by solving the coefficients during iterations, as follows:

y(x) = wp yp(x) + wr yr(x) + wk yk(x), (4) where x is the design variables, y(x) is the output of multiple surrogate model, yp(x), yr(x), and yk(x), respectively, are the output of PRS, RBFNN, and KRG, and wp, wr, and wk are coefficients. The key to multiple surrogates is to solve coefficients wp,wr, and wk. Viana et al. suggested that generalized mean square error (GMSE) is suitable for filtering out inaccurate surrogates [22], and it is as follows:

n 1 X 2 GMSE = ( f fa) , (5) n i − i=1 where n is the number of sample points, and fi and fa represent the prediction output and actual output at xi, respectively. Using the GMSE to identify the approximation accuracy of the multiple surrogates, means that it is smaller, and the outputs of the multiple surrogates are closer to the real responses. Therefore, solving coefficients in Equation (4) can be converted into an optimization problem aimed at the smallest GRMS. The optimization problem is as follows:   Find : w , w , w  p r k  L , (6)  = 1 P [ ( ) ( )]2  Min : G L y xi ya xi i=1 − Appl. Sci. 2020, 10, 8375 4 of 14

where G is the GMSE of multiple surrogates, y(xi) is the prediction output at xi using the multiple surrogate models, ya(xi) is the actual output at xi, and L is the number of sampling points. To find the partial derivatives of G with respect to wp, wr, and wk, respectively,    XL XL XL XL XL XL XL  ∂G 2  2  = wp yp (xi) + wr yr(xi) yp(xi) + wk yk(xi) yp(xi) yp(xi) ya(xi), (7) ∂wp L −  i=1 i=1 i=1 i=1 i=1 i=1 i=1    XL XL XL XL XL XL XL  ∂G 2  2  = wp yp(xi) yr(xi) + wr yr (xi) + wk yk(xi) yr(xi) yr(xi) ya(xi), (8) ∂wr L −  i=1 i=1 i=1 i=1 i=1 i=1 i=1   XL XL XL XL XL XL XL ∂G 2  2  = wp yp(x ) y (x ) + wr yr(x ) y (x ) + w y (x ) y (x ) ya(x ), (9) ∂w L i k i i k i k k i − k i i  k i=1 i=1 i=1 i=1 i=1 i=1 i=1 h iT By setting the partial derivatives to 0, the coefficients, w = wp wr wk , is solved as follows:

1 w = A− B, (10)

L T X PL PL PL PL PL B = [ yp(xi) ya(xi) yr(xi) yp(xi) yk(xi) yp(xi) ] , (11) = = = = = i=1 i 1 i 1 i 1 i 1 i 1  L L L L L   P 2( ) P ( ) P ( ) P ( ) P ( )   yp xi yr xi yp xi yk xi yp xi   i=1 i=1 i=1 i=1 i=1     PL PL PL PL PL  A =  y (x ) y (x ) y 2(x ) y (x ) y (x ) , (12)  p i r i r i k i r i   i=1 i=1 i=1 i=1 i=1   L L L L L   P P P P P 2   yp(xi) yk(xi) yr(xi) yk(xi) yk (xi)  i=1 i=1 i=1 i=1 i=1 After the multiple surrogates are built, it is necessary to estimate approximation accuracy. Generally, the determination coefficient, R2, is a measure used to assess how well a model can explain and predict outcomes. It ranges from 0 to 1 and will be closer to 1 when the model is more precise. In this work, the multiple surrogates with R2 greater than 0.9 can replace the simulation model. R2 can be obtained as follows: p P 2 (yr(x ) ya(x )) k − k 2 = k=1 R 1 p , (13) − P 2 (yr(xk) y) k=1 − where yr(xk) and ya(xk) are the output of multiple surrogates and the actual model at xk, respectively, p is the number of test points, and y is the average of the actual output.

2.2. Procedure of Multiple-Surrogate-Based Optimization A stepwise procedure of the proposed multiple-surrogate-based optimization methodology is explained as follows, also with the help of Figure1. Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 15

3. Build the surrogates, including PRS, RBFNN, and KRG (reference for the method of construction [9, 20, 21]), respectively. With the minimum GMSE of the test points, solve the coefficients in Equation (10) and construct the multiple surrogates. 2 2 4. According to Equation (13), solve the determination coefficient R . Check whether R is greater than 0.9. If it is greater than 0.9, go to the next step. Otherwise add the current training points, and return to step 2 to rebuild the multiple surrogates. 5. Optimize the multiple surrogates and check the boundary conditions by numerical simulation. If Appl. Sci.not,2020 add, 10 ,the 8375 current training points, and return to step 2 to rebuild the multiple surrogates.5 of 14 Otherwise, the optimization is completed.

