Optimization of fixture layouts of glass laser optics using multiple kernel regression Jianhua Su,* Enhua Cao, and Hong Qiao The State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Science, 100190 Beijing, China *Corresponding author: [email protected] Received 28 January 2014; revised 27 March 2014; accepted 28 March 2014; posted 1 April 2014 (Doc. ID 205461); published 5 May 2014

We aim to build an integrated fixturing model to describe the structural properties and thermal properties of the support frame of glass laser optics. Therefore, (a) a near global optimal set of clamps can be computed to minimize the surface shape error of the glass laser optic based on the proposed model, and (b) a desired surface shape error can be obtained by adjusting the clamping forces under various environmental temperatures based on the model. To construct the model, we develop a new multiple kernel learning method and call it multiple kernel support vector functional regression. The proposed method uses two layer regressions to group and order the data sources by the weights of the kernels and the factors of the layers. Because of that, the influences of the clamps and the temperature can be evaluated by grouping them into different layers. © 2014 Optical Society of America OCIS codes: (220.4880) Optomechanics; (220.1080) Active or adaptive optics. http://dx.doi.org/10.1364/AO.53.002988

1. Introduction

Glass laser systems represent one of the most important research topics for application of high-power lasers, and research is conducted not only in energy production but also in various new fields, such as x-ray lasers and medical applications. A glass laser system uses several thousand large optics, which are usually mounted on a support frame, to produce high energy. Surface shape errors of these large optics, which are mainly affected by gravity, fixture supports, and temperature gradients, will create perturbations in laser beam transmission and cause aberrations and even failure of laser systems [1,2]. In general, the surface shape errors include: (a) thermal deformation of the optical material under laser heating or temperature gradient distribution, and (b) mechanical deformation introduced by the clamping forces of the support frame. Thermal deformation of glass laser systems is usually caused by thermal 1559-128X/14/142988-10$15.00/0 © 2014 Optical Society of America 2988

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stress, which is generated by temperature changes and not free expansion due to the difference in the coefficient of thermal expansion between the optic and clamps. In the optical community, much effort has been devoted to reducing the deformation of the optics with the selection and cooling of optical materials or by optimal design of the support frame [1–4]. However, our paper does not discuss the choice of optical material, which is purely an optical material application problem, but focuses on the support frame design to eliminate or reduce surface shape errors of the large glass laser optics caused by clamping forces or the thermal residual stress of the support frame. Overall, our problem is the optimal design of the optical-mechanical system, which is similar to the fixture design of the mechanical fixture system in manufacturing. To eliminate the effect of deformation or reduce it to an acceptable level is difficult but of great important for the design of optical systems. When the size and the structure of the support frame are determined, the layout of the clamps and the selection of clamping

forces are the key issues in optimization of the support frame. Traditionally, the set of clamps of the support frame is chosen to minimize the deformation of the optical glass. In addition, the support frame’s ability to correct environmental influences such as temperature gradients also needs to be evaluated [5]. In the following, the related work on the analysis of surface errors and thermal deformation of optics is introduced. 2. Related Work

