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Enhanced two-point diagonal quadratic approximation methods for design optimization
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Comput. Methods Appl. Mech. Engrg. 197 (2008) 846–856 www.elsevier.com/locate/cma
Enhanced two-point diagonal quadratic approximation methods for design optimization Jong-Rip Kim
a,1
, Dong-Hoon Choi
b,*
a
CAE Group, Corporate R&D Institute, Samsung Electro-Mechanics Co., Ltd., 314, Maetan3-Dong, Yeongtong-Gu, Suwon, Gyunggi-Do 443-743, Republic of Korea b FRAMAX Co., Ltd. and the Center of Innovative Design Optimization Technology, Hanyang University, Haendang-Dong, Sungdong-Ku, Seoul 133-791, Republic of Korea Received 20 June 2007; received in revised form 23 September 2007; accepted 24 September 2007 Available online 29 September 2007
Abstract Based on two-point diagonal quadratic approximation (referred to as TDQA), developed by Kim et al. [Min-Soo Kim, Jong-Rip Kim, Jae-Young Jeon, Dong-Hoon Choi, Efficient mechanical system optimization using two-point diagonal approximation in the nonlinear intervening variable space, J. KSME 15 (2001) 1257–126], enhanced two-point approximation methods are proposed in this paper. The suggested methods reinforce TDQA with new quadratic correction terms using the concept of TANA-3. These methods overcome the disadvantage of TDQA when the derivatives at two design points have the same sign. In addition, the values and derivatives of the proposed approximation functions are completely equal to those of an original function at the two design points whether the derivatives at those points have the same sign or not. Several examples show the numerical performance and accuracy of the proposed methods compared to previous work, and optimization examples show that these methods can successfully reach the optimum via sequential approximation optimization. 2007 Elsevier B.V. All rights reserved. Keywords: Function approximation; Two-point approximation; Quadratic approximation; Sequential approximate optimization
1. Introduction Function approximation is one of the most important and active research fields in design optimization. In an optimization procedure, model analysis is completed repetitively to find the search direction and to perform the line search. If the model analysis cost is high, optimization will be inefficient. To reduce the repetitive cost of model analysis, Schmit and coworkers introduced suitable approximation concepts in the 1970s [1–3]. They combined the now-familiar techniques of intervening variable definition,
*
Corresponding author. Tel.: +82 2 2220 0478; fax: +82 2 2291 4070. E-mail addresses: [email protected] (J.-R. Kim), dhchoi@ hanyang.ac.kr (D.-H. Choi). 1 Tel.: +82 31 210 6776; fax: +82 31 300 7900x0117. 0045-7825/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2007.09.014
explicit approximation, reduced basis and design variable linking as well as constraint deletion and regionalization. In the 1980s, most approximations were based on function and gradient information at a single point and constructed by a first-order Taylor series expansion at this point, which are the linear, reciprocal and conservative approximations in Schmit and Fleury [3]. These are very popular because the function and its derivative values are required in most optimization algorithms, so no additional computation is involved in constructing the approximation functions. Although these approximations work effectively for stress and displacement functions, their truncated error sometimes increases and they are only valid in the vicinity of the expanding point. Haftka et al. [4] suggested several methods that include a modified reciprocal, a two-point projection, and an exponential approximation. The CONLIN scheme developed by Fleury and Braibant [5] is a
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convex approximation based on the first-order Taylor series expansion in terms of direct and reciprocal design variables and has shown good convergence properties in dealing with structural optimization problems. It was also independently developed by Starnes and Haftka [6] as the conservative approximation. Svanberg [7] proposed a modification of the CONLIN scheme called the Method of Moving Asymptotes (MMA). In this method, the linearization variables can be used to adjust the degree of convexity and conservativeness depending on the problem. Chickermane and Gea [8] extended a convex approximation algorithm for solving structural optimization problems. This algorithm – the Generalized Convex Approximation (GCA) method – generates a high-fidelity function approximation using the design sensitivity information in the current and previous design points. In order to make full use of the known information to construct approximation functions, many multi-point approximations have been developed since 1990 [9–14]. Among them, two-point approximation methods have been widely used for their simplicity. Fadel and his coworkers [9] were the first to propose Two-Point Exponential Approximation (TPEA) – an extension of the socalled projection method proposed by Haftka et al. [4]. They employed intervening variables with an exponential form as proposed by Prasad [15]; the exponents were then computed by matching the derivatives of the approximation function with those of the exact function at previous design point. Based on TPEA, Wang and Grandhi [10,11] developed a series of improved two-point approximations using both function and gradient information at two design points, which were called TPEA-change, TANA, TANA-1 and TANA-2. Xu and Grandhi [14] further developed TANA-3 which employs a diagonal and changeable Hessian matrix in order to avoid the computational burden of solving non-linear equations for each function as TANA-2 requires. However, TANA-3 still gives some inaccuracies when the derivatives of the two design points have values with different signs. Salajegheh [16] proposed a three-point approximation method based on the function and gradient information at three consecutive design points. He also applied this method to optimize the plate structure subjected to stress and frequency constraints. Sui [17] has also proposed a two-point approximation approach by employing the information obtained at two design points. Xu and Yamazaki [18] developed a new two-point approximation using the linear combination of Taylor expansions in terms of both the direct and reciprocal variables, and extended to a three-point approximation [19]. The expression of that scheme consists of a linear combination of the direct and reciprocal linear Taylor expansions as well as the lumped diagonal terms of the second-order direct and inverse terms. Recently, Groenwold et al. [20] presented an incomplete series expansion (ISE) as a basis for function approximation. They also presented an example of intervening variables for the ISE function. Another multi-point approximation scheme is a cumulative
847
approximation by combining a one-point or two-point approximation and a blending function [21–23]. In 2001, Kim et al. developed a two-point diagonal quadratic approximation which is abbreviated as TDQA [24]. Shifting coefficients were introduced into the intervening variables in order to avoid the singularity of the approximation gradients in TDQA. To overcome the critical difficulty, due to which other two-point approximations cannot be used because the derivatives of two design points have values with different signs, a new quadratic term with respect to the intervening variables was introduced. The components of the new quadratic term are determined in order to match the derivatives of the approximation function with those of the previous design point along each intervening variable axis, respectively. Also, a uniquely determined correction coefficient is multiplied with the quadratic term to match the approximate function value with that of the previous design point. When the derivatives of two design points have the same sign, TDQA gives less accurate results compared with other two-point approximations because it is constructed as a two point exponential approximation (TPEA) for the current design point in this case. In order to overcome this disadvantage of TDQA, this paper presents enhanced two-point diagonal quadratic approximation methods. The new quadratic correction terms, which are made using the concept of TANA-3, are added to TDQA. Then, a uniquely determined correction coefficient is multiplied with the new quadratic correction terms to match the function values at the previous design point. Therefore, the proposed methods overcome the demerit of TDQA and can match the function and derivative values of the approximation with the exact ones at the two design points. Section 2 reviews typical two-point approximation methods. Section 3 fully describes the proposed methods. Section 4 compares the results of the proposed methods with those of other approximation methods. 2. Review of two-point approximations In this section, we describe the mathematical details of the previous two-point approximations such as TPEA, TANA series and TDQA in order to better explain the proposed method. The known design points are denoted as x1(x1,1, x2,1, . . . , xn,1) and x2(x1,2, x2,2, . . . , xn,2) where the function and gradient information are available. Here n is the number of design variables. The function g~ðxÞ denotes the approximation function based on two-point approximation, which is expanded at the current design point x2 and uses the values of the function and/or derivatives at two design points. 2.1. Two-point exponential approximation (TPEA) Fadel et al. [9] first developed a two-point exponential approximation. It is a linear Taylor series approximation in terms of the intervening variables
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y i ¼ xi i ;
i ¼ 1; 2; . . . ; n;
ð1Þ
where the exponent pi for each design variable was evaluated by matching the derivatives of the approximation function with those at the previous design point. The approximation function is given in terms of the original variables xi as 1p n X ogðx2 Þ xi;2 i pi p ðxi xi;2i Þ: g~ðxÞ ¼ gðx2 Þ þ ox p i i i¼1
ð2Þ
In this approximation, the value of pi is limited from 1 to +1. However, Wang and Grandhi [10] removed the limitation of pi for better adaptability to different structural problems, which was called the TPEA-change method. 2.2. Two-point adaptive non-linear approximations (TANA series) Wang and Grandhi [10] proposed the TANA method using adaptive intervening variables y i ¼ xri , i = 1, 2, . . . , n where r represents the non-linearity index, which is updated during each iteration but is the same for all variables. The non-linearity index was determined by matching the function value at the previous point. Also, in order to utilize more information to construct a better approximation, Wang and Grandhi [11] proposed the following two approximation methods (TANA-1 and TANA-2) to combine TPEA-change and TANA methods, and Xu and Gradhi [14] developed TANA-3. These approximation methods are written by expanding the function at x2 except TANA-1 at x1 TANA-1 :
1p n X ogðx1 Þ xi;1 i g~ðxÞ ¼ gðx1 Þ þ oxi pi i¼1 p
p
ðxi i xi;1i Þ þ e1 ; TANA-2 :
TANA-3 :
ð3Þ 1p xi;2 i
n X ogðx2 Þ p p ðxi i xi;2i Þ ox p i i i¼1 n X 1 p p þ e2 ðx i xi;2i Þ2 ; ð4Þ 2 i¼1 i
g~ðxÞ ¼ gðx2 Þ þ
1p n X ogðx2 Þ xi;2 i pi p ðxi xi;2i Þ g~ðxÞ ¼ gðx2 Þ þ ox p i i i¼1 n X 1 p p 2 þ e3 ðxÞ ðxi i xi;2i Þ ; ð5Þ 2 i¼1
where ," e3 ðxÞ ¼ H
# n n X X pi pi 2 pi pi 2 ðxi xi;1 Þ þ ðxi xi;2 Þ : i¼1
ð6Þ
i¼1
In TANA-1, the correction term e1 is a constant in order to compensate for approximation error, which is computed by matching the function value of the approximation with the exact function value at the previous design point. TANA-2 and TANA-3 include second-order Taylor series
effects. In TANA-2, the Hessian matrix has only diagonal elements of the same value e2. Unlike the original secondorder approximation, the approximation is expanded in terms of the intervening variables y, so the error from the approximation Hessian matrix is partially corrected by adjusting the non-linearity index pi. TANA-2 has n + 1 unknown constants, so n + 1 equations are required. The n equations are obtained by matching the derivative values of the approximation function with those of the original function at the previous point. The (n + 1)th equation is obtained by matching the function value of the approximation function with the exact value at the previous point. Then, n + 1 unknown constants can be obtained by solving these n + 1 equations simultaneously. In TANA-3, in order to avoid the computational burden of TANA-2, a function e3(x) is employed, which uncouples pi and H in the equations in order to compute them. Then the unknown parameters are obtained in a closed-form solution to match the derivative and values of the approximation function with the exact values at the previous point. So, though TANA-2 and TANA-3 have almost the same accuracy, TANA-3 has the advantage of being less computationally intensive than TANA-2. 2.3. Two-point diagonal quadratic approximation, TDQA In 2001, Kim et al. developed a two-point diagonal quadratic approximation method [24]. This approximation method used the intervening variables as in p
y i ¼ ðxi þ ci Þ i ;
i ¼ 1; 2; . . . ; n;
ð7Þ
where ci is the shifting level for the ith design variable. This shifting level can avoid the singularity of the approximation derivatives and the associated fundamental difficulties. In order to match the derivatives of the approximation function with those of the exact function at the previous point, the exponent pi is determined without considering the quadratic term. The approximation is created by expanding the function at x2 g~ðxÞ ¼ gðx2 Þ þ
n n X ogðx2 Þ 1 X 2 ðy i y i;2 Þ þ g Gi ðy i y i;2 Þ ; oy 2 i i¼1 i¼1
ð8Þ where Gi, which is the ith component of the diagonal Hessian matrix, is predefined as 1 ogðx1 Þ ogðx2 Þ Gi ¼ ð9Þ ðy i;1 y i;2 Þ oy i oy i and the correction coefficient g is uniquely determined in order to match the function value of the approximation function with the exact value at the previous point. Unlike TANA-2 and TANA-3, this method determines the unknown parameters pi, Gi and g sequentially and uniquely. Numerical considerations which occur when the unknown parameters are determined are also presented [24].
