Estimation of optimum design of structural systems via machine learning

doi:10.1007/s11709-021-0774-0

Front. Struct. Civ. Eng.

2021, Vol. 15

Issue (6) : 1441-1452 https://doi.org/10.1007/s11709-021-0774-0

RESEARCH ARTICLE

Estimation of optimum design of structural systems via machine learning

Gebrail BEKDAŞ, Melda YÜCEL, Sinan Melih NIGDELI(

)

Department of Civil Engineering, Istanbul University-Cerrahpaşa, Istanbul 34320, Turkey

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Abstract

Three different structural engineering designs were investigated to determine optimum design variables, and then to estimate design parameters and the main objective function of designs directly, speedily, and effectively. Two different optimization operations were carried out: One used the harmony search (HS) algorithm, combining different ranges of both HS parameters and iteration with population numbers. The other used an estimation application that was done via artificial neural networks (ANN) to find out the estimated values of parameters. To explore the estimation success of ANN models, different test cases were proposed for the three structural designs. Outcomes of the study suggest that ANN estimation for structures is an effective, successful, and speedy tool to forecast and determine the real optimum results for any design model.

Keywords optimization metaheuristic algorithms harmony search structural designs machine learning artificial neural networks

Corresponding Author(s): Sinan Melih NIGDELI

Just Accepted Date: 19 October 2021 Online First Date: 25 November 2021 Issue Date: 21 January 2022

Cite this article:

Gebrail BEKDAŞ,Melda YÜCEL,Sinan Melih NIGDELI. Estimation of optimum design of structural systems via machine learning[J]. Front. Struct. Civ. Eng., 2021, 15(6): 1441-1452.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-021-0774-0
https://academic.hep.com.cn/fsce/EN/Y2021/V15/I6/1441

name of metric	abbreviation	meaning	formulation
mean absolute error	MAE	averaging of absolute value of difference between actual and predicted results for all samples/ margin of deviation	$∑ i = 1 n A i ? P i n$
mean squared error	MSE	average value of squared of deviation amount of estimations from actual values/ errors for n samples/ magnified error	$∑ i = 1 n (A i ? P i) 2 n$
root mean squared error	RMSE	root of mean squared error value/measure of distance between actual and predicted values	$∑ i = 1 n (A i ? P i) 2 n$

Tab.1 Properties of error metrics

Tab.2 Parameter information and properties of applied cases

Fig.1 3-bar truss structure.

Fig.2 Representation of change for minimum volume according to different values of FW and HMCR.

$A 1 = A 3$ (cm²)	$A 2$ (cm²)	Min_f(v) (cm³)	mean of volume (cm³)	Std. Dev. of volume	HMCR	FW
0.7885	0.4087	263.8959	263.9033	0.0058	0.55	0.05

Tab.3 The best solution and statistical evaluations for case 1

Fig.3 Analysis of the minimum volume of trusses by using several combinations of iteration-population numbers.

$A 1 = A 3$ (cm²)	$A 2$ (cm²)	Min_f(v) (cm³)	mean of volume	Std. Dev. of volume	iterationnumber	populationNumber
0.7886	0.4083	263.8959	263.8985	0.0031	25000	15

Tab.4 The best solution and statistical evaluations for case 2

Fig.4 Error metrics of estimated results for training data (model 1).

Tab.5 Estimation results for

A 1 = A 3

A 2

and minimum volume (

M i n f (v)

) of test models

Fig.5 10-bar truss structure.

property	parameter	notation	value/limit	unit
design constants	elasticity modulus	E_s	10000	ksi
design constants	weight per unit of volume of bars	Ρ	0.1	lb/in³
design variables	section areas	$A$	0.1–35	in²
design constraints	displacements for whole nodes in all directions	δ	±2	in
design constraints	tensile/compression stresses for each bar	σ	±25	ksi

Tab.6 Properties of optimization design for 10-bar truss model

Fig.6 Minimum weights for 10-bar truss obtained via FW and HMCR combinations.

parameter	value
$A 1$ (in²)	23.9436
$A 2$ (in²)	0.1
$A 3$ (in²)	25.2889
$A 4$ (in²)	14.053
$A 5$ (in²)	0.1
$A 6$ (in²)	1.9785
$A 7$ (in²)	12.5091
$A 8$ (in²)	12.9271
$A 9$ (in²)	20.1141
$A 10$ (in²)	0.1
$M i n f (v)$ (lb)	4680.8384
mean of weight	4693.8804
Std. Dev. of weight	4.9906
HMCR	0.25
FW	0.05

Tab.7 Optimum solution, design variables, and the best parameters belonging to case 1

Fig.7 Analysis of minimum weight of truss by using several combinations of iteration-population number.