Typical optical deformations are derived from Zernike polynomials or their Karhunen–Loéve transformation [6,7]. Unfortunately, mechanical constraints at the outer edge of faceplate deformable mirrors often do not allow for the high gradients prescribed by the Zernike basis functions [8]. Therefore, a common technique is to approximate the deformations as accurately as possible through finite element analysis (FEA) results. Michels and Genberg [9,10] introduced support frame analysis and design techniques for the development of adaptive optics. Through careful treatment of the nodal displacement predictions from FEA and the use of polynomial fitting, the deformations of grazing incidence optical surfaces may be characterized for subsequent optical analysis. In Cho’s work [11], FEA and optomechanical calculations were performed to optimize the Thirty Meter Telescope tertiary mirror, in which iterative parametric analyses were utilized to achieve the minimum global RMS surface error. Martin et al. [12] described the optimization of support forces for a 6.5 m primary mirror of the Multiple Mirror Telescope Conversion, in which the fixturing functions were determined by FEA, and then the optimization was performed by singular value decomposition of the fixturing functions into normal modes. Lee et al. [13] estimated the fixturing function of optics based on the analytic calculation and FEA, and their model was used in the continuous thin-plate type with discrete actuators in the medium size range from about 10 cm up to 2 m in diameter. Daudeville and Carre [14] discussed the thermal tempering of soda-lime silicate glass plates with Narayanaswamy’s model, in which the tridimensional residual stress state was computed using FEA. Chang et al. [15] analyzed the thermal deformation of optical surface and refractive index distribution caused by temperature gradient distribution, and a simplified mathematical model was built to describe the impact of radial temperature gradients on optical parameters. In their paper, the model was finally validated using FEA. Yu et al. [16] analyzed the thermal behaviors of three different support systems, which they called multi-points support, four edge support, and side face support, for fused silica optics with the finite element method, so that a suitable support scheme to reduce the residual stress was designed based on the analysis. Ning et al. [17] built a finite element model to analyze the influence of thickness, thermal expansion coefficient, and Young’s modulus of materials on thermal stability.

To enable high performance optical systems, integration of the thermal analysis and structural analysis is an essential task [18]. Cho et al. [19] conducted FEA and optical analyses for support frame design. They analyzed the static deformation induced by gravity and temperature, and then established a fixturing matrix to compensate potential errors using an active optics system. However, in the previous integration model, the thermal and structural analyses are conducted independently, and it is difficult to obtain a nearly globally optimal set of clamps. This paper aims to construct approximations for an unknown integrated fixturing function, which is better for describing the mechanical properties and the thermal properties of the support frame. Therefore, (a) we can search for the set of clamps that minimizes the shape error via the influence function, which is capable of obtaining a nearly globally optimal solution to the problem, and (b) we also can obtain a special shape deformation by adjusting the clamping forces or the temperature with the knowledge of the influence function. It should be noted that this second task is not sufficiently realized when using only optimization algorithms. The support vector machine (SVM) has been introduced as a powerful method for classification and regression problems {(called support vector regression (SVR) [20,21]}. The approximation of a function using SVM has some attractive properties [22]. For example, it does not suffer from the overfitting problem and it has good generalization ability. These features suggest that SVR may be a good candidate for constructing approximations for unknown nonlinear functions in the area of mechatronics, such as dynamics identification, friction modeling, and electromyography classification [22–26. In the robotic grasping field, researchers also use SVR for building a mapping between object shape and grasp parameters [27]. In this paper, a new multiple kernel learning (MKL) method is developed to build the nonlinear coupling among the clamping force, temperature, and the deformation of the optic. Then, based on the proposed model, the optimal magnitude and positions of clamping forces can be obtained by some classic optimization algorithms. The rest of the paper is arranged as follows. Section 3 introduces some statements related to this work. Section 4 presents a multiple kernel regression method to construct the integrated fixturing model, which is able to describe the nonlinear coupling among the clamping force, temperature, and the deformations of the optic. Section 5 describes the optimization processing of the optic deformation using a branch-and-bound algorithm. In Section 6, experiments and simulation are presented to illustrate the efficiency of the proposed method. 3. Problem Statements

In this section, we first describe the support structure of the large-scale optics. 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS

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A.