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3. Enhanced two-point diagonal quadratic approximation This study presents enhanced two-point diagonal quadratic approximation methods in order to guarantee the approximation accuracy whether the derivatives at two design points have the same sign or not. The proposed methods reinforce TDQA with additional quadratic correction terms. These quadratic correction terms consist of a correction coefficient as a numerator and a decoupling factor as the denominator. The denominators are defined so that approximation errors are zero at two design points as in TANA-3. These methods use the shifted intervening variables in Eq. (7) like TDQA. The proposed methods are represented by expanding the function at x2 TDQA I: g~ðxÞ ¼ gðx2 Þ þ
n X ogðx2 Þ ðy i y i;2 Þ oy i i¼1
n 1X 2 Gi ðy i y i;2 Þ 2 i¼1 " #, n n X X 1 2 2 þ ge ðy i y i;2 Þ ðy i y i;1 Þ 2 i¼1 i¼1 ! n X 2 ðy i y i;2 Þ ; þ ð10Þ
þ
849
posed methods uniquely determine the unknown parameters Gi, Hi, ge and pi. We predefine Gi as Eq. (9) to match the derivatives of the approximation function with the exact values at the previous point along each intervening variable axis when those at two design points have different signs. We also define Hi with the same process for Gi. But when the derivatives at two points have the same sign, Gi is equal to zero. In order to prevent the denominator of the quadratic correction term in TDQA II from being equal to 0, we reset Hi = 1 in this case. This is because Gi and Hi may be exact second-order derivatives in the intervening variable axis. Therefore, Hi can be defined as Gi if ½ogðx1 Þ=oxi ½ogðx2 Þ=oxi 6 0 Hi ¼ ð12Þ 1 otherwise: In order to match the approximate function value with that of the exact function at the previous point, the correction coefficient ge is determined to be " n X ogðx2 Þ ge ¼ 2 gðx1 Þ gðx2 Þ ðy i;1 y i;2 Þ oy i i¼1 # n 1X 2 Gi ðy i;1 y i;2 Þ : ð13Þ 2 i¼1
i¼1
TDQA II:
n X ogðx2 Þ g~ðxÞ ¼ gðx2 Þ þ ðy i y i;2 Þ oy i i¼1 n 1X Gi ðy i y i;2 Þ2 2 i¼1 " #, n n X X 1 2 þ ge H i ðy i y i;2 Þ H i ðy i y i;1 Þ2 2 i¼1 i¼1 ! n X 2 H i ðy i y i;2 Þ ; ð11Þ þ
þ
TDQA I and TDQA II have the same correction coefficient ge in Eq. (13). So Eqs. (10), (11) and (13) show that function values of the approximation function are equal to those of the exact function at two design points. The exponent pi is determined to match the derivative values of the approximation function with those of the exact function at the previous point in the intervening variable space when the values at two design points have the same sign. Then, the derivatives of Eqs. (10) and (11) with respect to xi can be derived as
i¼1
2
2
Pn
2
Pn
2
3
3
o~ gðxÞ 6 6ðy i y i;2 Þ j¼1 ðy j y j;1 Þ ðy i y i;1 Þ j¼1 ðy j y j;2 Þ 7 ogðx2 Þ 7 p 1 ¼ 4ge 4 þ Gi ðy i y i;2 Þ5 pi ðxi þ ci Þ i ; 5þ nP o2 Pn oxi oy i n 2 2 j¼1 ðy j y j;1 Þ þ j¼1 ðy j y j;2 Þ
ð14Þ
2 3 3 Pn Pn 2 2 o~ gðxÞ 6 6ðy i y i;2 Þ j¼1 H j ðy j y j;1 Þ ðy i y i;1 Þ j¼1 H j ðy j y j;2 Þ 7 ogðx2 Þ 7 p 1 ¼ 4ge H i 4 þ Gi ðy i y i;2 Þ5 pi ðxi þ ci Þ i 5þ nP o2 P oxi oy n 2 n 2 i j¼1 H j ðy j y j;1 Þ þ j¼1 H j ðy j y j;2 Þ 2
ð15Þ
where Gi and Hi are the ith components of the diagonal Hessian matrix and ge is the correction coefficient. The pro-
Eqs. (14) and (15) show that the derivative values of the approximation function are equal to those of the exact
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function at the current point. When those at two design points have the same sign, Gi is equal to 0. Therefore, at the previous point, the exponent pi is determined to be ogðx1 Þ ogðx2 Þ pi ¼ 1 þ ln ln½ðxi;1 þ ci Þ=ðxi;2 þ ci Þ: oxi oxi ð16Þ When the derivative values at two design points have different signs, pi is set as a special value (1 or 1), but the derivative values of the approximation function are nevertheless equivalent to those of the exact function at those points by the term Gi. Therefore, the function and derivative values of the approximation function are completely equal to the exact values at two design points in the proposed methods, unlike other two-point approximation methods. Despite the fact that Hi is added to TDQA II, unlike TDQA I, both TDQA I and TDQA II exhibit the same or very similar accuracy. A significant difference exists between the derivative values of TDQA I and TDQA II. Like TANA-3, TDQA I may falsely give non-zero derivative values with respect to some design variables on which the original function is not dependent [13]. In other words, although g(x) does not depend on xi, Eq. (14) may not equal zero. But the approximate derivatives of TDQA II with respect to design variables, of which the original function is independent, are always zero because Gi = 0 in Eq. (9) and Hi = 0 in Eq. (12). Therefore, when the original function is independent of any design variables, we highly recommend that TDQA II should be used instead of TDQA I. 3.1. Numerical considerations Now, we describe three guidelines in order to avoid the numerical difficulties that occur when determining the three parameters of pi, Gi and Hi. • First, pi cannot be numerically determined when [og(x1)/ oxi] Æ [og(x2)/oxi] 6 0 or ðxi;1 þ ci Þ=ðxi;2 þ ci Þ 1. Thus, if [og(x1)/o xi] Æ [og(x2)/oxi] 6 0, we reset pi = 1 or 1. Also, if j ðxi;1 þ ci Þ=ðxi;2 þ ci Þ 1 j6 e for a small positive real number e, we reset pi = 1. The suggested value of e is 0.001. If the value of the exponent jpij becomes too large, the approximation function g~ðxÞ can be numerically ill-conditioned. Thus, we bound jpij 6 pmax with limit value pmax = 5. If pi evaluated by Eq. (16) exceeds the range (pmax, pmax), it is rounded up to pmax or down to pmax. • Second, if jyi,2 – yi,1j 6 e, the values of Gi and Hi may go to infinity. Therefore, we reset Gi = 0 and Hi = 0. This means that a linear approximation is employed for the intervening variable yi. P Pn n 2 • Finally, in TDQA II, if i¼1 H i ðy i y i;1 Þ þ i¼1 H i ðy i y i;2 Þ2 j 6 e, we reset Hi = 1. It is necessary in order to prevent the denominator of the new quadratic term from being zero. In this case, TDQA II is equal to TDQA I.
4. Numerical results This section is divided into two parts. The first part is for simple problems with closed-form solutions, whereas the second part is for applications in engineering optimization. 4.1. Part 1: comparisons using closed-form problems The following five examples are selected to examine the difference among two-point approximation methods such as TANA-3, TDQA, TDQA I and TDQA II. We use the relative percentage error as the criterion in evaluating the accuracy. For all of the test examples, the relative percentage error index is defined as Relative percentage error ¼
Exact Approximation Exact 100ð%Þ ð17Þ
and the test points are derived using x ¼ x2 þ aD;
ð18Þ
where x2 is an expanding point or a current point, a is a step length, and D is a search direction vector. Example 1. This example is used in Ref. [11] in order to compare the two-point approximations mentioned above. The function to be approximated is defined as gðxÞ ¼
10 30 15 2 25 108 40 47 þ þ þ þ þ 3 þ þ 3 1:0: x1 x31 x2 x32 x3 x4 x4 x3 ð19Þ
The expanding points for all methods are selected to be x2 = (1.0, 1.0, 1.0, 1.0)T, and the previous points are selected to be x1 = (1.2, 1.2, 1.2, 1.2)T. By using D = (1.0, 1.0, 1.0, 1.0)T the relative percentage errors are determined in Eq. (17). In TANA-3, H is 0.242 and p = (2.76, 1.50, 2.83, 2.49)T. For this example, the derivative values at all points are negative. That is, with two arbitrary design points selected, the derivative values have the same sign. In this case, TANA-3 guarantees better accuracy than TDQA, which becomes TPEA. When the derivatives at two points have the same sign with Eqs. (9) and (12), we acquire Gi = 0, i = 1, . . . , n and Hi = 1, i = 1, . . . , n in TDQA I and TDQA II. By Eq. (13), the correction coefficient ge is equal to H of TANA-3 mathematically. Therefore, when the derivatives at two points have the same sign, the proposed methods execute approximation similar to TANA-3. Fig. 1 shows the results of the relative percentage errors of all methods. Comparing the relative percentage errors, TANA-3 has better accuracy than TDQA for this example and the proposed methods produce the same approximation function as TANA-3. Consequently, TDQA I and TDQA II have the same relative percentage error as TANA-3.
