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A dynamic surrogate-assisted evolutionary algorithm framework for expensive structural optimization

Abstract

In the expensive structural optimization, the data-driven surrogate model has been proven to be an effective alternative to physical simulation (or experiment). However, the static surrogate-assisted evolutionary algorithm (SAEA) often becomes powerless and inefficient when dealing with different types of expensive optimization problems. Therefore, how to select high-reliability surrogates to assist an evolutionary algorithm (EA) has always been a challenging task. This study aimed to dynamically provide an optimal surrogate for EA by developing a brand-new SAEA framework. Firstly, an adaptive surrogate model (ASM) selection technology was proposed. In ASM, according to different integration criteria from the strategy pool, elite meta-models were recombined into multiple ensemble surrogates in each iteration. Afterward, a promising model was adaptively picked out from the model pool based on the minimum root of mean square error (RMSE). Secondly, we investigated a novel ASM-based EA framework, namely ASMEA, where the reliability of all models was updated in real-time by generating new samples online. Thirdly, to verify the performance of the ASMEA framework, two instantiation algorithms are widely compared with several state-of-the-art algorithms on a commonly used benchmark test set. Finally, a real-world antenna structural optimization problem was solved by the proposed algorithms. The results demonstrate that the proposed framework is able to provide a high-reliability surrogate to assist EA in solving expensive optimization problems.

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Abbreviations

FEA:

finite element analysis

CFD:

computational fluid dynamics

EA:

evolutionary algorithm

SAEA:

surrogate-assisted evolutionary algorithm

ASM:

adaptive surrogate model

ASMEA:

ASM-based evolutionary algorithm

PSO:

particle swarm optimization

DE:

differential evolution

ASMPSO:

particle swarm optimization based on ASM

ASMDE:

differential evolution based on ASM

GP:

Gauss process model

KRG:

Kriging model

PRS:

polynomial response surface model

RBF:

radial basis function model

SHEP:

Shepard model

ANN:

artificial neural network

SVM:

support vector machine

RBNN:

radial basis neural network

ELM:

extreme learning machine

HFSS:

high-frequency simulation software

WAS:

weighted average surrogate

OWS:

optimum weight surrogate

DOE:

design of experiment

LHS:

Latin hypercube sampling

RMSE:

root of mean square error

AE:

absolute error

PRESS:

prediction residual error sum of square

SR:

successful run

VTR:

value to reach

n :

the number of meta-models

k :

the number of elite meta-models

D :

the number of variables

NP :

the size of the population

S t :

the size of the strategy pool

T 1 :

the train set in the database

T 2 :

the test set in the database

G or G m :

the evolutional generation or maximum G

FES or FESmax :

the function evaluations or maximum FES

c1 and c2 :

learn rates in PSO

F :

scale factor in DE

Cr :

crossover rate in DE

xmin and xmax :

variable space

fun ASM :

evaluation function based on ASM

fun real :

evaluation function based on simulation

G-opt:

global optimum

L-opt:

local optimum

#G:

the number of G-opt

#L:

the number of L-opt

ASM1:

the ASMPSO algorithm

ASM2:

the ASMDE algorithm

N :

the number of sampling points

S 11 :

return less

L, W, h, :

the length, width, height of the patch

L 1 :

edge distance

τ:

scale factor in T1 and T2

ω:

inertia weight in PSO

λ :

wavelength

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Funding

This research was supported by the National Natural Science Foundation of China (Grant Nos. 61803054 and 61876169) and the State Education Ministry and Fundamental Research Funds for the Central Universities (2019 CDJSK 04 XK 23).

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Authors

Corresponding author

Correspondence to Jing Liang.

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The authors declare that they have no conflict of interest.

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Responsible Editor: Raphael Haftka

Appendices

Appendix1

(1) Kriging (KRG)

A general form of the Kriging model is as follows:

$$ f(x)=g(x)+z(x) $$
(1)

where is an unknown prediction function, is a known approximate function, and is the realization of a stochastic process with a zero mean and a nonzero covariance. The (i,j)th element of the covariance matrix z(x) is given as

$$ \mathrm{Cov}\left[z\left({x}^i\right),z\left({x}^j\right)\right]=\sigma \frac{2}{s}{R}_{ij} $$
(2)

where \( \sigma \frac{2}{z} \) is the process variance, and Rij is the correlation function between the ith and the jth data points. A Gaussian function is used as the correlation function in this paper, defined by

$$ {R}_{ij}=R\left({X}^i,{X}_j\right)=\exp \left\{-\sum \limits_{k-1}^n{\theta}_k{\left({x}_k^i-{x}_k^i\right)}^2\right\} $$
(3)

where θk is distinct for each dimension, and these unknown parameters are generally obtained by solving a nonlinear optimization problem. In this paper, the order of the global polynomial trend function is specified to be zero.

(2) Polynomial response surface (PRS)

It is relatively easy to establish a polynomial regression model using least squares. In general, the second-order polynomial regression model can solve most problems well, which can be expressed as

$$ {y}_i\left({x}_i\right)={\beta}_o+\sum \limits_{j=1}^k{\beta}_j{x}_{ij}+\sum \limits_{j=1}^k{\beta}_j{x}_{ij}^2+\sum \limits_{j_1=1}^k\sum \limits_{\underset{J_2\ne {j}_1}{j_2=1}}^k{\beta}_{j_1{j}_2}{x}_{i,{j}_1}{x}_{i,{j}_2}+{\in}_i $$
(4)

where xi(i = 1, 2, …, n) is design variable. yi is the corresponding predictive response. k is the number of variables, and is the fitting error. β1 is the parameter to be determined, which can be calculated by the least square method

$$ \beta ={\left({\mathbf{F}}^{\mathrm{T}}\mathbf{F}\right)}^{-1}{\mathbf{F}}^{\mathrm{T}}y $$
(5)

where F is a Gram-matrix based on sample x.

(3) Radial basis function (RBF)

RBF has been developed for the interpolation of scattered multivariate data. The model is defined by

$$ \overset{\sim }{f}(x)=\sum \limits_{i=1}^N{\sigma}_i\psi \left(\left\Vert x-{x}_i\right\Vert \right) $$
(6)

where N denotes the number of training points, is the unknown coefficients, and is the multi-quadric function. The multi-quadric function is defined as

$$ \psi (r)=\sqrt{r^2+{c}^2} $$
(7)

where r = ‖x − xi‖ is the Euclidean distance of a point x from a given data point xi, and the parameter c > 0 is a constant in the multi-quadric.

Appendix 2

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Appendix 3

figure b

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Yu, M., Li, X. & Liang, J. A dynamic surrogate-assisted evolutionary algorithm framework for expensive structural optimization. Struct Multidisc Optim 61, 711–729 (2020). https://doi.org/10.1007/s00158-019-02391-8

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Keywords

  • Evolutionary algorithm
  • Adaptive surrogate model
  • Expensive optimization
  • Reliability