Description of the Support Frame

B. Deformations of the Optics

A support frame for the large-scale optics in a highpower laser system is generally composed of two types of elements: a carriage and some clamps for firmly holding the optics during laser operation. The process of designing the support frame for a high-power laser is similar to the design of fixtures in manufacturing, where we all try to find a suitable set of clamps and select appropriate clamping forces to minimize target deformation. As we know, optical properties are extremely sensitive to thermal and residual stress, so much effort needs to be applied to reduce stress while the laser is running. In comparison with fixture design in the manufacturing industry (a review of the research on optimal fixture design in manufacturing can be found in [28]), the major design constraints of the support structure in a high-power laser system is that the surface shape errors induced by thermal residual stress during laser operation is critical for the placement of the clamps. That is, the fixturing model needs to describe the relationship among the placement of the clamps, the clamping forces, and the temperature. In practical application, an entirely different method is applied to fixturing the optics, in which the clamps provide clamping forces to ensure that the optics cannot slip from the carriage and to support against gravity. To totally restrain the optic on the narrow side surface, and to make the optical glass force uniform with low stress clamping, plastic gel nails are usually used to mitigate the interfacial friction between the support frame and the optics. Figure 1 shows the mechanical structure of the support frame. As shown in Fig. 1(a), the support carriage is comprised of a precisely machined frame with threaded holes and a set of plastic gel nails. In Fig. 1(b), the clamps are at the carriage of the frame, which would apply clamping force to the optics. The design of the support carriage is the layout of the clamps on the threaded holes and the actuating force applied on the optic, which ensure the optics is firmly held by the frame with minimal shape deformation under different working temperature. The nonlinear fixturing model is critical for optimal design of the support frame.

(a)

(b)

Fig. 1. Structure of the support frame. 2990

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There are three perpendicular forces existing on a contact point: a clamping force perpendicular to the contact surface and two shear forces parallel to the contact surface. The clamping force is likely to create shape deformation at the contact area. Meanwhile, the behavior of the optics varies with temperature, i.e., variations of temperature will finally lead to thermal deformation of the optic. Understanding of the relationship between thermal deformation and temperature gradient is not only helpful to thermal control but also to clamping force adjustment. At the present time, determination of the thermal deformation of an optical system under temperature distribution always uses the finite element model. Currently, we compute the shape deformations caused by gravity and the clamping forces with the structure module, and then compute thermal deformations with the thermal module in ANSYS. In this process, the impact of the mechanical structure on thermal deformation is difficult to determine. That is, it is difficult to obtain the optimal support system to decrease or eliminate shape deformation. Intuitively, it is helpful to construct an integrated fixturing model of the support frame for minimizing the clamping force and optimal placement of the clamps. Furthermore, we can dynamically adjust the clamping forces with the various temperatures for decrease of shape deformation. As a powerful method for unknown nonlinear approximation, SVM finds wide use in regression, including in the area of mechatronics. In this paper, we will present a novel support vector regression method for the construction of the nonlinear fixturing model. 4. Multiple Kernel Support Vector Functional Regression

In this section, we will present the new regression method for the construction of the integrated fixturing model for the design of the support frame. Multiple kernel support vector regression (MKSVR) methods have recently been proposed [29–38]. However, in the canonical MKL method, the sample data is translated via a mapping from their input spaces into the three feature spaces. For each feature map there will be a separate weight decided by the training data from different sources. Obviously, the relative importance of the information source is represented by the kernels, in which the choice of kernels is vital for the regression. As we know, in most mechanical applications, the role of different sources can be known a priori. In such a case, the hierarchical method is a possible way to represent the relative importance of the information sources. In the hierarchical method, the importance of the sources are grouped and ordered in different layers depending on the role of the sources. In practice, we know that temperature plays a more important role for the influence function than the

set of the support layout and the set of the clamping forces. In the following, a hierarchical method we call multiple kernel support vector functional regression (MKSVFR) is developed to construct the fixturing model. A.