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851
better accuracy than TDQA. The proposed methods produce the same approximation function as TANA-3. Therefore, they have the same relative percentage errors. Example 3. This example has also been borrowed from Ref. [11] and the function to be approximated is defined as 2 3 0:125 2 1 0:5 þ 15x1 gðxÞ ¼ 10x1 x1 2 x4 x6 x7 1 x2 x3 x4 x5 x7 1 2 2 2 1 0:5 2 þ 20x2 1 x2 x4 x5 x6 þ 25x1 x2 x3 x5 x6 x7 :
Fig. 1. Example 1.
Example 2. This example also has been borrowed from Ref. [11] in order to make a comparative study of the accuracy of approximation methods. The function to be approximated is defined as gðxÞ ¼ 180x1 þ 20x2 3:1x3 þ 0:2x4 5x1 x2 þ 37x1 x3 þ 8:7x2 x4 3x3 x4 0:1x21 x2 þ 0:001x22 x3 þ 95x1 x24 81x4 x23 þ x31 6:2x32 þ 0:48x33 þ 22x34 1:0:
ð20Þ
The current design point is selected to be x2 = (1.5, 1.5, 1.5, 1.5)T. The previous design point is selected to be x1 = (2.0, 2.0, 2.0, 2.0)T. As a search direction, D = (1.0, 1.0, 1.0, 1.0)T is used. Under these circumstances, the exponent indices are p = (2.214, 4.681, 3.124, 2.981)T. In TDQA, Gi = 0, i = 1, . . . , n. In TANA-3, H = 4.729. In the proposed methods, Hi = 1, i = 1, . . . , n. The correction coefficient ge is equal to 4.729, which is the same value as TANA-3’s H. Fig. 2 shows the results of the relative percentage errors of all methods. In this example, the derivatives have the same sign at two points and TANA-3 has
Fig. 2. Example 2.
ð21Þ
In this example, in order to examine the mixed condition that includes the case, in which the derivatives have the same sign, and the case in which the derivatives have different signs, two design points are selected to be x1 = (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)T and x2 = (0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9)T. At two design points, design variables x1 and x3 satisfy the condition in which the values of [og(x1)/ oxi] Æ [og(x2)/oxi] are negative signed values. As a search direction, D = (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)T is used. Under these circumstances, the exponent indices are p = (1, 6.055, 1, 0.643, 4.001, 2.969, 5)T. In TDQA, G = (247.520, 0, 103.438, 0, 0, 0, 0)T, and the correction coefficient g is 0.938. In TANA-3 H is 1.895. In the proposed methods, H = (247.520, 1, 103.426, 1, 1, 1, 1)T and the correction coefficient ge is 0.126. Fig. 3 shows the results of the relative percentage errors of all methods. When the derivatives at two points have different signs, TDQA has better accuracy than TANA-3 and the proposed methods have similar accuracy to TDQA because the original quadratic term Gi, which is also in the proposed methods, dominates the new quadratic term in this case. Example 4. This example was examined in Ref. [13] and the function to be approximated is defined as pffiffiffi 2 3 gðxÞ ¼ 1 þ : ð22Þ 3x1 x2 þ 0:25x1
Fig. 3. Example 3.
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Fig. 5. Example 5.
Fig. 4. Example 4.
The expanding design point is selected to be x2 = (1.5, 0.5)T. The previous design point is selected to be x1 = (1.0, 0.75)T. As a search direction, D = (0.5, 0.25)T is used. At two design points, the derivatives with respect to x1 have different signs but those with respect to x2 have the same sign. Under these circumstances, the exponent indices pi are 1 and 0.341. In TDQA, Gi are 0.977 and 0, and the correction coefficient g is 0.915. In TANA-3, H is 0.217. In the proposed methods, Hi are 0.977 and 1, and the correction coefficient ge is -0.020. In Fig. 4, the relative percentage errors of all methods are shown. In this example, when the derivatives at two points have different signs, TDQA has better accuracy than TANA-3 and the proposed methods have accuracy similar to TDQA. Example 5. This example is from Ref. [25] in order to illustrate the ‘Saddle Point’. The function to be approximated is defined as 2
2
gðxÞ ¼ ðx1 1Þ ðx2 1Þ þ x1 x2 :
ð23Þ
In TDQA, when at least one component of the diagonal Hessian matrix is not zero and g is equal to 1, theoretically, at two design points the function and the derivative values of TDQA are equal to those of the exact function. In order for g to be 1 in TDQA, the current design point is selected as x2 = (0.5, 1.5)T and the previous design point is selected as x1 = (1.5, 0.5)T. Under these circumstances, the exponent indices pi are 1 and 2. In TDQA Gi are 3.0 and 0. In TANA-3, H is 3. In the proposed methods, Hi are 3 and 1, and the correction coefficient ge is 0. In this case, mathematically, the proposed methods are completely equal to TDQA. Using D = (1.0, 1.0)T as the search direction, the relative percentage errors of all methods are shown in Fig. 5. Considering the signs of derivatives at two points, the derivatives with respect to x1 have different signs but those with respect to x2 have the same sign. Fig. 5 shows that TDQA has better accuracy than TANA-3 for this case and the relative percentage error of TDQA is almost zero. TDQA and the proposed methods have the same relative percentage errors.
4.2. Part 2: applications in engineering optimization In order to examine the numerical performance of the enhanced two-point approximation methods, a sequential approximation optimizer, which has various two-point approximations as options, is developed. Fig. 6 shows a flow chart of this process. This sequential approximation optimization procedure was described in Kim et al. [24]. The optimization results of TANA-3 as well as TDQA I and TDQA II are presented. The first example is the Rosen–Suzuki problem. The other examples are applications in structural optimization. This study uses the modified feasible direction method in DOT 4.20 [26]. In the sequential approximation optimization procedure, we used conservative approximation with a 40 percent move limit for the first iteration, and various move limits are applied after the second iteration. Five move limit strategies are selected. These are 40%, 75%, 100%, 150% and no move limits. The move limit strategy is defined for the conservative approximation method as xL ¼ maxðxk maxðml jxk j; 0:3Þ; xL0 Þ; xU ¼ minðxk þ maxðml jxk j; 0:3Þ; xU 0 Þ
Fig. 6. Flow chart of SAO process.
ð24Þ
J.-R. Kim, D.-H. Choi / Comput. Methods Appl. Mech. Engrg. 197 (2008) 846–856
and for the two-point approximation methods as L
x ¼ maxðminðxk1 maxðml jxk1 j; 0:3Þ; xk maxðml jxk j; 0:3ÞÞ; xL0 Þ xU ¼ minðmaxðxk1 þ maxðml jxk1 j; 0:3Þ; xk þ maxðml jxk j; 0:3ÞÞ; xU 0 Þ
ð25Þ
Here k is the number of iterations in SAO. xL0 and xU 0 are the initial lower and upper limits of design variables. The move limit ratio is abbreviated as ml. Example 6. Rosen–Suzuki Problem. This problem has been extensively used in Rosen and Suzuki [27] to verify the algorithm in the optimization procedure because it has many local optima. There are four design variables and three constraints. Originally the lower and upper limit of design variables did not exist. But in this study, they are taken as [10, 10]. The initial design is taken as x = (0.0, 0.0, 0.0, 0.0)T. Constraints 1 and 3 are active at the optimum. The optimum is known as f(x*) = 56.0 and x* = (0.0, 1.0, 2.0, 1.0)T. The mathematical formulation is minimize
f ðxÞ ¼ x21 þ x22 þ 2x23 þ x24 5x1 5x2
move limit, TANA-3, TDQA I, and TDQA II all need 8 iterations to reach similar optimum points. As the move limit increases, all three methods converge to similar optimum points and the number of iterations to obtain the solutions decreases in this example. When no artificial move limit is imposed, both TDQA I and TDQA II need only 5 iterations to reach the same optimum point, whereas TANA-3 fails to obtain the solution. These results demonstrate that the proposed methods work very well without regard to the move limit strategy and that they are also effective in wider ranges. Example 7. Cantilevered Beam. This problem is presented in the DOT User Manual [26]. The cantilevered beam shown in Fig. 7 is to be designed for minimum material volume. The design variables are the width b and height h at each of N segments. This beam must be designed subject to limits on stress (calculated at the left end of each segment), deflection under the load, and the geometric requirement that the height of any segment does not exceed twenty times the width. The data information is listed in Fig. 7. This design task is defined as
21x3 þ 7x4 þ 100 Minimize V ¼
subject to g1 ðxÞ ¼ g2 ðxÞ ¼
ðx21 ðx21
þ
x22
þ
þ 2x22 ð2x21 þ x22
x23
þ
þ
x23 x23
x24
þ x1 x2 þ x3 x4 Þ=8 1 6 0
þ 2x24 x1 x4 Þ=10 1 6 0
N X
b i h i li
i¼1
subject to ðM i hi =2I i Þ=ra 1 6 0
g3 ðxÞ ¼ þ þ 2x1 x2 x4 Þ=5 1 6 0 10 6 xi 6 10 for i ¼ 1; . . . ; 4 ð26Þ Table 1 shows the optimization results obtained by using TANA-3, TDQA I, and TDQA II. For this problem, the shifted intervening variables used in TDQA I and TDQA II is also applied to TANA-3 in order to avoid the singularity due to the side constraints. Under a 40%
853
i ¼ 1; . . . ; N
hi 20bi 6 0 dN =da;y 1 6 0
i ¼ 1; . . . ; N
1:0 6 bi 6 100
i ¼ 1; . . . ; N
5:0 6 hi 6 100
i ¼ 1; . . . ; N :
ð27Þ
The deflection di at the right end of segment i is calculated by recursion formulas. These recursion formulas were written in DOT User Manual [26]. The initial values of bi and hi are 5.0 and 40.0, respectively. This is a design
Table 1 Comparison of optimization results for the Rosen–Suzuki problem
x1 x2 x3 x4 f gmax Iterations
Initial
40% move limit TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
0 0 0 0 100 1
0.004 0.997 1.998 1.005 56.027 0.000 8
0.002 0.999 1.999 1.002 55.997 0.000 8
0.002 0.999 1.999 1.002 55.997 0.000 8
0.005 1.002 1.996 1.004 56.000 0.000 7
0.008 1.003 1.994 1.007 55.998 0.000 7
0.008 1.003 1.994 1.007 55.998 0.000 7
100 % move limit
75% move limit
150 % move limit
No move limits
TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
0.001 0.999 2.000 1.000 55.999 0.000 7
0.003 0.998 1.998 1.003 55.997 0.000 6
0.003 0.998 1.998 1.003 55.997 0.000 6
0.000 1.000 2.000 1.000 56.000 0.000 6
0.000 1.001 2.000 1.000 56.000 0.000 6
0.000 1.001 2.000 1.000 56.000 0.000 6
fail
0.001 1.001 1.999 1.001 56.000 0.000 5
0.001 1.001 1.999 1.001 56.000 0.000 5
854
J.-R. Kim, D.-H. Choi / Comput. Methods Appl. Mech. Engrg. 197 (2008) 846–856
Fig. 7. The cantilevered beam.