parameter	value
$A 1$ (in²)	24.1027
$A 2$ (in²)	0.1
$A 3$ (in²)	26.0599
$A 4$ (in²)	14.0816
$A 5$ (in²)	0.1
$A 6$ (in²)	2.0014
$A 7$ (in²)	12.258
$A 8$ (in²)	12.5361
$A 9$ (in²)	20.0621
$A 10$ (in²)	0.1
$M i n f (v)$ (lb)	4680.852
mean of weight	4694.1209
Std. Dev. of weight	4.5771
iteration number	35000
population number	35

Tab.8 Optimum results with best numbers of iteration and population for case 2

Fig.8 Error metrics of estimated results for training data (model 2).

parameter	P₁ (kip)	P₂ (kip)	HS optimization result	ANN estimation	error calculations for HS	average		RMSE
parameter	P₁ (kip)	P₂ (kip)	HS optimization result	ANN estimation	error calculations for HS	MAE	MSE	RMSE
A₁	143	48	22.5921	22.3536	0.2385	0.7326	0.7398	0.8601
	150.4	43.5	27.6794	26.6037	1.0758
	154	50.5	24.151	24.5216	–0.3707
	145	42	24.401	25.8323	–1.4313
	157.25	55.25	22.8998	23.4463	–0.5465
A₂	143	48	0.1	0.1254	–0.0254	0.0206	0.0006	0.0247
	150.4	43.5	0.1	0.1419	–0.0419
	154	50.5	0.1	0.0995	0.0005
	145	42	0.1	0.1143	–0.0143
	157.25	55.25	0.1	0.121	–0.021
A₃	143	48	23.9015	24.3014	–0.3999	0.3782	0.1464	0.3826
	150.4	43.5	26.4858	26.764	–0.2782
	154	50.5	26.4822	26.1031	0.3792
	145	42	26.4865	26.0293	0.4572
	157.25	55.25	26.2506	25.8739	0.3767
A₄	143	48	12.7495	136,382	–0.8887	0.4566	0.3336	0.5776
	150.4	43.5	15.6635	155,987	0.0649
	154	50.5	14.1376	149,142	–0.7766
	145	42	15.5657	150,464	0.5193
	157.25	55.25	14.6318	146,655	–0.0337
A₅	143	48	0.1	0.094	0.006	0.0071	7.01e–05	0.0084
	150.4	43.5	0.1	0.1146	–0.0146
	154	50.5	0.1	0.1091	–0.0041
	145	42	0.1	–0.009	–0.009
	157.25	55.25	0.1	0.0982	0.0018
A₆	143	48	1.9183	1.9275	–0.0092	0.0361	0.0017	0.0415
	150.4	43.5	1.7466	1.8007	–0.0541
	154	50.5	1.9958	2.0378	–0.0419
	145	42	1.72	1.7802	–0.0602
	157.25	55.25	2.1835	2.1984	–0.0149
A₇	143	48	11.9171	11.8335	0.0836	0.1305	0.0411	0.2027
	150.4	43.5	12.2539	12.2239	0.0299
	154	50.5	12.6766	12.7335	–0.0569
	145	42	11.522	11.9607	–0.4386
	157.25	55.25	12.9318	12.9754	–0.0436
A₈	143	48	12.2143	12.046	0.1683	0.2253	0.08	0.2829
	150.4	43.5	15.1782	15.4772	–0.299
	154	50.5	13.5386	13.4783	0.0603
	145	42	14.8597	14.936	–0.0763
	157.25	55.25	11.8532	12.3758	–0.5226
A₉	143	48	19.774	19.2167	0.5573	0.5299	0.3321	0.5762
	150.4	43.5	21.542	22.1308	–0.5889
	154	50.5	21.7187	21.1551	0.5636
	145	42	21.4216	21.3008	0.1208
	157.25	55.25	21.4534	20.6344	0.819
A₁₀	143	48	0.1	0.0784	0.0216	0.0168	0.0004	0.0205
	150.4	43.5	0.1	0.0896	0.0104
	154	50.5	0.1	0.1082	–0.0082
	145	42	0.1	0.0622	0.0378
	157.25	55.25	0.1	0.106	–0.006
Min_f(w) (lb)	143	48	4449.3922	4449.0023	0.0216	1.2065	1.841	1.3469
	150.4	43.5	5082.3537	5083.7904	0.0104
	154	50.5	4856.2858	4857.0534	–0.0082
	145	42	4900.2784	4898.137	0.0378
	157.25	55.25	4741.1288	4742.4228	–0.006

Tab.9 Estimation results for

A 1 ? A 10

and minimum weight (

M i n f (w)

) of test models

Fig.9 Simply-supported RC beam.

Tab.10 The best solution and statistical evaluations for case 1

Fig.10 Variation of minimum costs according to different values of FW and HMCR.

Tab.11 Optimum design results together with cost statistics for case 2

Fig.11 Detection of minimum cost for RC beam through the usage of various iteration-population numbers.

Fig.12 Error metrics of estimated results for training data (model 3).

Tab.12 Estimation results for

h b

A s

and minimum cost (

M i n f (c)

) of test models

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