Structure of MKSVFR

Without loss of generality, suppose there is a set of training data f
xi ;yi gni1 ⊂ RD × R, xi 
f
Sik ; Cik gm k1 ; i . S is the position of the T i , and yi  f
yik gm k1 k i clamps and Ck defines the clamping forces under temperature T i . We define the MKSVFR model, which is composed of two layers, as follows: (a) In the first layer, a sub-regression function f i represents the relationship between the output yik with respect to the input vector
Sik ; Cik . (b) In the second layer, a total regression function f describes the relationship between the sub-function f i with respect to input ti . It should be noted that in the higher layer, the regression represents the relationship of a function and a set of input data, i.e., the relationship of f i and the sample data T i . The mapping from a function to the input data is different with the mapping of two vectors. Because of that, we use a new notation, functional regression, to represent the model. The output of the MKSVFR model is obtained by the mapping as follows: f
x  F
f 1
x1 ; …; f n
xn ;

(1)

where f i
xi  is the ith sub-egression function in the i first layer, xi 
f
Sik ; Cik gm k1 ; T  denotes the input vector, and F is usually a nonlinear map. B.

i

Sub-Regression Function f in the First Layer

We assume all the sub-regressions have the same functional form, as expressed by f i
x 

m X k1

Thus, the ith sub-regression function is f i
xi  

Mi X mi 1

Similar to the parameter learning in localized MKL [29,31,21], we can obtain the resulting discriminant function f i
xi  in the lower layer. In the next step, two methods are proposed to get the total regression function. C.

Functional Regression in the Second Layer

In the lower layer, we build the relationship between the input vector
Sik ; Cik  and the output yik to get the sub-regression function f i
xi . In the top layer, the input vectors become fT i ; f i
xi g; hence, we establish a new regression relating to the input data fT i ; f i
xi g. Suppose that the total function is described by F
X  F
f i
xi ; T i : Then, we can compute the total function via the parameters of f i
xi  or by the kNN method as illustrated in the following. 1. Computation of the Top Function by Treating the Parameters aik and bi of the Sub-Regression Functions as Input Data We treat the parameters aik and bi of the subregression functions and the temperature T i as the input vector of the top regression function, i.e., the input data are X  f
aik ; bi ; T i gni1 : We then can train the parameters A
X and B
X of the top function by A
X 

n X

βi k
X; X i   ν;

(4)

γ i k
X; X i   λ;

(5)

i1

aik K iM
xi ; xik   bi ;

(2) B
X 

xik

is the data point related to the subwhere regression function, aik  αik − αi k is the weight of kth support vector (which is the vector of dual variables corresponding to each separation constraint), and bi is the bias term of the kth regression function., K iM is chosen to be a set of convex combination of M i predefined base kernels: K iM
xi ; xik  

μimi hωimi ; Φimi
xi i  bi :

Mi X mi 1

μimi kimi
xi;mi ; xi;mi k ;

i1

where βi ; γ i is the weight of ith support vector of the top regression function, i.e., they are the vectors of the dual variables corresponding to each separation constraint. ν; λ are the bias terms, so the top regression can be described by F
X 

(3)

where mi indexes the feature spaces of the ith subregression function.

n X

n X

Ai
XK M
X; X i   B
X;

(6)

i1

where Ai
X is the weight of the top regression function, and B is the bias term of the regression function; X is the input data point. 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS

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2. Computation of the Top Function by the kNN Method The kNN regression [39] simply assigns the weight for the regression function to be the average of the values of its k nearest neighbors, in which the nearer neighbors contribute more to the average than the more distant ones. We define the top kNN regression functional form by F
X 

n X

ϖ i
T i  · f i
xi   θ
xi ;

(7)

i1 i

where ϖ i
T  indexes the weight of the ith subregression function, and θ
xi  is a bias term under the condition of xi. To calculate the weight ϖ i
T i , we use a Gaussian kernel to establish the relationship between the distance of nearest neighbors and the weight ϖ i
T i  of the ith sub-regression function: ϖ i
T i   g
‖T − T i ‖2 ;
g
•  κ  exp n X

 ‖T − T i ‖22  λ; σ2

ϖ i  0:

(8) (9)

(10)

i1

In this section, we introduced methods to calculate the fixturing model of the optical system, in which we group and order the input sources into two layers to construct the regression function. In the next section, we will describe the optimization processing of the optic deformation using a branch-and-bound algorithm. 5. Optimization of the Optical Deformation