Table 2 Comparison of optimization results for cantilevered beam Initial
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 f gmax Iterations
5 5 5 5 5 40 40 40 40 40 100 1
40% move limit
75 % move limit
TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
3.134 2.882 2.580 2.205 1.750 62.684 57.641 51.609 44.091 34.995 65419.7 0.000 9
3.126 2.882 2.590 2.205 1.750 62.514 57.648 51.805 44.091 34.995 65419.0 0.000 8
3.123 2.893 2.582 2.205 1.750 62.457 57.861 51.643 44.091 34.995 65422.3 0.000 8
3.146 2.874 2.575 2.205 1.750 62.925 57.480 51.502 44.091 34.995 65422.7 0.000 7
3.131 2.886 2.581 2.205 1.750 62.610 57.714 51.613 44.090 34.995 65417.4 0.000 8
3.134 2.883 2.580 2.205 1.750 62.684 57.654 51.595 44.090 34.995 65419.3 0.000 8
100% move limit
150% move limit
No move limits
TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
3.134 2.883 2.579 2.205 1.750 62.674 57.669 51.590 44.091 34.995 65419.6 0.000 9
3.135 2.884 2.580 2.205 1.750 62.698 57.675 51.595 44.091 34.997 65441.4 0.000 7
3.135 2.884 2.580 2.205 1.750 62.696 57.676 51.597 44.091 34.996 65441.3 0.000 7
3.140 2.878 2.579 2.205 1.750 62.796 57.554 51.572 44.091 34.995 65420.4 0.000 7
3.134 2.883 2.580 2.205 1.750 62.674 57.659 51.600 44.091 34.995 65419.5 0.000 8
3.143 2.872 2.580 2.205 1.750 62.867 57.447 51.603 44.091 34.995 65419.5 0.000 9
3.139 2.878 2.579 2.205 1.750 62.776 57.565 51.583 44.091 34.995 65420.2 0.000 7
3.131 2.886 2.579 2.205 1.750 62.628 57.727 51.578 44.091 37.995 65418.5 0.000 8
3.134 2.883 2.580 2.205 1.750 62.680 57.657 51.596 44.091 34.995 65419.5 0.000 8
problem with n = 2N variables. In this study, this problem was solved using five segments (10 design variables). The optimum is known as f(x*) = 65, 368 in the DOT User Manual. Also the optimum point is known as x* = (3.11, 2.88, 2.62, 2.20, 1.75, 61.20, 57.55, 52.31, 44.09, 35.00)T. Table 2 shows the optimization results obtained by using TANA-3, TDQA I, and TDQA II. All three methods succeed to converge to similar optimum points in almost
the same number of iterations under various move limits. These results show that the proposed methods work very well regardless of the move limit strategy.
Example 8. 25-Bar Truss with Stress and Displacement constraints. This problem, which is often seen in the literature, calls for a minimum weight structure subject to
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855
Table 3 Loading conditions for the 25 bar truss structure problem Loading conditions
Loaded node
Load, kN X
y
z
1
1 2 3 6 1 2
4.45 0 2.225 2.225 0 0
44.5 44.5 0 0 89 89
22.25 22.25 0 0 22.5 22.5
2
Fig. 8. Twenty-five bar structure.
member stress, Euler buckling and joint displacement constraints. This structure is subjected to two load cases. The geometry and data information is shown in Fig. 8. There are eight design variables for considering symmetry: x1 = A1, x2 = A2 = A3 = A4 = A5, x3 = A6 = A7 = A8 = A9, x4 = A10 = A11, x5 = A12 = A13, x6 = A14 = A15 = A16 = A17, x7 = A18 = A19 = A20 = A21 and x8 = A22 = A23 = A24 = A25. All design variables are limited within [0.001, 50.0], and all initial design points are 2.0. Each load-
ing condition is shown in Table 3. MSC/NASTRAN 70.7 [28] is used in this study. Table 4 shows the optimization results obtained by using TANA-3, TDQA I, and TDQA II. All three methods succeed to converge to similar optimum points under various move limits. Without regard to the size of the move limit, TDQA I and TDQA II require a similar number of iterations, whereas TANA-3 requires about 1.5 times more iterations in case of a 40% move limit than those in other cases. These results illustrate that the proposed methods work very well regardless of the move limit strategy. 5. Conclusions Function approximation is one of the most important and active fields of research in engineering optimization. Accurate function approximations can reduce repetitive computational efforts. This study presented enhanced
Table 4 Comparison of optimization results for 25-Bar Truss with stress and displacement constraints
x1 x2 x3 x4 x5 x6 x7 x8 f gmax Iterations
Initial
40% move limit TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
2 2 2 2 2 2 2 2 100 1
0.010 1.988 3.004 0.026 0.051 0.686 1.670 2.656 545.9 0.000 12
0.074 2.037 3.027 0.026 0.052 0.679 1.617 2.669 546.2 0.000 8
0.073 2.044 3.030 0.026 0.055 0.676 1.610 2.674 546.3 0.000 8
0.275 2.041 3.014 0.026 0.048 0.679 1.619 2.671 547.6 0.000 7
0.297 2.039 3.023 0.025 0.054 0.678 1.618 2.673 547.9 0.000 7
0.079 2.045 3.024 0.023 0.055 0.711 1.609 2.633 546.4 0.000 7
100% move limit
75% move limit
150% move limit
No move limit
TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
0.010 2.047 3.001 0.026 0.118 0.681 1.621 2.673 546.7 0.000 7
0.255 2.032 3.060 0.025 0.112 0.672 1.606 2.669 548.4 0.000 7
0.010 2.057 3.026 0.026 0.165 0.676 1.601 2.672 547.2 0.000 6
0.010 2.035 3.005 0.025 0.054 0.690 1.628 2.659 545.8 0.000 7
0.256 2.052 3.048 0.025 0.098 0.684 1.594 2.660 548.2 0.000 8
0.010 2.054 3.027 0.026 0.155 0.676 1.603 2.671 547.1 0.000 7
0.377 2.056 3.014 0.025 0.132 0.676 1.608 2.675 549.5 0.000 7
0.019 2.037 3.062 0.026 0.106 0.674 1.601 2.667 546.5 0.000 8
0.286 2.075 3.030 0.024 0.258 0.687 1.584 2.656 550.5 0.000 5
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two-point diagonal quadratic approximation methods to guarantee the approximation accuracy whether the derivatives at two design points have the same sign or not. The approximation accuracy of the proposed methods is as good as that of TDQA when the derivatives of two design points have different signs. In addition, when those at two points have the same sign, they are as good as TANA-3. Therefore, considering all factors, the enhanced two-point diagonal quadratic approximation methods are more accurate than TANA-3 and TDQA. The proposed methods, which are abbreviated as TDQA I and TDQA II, exhibit the same or very similar accuracy. When the original function is independent of any design variables, we highly recommend that TDQA II be used. In the proposed methods, both the approximation function and derivative values at two design points are equal to their exact counterparts on all occasions. The unknown parameters of the approximated function can be obtained in a closed form. In part 2 of Section 4, numerical examples show that the proposed methods can successfully converge to a similar optimum under various move limit strategies in sequential approximation optimization. They also show that the proposed methods work very well regardless of the move limit strategy. Therefore, the proposed methods can be applied in reliability analysis and reliability based design optimization. When applied, the proposed methods will prove to be more effective than other two-point approximation methods. Acknowledgements This research work was supported by the International Joint R&D Program, the Ministry of Commerce, Industry, and Energy, and the center of Innovative Design Optimization Technology (iDOT), Korea Science and Engineering Foundation. References [1] L.A. Schmit, B. Farshi, Some approximation concepts for structural synthesis, AIAA J. 12 (1974) 692–699. [2] L.A. Schmit, H. Miura, A new structural analysis/synthesis capability – ACCESS 1, AIAA J. 14 (1976) 661–671. [3] L.A. Schmit, C. Fleury, Structural synthesis by combining approximation concepts and dual methods, AIAA J. 18 (1980) 1252–1260. [4] R.T. Haftka, J.A. Nachlas, I.T. Watson, T. Rizzo, R. Desai, Twopoint constraint approximation in structural optimization, Comput. Meth. Appl. Mech. Engrg. 60 (1987) 289–301. [5] C. Fleury, V. Braibant, Structural optimization: a new dual method using mixed variables information, Int. J. Numer. Meth. Engrg. 23 (1986) 409–428. [6] J.H. Starnes Jr., R.T. Haftka, Preliminary design of composite wings for buckling, stress and displacement constraints, J. Air Craft 16 (1979) 564–570.
[7] K. Svanberg, Method of moving asymptotes: a new method for structural optimization, Int. J. Numer. Meth. Engrg. 24 (1987) 359– 373. [8] H. Chickermane, H.C. Gea, Structural optimization using a new local approximation method, Int. J. Numer. Meth. Engrg. 39 (1996) 829– 846. [9] G.M. Fadel, M.F. Riley, J.F.M. Barthelemy, Two point exponential approximation method for structural optimization, Struct. Opt. 2 (1990) 117–124. [10] L.P. Wang, R.V. Grandhi, Efficient safety index calculation for structural reliability analysis, Comput. Struct. 52 (1994) 103–111. [11] L.P. Wang, R.V. Grandhi, Improved two-point function approximation for design optimization, AIAA J. 33 (1995) 1720–1727. [12] L.P. Wang, R.V. Grandhi, Multipoint approximations: comparisons using structural size, configuration and shape design, Struct. Opt. 12 (1996) 177–185. [13] L.P. Wang, R.V. Grandhi, R.A. Canfield, Multivariate hermite approximation for design optimization, Int. J. Numer. Meth. Engrg. 39 (1996) 787–803. [14] S. Xu, R.V. Grandhi, Effective two-point function approximation for design optimization, AIAA J. 36 (1998) 2269–2275. [15] B. Prasad, Explicit constraint approximation forms in structural optimization. part 1; analysis and projections, Comput. Meth. Appl. Mech. Engrg. 40 (1983) 1–26. [16] E. Salajegheh, Optimum design of plate structures using three-point approximation, Struct. Opt. 13 (1997) 142–147. [17] Y.K. Sui, The Advances of Modeling Transformation Optimization and Synthesis, Dalian University of Technology Press, Dalian, 1998. [18] G. Xu, K. Yamazaki, G.D. Cheng, A new two-point approximation approach for structural optimization, Struct. Multidisciplinary Opt. 20 (2000) 22–28. [19] G. Xu, K. Yamazaki, G.D. Cheng, A new three-point approximation approach for design optimization problems, Int. J. Numer. Meth. Engrg. 50 (2001) 869–884. [20] A.A. Groenwold, L.F.P. Etman, J.A. Snyman, J.E. Rooda, Incomplete series expansion for function approximation, Struct. Multidisciplinary Opt. 34 (2007) 21–40. [21] J. Rasmussen, Accumulated approximations: a new method for structural optimization, in: Proceedings of the 3rd Air Force/NASA Symposium on Recent Advances in Multidisciplinary Analysis and Optimization, San Francisco, CA, September 24–26, 1990, pp. 253– 258. [22] J. Rasmussen, Nonlinear programming by cumulative approximation refinement, Struct. Opt. 15 (1998) 1–7. [23] S. Xu, R.V. Grandhi, Multipoint approximation development: thermal structural optimization case study, Int. J. Numer. Meth. Engrg. 48 (2000) 1151–1164. [24] Min-Soo Kim, Jong-Rip Kim, Jae-Young Jeon, Dong-Hoon Choi, Efficient mechanical system optimization using two-point diagonal approximation in the nonlinear intervening variable space, J. KSME 15 (2001) 1257–1267. [25] G.N. Vanderplaats, Numerical Optimization Techniques for Engineering with Applications, McGraw-Hill, New York, 1984. [26] G.N. Vanderplaats, DOT Users Manual, version 4.20, VR& D Inc., 1995. [27] J.B. Rosen, S. Suzuki, Construction of nonlinear programming test problem, Commun. Assoc. Comput. Machin. 8 (1965) 113. [28] J.M. Gregory, MSC/NASTRAN Design Sensitivity and Optimization: User’s Guide version 68, The MacNeal-Schwendler Corporation.
