Paper review

The history of structural optimization can be traced back to the 1904s. Mitchell derived the theoretical lower bounds of the weight of truss structures with stress constraints [1]. Although it is based on ideal and simple structures, it is still an important inspiration for structural optimization. In 1974, Schmit and Farshi applied approximation concepts to convert structural behaviors into explicit functions of design variables [2]. This method turns time-consuming structural analyses into simple approximate functions, and it greatly reduces the time of structural optimization process.

So far, a lot of local approximation methods have been developed. Among these schemes, the direct linear approximation method, which performs the first-order Taylor series expansion in terms of design variables, is the simplest one. However, most structure characteristics are nonlinear, and therefore this method may not be reliable. In order to approximate structural behavior more accurately, some scholars proposed reciprocal approximation method, which uses the first-order Taylor series expansion in reciprocal variables [3]. Although this method can construct more accurate approximations for simple truss problems with stress and displacement constraints, the

function value tends to infinity when the design variable approach zero that may cause inappropriate approximation. To overcome this problem, Haftka and Shore proposed modified reciprocal approximation method to shift the singular point in reciprocal approximation [4].

In 1979, Starnes and Haftka proposed conservative approximation method, which is also known as convex linearization (COLIN) presented by Fleury and Braibant [5, 6].

This method adopts either direct linear or reciprocal approximation for each design variable according to which approximate function value is estimated higher. It means that this method adopts the more conservative one between direct linear and reciprocal approximation for each design variable. In 1987, Svanberg presented the method of moving asymptotes (MMA), which can be regarded as the generalized form of conservative approximation [7]. This method calculates the moving asymptotes by heuristic rules and utilizes the moving asymptotes to adjust the conservatism of approximations, so that the approximation becomes more adaptive and more efficient.

The efficiency of this method depends on the asymptote locations strongly, and therefore some scholars proposed some methods which utilize the first-order or second-order information to obtain the appropriate moving asymptotes [8, 9].

The above methods have been shown to be successful in a lot of applications, but they are all single-point schemes. Therefore, in order to improve the approximation quality, multi-point information can be used to construct reliable mid-range approximations that are valid in a relatively larger region near the design point. In 1987, Haftka et al. proposed two-point modified reciprocal approximation with a strategy to decide the indeterminate coefficients in modified reciprocal approximation [10].

However, it would fail when the sensitivities of two successive points have the different signs. In 1990, Fadel et al. performed the first-order Taylor series expansion with

exponential intervening variables, called two-point exponential approximation (TPEA) [11]. Although this method can determine the exponent by the sensitivity of the previous design point, it would fail when the sensitivities of two successive points have the different signs. In 1994 and 1995, Wang and Grandhi proposed a series of two-point adaptive nonlinear approximations (TANA) based on TPEA, which enhance TPEA by matching the function value of previous design point [12, 13].

However, the above methods are all monotonic schemes, and therefore are not suitable to approximate non-monotonic structural behaviors. It means that these methods have poor convergence and cannot even converge on certain problems. Hence, some scholars have proposed second-order approximation based on Taylor series expansion to achieve the best compromise between conservativeness and accuracy. In 1994, Snyman and Stander presented spherical approximation method (SAM), which appends a quadratic term to direct linear approximation for correcting the function value of the previous design point [14]. In 1997, Zhang and Fleury proposed modified convex approximation (MCA) based on CONLIN to increase the convexity of approximation [15]. In 1998, Xu and Grandhi proposed two-point adaptive nonlinear approximation-3 (TANA-3), which appends a term to TPEA for matching function value of the previous design point [16]. In 2001, Kim et al. presented two-point diagonal quadratic approximation (TDQA), which adds shifting level into exponential intervening variables to avoid the singularity of the sensitivities in TPEA [17]. In 2007, Groenwold et al. proposed incomplete series expansion (ISE), which includes a series of approximations [18]. ISE uses quadratic, cubic, and even higher order diagonal terms to construct the approximate functions. In 2008, Kim and Choi proposed enhanced two-point diagonal quadratic approximation to reinforce TDQA with new quadratic correction terms by the concept of TANA-3 [19].

In fact, the constraint functions are generally neither purely monotonic nor purely non-monotonic. Therefore, in 1995, Fadel considered the monotonicity of the structural behavior to construct the approximate functions [20]. Then the mixed method named DQA-GMMA is proposed, which adopts monotonic approximation for design variable when the sensitivities of two successive design points have the same signs, and vice versa. Furthermore, several approximation schemes have the approximate function convex to ensure stability of the optimization process, such as GCMMA [21, 22]. In 2015, Li made the approximate functions be strictly convex to improve the robustness and convergence performance of the optimization process, called adaptive quadratic approximation (AQA) [23]. However, this enforcement would cause inconsistency and may lower the efficiency of optimization process.

Moreover, in 2000, Chiou proposed two new convex approximation methods, including self-adjusted convex approximation (SACA) and two-point convex approximation (TPCA) [24]. Both methods are developed based on conservative approximation to achieve numerical stability. In 2002, Chen proposed improved two-point approximation (ITPA), which can be seen as the combination of linear-reciprocal and TPEA [25]. In 2007, Chang proposed quasi-quadratic two-point conservative approximation (QTCA) to improve the conventional conservative approximation methods [26]. In 2010, Chen proposed exponential MMA (EMMA), which makes the order of intervening variables in MMA adjustable for more flexibility [27]. In 2012, Chen proposed a new mixed two-point approximation method, which is the combination of TPEA and GBMMA [28]. When the sensitivities of two successive design points have the same signs, the TPEA scheme is used. Otherwise, the GBMMA scheme is considered. In 2013, Jiang proposed enhanced two-point exponential approximation (ETPEA) that uses intervening variable which is the second order Taylor

series expansion of the original variable to deduce the new formula as the remedy of TPEA [29]. This method can still construct the approximate function when the exponent in TPEA cannot be calculated. In 2016, Wang proposed quasi-quadratic method of moving asymptotes approximation (QMMA), which adds a non-spherical second-order term to MMA to improve the accuracy and efficiency [30]. In the same year, Ke proposed two-point piecewise adaptive approximation (TPPAA), which utilizes the piecewise approximate functions to consider the monotonicity of structural behavior to ensure the quality [31].