Once the fixturing model is constructed as in Section 4, we are able to detect the optimal clamp placement and the clamping forces under different temperatures. Without loss of generality, we use the fixturing function obtained by Eq. (7) to discuss the optimization processing. The problem is to minimize the fixturing function F
X of variables
X 1 ; …; X n  over a region of the feasible solution, Ω: Dmin  min
maxF
X; X∈Ω

In this paper, we use the branch-and-bound optimization method [40] to solve the optimal problem [Eq. (11)], as it is a general algorithm for finding global optimal solutions of optimization problems. The description of the branch-and-bound algorithm is shown in Algorithm 1, of which the goal is to find the minimum value of the function F
X, where X ∈ Ω and glb is a bounding function. Algorithm 1. Branch-and-Bound Algorithm

Input: Variables X i  fT i ;
Sik ; Cik g, and the set of feasible solutions Ω. Output: The optimal shape deformation Dmin , and the corresponding X opt Step 1. Initialize: Dmin  ∞; L0  glb
Ω; S  Ω; Step 2. Pick P from S to be processed; S  S − fPg; Step 3. Branch on P generating P1 ; …Pk ; For 0 < i < k, do Bound Pi : Li  glb
Pi ; If Li  F
X for a feasible solution X and F
X < Dmin , then Dmin  F
X, X opt  arg F
X; Go to End; If Li > Dmin, then change Pi ; Else S  S∪f
Li ; Pi g; Step 4. Repeat Step 2 and Step 3 until S  ∅. End.

The key idea of the branch-and-bound algorithm is: if the lower bound for node P is greater than the upper bound for some other node, then P may be safely discarded from the search. In next section, several experiments and a discussion are given. Comparisons between the proposed methods and some classical SVR methods are also given to illustrate the efficiency of the proposed methods. 6. Experiments and Discussion

In this section, we perform several experiments to discuss the influence of the input data T. We first describe the creation of the input datasets. Then we compare the proposed method and the canonical MKL, MKSVR, SVR, and MKSVFR methods. Finally, the optimal deformation of the optic is calculated using the branch-and-bound algorithm. Figure 2 shows a block diagram of the process to implement of the fixturing model and the optimization method into the analysis by ANSYS. The core

(11)

where the set of feasible solutions Ω is determined by the feasible set of clamps and the clamping forces. The set of optimal clamps and the temperature related to the minimal deformation are obtained by X opt  arg min
maxF
X: X∈Ω

(12) Fig. 2. Block diagram of the optimization process.

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apply clamping forces on the optic. Under different temperature conditions, even the same clamping force will cause various shape deformation. The training data, i.e., the structural analysis data and the thermal analysis data, are obtained in ANSYS:

Fig. 3. ANSYS software environment to created datasets.

code is implemented in MATLAB, and interacts with ANSYS FEA. The typical calculation process begins in ANSYS, and the saving of the data to a results file. Then the integrated fixturing model is constructed by the proposed MKSVFR method in MATLAB. The optimization processing with the branch-and-bound algorithm is realized in MATLAB, and can be invoked by ANSYS or can be run interactively from MATLAB. A.

Datasets

The optic used in the experiment is shown in Fig. 3. Its size is 430 mm × 430 mm × 20 mm. As the optic is vertically mounted on the support frame, its optical axis is perpendicular to the support plane. The basic data of the structural materials, i.e., the clamps and the frame, and the optic material are given in Table 1. In general, the available data of the support frame are:

(a) Thermal analysis data: set the temperature by T i , the position of the clamps by Sik 
sik;1 ; …; sik;L , and the clamping forces by Cik 
cik;1 ; …; cik;L , where L is the total number of clamps. Then, we can obtain the shape deformation yik , which describes the thermal residual stress of the optic. (b) Thermal analysis data: change the temperature from T i to T i1 , but do not change the clamp place Sik 
sik;1 ; …; sik;L . Also, do not ment, i.e., Si1 k  Cik 
cik;1 ; …; cik;L . change the clamping force Ci1 k Then, we can obtain a new shape deformation yi1 k . (c) Structural analysis data: denote the temperature by T i, but change the clamps from Sik 
sik;1 ; …; sik;L  to Sik1 
sik1;1 ; …; sik1;L  and clamping force from Cik 
cik;1 ; …; cik;L  to Cik1 
cik1;1 ; …; cik!;L . Then, we can obtain a new output yik1 . Based on the structural analysis and the thermal analysis, we can get the training data as in Table 2, in which the total number of the training data is nearly 1800, and T. denotes the temperature. Pos. denotes the position of the clamps. F. denotes the clamping force. Deform. represents the deform of the optic. B. Calculate the Fixturing Model in MATLAB

We first calculate the fixturing model in MATLAB. The Gaussian kernel used in the fixturing model is

• the temperature around the support frame, and • the position of the clamps and the clamping forces applied on the optic. The simulations on ANSYS are shown in Fig. 3. There are four clamps on each side of the support frame, and each of the 16 clamps can individually Table 1.

Variable Young’s modulus Poisson’s ratio Density

Optic and Support Element Properties

Optic

Plastic Nails

Frame

7.92 GPa 0.25 2530 kg∕m3

0.023 GPa 0.46 1100 kg∕m3

206 GPa 0.28 7850 kg∕m3

Table 2.

T. 22 22 25 25 … 28

km
x; xm i 

For SVR, MKSVR, and MKSVFR, we take the tube width, ε, as 0.01 and the regularization parameter, C, as 100,000. We then choose the best performing σ by a cross-validation procedure. It should be noted that the linear MKSVR (L-MKSVR) is only a linear combination of the sub-regression. For all performed experiments, we quantified the prediction performance with root mean square error (RMSE) and mean absolute percentage error (MAPE). The performance of MKSVFR-function

Input Vector and the Output

Pos. (mm) (40,40,40.40; (40,40,40.40; (40,40,40.40; (40,40,40.40;


 ‖x − xi ‖22  exp : σ2

40,0,0,40; 40,0,0,40;40,40,40.40;) 40,0,0,40; 40,0,0,40;40,40,40.40;) 40,0,0,40; 40,0,0,40;40,40,40.40;) 40,0,0,40; 40,0,0,40;40,40,40.40;) … (0,70,70.0; 0,70,70,0; 0,70,70,0;0,70,70.0;)

F. (N)

Deform. (μm)

(40,40,40,40) (20,30,30,40) (20,30,30,40) (20,30,30,40) … (70,70,70,70)

1.2294 1.3402 2.5915 2.2915 … 1.6131

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Table 4.

Minimal Deformation and the Related Clamping Forces

T. MKSVFR-PR

MKSVFR-FR

Pos. (mm)

26 (54,54,54.54; 0,62,62,0;0,62,62,0; 60,60,60.60;) 26 (54,54,54.54; 0,62,62,0;0,62,62,0; 60,60,60.60;)

F. (N)

Deform. (μm)

(54,62,62,60)

0.8679

(54,62,62,60)

0.8654

of training data increases, the RMSEs shown in Fig. 4(a) and the MAPEs shown in Fig. 4(b) of the proposed MKSVFR-PR and MKSVFR-FR are much better than the results of MKSVR and SVR. In conclusion, the proposed MKSVFR algorithm is suitable to represent the coupling among the clamps, the clamping forces, and the temperature with the shape deformation of the optic. Moreover, the performance, e.g., the MAPE and RMSE, of the proposed MKSVFR algorithm is much better than the results of MKSVR and SVR. C.