Comput. Methods Appl. Mech. Engrg. 197 (2008) 846–856 www.elsevier.com/locate/cma
Enhanced two-point diagonal quadratic approximation methods for design optimization Jong-Rip Kim
a,1
, Dong-Hoon Choi
b,*
a
CAE Group, Corporate R&D Institute, Samsung Electro-Mechanics Co., Ltd., 314, Maetan3-Dong, Yeongtong-Gu, Suwon, Gyunggi-Do 443-743, Republic of Korea b FRAMAX Co., Ltd. and the Center of Innovative Design Optimization Technology, Hanyang University, Haendang-Dong, Sungdong-Ku, Seoul 133-791, Republic of Korea Received 20 June 2007; received in revised form 23 September 2007; accepted 24 September 2007 Available online 29 September 2007
Abstract Based on two-point diagonal quadratic approximation (referred to as TDQA), developed by Kim et al. [Min-Soo Kim, Jong-Rip Kim, Jae-Young Jeon, Dong-Hoon Choi, Efficient mechanical system optimization using two-point diagonal approximation in the nonlinear intervening variable space, J. KSME 15 (2001) 1257–126], enhanced two-point approximation methods are proposed in this paper. The suggested methods reinforce TDQA with new quadratic correction terms using the concept of TANA-3. These methods overcome the disadvantage of TDQA when the derivatives at two design points have the same sign. In addition, the values and derivatives of the proposed approximation functions are completely equal to those of an original function at the two design points whether the derivatives at those points have the same sign or not. Several examples show the numerical performance and accuracy of the proposed methods compared to previous work, and optimization examples show that these methods can successfully reach the optimum via sequential approximation optimization. 2007 Elsevier B.V. All rights reserved. Keywords: Function approximation; Two-point approximation; Quadratic approximation; Sequential approximate optimization
1. Introduction Function approximation is one of the most important and active research fields in design optimization. In an optimization procedure, model analysis is completed repetitively to find the search direction and to perform the line search. If the model analysis cost is high, optimization will be inefficient. To reduce the repetitive cost of model analysis, Schmit and coworkers introduced suitable approximation concepts in the 1970s [1–3]. They combined the now-familiar techniques of intervening variable definition,
*
Corresponding author. Tel.: +82 2 2220 0478; fax: +82 2 2291 4070. E-mail addresses: [email protected] (J.-R. Kim), dhchoi@ hanyang.ac.kr (D.-H. Choi). 1 Tel.: +82 31 210 6776; fax: +82 31 300 7900x0117. 0045-7825/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2007.09.014
explicit approximation, reduced basis and design variable linking as well as constraint deletion and regionalization. In the 1980s, most approximations were based on function and gradient information at a single point and constructed by a first-order Taylor series expansion at this point, which are the linear, reciprocal and conservative approximations in Schmit and Fleury [3]. These are very popular because the function and its derivative values are required in most optimization algorithms, so no additional computation is involved in constructing the approximation functions. Although these approximations work effectively for stress and displacement functions, their truncated error sometimes increases and they are only valid in the vicinity of the expanding point. Haftka et al. [4] suggested several methods that include a modified reciprocal, a two-point projection, and an exponential approximation. The CONLIN scheme developed by Fleury and Braibant [5] is a
J.-R. Kim, D.-H. Choi / Comput. Methods Appl. Mech. Engrg. 197 (2008) 846–856
convex approximation based on the first-order Taylor series expansion in terms of direct and reciprocal design variables and has shown good convergence properties in dealing with structural optimization problems. It was also independently developed by Starnes and Haftka [6] as the conservative approximation. Svanberg [7] proposed a modification of the CONLIN scheme called the Method of Moving Asymptotes (MMA). In this method, the linearization variables can be used to adjust the degree of convexity and conservativeness depending on the problem. Chickermane and Gea [8] extended a convex approximation algorithm for solving structural optimization problems. This algorithm – the Generalized Convex Approximation (GCA) method – generates a high-fidelity function approximation using the design sensitivity information in the current and previous design points. In order to make full use of the known information to construct approximation functions, many multi-point approximations have been developed since 1990 [9–14]. Among them, two-point approximation methods have been widely used for their simplicity. Fadel and his coworkers [9] were the first to propose Two-Point Exponential Approximation (TPEA) – an extension of the socalled projection method proposed by Haftka et al. [4]. They employed intervening variables with an exponential form as proposed by Prasad [15]; the exponents were then computed by matching the derivatives of the approximation function with those of the exact function at previous design point. Based on TPEA, Wang and Grandhi [10,11] developed a series of improved two-point approximations using both function and gradient information at two design points, which were called TPEA-change, TANA, TANA-1 and TANA-2. Xu and Grandhi [14] further developed TANA-3 which employs a diagonal and changeable Hessian matrix in order to avoid the computational burden of solving non-linear equations for each function as TANA-2 requires. However, TANA-3 still gives some inaccuracies when the derivatives of the two design points have values with different signs. Salajegheh [16] proposed a three-point approximation method based on the function and gradient information at three consecutive design points. He also applied this method to optimize the plate structure subjected to stress and frequency constraints. Sui [17] has also proposed a two-point approximation approach by employing the information obtained at two design points. Xu and Yamazaki [18] developed a new two-point approximation using the linear combination of Taylor expansions in terms of both the direct and reciprocal variables, and extended to a three-point approximation [19]. The expression of that scheme consists of a linear combination of the direct and reciprocal linear Taylor expansions as well as the lumped diagonal terms of the second-order direct and inverse terms. Recently, Groenwold et al. [20] presented an incomplete series expansion (ISE) as a basis for function approximation. They also presented an example of intervening variables for the ISE function. Another multi-point approximation scheme is a cumulative
847
approximation by combining a one-point or two-point approximation and a blending function [21–23]. In 2001, Kim et al. developed a two-point diagonal quadratic approximation which is abbreviated as TDQA [24]. Shifting coefficients were introduced into the intervening variables in order to avoid the singularity of the approximation gradients in TDQA. To overcome the critical difficulty, due to which other two-point approximations cannot be used because the derivatives of two design points have values with different signs, a new quadratic term with respect to the intervening variables was introduced. The components of the new quadratic term are determined in order to match the derivatives of the approximation function with those of the previous design point along each intervening variable axis, respectively. Also, a uniquely determined correction coefficient is multiplied with the quadratic term to match the approximate function value with that of the previous design point. When the derivatives of two design points have the same sign, TDQA gives less accurate results compared with other two-point approximations because it is constructed as a two point exponential approximation (TPEA) for the current design point in this case. In order to overcome this disadvantage of TDQA, this paper presents enhanced two-point diagonal quadratic approximation methods. The new quadratic correction terms, which are made using the concept of TANA-3, are added to TDQA. Then, a uniquely determined correction coefficient is multiplied with the new quadratic correction terms to match the function values at the previous design point. Therefore, the proposed methods overcome the demerit of TDQA and can match the function and derivative values of the approximation with the exact ones at the two design points. Section 2 reviews typical two-point approximation methods. Section 3 fully describes the proposed methods. Section 4 compares the results of the proposed methods with those of other approximation methods. 2. Review of two-point approximations In this section, we describe the mathematical details of the previous two-point approximations such as TPEA, TANA series and TDQA in order to better explain the proposed method. The known design points are denoted as x1(x1,1, x2,1, . . . , xn,1) and x2(x1,2, x2,2, . . . , xn,2) where the function and gradient information are available. Here n is the number of design variables. The function g~ðxÞ denotes the approximation function based on two-point approximation, which is expanded at the current design point x2 and uses the values of the function and/or derivatives at two design points. 2.1. Two-point exponential approximation (TPEA) Fadel et al. [9] first developed a two-point exponential approximation. It is a linear Taylor series approximation in terms of the intervening variables
848
J.-R. Kim, D.-H. Choi / Comput. Methods Appl. Mech. Engrg. 197 (2008) 846–856 p
y i ¼ xi i ;
i ¼ 1; 2; . . . ; n;
ð1Þ
where the exponent pi for each design variable was evaluated by matching the derivatives of the approximation function with those at the previous design point. The approximation function is given in terms of the original variables xi as 1p n X ogðx2 Þ xi;2 i pi p ðxi xi;2i Þ: g~ðxÞ ¼ gðx2 Þ þ ox p i i i¼1
ð2Þ