FigureFigure 1.1. FlowFlow chartchart ofof thethe multiple-surrogate-basedmultiple-surrogate-based optimization.optimization.

1.3. DesignDetermine of Segmented the optimization Mirror with problem, a GRoC including Actuation objectives, System constraints, and design variables, etc., and generate sample points of design variables as training points and testing points with the help 3.1. Designof Latin Requirements hypercube sampling (LHS) [23]. LHS can more efficiently estimate the overall mean of response than estimation based on random sampling. The primary mirror of an on-orbit assembled telescope consists of six hexagonal segmented 2. Perform the numerical simulation at the training points and testing points to obtain the mirrors made of Zerodur (point-to-point distance is 200 mm and the GRoC is 2406 mm). Using the interest responses. JWST for reference, as shown in Figure 2, there is a GRoC on each segmented mirror’s back. The 3. Build the surrogates, including PRS, RBFNN, and KRG (reference for the method of construction GRoC actuation system comprises an actuator that pushes and pulls against the center of the mirror [9, 20, 21]), respectively. With the minimum GMSE of the test points, solve the coefficients in Equation (10) and construct the multiple surrogates. 4. According to Equation (13), solve the determination coefficient R2. Check whether R2 is greater than 0.9. If it is greater than 0.9, go to the next step. Otherwise add the current training points, and return to step 2 to rebuild the multiple surrogates. 5. Optimize the multiple surrogates and check the boundary conditions by numerical simulation. If not, add the current training points, and return to step 2 to rebuild the multiple surrogates. Otherwise, the optimization is completed. Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 15 Appl. Sci. 2020, 10, 8375 6 of 14 and six struts to react the force at the six corners of the mirror. The GRoC system can effectively change the GRoC by introducing undesired deformation called GRoC actuation residual. 3. Design of Segmented Mirror with a GRoC Actuation System

3.1. Design Requirements The primary mirror of an on-orbit assembled telescope consists of six hexagonal segmented mirrors made of Zerodur (point-to-point distance is 200 mm and the GRoC is 2406 mm). Using the Appl.JWST Sci. for2020 reference,, 10, x FOR PEER as shown REVIEW in Figure2, there is a GRoC on each segmented mirror’s back. The6 GRoCof 15 actuation system comprises an actuator that pushes and pulls against the center of the mirror and six andstruts six tostruts react to the react force the at force the six at corners the six ofcorner the mirror.s of the The mirror. GRoC The system GRoC can system effectively can effectively change the changeGRoC the by introducingGRoC by introducing undesired undesired deformation def calledormation GRoC called actuation GRoC residual.actuation residual.

Figure 2. The James Webb Space Telescope (JWST) segmented mirror back [24].

A segmented mirror with a GRoC actuation system was designed in accordance with the process in Figure 3. The specific method of traditional mirror design can be referred to [25], and this work focuses on the support design and the design of the activation point’s position. Space mirrors are always fabricated on the ground and launched into space, and the surface accuracy could be degraded by gravity relief. Therefore, the disturbance of gravity should be considered in the support design. According to the optical requirement of cophase, the segmented mirror’s required surface accuracy should not exceed λ /50 including accuracy degradation FigureFigure 2. 2.TheThe James James Webb Webb Space Space Telescope Telescope (JWST) (JWST) segmented segmented mirror mirror back back [24]. [24]. caused by the disturbance of gravity, manufacturing error, and the GRoC actuation residual. The specificA segmented design requirements mirror with aare GRoC introduced actuation in systemFigure was3. In designed orbit, the in cophasing accordance process with the needs process the A segmented mirror± with a GRoC actuation system was designed in accordance with the processinGRoC Figure in actuation 3Figure. The 3. specific to The be specific method0.1mm method of, then traditional of the traditional surface mirror shape mirror design error design can RMS be can referred of be residual referred to [ should25 to], [25], and be thisand less workthis than λ λ= workfocuses/10 focusesper on themillimeter on support the support of design GRoC design and actuation theand design the ( design of632.8 the of activationnm the). activation point’s point’s position. position. Space mirrors are always fabricated on the ground and launched into space, and the surface accuracy could be degraded by gravity relief. Therefore, the disturbance of gravity should be considered in the support design. According to the optical requirement of cophase, the segmented mirror’s required surface accuracy should not exceed λ /50 including accuracy degradation caused by the disturbance of gravity, manufacturing error, and the GRoC actuation residual. The specific design requirements are introduced in Figure 3. In orbit, the cophasing process needs the GRoC actuation to be ±0.1mm , then the surface shape error RMS of residual should be less than λ /10per millimeter of GRoC actuation ( λ=632.8nm ).

