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

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## 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.

## Keywords

Evolutionary algorithm Adaptive surrogate model Expensive optimization Reliability## Nomenclature

## Abbreviation

- 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

## Symbols

*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 FES
_{max} the function evaluations or maximum FES

*c*_{1}and*c*_{2}learn rates in PSO

*F*scale factor in DE

*Cr*crossover rate in DE

*x*_{min}and*x*_{max}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

## Greek symbols

- τ
scale factor in

*T*_{1}and*T*_{2}- ω
inertia weight in PSO

*λ*wavelength

## Notes

### Funding information

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).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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