In this approximation, the value of pi is limited from 1 to +1. However, Wang and Grandhi [10] removed the limitation of pi for better adaptability to different structural problems, which was called the TPEA-change method. 2.2. Two-point adaptive non-linear approximations (TANA series) Wang and Grandhi [10] proposed the TANA method using adaptive intervening variables y i ¼ xri , i = 1, 2, . . . , n where r represents the non-linearity index, which is updated during each iteration but is the same for all variables. The non-linearity index was determined by matching the function value at the previous point. Also, in order to utilize more information to construct a better approximation, Wang and Grandhi [11] proposed the following two approximation methods (TANA-1 and TANA-2) to combine TPEA-change and TANA methods, and Xu and Gradhi [14] developed TANA-3. These approximation methods are written by expanding the function at x2 except TANA-1 at x1 TANA-1 :
1p n X ogðx1 Þ xi;1 i g~ðxÞ ¼ gðx1 Þ þ oxi pi i¼1 p
p
ðxi i xi;1i Þ þ e1 ; TANA-2 :
TANA-3 :
ð3Þ 1p xi;2 i
n X ogðx2 Þ p p ðxi i xi;2i Þ ox p i i i¼1 n X 1 p p þ e2 ðx i xi;2i Þ2 ; ð4Þ 2 i¼1 i
g~ðxÞ ¼ gðx2 Þ þ
1p n X ogðx2 Þ xi;2 i pi p ðxi xi;2i Þ g~ðxÞ ¼ gðx2 Þ þ ox p i i i¼1 n X 1 p p 2 þ e3 ðxÞ ðxi i xi;2i Þ ; ð5Þ 2 i¼1
where ," e3 ðxÞ ¼ H
# n n X X pi pi 2 pi pi 2 ðxi xi;1 Þ þ ðxi xi;2 Þ : i¼1
ð6Þ
i¼1
In TANA-1, the correction term e1 is a constant in order to compensate for approximation error, which is computed by matching the function value of the approximation with the exact function value at the previous design point. TANA-2 and TANA-3 include second-order Taylor series
effects. In TANA-2, the Hessian matrix has only diagonal elements of the same value e2. Unlike the original secondorder approximation, the approximation is expanded in terms of the intervening variables y, so the error from the approximation Hessian matrix is partially corrected by adjusting the non-linearity index pi. TANA-2 has n + 1 unknown constants, so n + 1 equations are required. The n equations are obtained by matching the derivative values of the approximation function with those of the original function at the previous point. The (n + 1)th equation is obtained by matching the function value of the approximation function with the exact value at the previous point. Then, n + 1 unknown constants can be obtained by solving these n + 1 equations simultaneously. In TANA-3, in order to avoid the computational burden of TANA-2, a function e3(x) is employed, which uncouples pi and H in the equations in order to compute them. Then the unknown parameters are obtained in a closed-form solution to match the derivative and values of the approximation function with the exact values at the previous point. So, though TANA-2 and TANA-3 have almost the same accuracy, TANA-3 has the advantage of being less computationally intensive than TANA-2. 2.3. Two-point diagonal quadratic approximation, TDQA In 2001, Kim et al. developed a two-point diagonal quadratic approximation method [24]. This approximation method used the intervening variables as in p
y i ¼ ðxi þ ci Þ i ;
i ¼ 1; 2; . . . ; n;
ð7Þ
where ci is the shifting level for the ith design variable. This shifting level can avoid the singularity of the approximation derivatives and the associated fundamental difficulties. In order to match the derivatives of the approximation function with those of the exact function at the previous point, the exponent pi is determined without considering the quadratic term. The approximation is created by expanding the function at x2 g~ðxÞ ¼ gðx2 Þ þ
n n X ogðx2 Þ 1 X 2 ðy i y i;2 Þ þ g Gi ðy i y i;2 Þ ; oy 2 i i¼1 i¼1
ð8Þ where Gi, which is the ith component of the diagonal Hessian matrix, is predefined as 1 ogðx1 Þ ogðx2 Þ Gi ¼ ð9Þ ðy i;1 y i;2 Þ oy i oy i and the correction coefficient g is uniquely determined in order to match the function value of the approximation function with the exact value at the previous point. Unlike TANA-2 and TANA-3, this method determines the unknown parameters pi, Gi and g sequentially and uniquely. Numerical considerations which occur when the unknown parameters are determined are also presented [24].
J.-R. Kim, D.-H. Choi / Comput. Methods Appl. Mech. Engrg. 197 (2008) 846–856
3. Enhanced two-point diagonal quadratic approximation This study presents enhanced two-point diagonal quadratic approximation methods in order to guarantee the approximation accuracy whether the derivatives at two design points have the same sign or not. The proposed methods reinforce TDQA with additional quadratic correction terms. These quadratic correction terms consist of a correction coefficient as a numerator and a decoupling factor as the denominator. The denominators are defined so that approximation errors are zero at two design points as in TANA-3. These methods use the shifted intervening variables in Eq. (7) like TDQA. The proposed methods are represented by expanding the function at x2 TDQA I: g~ðxÞ ¼ gðx2 Þ þ
n X ogðx2 Þ ðy i y i;2 Þ oy i i¼1
n 1X 2 Gi ðy i y i;2 Þ 2 i¼1 " #, n n X X 1 2 2 þ ge ðy i y i;2 Þ ðy i y i;1 Þ 2 i¼1 i¼1 ! n X 2 ðy i y i;2 Þ ; þ ð10Þ
þ
849
posed methods uniquely determine the unknown parameters Gi, Hi, ge and pi. We predefine Gi as Eq. (9) to match the derivatives of the approximation function with the exact values at the previous point along each intervening variable axis when those at two design points have different signs. We also define Hi with the same process for Gi. But when the derivatives at two points have the same sign, Gi is equal to zero. In order to prevent the denominator of the quadratic correction term in TDQA II from being equal to 0, we reset Hi = 1 in this case. This is because Gi and Hi may be exact second-order derivatives in the intervening variable axis. Therefore, Hi can be defined as Gi if ½ogðx1 Þ=oxi ½ogðx2 Þ=oxi 6 0 Hi ¼ ð12Þ 1 otherwise: In order to match the approximate function value with that of the exact function at the previous point, the correction coefficient ge is determined to be " n X ogðx2 Þ ge ¼ 2 gðx1 Þ gðx2 Þ ðy i;1 y i;2 Þ oy i i¼1 # n 1X 2 Gi ðy i;1 y i;2 Þ : ð13Þ 2 i¼1
i¼1
TDQA II:
n X ogðx2 Þ g~ðxÞ ¼ gðx2 Þ þ ðy i y i;2 Þ oy i i¼1 n 1X Gi ðy i y i;2 Þ2 2 i¼1 " #, n n X X 1 2 þ ge H i ðy i y i;2 Þ H i ðy i y i;1 Þ2 2 i¼1 i¼1 ! n X 2 H i ðy i y i;2 Þ ; ð11Þ þ
þ
TDQA I and TDQA II have the same correction coefficient ge in Eq. (13). So Eqs. (10), (11) and (13) show that function values of the approximation function are equal to those of the exact function at two design points. The exponent pi is determined to match the derivative values of the approximation function with those of the exact function at the previous point in the intervening variable space when the values at two design points have the same sign. Then, the derivatives of Eqs. (10) and (11) with respect to xi can be derived as
i¼1
2
2
Pn
2
Pn
2
3
3
o~ gðxÞ 6 6ðy i y i;2 Þ j¼1 ðy j y j;1 Þ ðy i y i;1 Þ j¼1 ðy j y j;2 Þ 7 ogðx2 Þ 7 p 1 ¼ 4ge 4 þ Gi ðy i y i;2 Þ5 pi ðxi þ ci Þ i ; 5þ nP o2 Pn oxi oy i n 2 2 j¼1 ðy j y j;1 Þ þ j¼1 ðy j y j;2 Þ
ð14Þ
2 3 3 Pn Pn 2 2 o~ gðxÞ 6 6ðy i y i;2 Þ j¼1 H j ðy j y j;1 Þ ðy i y i;1 Þ j¼1 H j ðy j y j;2 Þ 7 ogðx2 Þ 7 p 1 ¼ 4ge H i 4 þ Gi ðy i y i;2 Þ5 pi ðxi þ ci Þ i 5þ nP o2 P oxi oy n 2 n 2 i j¼1 H j ðy j y j;1 Þ þ j¼1 H j ðy j y j;2 Þ 2
ð15Þ
where Gi and Hi are the ith components of the diagonal Hessian matrix and ge is the correction coefficient. The pro-
Eqs. (14) and (15) show that the derivative values of the approximation function are equal to those of the exact
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function at the current point. When those at two design points have the same sign, Gi is equal to 0. Therefore, at the previous point, the exponent pi is determined to be ogðx1 Þ ogðx2 Þ pi ¼ 1 þ ln ln½ðxi;1 þ ci Þ=ðxi;2 þ ci Þ: oxi oxi ð16Þ When the derivative values at two design points have different signs, pi is set as a special value (1 or 1), but the derivative values of the approximation function are nevertheless equivalent to those of the exact function at those points by the term Gi. Therefore, the function and derivative values of the approximation function are completely equal to the exact values at two design points in the proposed methods, unlike other two-point approximation methods. Despite the fact that Hi is added to TDQA II, unlike TDQA I, both TDQA I and TDQA II exhibit the same or very similar accuracy. A significant difference exists between the derivative values of TDQA I and TDQA II. Like TANA-3, TDQA I may falsely give non-zero derivative values with respect to some design variables on which the original function is not dependent [13]. In other words, although g(x) does not depend on xi, Eq. (14) may not equal zero. But the approximate derivatives of TDQA II with respect to design variables, of which the original function is independent, are always zero because Gi = 0 in Eq. (9) and Hi = 0 in Eq. (12). Therefore, when the original function is independent of any design variables, we highly recommend that TDQA II should be used instead of TDQA I. 3.1. Numerical considerations Now, we describe three guidelines in order to avoid the numerical difficulties that occur when determining the three parameters of pi, Gi and Hi. • First, pi cannot be numerically determined when [og(x1)/ oxi] Æ [og(x2)/oxi] 6 0 or ðxi;1 þ ci Þ=ðxi;2 þ ci Þ 1. Thus, if [og(x1)/o xi] Æ [og(x2)/oxi] 6 0, we reset pi = 1 or 1. Also, if j ðxi;1 þ ci Þ=ðxi;2 þ ci Þ 1 j6 e for a small positive real number e, we reset pi = 1. The suggested value of e is 0.001. If the value of the exponent jpij becomes too large, the approximation function g~ðxÞ can be numerically ill-conditioned. Thus, we bound jpij 6 pmax with limit value pmax = 5. If pi evaluated by Eq. (16) exceeds the range (pmax, pmax), it is rounded up to pmax or down to pmax. • Second, if jyi,2 – yi,1j 6 e, the values of Gi and Hi may go to infinity. Therefore, we reset Gi = 0 and Hi = 0. This means that a linear approximation is employed for the intervening variable yi. P Pn n 2 • Finally, in TDQA II, if i¼1 H i ðy i y i;1 Þ þ i¼1 H i ðy i y i;2 Þ2 j 6 e, we reset Hi = 1. It is necessary in order to prevent the denominator of the new quadratic term from being zero. In this case, TDQA II is equal to TDQA I.