FigureFigure 3.3. TheThe designdesign processprocess andand designdesign requirementrequirement ofof thethe segmentedsegmentedmirror. mirror.

Figure 3. The design process and design requirement of the segmented mirror.

Appl. Sci. 2020, 10, 8375 7 of 14

Space mirrors are always fabricated on the ground and launched into space, and the surface accuracy could be degraded by gravity relief. Therefore, the disturbance of gravity should be considered in the support design. According to the optical requirement of cophase, the segmented mirror’s required surface accuracy should not exceed λ/50 including accuracy degradation caused by the disturbanceAppl. Sci. 2020, 10 of, x gravity, FOR PEER manufacturing REVIEW error, and the GRoC actuation residual. The specific design7 of 15 requirements are introduced in Figure3. In orbit, the cophasing process needs the GRoC actuation to be3.2. Support0.1mm, Design then the of surfacethe Segmented shape Mirror error RMS of residual should be less than λ/10 per millimeter of ± GRoC actuation (λ = 632.8nm). 3.2.1. Problem Description 3.2. Support Design of the Segmented Mirror The mirror is supported by three flexures located on its back. The supporting flexure’s 3.2.1.structural Problem parameters Description and the supportive position were optimized under the environment of 1 g gravity (direction parallel to the mirror surface). The location of the design variables is shown in FigureThe 4. mirror is supported by three flexures located on its back. The supporting flexure’s structural parameters and the supportive position were optimized under the environment of 1 g gravity minXftthhlp= ( , , , , , ) , (direction parallel to the mirror surface).RMS The location 1 1 2 of 1 the 2 1design variables is shown in Figure4(14). ≤≤ minXRMS1,5mm= f1 t(12t1 t, t2, h1 mm, h2, l1, p), (14)  ≤≤ 1,7mm h12 h mm st.. 1 mm t , t 5 mm,  1 2 (15)  20mm≤≤≤ l 90≤ mm  1 mm h1 , h2 7 mm s.t. ≤ ≤ , (15)  52020mm mm≤≤ pl1 90 mm mm  ≤ ≤  5 mm p 20 mm where X RMS is surface shape error RMS under a≤ gravity≤ of 1 g, h1 and h2 are the height of the upper where XRMS is surface shape error RMS under a gravity of 1 g, h1 and h2 are the height of the upper and lower grooves of the flexure, t1 and t2 are the width of the upper and lower grooves of the and lower grooves of the flexure, t1 and t2 are the width of the upper and lower grooves of the flexure, pflexure,is the height p is the of theheight flexure, of the and flexure,l1 is the and distance l1 is betweenthe distance the flexure between support the flexure position support and the position center ofand the the segmented center of the mirror. segmented mirror.

Figure 4. The optimization variables of support design. Figure 4. The optimization variables of support design. 3.2.2. Results of Optimization 3.2.2. Results of Optimization After 125 iterations, the design requirement in Section 3.1 was met by a multi-island genetic algorithmAfter (MIGA)125 iterations, [26] (the the population design requirement number is 5, in the Section island number3.1 was ismet 4, andby a the multi-island evolution numbergenetic algorithm (MIGA) [26] (the population number is 5, the island number is 4, and the evolution number is 10). At that time, the surface shape error RMS was 4.5 nm, and the basic frequency of the segmented mirror and the flexures was 150 Hz. The design variables before and after optimization are shown in Table 1. The displacement map and the surface shape error map of the optimum are shown in Figures 5 and 6, respectively.

