A developed surrogate-based optimization framework combining HDMR-based modeling technique and TLBO algorithm for high-dimensional engineering problems

Abstract

The traditional surrogate-based optimization techniques are facing severe challenges for high-dimensional engineering optimization problems. The main challenges are how to overcome metamodeling and optimization difficulties in high-dimensional design space. To solve the above difficulties, a developed surrogate-based optimization framework combining high-dimensional model representation (HDMR)-based modeling technique and teaching-learning-based optimization (TLBO) algorithm is developed. A high-dimensional model can be decomposed into a series of low-dimensional models by HDMR-based modeling technique, which can greatly reduce the difficulty of building high-dimensional model. The TLBO algorithm which has strong optimization ability and non-parameter setting characteristic is introduced to optimize the HDMR-based model to overcome the difficulty of optimization. Several representative functions are selected as examples to verify the developed optimization method for high-dimensional problems. In addition, The developed surrogate-based optimization is applied to solve a typical engineering optimization problem: high-dimensional aerodynamic shape optimization. It can be concluded that the optimization ability of the traditional surrogate-based optimization framework can be improved with assistance of HDMR-based modeling technique and TLBO algorithm for high-dimensional engineering problems.

Appendix. Aerodynamic shape parameterization method

With the rapid development of computational fluid dynamics (CFD), the aerodynamic shape optimization (ASO) which includes shape parameterization, mesh deformation, optimization algorithm, and aerodynamic analysis has been widely concerned in the aircraft design. The aerodynamic shape parameterization methods have a great influence on the optimal results of ASO. A good parameterization method should have the ability that covers more potential shapes with fewer design parameters in the design space. A variety of shape parameterization methods are developed and applied to ASO, such as, parametric section (PARSEC) method and its developed methods, the Hicks-Henne function methods, class-shape function transformation (CST) methods [9], free-form deformation (FFD) methods, and so on. These airfoil parametric methods represent the airfoil shape by summing a set of basic functions. It is well known that the more parameters are used to describe airfoil shape, the more potential airfoils are contained in the design space. The high-dimensionality of design parameters is theoretically beneficial for searching for the optimal design. But in reality, the increase of design parameters will inevitably increase the search difficulty of optimization algorithms. The high-dimensionality of design parameters inevitably restricts the ability of optimization algorithms and makes the convergence of optimization process slowly, which is not helpful to finding the best optimal design results. Therefore, how to overcome this contradiction should be concerned in the airfoil parameterization research field.

The class-shape function transformation (CST) is an airfoil parametric method. The parameterization proposed by Kulfan and Bussoletti represents a two-dimensional geometry by the product of a class function C(x/c) and a shape function S(x/c) plus a term that characterizes the trailing edge thickness:

$$ \frac{y}{c}=C\left(\frac{x}{c}\right)\kern0em S\left(\frac{x}{c}\right)+\frac{x}{c}\kern0em \frac{\varDelta {Z}_{te}}{c} $$

(13)

where C(x/c) is given in generic form by

$$ C\left(\frac{x}{c}\right)={\left(\frac{x}{c}\right)}^{N_1}\kern0em {\left(1-\frac{x}{c}\right)}^{N_2}\kern1em 0\le \frac{x}{c}\le 1 $$

(14)

The exponents N₁ and N₂ define the type of geometric to be represented. An airfoil, for example, is represented by N₁ = 0.5, N₂ = 1. The shape function is defined on the basis of Bernstein binomials, by the introduction of weight factors b_i as follows:

$$ S\left(\frac{x}{c}\right)={\sum}_{i=0}^n\left[{b}_i\cdotp \frac{n!}{i!\left(n-i\right)!}\cdotp {\left(\frac{\mathrm{x}}{c}\right)}^i{\left(1-\frac{x}{c}\right)}^{n-i}\right] $$

(15)

Taking NACA0012 airfoil as an example to verify the fitting ability of CST, Fig. 4 gives the geometric representation and error analysis by CST method. It can be observed that the fitting ability of CST is very good (Fig. 15).

Fig. 15

Geometric representation and error analysis

The 3D wing can be regarded as a combination of a series of airfoils along the span wise distribution. Therefore, the CST method introduced above can be expanded to describe the shape of different airfoils on different span wise sections to parameterize the shape of the entire wing. The parameter b_i in the (15) is represented as a weighted sum of m-order Bernstein polynomials along the wing span wise.

$$ {b}_i={\sum}_{j=0}^m{b}_{i,j}{B}_m^j\left(\eta \right) $$

(16)

where η = 2y/b_w, y represents the span wise coordinate, b_w represents the length of the wing span wise. ${B}_m^j\left(\eta \right)$ can be expressed by (17)

$$ {B}_m^j\left(\eta \right)={K}_m^i{\eta}^j{\left(1-\eta \right)}^{m-j} $$

(17)

$$ {K}_m^j=\frac{m!}{j!\left(m-j\right)!} $$

(18)

where j = 0, 1, ...m。The geometric shape of wing can be expressed by the Eq. (19):

$$ \xi \left(\phi, \eta \right)={C}_{N_2}^{N_1}\left(\psi \right){\sum}_{i=0}^n{\sum}_{j=0}^m{b}_{i.j}{B}_n^i\left(\psi \right){B}_m^j\left(\eta \right) $$

(19)

where b_i,j represents the unknown coefficients, which can be solved by least square method. Once the coefficients are known, the aerodynamic shape can be parameterized.

The fitting accuracy of 3D CST method is verified by taking the selected wing as an example. The parameter m has little influence on the fitting accuracy because the cross-section airfoils of the selected wing are the same. Therefore, the effect of parameter n on the fitting accuracy is considered in the paper. Figure 16 shows the fitting results of CST for the wing with different parameter n and m. It can be seen from comparison of the thickness contour that the CST parameterization method can fit the original wing well when the n reaches the second-order. Figure 17 gives the fitting error analysis of the cross-section airfoil at different span wise positions. It can be observed that the fitting error is less than 3e⁻⁴ when the n reaches the second-order, which indicate that the aerodynamic shape of the wing can be fitted well by the 3D CST parameterization method.

Fig. 16

The fitting results of CST for the wing with different parameter n and m

Fig. 17

The fitting error of different wing cross section