Once the fixturing model is constructed, we can search the clamp placement and the clamping forces with a classic branch-and-bound algorithm, e.g., BNB20 toolbox, in MATLAB. The minimal deformation of the optic under a special temperature (T  26) can then be computed via the fixturing model constructed by the MKSVFR-PR method or the MKSVFR-FR method. The results are listed in Table 4.

Fig. 4. (a) RMSEs. (b) MAPEs.

D.

regression (MKSVFR-FR) and MKSVFR-parameter regression (MKSVFR-PR) are compared with the SVR, MKSVR, and L-MKSVR models based on the data sources given in Table 2. This comparison is depicted in Fig. 4 and shown in Table 3. These results show that the MKSVFR-FR and MKSVFR-PR models outperform the SVR, MKSVR, and L-MKSVR models. Table 3 lists the RMSE and the MAPE of the proposed MKSVFR method, the MKSVR method, and the SVR method on the datasets (the datasets are shown in Table 2). We can see that both the RMSE and the MAPE of the MKSVFR method are better than those of the MKSVR and SVR methods. It can be seen from Fig. 4 that the RMSEs and the MAPE reduce as the number of training data increases. We can also know that when the number Table 3.

Show the Optimal Shape Deformation in ANSYS

The simulation of the deformation of the optic in ANASYS is shown in Fig. 5, in which the max deformation of the optic is 0.8738. That is, the MAPE of the fixturing model is 6.8% constructed by MKSVFR-PR, and is 9.7% constructed by MKSVFR-FR. We can also compute the optimal clamp placements, the clamping force, and the temperature, which correspond to the global minimal deformation of the optic. The minimal deformation of the optic can then be obtained via the fixturing model constructed

Comparison of the MKSVFR and Other Methods

Prediction Model MKSVFR-PR MKSVFR-FR MKSVR SVR L-MKSVR

2994

Implement the Optimization Processing in MATLAB

MAPE (%)

RMSE (μm)

5.67 5.80 6.32 9.46 10.62

0.1172 0.1082 0.1278 0.1745 0.1957

APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014

Fig. 5. Deformation of the optic under a given temperature.

Table 5.

Minimal Deformation and the Related Clamping Forces

T. MKSVFR-PR 21.761 MKSVFR-FR 21.732

Pos. (mm)

F. (N)

Deform (μm)

(0,15,15.0 0,20, (16,20,20,30) 0.71030 20,0; 0,20,20,0; 30,30,30.30;) (0,15,15.0 0,20, (15,20,20,30) 0.70756 20,0; 0,20,20,0; 30,30,30.30;)

measuring errors, show the superiority of the proposed method over SVR, MKSVR, and L-MKSVR. Moreover, by using a hierarchical approach, it is able to emphasize the role of some important regions. Meanwhile, the integrated clamping function constructed by the proposed method is better for describing the mechanical properties and thermal properties of the support frame of the optical system. It is also capable of searching a set of clamping forces and clamp placement to minimize the shape error by using a branch-and-bound algorithm, and it is also capable of obtaining a nearly globally optimal solution to the problem. We can also obtain a special shape deformation by adjusting the clamp layouts, clamping forces, or the temperature with knowledge of the clamping function. It should be noted that this second task is not sufficiently realized when using only optimization algorithms. Appendix A

Fig. 6. Deformation of the optic.

by the MKSVFR-PR method or the MKSVFR-FR method. The results are listed in Table 5. The simulation of the deformation in ANASYS is shown in Fig. 6, in which the maximum deformation of the optic is 0.7251. That is, the MAPE of the proposed model is 2.083% via MKSVFR-PR, and is 2.4789% via MKSVFR-FR. As discussed in Tables 4 and Table 5, the optimal shape deformation can be calculated using a branchand-bound algorithm. We can also compute the clamp placement, the clamping forces, and the environmental temperature corresponding to the minimal deformation of the optic by the branch-and-bound algorithm.