4. Numerical results This section is divided into two parts. The first part is for simple problems with closed-form solutions, whereas the second part is for applications in engineering optimization. 4.1. Part 1: comparisons using closed-form problems The following five examples are selected to examine the difference among two-point approximation methods such as TANA-3, TDQA, TDQA I and TDQA II. We use the relative percentage error as the criterion in evaluating the accuracy. For all of the test examples, the relative percentage error index is defined as Relative percentage error ¼
Exact Approximation Exact 100ð%Þ ð17Þ
and the test points are derived using x ¼ x2 þ aD;
ð18Þ
where x2 is an expanding point or a current point, a is a step length, and D is a search direction vector. Example 1. This example is used in Ref. [11] in order to compare the two-point approximations mentioned above. The function to be approximated is defined as gðxÞ ¼
10 30 15 2 25 108 40 47 þ þ þ þ þ 3 þ þ 3 1:0: x1 x31 x2 x32 x3 x4 x4 x3 ð19Þ
The expanding points for all methods are selected to be x2 = (1.0, 1.0, 1.0, 1.0)T, and the previous points are selected to be x1 = (1.2, 1.2, 1.2, 1.2)T. By using D = (1.0, 1.0, 1.0, 1.0)T the relative percentage errors are determined in Eq. (17). In TANA-3, H is 0.242 and p = (2.76, 1.50, 2.83, 2.49)T. For this example, the derivative values at all points are negative. That is, with two arbitrary design points selected, the derivative values have the same sign. In this case, TANA-3 guarantees better accuracy than TDQA, which becomes TPEA. When the derivatives at two points have the same sign with Eqs. (9) and (12), we acquire Gi = 0, i = 1, . . . , n and Hi = 1, i = 1, . . . , n in TDQA I and TDQA II. By Eq. (13), the correction coefficient ge is equal to H of TANA-3 mathematically. Therefore, when the derivatives at two points have the same sign, the proposed methods execute approximation similar to TANA-3. Fig. 1 shows the results of the relative percentage errors of all methods. Comparing the relative percentage errors, TANA-3 has better accuracy than TDQA for this example and the proposed methods produce the same approximation function as TANA-3. Consequently, TDQA I and TDQA II have the same relative percentage error as TANA-3.
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better accuracy than TDQA. The proposed methods produce the same approximation function as TANA-3. Therefore, they have the same relative percentage errors. Example 3. This example has also been borrowed from Ref. [11] and the function to be approximated is defined as 2 3 0:125 2 1 0:5 þ 15x1 gðxÞ ¼ 10x1 x1 2 x4 x6 x7 1 x2 x3 x4 x5 x7 1 2 2 2 1 0:5 2 þ 20x2 1 x2 x4 x5 x6 þ 25x1 x2 x3 x5 x6 x7 :
Fig. 1. Example 1.
Example 2. This example also has been borrowed from Ref. [11] in order to make a comparative study of the accuracy of approximation methods. The function to be approximated is defined as gðxÞ ¼ 180x1 þ 20x2 3:1x3 þ 0:2x4 5x1 x2 þ 37x1 x3 þ 8:7x2 x4 3x3 x4 0:1x21 x2 þ 0:001x22 x3 þ 95x1 x24 81x4 x23 þ x31 6:2x32 þ 0:48x33 þ 22x34 1:0:
ð20Þ
The current design point is selected to be x2 = (1.5, 1.5, 1.5, 1.5)T. The previous design point is selected to be x1 = (2.0, 2.0, 2.0, 2.0)T. As a search direction, D = (1.0, 1.0, 1.0, 1.0)T is used. Under these circumstances, the exponent indices are p = (2.214, 4.681, 3.124, 2.981)T. In TDQA, Gi = 0, i = 1, . . . , n. In TANA-3, H = 4.729. In the proposed methods, Hi = 1, i = 1, . . . , n. The correction coefficient ge is equal to 4.729, which is the same value as TANA-3’s H. Fig. 2 shows the results of the relative percentage errors of all methods. In this example, the derivatives have the same sign at two points and TANA-3 has
Fig. 2. Example 2.
ð21Þ
In this example, in order to examine the mixed condition that includes the case, in which the derivatives have the same sign, and the case in which the derivatives have different signs, two design points are selected to be x1 = (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)T and x2 = (0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9)T. At two design points, design variables x1 and x3 satisfy the condition in which the values of [og(x1)/ oxi] Æ [og(x2)/oxi] are negative signed values. As a search direction, D = (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)T is used. Under these circumstances, the exponent indices are p = (1, 6.055, 1, 0.643, 4.001, 2.969, 5)T. In TDQA, G = (247.520, 0, 103.438, 0, 0, 0, 0)T, and the correction coefficient g is 0.938. In TANA-3 H is 1.895. In the proposed methods, H = (247.520, 1, 103.426, 1, 1, 1, 1)T and the correction coefficient ge is 0.126. Fig. 3 shows the results of the relative percentage errors of all methods. When the derivatives at two points have different signs, TDQA has better accuracy than TANA-3 and the proposed methods have similar accuracy to TDQA because the original quadratic term Gi, which is also in the proposed methods, dominates the new quadratic term in this case. Example 4. This example was examined in Ref. [13] and the function to be approximated is defined as pffiffiffi 2 3 gðxÞ ¼ 1 þ : ð22Þ 3x1 x2 þ 0:25x1
Fig. 3. Example 3.
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Fig. 5. Example 5.
Fig. 4. Example 4.
The expanding design point is selected to be x2 = (1.5, 0.5)T. The previous design point is selected to be x1 = (1.0, 0.75)T. As a search direction, D = (0.5, 0.25)T is used. At two design points, the derivatives with respect to x1 have different signs but those with respect to x2 have the same sign. Under these circumstances, the exponent indices pi are 1 and 0.341. In TDQA, Gi are 0.977 and 0, and the correction coefficient g is 0.915. In TANA-3, H is 0.217. In the proposed methods, Hi are 0.977 and 1, and the correction coefficient ge is -0.020. In Fig. 4, the relative percentage errors of all methods are shown. In this example, when the derivatives at two points have different signs, TDQA has better accuracy than TANA-3 and the proposed methods have accuracy similar to TDQA. Example 5. This example is from Ref. [25] in order to illustrate the ‘Saddle Point’. The function to be approximated is defined as 2
2
gðxÞ ¼ ðx1 1Þ ðx2 1Þ þ x1 x2 :
ð23Þ
In TDQA, when at least one component of the diagonal Hessian matrix is not zero and g is equal to 1, theoretically, at two design points the function and the derivative values of TDQA are equal to those of the exact function. In order for g to be 1 in TDQA, the current design point is selected as x2 = (0.5, 1.5)T and the previous design point is selected as x1 = (1.5, 0.5)T. Under these circumstances, the exponent indices pi are 1 and 2. In TDQA Gi are 3.0 and 0. In TANA-3, H is 3. In the proposed methods, Hi are 3 and 1, and the correction coefficient ge is 0. In this case, mathematically, the proposed methods are completely equal to TDQA. Using D = (1.0, 1.0)T as the search direction, the relative percentage errors of all methods are shown in Fig. 5. Considering the signs of derivatives at two points, the derivatives with respect to x1 have different signs but those with respect to x2 have the same sign. Fig. 5 shows that TDQA has better accuracy than TANA-3 for this case and the relative percentage error of TDQA is almost zero. TDQA and the proposed methods have the same relative percentage errors.
4.2. Part 2: applications in engineering optimization In order to examine the numerical performance of the enhanced two-point approximation methods, a sequential approximation optimizer, which has various two-point approximations as options, is developed. Fig. 6 shows a flow chart of this process. This sequential approximation optimization procedure was described in Kim et al. [24]. The optimization results of TANA-3 as well as TDQA I and TDQA II are presented. The first example is the Rosen–Suzuki problem. The other examples are applications in structural optimization. This study uses the modified feasible direction method in DOT 4.20 [26]. In the sequential approximation optimization procedure, we used conservative approximation with a 40 percent move limit for the first iteration, and various move limits are applied after the second iteration. Five move limit strategies are selected. These are 40%, 75%, 100%, 150% and no move limits. The move limit strategy is defined for the conservative approximation method as xL ¼ maxðxk maxðml jxk j; 0:3Þ; xL0 Þ; xU ¼ minðxk þ maxðml jxk j; 0:3Þ; xU 0 Þ
Fig. 6. Flow chart of SAO process.
ð24Þ
J.-R. Kim, D.-H. Choi / Comput. Methods Appl. Mech. Engrg. 197 (2008) 846–856
and for the two-point approximation methods as L
x ¼ maxðminðxk1 maxðml jxk1 j; 0:3Þ; xk maxðml jxk j; 0:3ÞÞ; xL0 Þ xU ¼ minðmaxðxk1 þ maxðml jxk1 j; 0:3Þ; xk þ maxðml jxk j; 0:3ÞÞ; xU 0 Þ
ð25Þ
Here k is the number of iterations in SAO. xL0 and xU 0 are the initial lower and upper limits of design variables. The move limit ratio is abbreviated as ml. Example 6. Rosen–Suzuki Problem. This problem has been extensively used in Rosen and Suzuki [27] to verify the algorithm in the optimization procedure because it has many local optima. There are four design variables and three constraints. Originally the lower and upper limit of design variables did not exist. But in this study, they are taken as [10, 10]. The initial design is taken as x = (0.0, 0.0, 0.0, 0.0)T. Constraints 1 and 3 are active at the optimum. The optimum is known as f(x*) = 56.0 and x* = (0.0, 1.0, 2.0, 1.0)T. The mathematical formulation is minimize
f ðxÞ ¼ x21 þ x22 þ 2x23 þ x24 5x1 5x2
move limit, TANA-3, TDQA I, and TDQA II all need 8 iterations to reach similar optimum points. As the move limit increases, all three methods converge to similar optimum points and the number of iterations to obtain the solutions decreases in this example. When no artificial move limit is imposed, both TDQA I and TDQA II need only 5 iterations to reach the same optimum point, whereas TANA-3 fails to obtain the solution. These results demonstrate that the proposed methods work very well without regard to the move limit strategy and that they are also effective in wider ranges. Example 7. Cantilevered Beam. This problem is presented in the DOT User Manual [26]. The cantilevered beam shown in Fig. 7 is to be designed for minimum material volume. The design variables are the width b and height h at each of N segments. This beam must be designed subject to limits on stress (calculated at the left end of each segment), deflection under the load, and the geometric requirement that the height of any segment does not exceed twenty times the width. The data information is listed in Fig. 7. This design task is defined as
21x3 þ 7x4 þ 100 Minimize V ¼
subject to g1 ðxÞ ¼ g2 ðxÞ ¼
ðx21 ðx21
þ
x22
þ
þ 2x22 ð2x21 þ x22
x23
þ
þ
x23 x23
x24
þ x1 x2 þ x3 x4 Þ=8 1 6 0
þ 2x24 x1 x4 Þ=10 1 6 0
N X
b i h i li
i¼1
subject to ðM i hi =2I i Þ=ra 1 6 0
g3 ðxÞ ¼ þ þ 2x1 x2 x4 Þ=5 1 6 0 10 6 xi 6 10 for i ¼ 1; . . . ; 4 ð26Þ Table 1 shows the optimization results obtained by using TANA-3, TDQA I, and TDQA II. For this problem, the shifted intervening variables used in TDQA I and TDQA II is also applied to TANA-3 in order to avoid the singularity due to the side constraints. Under a 40%
853
i ¼ 1; . . . ; N
hi 20bi 6 0 dN =da;y 1 6 0
i ¼ 1; . . . ; N
1:0 6 bi 6 100
i ¼ 1; . . . ; N
5:0 6 hi 6 100
i ¼ 1; . . . ; N :
ð27Þ
The deflection di at the right end of segment i is calculated by recursion formulas. These recursion formulas were written in DOT User Manual [26]. The initial values of bi and hi are 5.0 and 40.0, respectively. This is a design
Table 1 Comparison of optimization results for the Rosen–Suzuki problem
x1 x2 x3 x4 f gmax Iterations
Initial
40% move limit TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
0 0 0 0 100 1
0.004 0.997 1.998 1.005 56.027 0.000 8
0.002 0.999 1.999 1.002 55.997 0.000 8
0.002 0.999 1.999 1.002 55.997 0.000 8
0.005 1.002 1.996 1.004 56.000 0.000 7
0.008 1.003 1.994 1.007 55.998 0.000 7
0.008 1.003 1.994 1.007 55.998 0.000 7
100 % move limit
75% move limit
150 % move limit
No move limits
TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
0.001 0.999 2.000 1.000 55.999 0.000 7
0.003 0.998 1.998 1.003 55.997 0.000 6
0.003 0.998 1.998 1.003 55.997 0.000 6
0.000 1.000 2.000 1.000 56.000 0.000 6
0.000 1.001 2.000 1.000 56.000 0.000 6
0.000 1.001 2.000 1.000 56.000 0.000 6
fail
0.001 1.001 1.999 1.001 56.000 0.000 5
0.001 1.001 1.999 1.001 56.000 0.000 5
854
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Fig. 7. The cantilevered beam.