Appl. Sci. 2020, 10, 8375 8 of 14 is 10). At that time, the surface shape error RMS was 4.5 nm, and the basic frequency of the segmented mirror and the flexures was 150 Hz. The design variables before and after optimization are shown in

TableAppl.1 Sci.. The 2020 displacement, 10, x FOR PEER map REVIEW and the surface shape error map of the optimum are shown in Figures8 of5 15 and6, respectively. Table 1. Design variables before and after optimization. Appl. Sci. 2020, 10, x FOR PEERTable REVIEW 1. Design variables before and after optimization. 8 of 15 Variable Initial Value Optimization Result Table 1. Design variables before and after optimization. Variablel1 Initial50 mm Value Optimization 52.5 mm Result Variable1 Initial Value Optimization Result l1 t 503 mm mm 3.9 52.5 mm mm l1 50 mm 52.5 mm t1 t2 43 mm mm 3.5 3.9 mm mm t1 3 mm 3.9 mm t2 h1 54 mm mm 4.5 3.5 mm mm t2 4 mm 3.5 mm h1 h2 55 mm mm 4.5 4.5 mm mm h1 5 mm 4.5 mm h2 p 155 mmmm 12 4.5 mm mm ph2 5 mm15 mm 4.5 mm 12 mm p 15 mm 12 mm

Figure 5. The displacement map of the optimum. FigureFigure 5. 5.The The displacement displacement map map of of the the optimum. optimum.

Figure 6. The surface shape error map of the optimum.

3.3. Actuation Point’s Position Design of Segmented Mirror FigureFigure 6. 6.The The surface surface shape shape error error map map of of the the optimum. optimum. 3.3.1. Problem Description 3.3. ActuationAfter the Point’s segmented Position mirror Design support of Segmented design was Mirror completed, we optimized the actuation point’s position of the GRoC actuation system on the segmented mirror’s back with the help of proposed 3.3.1.multiple Problem surrogates. Description Due to the surface shape error RMS of residual, the stress and the amount of After the segmented mirror support design was completed, we optimized the actuation point’s position of the GRoC actuation system on the segmented mirror’s back with the help of proposed multiple surrogates. Due to the surface shape error RMS of residual, the stress and the amount of Appl. Sci. 2020, 10, 8375 9 of 14

Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 15 3.3. Actuation Point’s Position Design of Segmented Mirror GRoC actuation are proportional to the actuator’s output. The influence of the actuator on the 3.3.1.optimization Problem object Description can be eliminated by division. In other words, the surface shape error RMS per millimeter of GRoC actuation has nothing to do with the actuator’s output, and the same is true of After the segmented mirror support design was completed, we optimized the actuation point’s the maximum local stress per millimeter of GRoC actuation. In this optimization, the output of the position of the GRoC actuation system on the segmented mirror’s back with the help of proposed actuator was set as 1 N. The design variables of the actuation point’s position are shown in Figure 7, multiple surrogates. Due to the surface shape error RMS of residual, the stress and the amount of GRoC actuationwhere h are is the proportional depth of the to theactuation actuator’s point, output. l2 is The the influencedistance between of the actuator the actuation on the optimization point of the object can be eliminated by division. In other words, the surface shape error RMS per millimeter of strut and the actuation point of the actuator, and d1 and d2 are the diameters of the actuation GRoC actuation has nothing to do with the actuator’s output, and the same is true of the maximum point of the actuator and the strut, respectively. Due to the fact that the range of GRoC actuation is local stress per millimeter of GRoC actuation. In this optimization, the output of the actuator was set as required to be greater than 2 mm, in order to ensure the normal work of the mirror, the maximum 1 N. The design variables of the actuation point’s position are shown in Figure7, where h is the depth local stress should be checked for strength. The objective functions of the activation point’s position of the actuation point, l is the distance between the actuation point of the strut and the actuation point are constructed as follows:2 d d of the actuator, and 1 and 2 are the diameters= of the actuation point of the actuator and the strut, respectively. Due to the fact that themin rangeYfddlhRMS of GRoC ( 1 , actuation 2 , 2 , ) , is required to be greater than 2 mm,(16) in order to ensure the normal work of the mirror, the maximum local stress should be checked for strength. The objective functions of the activation010<

Figure 7. The optimization variables of the actuation point’s position design. Figure 7. The optimization variables of the actuation point’s position design.

3.3.2. Procedure of the Actuation Point’s Position Optimization Appl. Sci. 2020, 10, 8375 10 of 14

Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 15 3.3.2. Procedure of the Actuation Point’s Position Optimization With thethe help help of of Figure Figure8, the 8, procedurethe procedure of the of actuation the actuation point’s positionpoint’s position optimization optimization is outlined is inoutlined the following in the following steps. steps.