Here we use an optic with a hemisphere-like shape to discuss the proposed regression and optimization model. In the experiment, the clamp layout and the clamping force should be searched in threedimensional space, as discussed in the following. The optic used in the experiment is shown in Fig. 7. Its diameter is 100 mm, and its height is 30 mm. The tilted angle of the support plane with the ground is 45°. The basic data of the structural materials are given in Table 6. The simulations in ANSYS are shown in Fig. 7. Four clamps are on the upper side of the optic, and eight clamps are on the outside of the optic. Each of the 12 clamps can individually apply clamping forces on the optic. Similarly, under different temperature conditions, even the same clamping force will cause various shape deformation. Based on the structural analysis and the thermal analysis, we can get the training data, as in Table 7, in which the total number of training data is nearly 180, and T. denotes the temperature.

7. Conclusion

In this paper, we have proposed a new approach that can be used to solve the optimal design of the support frame of an optical system. The approach uses twolayer regression to construct the fixturing model of the frame. The first layer is used to obtain a set of sub-regression functions f i, then the second layer establishes a top-regression function f , in which the set of sub-function f i is taken as input data. We propose two methods to compute the top function. One method computes the top function by treating the parameters of the sub-regression functions as input data, and the other directly computes the top regression function by the kNN method. A dataset obtained from the ANSYS software has been used to evaluate the performance of the proposed model. The proposed model has been compared with the SVR, MKSVR, and L-MKSVR methods. The numerical results, achieved on the basis of different

Fig. 7. Layout of the clamps.

Table 6.

Variable Young’s modulus

Properties of the Optic

Young’s Modulus Poisson’s Ratio 10.23 GPa

0.27

Density 2560 kg∕m3

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Table 7.

Input Vector and the Output

T.

Force

Deform. (μm)

22 22 24 … 28

(10,15,15,20; 50,50,50,50; 60,60,60,60) (15,15,15,15;50,50,50,50; 60,50,50,60;) (20,15,15.30; 60,70,70,80; 80,80,80,80;) … (20,25,25.30; 60,80,80,100; 70,80,80,100;)

3.42 4.02 2.98 … 4.62

Table 8.

Minimal Deformation and the Related Clamping Forces

MKSVFR-FR MKSVFR-PR

T.

Force

23.7 23.7

(18,15,15.26; 60,72,72,80; 70,75,75,80;) (18,15,15.26; 61,72,72,80; 70,75,75,80;)

Fig. 8. Deformation of the optic.

Under a given temperature, we can obtain the clamp placements and the clamping forces with the BNB20 toolbox in MATLAB. We can also compute the clamping forces and the temperature, which correspond to the global minimal deformation of the optic, as listed in Table 8. The simulation of the deformation of the optic in ANSYS is shown in Fig. 8, in which maximal deformation of the optic is 2.23. This work was supported in part by the National Natural Science Foundation of China under Grants 61105085, 61033011, 61100098, and 61210009, and by the Beijing Natural Science Foundation (4142056). The authors would like to thank Prof. Zhiyong Liu for his help and valuable suggestions. References 1. L. P. Zhou and D. W. Tang, “A functionally graded structural design of mirrors for reducing their thermal deformations in high-power laser systems by finite element method,” Opt. Laser Technol. 39, 980–986 (2007). 2. G. L. Herrit and H. E. Reedy, “Advanced figure of merit evaluation for CO2 laser optics using finite element analysis,” Proc. SPIE 1047, 33–42 (1989). 3. Y. F. Peng, J. L. Cui, Z. H. Cheng, D. L. Zuo, and Y. N. Zhang, “Characteristics of thermal distortions of the laser mirror substrates filled with phase change materials,” Opt. Laser Technol. 38, 594–598 (2006). 4. Y. Miyamoto, W. A. Kaysser, B. H. Rabin, A. Kawasaki, and R. G. Ford, Functionally Graded Materials: Design, Processing and Applications (Kluwer, 1999). 2996

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