Table 2 Comparison of optimization results for cantilevered beam Initial
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 f gmax Iterations
5 5 5 5 5 40 40 40 40 40 100 1
40% move limit
75 % move limit
TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
3.134 2.882 2.580 2.205 1.750 62.684 57.641 51.609 44.091 34.995 65419.7 0.000 9
3.126 2.882 2.590 2.205 1.750 62.514 57.648 51.805 44.091 34.995 65419.0 0.000 8
3.123 2.893 2.582 2.205 1.750 62.457 57.861 51.643 44.091 34.995 65422.3 0.000 8
3.146 2.874 2.575 2.205 1.750 62.925 57.480 51.502 44.091 34.995 65422.7 0.000 7
3.131 2.886 2.581 2.205 1.750 62.610 57.714 51.613 44.090 34.995 65417.4 0.000 8
3.134 2.883 2.580 2.205 1.750 62.684 57.654 51.595 44.090 34.995 65419.3 0.000 8
100% move limit
150% move limit
No move limits
TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
3.134 2.883 2.579 2.205 1.750 62.674 57.669 51.590 44.091 34.995 65419.6 0.000 9
3.135 2.884 2.580 2.205 1.750 62.698 57.675 51.595 44.091 34.997 65441.4 0.000 7
3.135 2.884 2.580 2.205 1.750 62.696 57.676 51.597 44.091 34.996 65441.3 0.000 7
3.140 2.878 2.579 2.205 1.750 62.796 57.554 51.572 44.091 34.995 65420.4 0.000 7
3.134 2.883 2.580 2.205 1.750 62.674 57.659 51.600 44.091 34.995 65419.5 0.000 8
3.143 2.872 2.580 2.205 1.750 62.867 57.447 51.603 44.091 34.995 65419.5 0.000 9
3.139 2.878 2.579 2.205 1.750 62.776 57.565 51.583 44.091 34.995 65420.2 0.000 7
3.131 2.886 2.579 2.205 1.750 62.628 57.727 51.578 44.091 37.995 65418.5 0.000 8
3.134 2.883 2.580 2.205 1.750 62.680 57.657 51.596 44.091 34.995 65419.5 0.000 8
problem with n = 2N variables. In this study, this problem was solved using five segments (10 design variables). The optimum is known as f(x*) = 65, 368 in the DOT User Manual. Also the optimum point is known as x* = (3.11, 2.88, 2.62, 2.20, 1.75, 61.20, 57.55, 52.31, 44.09, 35.00)T. Table 2 shows the optimization results obtained by using TANA-3, TDQA I, and TDQA II. All three methods succeed to converge to similar optimum points in almost
the same number of iterations under various move limits. These results show that the proposed methods work very well regardless of the move limit strategy.
Example 8. 25-Bar Truss with Stress and Displacement constraints. This problem, which is often seen in the literature, calls for a minimum weight structure subject to
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855
Table 3 Loading conditions for the 25 bar truss structure problem Loading conditions
Loaded node
Load, kN X
y
z
1
1 2 3 6 1 2
4.45 0 2.225 2.225 0 0
44.5 44.5 0 0 89 89
22.25 22.25 0 0 22.5 22.5
2
Fig. 8. Twenty-five bar structure.
member stress, Euler buckling and joint displacement constraints. This structure is subjected to two load cases. The geometry and data information is shown in Fig. 8. There are eight design variables for considering symmetry: x1 = A1, x2 = A2 = A3 = A4 = A5, x3 = A6 = A7 = A8 = A9, x4 = A10 = A11, x5 = A12 = A13, x6 = A14 = A15 = A16 = A17, x7 = A18 = A19 = A20 = A21 and x8 = A22 = A23 = A24 = A25. All design variables are limited within [0.001, 50.0], and all initial design points are 2.0. Each load-
ing condition is shown in Table 3. MSC/NASTRAN 70.7 [28] is used in this study. Table 4 shows the optimization results obtained by using TANA-3, TDQA I, and TDQA II. All three methods succeed to converge to similar optimum points under various move limits. Without regard to the size of the move limit, TDQA I and TDQA II require a similar number of iterations, whereas TANA-3 requires about 1.5 times more iterations in case of a 40% move limit than those in other cases. These results illustrate that the proposed methods work very well regardless of the move limit strategy. 5. Conclusions Function approximation is one of the most important and active fields of research in engineering optimization. Accurate function approximations can reduce repetitive computational efforts. This study presented enhanced
Table 4 Comparison of optimization results for 25-Bar Truss with stress and displacement constraints
x1 x2 x3 x4 x5 x6 x7 x8 f gmax Iterations
Initial
40% move limit TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
2 2 2 2 2 2 2 2 100 1
0.010 1.988 3.004 0.026 0.051 0.686 1.670 2.656 545.9 0.000 12
0.074 2.037 3.027 0.026 0.052 0.679 1.617 2.669 546.2 0.000 8
0.073 2.044 3.030 0.026 0.055 0.676 1.610 2.674 546.3 0.000 8
0.275 2.041 3.014 0.026 0.048 0.679 1.619 2.671 547.6 0.000 7
0.297 2.039 3.023 0.025 0.054 0.678 1.618 2.673 547.9 0.000 7
0.079 2.045 3.024 0.023 0.055 0.711 1.609 2.633 546.4 0.000 7
100% move limit
75% move limit
150% move limit
No move limit
TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
TANA-3
TDQA I
TDQA II
0.010 2.047 3.001 0.026 0.118 0.681 1.621 2.673 546.7 0.000 7
0.255 2.032 3.060 0.025 0.112 0.672 1.606 2.669 548.4 0.000 7
0.010 2.057 3.026 0.026 0.165 0.676 1.601 2.672 547.2 0.000 6
0.010 2.035 3.005 0.025 0.054 0.690 1.628 2.659 545.8 0.000 7
0.256 2.052 3.048 0.025 0.098 0.684 1.594 2.660 548.2 0.000 8
0.010 2.054 3.027 0.026 0.155 0.676 1.603 2.671 547.1 0.000 7
0.377 2.056 3.014 0.025 0.132 0.676 1.608 2.675 549.5 0.000 7
0.019 2.037 3.062 0.026 0.106 0.674 1.601 2.667 546.5 0.000 8
0.286 2.075 3.030 0.024 0.258 0.687 1.584 2.656 550.5 0.000 5
856
J.-R. Kim, D.-H. Choi / Comput. Methods Appl. Mech. Engrg. 197 (2008) 846–856
two-point diagonal quadratic approximation methods to guarantee the approximation accuracy whether the derivatives at two design points have the same sign or not. The approximation accuracy of the proposed methods is as good as that of TDQA when the derivatives of two design points have different signs. In addition, when those at two points have the same sign, they are as good as TANA-3. Therefore, considering all factors, the enhanced two-point diagonal quadratic approximation methods are more accurate than TANA-3 and TDQA. The proposed methods, which are abbreviated as TDQA I and TDQA II, exhibit the same or very similar accuracy. When the original function is independent of any design variables, we highly recommend that TDQA II be used. In the proposed methods, both the approximation function and derivative values at two design points are equal to their exact counterparts on all occasions. The unknown parameters of the approximated function can be obtained in a closed form. In part 2 of Section 4, numerical examples show that the proposed methods can successfully converge to a similar optimum under various move limit strategies in sequential approximation optimization. They also show that the proposed methods work very well regardless of the move limit strategy. Therefore, the proposed methods can be applied in reliability analysis and reliability based design optimization. When applied, the proposed methods will prove to be more effective than other two-point approximation methods. Acknowledgements This research work was supported by the International Joint R&D Program, the Ministry of Commerce, Industry, and Energy, and the center of Innovative Design Optimization Technology (iDOT), Korea Science and Engineering Foundation. References [1] L.A. Schmit, B. Farshi, Some approximation concepts for structural synthesis, AIAA J. 12 (1974) 692–699. [2] L.A. Schmit, H. Miura, A new structural analysis/synthesis capability – ACCESS 1, AIAA J. 14 (1976) 661–671. [3] L.A. Schmit, C. Fleury, Structural synthesis by combining approximation concepts and dual methods, AIAA J. 18 (1980) 1252–1260. [4] R.T. Haftka, J.A. Nachlas, I.T. Watson, T. Rizzo, R. Desai, Twopoint constraint approximation in structural optimization, Comput. Meth. Appl. Mech. Engrg. 60 (1987) 289–301. [5] C. Fleury, V. Braibant, Structural optimization: a new dual method using mixed variables information, Int. J. Numer. Meth. Engrg. 23 (1986) 409–428. [6] J.H. Starnes Jr., R.T. Haftka, Preliminary design of composite wings for buckling, stress and displacement constraints, J. Air Craft 16 (1979) 564–570.
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