Figure 8. Framework of the actuation point’s position optimization.

1.1. GenerateGenerate 20 20 training points and 10 test testinging points of design variables by LHS. 2. TakeTake thethe sampled sampled points points into into the numericalthe numerical simulation. simulation. In the numericalIn the numerical simulation, simulation, a geometric a geometricmodel is builtmodel at is a sampledbuilt at pointa sampled by computer-aided point by computer-aided design (CAD) design software, (CAD) Solidworks. software, Solidworks.Then analyze Then the geometric analyze the model geometric to get figure model deformation to get figure and deformation the maximum and local the stress maximum of the localmirror stress with of the the finite mirror element with analysisthe finite (FEA) elemen softwaret analysis Patran (FEA) and software Nastran. Patran Finally, and the Nastran. surface Finally,shape error the surface RMS and shape the amounterror RMS of GRoCand the actuation amount areof GRoC evaluated actuation by using are MATLAB.evaluated by using 3. MATLAB.Divide the surface shape error RMS and the maximum local stress by the amount of GRoC 3. Divideactuation the to surface get YRMS shapeand errorδMAX .RMS Build and RBFNN, the ma PRS,ximum and KRG,local stress respectively, by the usingamountYRMS of ,GRoCδMAX, and training points, then constructδ multiple surrogates according to Section 2.1. actuation to get YRMS and MAX . Build RBFNN, PRS, and KRG, respectively, using YRMS , 2 4. δSolve the determination coefficient R of multiple surrogates and check whether it is greater than 0.9.MAX If,not, andadd training 5 training points, points then andconstruct return multiple to step2. surrogates according to Section 2.1. 2 5.4. SolveSearch the the determination optimum of coefficient the multiple R surrogates of multiple through surrogates MIGA and check (population whether number it is greater is 5, thanisland 0.9. number If not, isadd 4, evolution5 training numberpoints and is 10). return to step 2. 6.5. SearchVerify whetherthe optimum the optimum of the multiple meets boundary surrogates conditions. through MIGA If not, add(population 5 training number points is and 5, returnisland numberto step 2. is 4, evolution number is 10). 6. Verify whether the optimum meets boundary conditions. If not, add 5 training points and 3.3.3.return Results to ofstep Optimization 2. After 156 iterations of MIGA, the optimum was obtained. At that time, the number of training 3.3.3. Results of Optimization points was 55, YRMS was 0.09λ, and δMAX was 1.51MPa, which met the design requirements in SectionAfter 3.1 .156 The iterations displacement of MIGA, map andthe theoptimum surface was shape obtained. error map At ofthat the time, optimal the designnumber are of shown training in Figures9 and 10, respectively. Theλ changesδ in design variables before and after the optimization are points was 55, YRMS was 0.09 , and MAX was 1.51MPa , which met the design requirements shown in Table2 and the coe fficients of each reconstruction of multiple surrogates are shown in Table3. in Section 3.1. The displacement map and the surface shape error map of the optimal design are shown in Figures 9 and 10, respectively. The changes in design variables before and after the optimization are shown in Table 2 and the coefficients of each reconstruction of multiple surrogates are shown in Table 3. Appl. Sci. 2020, 10, 8375 11 of 14

Appl.Appl. Sci. Sci. 2020 2020, ,10 10, ,x x FOR FOR PEER PEER REVIEW REVIEW 1111 ofof 1515

FigureFigureFigure 9. 9.9. The The displacement displacement map map of of the thethe optimum. optimum.optimum.

FigureFigureFigure 10. 10.10. The The surface surface shape shape error error map map of ofof the thethe optimum. optimum.optimum. Table 2. Design variables and optimization results. TableTable 2. 2. Design Design variables variables and and optimization optimization results. results. Variable Initial Value Optimization Result VariableVariable InitialInitial Value Value OptimizationOptimization Result Result hhh 5 5 5mm mm mm 7.5 7.5 7.5mm mm mm d1d1d1 1515 15 9.4 9.4 9.4mm mm mm d2 10 8.7 mm d2d2 1010 8.7 8.7 mm mm l2 80 65 mm l2l2 8080 65 65 mm mm

Table 3. The coefficients of multiple surrogates. TableTable 3. 3. The The coefficients coefficients of of multiple multiple surrogates. surrogates.

