Paper review

In 1904, Michell calculated the theoretical lower bound of the weight of truss structures with stress constraints [1]. The theoretical derivation of ideal structures was an important inspiration for structural optimization. After finite element method was proposed and developed maturely, structural optimization is valid for designing complicate structures.

In 1974, Schmit and Farshi applied approximation concepts to convert structural behavior into explicit functions of design variables [2]. This method turned limited information from structural analyses into simple approximate functions, and greatly reduced the time of structural optimization process.

So far, a lot of local approximation methods have been developed. Among these schemes, direct linear approximation is the most fundamental approach which performs the 1st order Taylor series expansion in terms of design variables. However, most structural characteristics are nonlinear, this method may not be reliable for therefore. To enhance the approximation quality, some scholars proposed reciprocal approximation method, which adopting the reciprocals of original variables as intervening variables in

1st order Taylor series expansion [3]. This approximation method is quite suitable for stress and displacement constraint in simple truss problems when cross-section area is selected as design variable. However, 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 [5] which is also known as convex linearization (COLIN) presented by Fleury and Braibant [6]. This method adopts either direct linear or reciprocal approximation for each design variable, according to which approximate function is estimated higher. In other words, conservative approximation adopts the more conservative one between direct linear and reciprocal approximation for every design variable. In 1987, Svanberg presented the method of moving asymptotes (MMA), which can be regarded as the generalization of CONLIN [7].

To improve the approximation quality of single-point approximations, a lot of approximation schemes developed subsequently with utilizing the information of previous design point to construct approximate functions. These approximations are classified as two-point approximations. In 1987, Haftka et al. proposed two-point modified reciprocal approximation which has the strategy to decide the indeterminate coefficients in modified reciprocal approximation [8]. However, this strategy is undefined when the derivatives of two successive points have the different signs. In 1990, Fadel et al. proposed two-point exponential approximation (TPEA) [9]. TPEA performs Taylor series expansion with exponential intervening variables. The derivative of the previous design point is used to determine the exponent. But this approximation lacks definition for two-point approximation when the derivatives of the variable 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 [10][11]. 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 [12]. In 1995, Fadel classified the approximate functions into monotonic and non-monotonic functions [13]. It is suggested that the selection the approximations should consider the characteristic of monotonicity of the structural behavior. Then the mixed method is proposed named DQA-GMMA, which adopts monotonic approximation for design variable when the derivatives of two successive design points have the same signs, and vice versa.

In 1997, Zhang and Fleury proposed modified convex approximation (MCA) based on CONLIN [14]. MCA increases the convexity of approximation to avoid non-convergent process. In 1998, Xu and Grandhi proposed two-point adaptive nonlinear approximation-3 (TANA-3), which appends a term to TPEA for additionally matching the function value of previous design point [15]. In 2000, Xu et al. presented a new two-point approximation approach which uses the linear combination of linear and reciprocal approximations to match the derivatives of previous design point [16]. In 2001, Kim et al. presented two-point diagonal quadratic approximation (TDQA) based on TPEA [17]. TDQA adds shifting level into exponential intervening variables to avoid the singularity of the derivatives. In 1996, Chickermane and Gea proposed generalized convex approximation (GCA) [18]. GCA uses the derivatives of two points to construct approximate functions without lacking definition in TPEA. In 2007, Groenwold proposed incomplete series expansion (ISE) which includes a series of approximations [19]. ISE uses quadratic, cubic, and even higher order diagonal terms to construct the approximate functions.

Several approximation schemes have the approximate function convex to ensure stability of the optimization process such as GCMMA [20][21]. In 2015, Li proposed an adaptive quadratic approximation (AQA) which enforces the approximate functions to be strictly convex to improve the robustness and convergence performance of the optimization process [22]. However, this enforcement would cause inconsistency and may lower the efficiency of optimization process.

Moreover, Chiou proposed two new convex approximation methods in 2000, including self-adjusted convex approximation (SACA) and two-point convex approximation (TPCA) [23]. In 2002, Chen proposed improved two-point approximation (ITPA) which can be seen as the combination the linear-reciprocal and TPEA [24]. In 2007, Chang proposed quasi-quadratic two-point conservative approximation (QTCA) [25]. In 2010, Chen proposed exponential MMA (EMMA), which makes the order of intervening variables in MMA adjustable for more flexibility [26]. In 2012, Chen proposed a new mixed two-point approximation method which is the combination of TPEA and GBMMA [27]. In 2013, Jiang proposed enhanced two-point exponential approximation (ETPEA) to conquer the problem of lack of definition in TPEA [28]. ETPEA use intervening variable which is the second order Taylor series expansion of the original variable to deduce the new formula as the remedy of TPEA.