Output Response YRMS δ δMAX OutputOutput Response Response YYRMS δ MAX Surrogate RBF PRSRMS KRG RBFMAX PRS KRG SurrogateSurrogate RBF RBF PRSPRS KRGKRG RBFRBF PRSPRS KRGKRG 20 0 0 1 1 0 0 20 0 0 1 1 0 0 2520 0 0 00 11 1 10 00 0 302525 0.25 0 0.55 0 00 0.2011 11 100 000 0 353030 00.250.25 1 0.550.55 00.200.20 1 0.821 00 0.1800 0 403535 0 0 0.29 0 1 1 0.71 0 0 0.820.82 0.78 0.180.18 0.2200 0 4540 10 00.29 00.71 0.78 0.69 0.22 0.310 0 5040 0.690 0.31 0.29 00.71 0.78 0.65 0.22 0.350 0 45 1 0 0 0.69 0.31 0 5545 0.47 1 0 0 0.53 0 0.69 0.33 0.31 0.330 0.24 5050 0.69 0.69 0.310.31 00 0.650.65 0.350.35 00 5555 0.470.47 00 0.530.53 0.330.33 0.330.33 0.240.24 3.3.4. Approximation Ability Analysis

3.3.4.3.3.4. AApproximation Approximation comparison of Ability determinationAbility Analysis Analysis coe fficient R2 between the multiple surrogates and the surrogates (RBFNN, PRS, and KRG) is shown in Figure 11 to verify2 the approximate ability of the multiple A comparison of determination coefficient R2 between the multiple surrogates and the surrogates.A comparison It can beof seendetermination that the R 2 coefficientof the multipleR between surrogates the ismultiple always thesurrogates largest eachand timethe surrogates (RBFNN, PRS, and KRG) is shown in Figure 11 to verify the approximate ability of the surrogatesthe surrogates (RBFNN, are rebuilt. PRS, and When KRG) the is shown multiple in surrogatesFigure 11 to met verify the the decision approximate condition ability (R2 of> the0.9) 22 multiplewithmultiple 55 samplingsurrogates. surrogates. points, It It can can the be be coeseen seenffi cientthat that the ofthe determination RR of of the the multiple multiple is shown surrogates surrogates in Table 4is is. always always the the largest largest each each 22 > timetime the the surrogates surrogates are are rebuilt. rebuilt. When When the the mult multipleiple surrogates surrogates met met the the decision decision condition condition ( (RR > 0.90.9) ) withwith 55 55 sampling sampling points, points, the the coefficient coefficient of of determination determination is is shown shown in in Table Table 4. 4. Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 15

Table 4. The determination coefficient of surrogates.

Surrogate MS PRS RBFNN KRG Y 0.92 0.87 0.90 0.90 Appl. Sci. 2020, 10, 8375 RMS 12 of 14 δ 0.91 0.85 0.85 0.64 Appl. Sci. 2020, 10, x FOR PEER REVIEW MAX 12 of 15

Table 4. The determination coefficient of surrogates.

Surrogate MS PRS RBFNN KRG Y 0.92 0.87 0.90 0.90 RMS δ 0.91 0.85 0.85 0.64 MAX

(a) (b)

Figure 11. Determination coefficient of multiple surrogates: (a) Y and (b) δ . Figure 11. Determination coefficient of multiple surrogates: (a) RMYRMSS and (b)MAXδMAX .

3.3.5. Computational EfficiencyTable 4. TheAnalysis determination coefficient of surrogates. Since the computationalSurrogate cost of MSthe solvingPRS coefficient RBFNN and the KRGiteration of the MIGA were

negligible compared withY(aRMS) the numerical0.92 simulati 0.87on running, 0.90 the number 0.90(b) of iterations of the numerical simulation canδMAX be used to0.91 characterize 0.85 the computational 0.85 0.64efficiency in this work. The Figure 11. Determination coefficient of multiple surrogates: (a) Y and (b) δ . comparison between the direct optimization by numericalRMS simulationMAX and the 3.3.5. Computationalmultiple-surrogate-based Efficiency optimization Analysis is shown in Figure 12. Both optimizations used the same 3.3.5.MIGA Computational (the population Efficiency number isAnalysis 5, the number of islands is 4, and the number of evolution is 10). SinceSince the computational the computational cost ofcost the of solving the solving coeffi cientcoefficient and the and iteration the iteration of the of MIGA the MIGA were negligiblewere comparednegligible with thecompared numerical with simulation the numerical running, simulati the numberon running, of iterations the number of the of numericaliterations simulationof the can benumerical used to characterizesimulation can the be computational used to characterize efficiency the computational in this work. efficiency The comparison in this work. between The the directcomparison optimization between by numerical the simulationdirect optimization and the multiple-surrogate-based by numerical simulation optimization and isthe shown in Figuremultiple-surrogate-based 12. Both optimizations optimization used the is same shown MIGA in Figure (the population12. Both optimizations number is used 5, the the number same of islandsMIGA is 4, and(the population the number number of evolution is 5, the isnumber 10). of islands is 4, and the number of evolution is 10).

(a) (b)

Figure 12. Iterative process of optimization: (a) the optimization by numerical simulation and (b) the multiple-surrogate-based optimization.

It can be seen that after 200 iterations of the direct optimization by the numerical simulation, the convergence condition was still not satisfied in Figure 12a. In comparison, the multiple-surrogate-based(a )optimization had been iterated 158 times (tob) meet the convergence condition and the design requirement (a total of 55 iterations of the numerical simulation) in Figure Figure 12. Iterative process of optimization: (a) the optimization by numerical simulation and (b) the Figure 12. Iterative process of optimization: (a) the optimization by numerical simulation and 12b. multiple-surrogate-based optimization. (b) the multiple-surrogate-based optimization. 4. DiscussionIt can be seen that after 200 iterations of the direct optimization by the numerical simulation, It can be seen that after 200 iterations of the direct optimization by the numerical simulation, the convergence condition was still not satisfied in Figure 12a. In comparison, the the convergencemultiple-surrogate-based condition was stilloptimization not satisfied had in Figurebeen iterated 12a. In comparison,158 times to the meet multiple-surrogate-based the convergence optimizationcondition had and been the iterateddesign requirement 158 times to (a meettotal of the 55 convergence iterations of conditionthe numerical and simulation) the design in requirement Figure (a total12b. of 55 iterations of the numerical simulation) in Figure 12b.

4. Discussion4. Discussion In this paper, a segmented mirror with a GRoC actuation system was designed. To solve the problem of insufficient approximation ability of existing surrogates, we proposed multiple surrogates to optimize the actuation point’s position on the segmented mirror’s back. Appl. Sci. 2020, 10, 8375 13 of 14

The comparison of approximation ability between the multiple surrogates and some existing surrogates (PRS, RBFNN, and KRG) in Figure 11 shows that the multiple surrogates’ determination coefficient remained the highest during the iteration of the algorithm. It was the first to reach the decision condition (R2 > 0.9) with 55 sampling points, and the determination coefficients of all surrogates at that time are shown in Table4. The comparison of computational e fficiency is shown in Figure 12, and it can be seen that the multiple surrogates needed 55 iterations of the numerical simulation to satisfy the convergence condition. In comparison, the direct optimization cannot fully converge after 200 iterations of numerical simulation. This might be because the numerical simulation is too complicated, and the optimum is affected by the local minima. The approximation of the numerical simulation is equivalent to smooth filtering, which reduces the local minima’s influence and makes the optimization easier to converge. Some conclusions can be drawn: 1. The optimization methodology used in this work provides a solution for designing a segmented mirror with a GRoC actuation system. 2. The proposed multiple surrogates have higher approximation ability than a single surrogate (PRS, RBFNN, and KRG), and at the same time, solve the problem of difficulty in choosing a reasonable surrogate in an approximation optimization. 3. The proposed multiple-surrogate-based optimization methodology effectively reduces the computational cost of optimization problems and makes the optimization problems easier to converge. In addition, this methodology is also suitable to be extended to other complex optimization problems. It should be noted that although our multiple surrogates achieved good results in this work, other surrogates such as deep learning could also be added to them. Hence, more surrogates will be added to our multiple-surrogate-based optimization methodology in the future.

Author Contributions: Project administration, S.W.; methodology, J.D. and S.W.; software, Z.L.; writing—original draft, S.W.; writing—review and editing, S.X. and B.X. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National key research and development plan of China, grant number 2016YFE020500 and the National Natural Science Foundation of China (NSFC), grant number 61905241. Conflicts of Interest: The authors declare no conflict of interest.

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