Estimation of optimum design of structural systems via machine learning

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.

Graphical abstract

Keywords

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

1 Introduction

Structural engineering, which is a branch of civil engineering, employs a great variety of structural designs and models. These must deal with safety, with extreme conditions, with economics, as well as with usability and aesthetics. To bring these requirements together, it is essential to design correctly, efficiently and sensitively, incorporating controllable and renewable solutions. For this, classical analysis programs with advanced coding software have become more important in recent years. In particular, coding applications can be more practical for optimization in comparison with analysis programs because they are more updatable, portable, and can be tuned by any designer/engineer.

Nonetheless, if optimization issues are evaluated, the determination of the best solution or choice for any problem, design, and model needs to be considered. This is because the methodology can be used to solve and converge optimally to meet the demands of the specific problem. There are many types of optimization, but metaheuristic methods are the most researched, and frequently preferred. Metaheuristics has power and capability to solve various problems in providing optimization. There are several different types of metaheuristic algorithms. First, there are algorithms based on biological science and principles of genetics, including genetic algorithm (GA) [1], differential evolution (DE) [2], big-bang big-crunch (BB-BC) algorithm [3], and biogeography-based optimization (BBO) [4]. Secondly, there are algorithms which are inspired by living population behaviors, such as particle swarm optimization (PSO) [5], artificial fish school algorithm (AFSA) [6], artificial bee colony (ABC) algorithm [7], cuckoo search (CS) [8], flower pollination algorithm (FPA) [9], gray wolf optimization (GWO) [10] and whale optimization algorithm (WOA) [11]. Thirdly, there are algorithms based on memory-based methods, like tabu search (TS) [12] and harmony search (HS) [13]. Finally, the algorithms which are designed by considering the laws and processes of physics and mechanics, are simulated annealing (SA) [14], central force optimization (CFO) [15], gravitational search algorithm (GSA) [16], and water evaporation optimization (WEO) [17].

Many previous studies have included examples such as the determination of the best/optimum section area/size for beams by using DE [18], BB-BC algorithm [19], modified versions of bat algorithm (BA), and ABC [20]. The studies for columns were performed by GA [21] and HS [22]. The design of retaining wall prioritizing minimum cost was conducted by using BBO [23], GA [24,25], shuffled shepherd optimization algorithm (SSOA) [26], and teaching-learning-based optimization (TLBO) [27]. In addition, ant colony optimization (ACO) [28], TLBO [29], FPA [30], PSO [31], ant lion optimizer (ALO) [32], HS [33], symbiotic organisms search (SOS) [34], and interactive fuzzy search algorithm (IFSA) [35] have been used to find the best truss structure with the aim of optimization for sizing, layout, topology by considering factors such as minimum weight, cost, as well as conforming to different design conditions and laws. Charged system search (CSS) [36], AFSA [37], BA [38], FPA [39], and ACO [40] were employed for optimal design of different formed tuned mass dampers intended for various purposes. Also in computational mechanics for structural engineering, new approaches have been considered and applied by using various machine learning algorithms [41–48].

This study used three different models for various benchmark structural designs. These were two optimizations and a machine learning estimation process. The designs contained three models: 3-bar and 10-bar truss and simply supported reinforced concrete (RC) beams. In detail, the first and second optimization cases were related to evaluating and comparing two specific parameters of HS and general parameters (iteration and population numbers), respectively. The third case was based on the estimation of optimum results of structural models including design variables and the best target functions, and was conducted with artificial neural networks (ANN). Three test models were analyzed to find out the optimum results under different design conditions and without the use of optimization stages.

2 Harmony search

The effectiveness and success of musical performance are based on some activities such as presenting songs and singing, producing notes or timbres perfectly by the musician. To make an exceptional performance, musicians study and develop notes of music until they find notes with perfect harmony. This musical effort can be characterized as an optimization process, application, or tool.

Geem et al. [13] proposed HS that is inspired by the musical process. While the HS metaheuristic algorithm is developed, musicians can perform any natural musical performance according to two different actions, as follows [49].

1) A musician may present any musical performance by benefiting from notes and harmonies within the own recorded harmony memory. During realization of this process, the musician uses a parameter called fret width (FW), so that notes can be played as similar to known harmonies (Eq. (1)). In this study, HS is modified by combining pitch adjustment and memory consideration.

2) A musician may play new notes randomly when he/she cannot remember harmonies (Eq. (2)).

(1)

(2)X_{i,\mathrm{n}\mathrm{e}\mathrm{w}}=HMCR>\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d},X_{i,\mathrm{m}\mathrm{i}\mathrm{n}}+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\left(X_{i,\mathrm{m}\mathrm{a}\mathrm{x}}-X_{i,\mathrm{m}\mathrm{i}\mathrm{n}}\right).

The classical equations of HS are modified by combining pitch adjusting and memory consideration in Eq. (1). HMCR is one of the HS parameters known as harmony memory consideration rate and is related to the possibility of remembering and usage of harmony memory. X_{i,\mathrm{n}\mathrm{e}\mathrm{w}}, X_{i,\mathrm{m}\mathrm{i}\mathrm{n}}, and X_{i,\mathrm{m}\mathrm{a}\mathrm{x}} are new, minimum (lower limit), and maximum (upper limit) values for the ith design variable. On the other hand, X_{i,n} is the value within the nth candidate vector of the ith design variable.

3 Artificial neural networks

ANN provide one machine-learning method. It is widely used in artificial intelligence technologies and applications such as estimation, pattern recognition, detection of objects, and facial recognition.

ANN models are based on the human nervous system to sense and to learn. This network learning is the training of the network. A rule called generalized delta is used for errors’ back-propagation, which is one of the learning algorithms that is operated for ANN training [50].

Three independent ANN estimation and decision-making models were created by using an application called “Neural Net Fitting” within the machine learning toolbox of the MATLAB R2018a computer program, which concerns the presented structural designs to determine optimal design results directly and rapidly [51]. Also, this estimation model benefited from obtaining actual results of optimum values. Afterward, unknown and untested structural designs without optimization processes were operated for verification. Moreover, some calculations for error rates between actual and predicted values of both training and test data were made. The considered error metrics are comprised of mean absolute error (MAE), mean squared error (MSE) and root mean squared error (RMSE). All of these error metrics were utilized for evaluation of estimations’ accuracy and validation of the reliability of estimation results. The properties with their formulations are given in Table 1.

4 Structural models and cases

In this study, three different benchmark models of structural optimization were used: a 3-bar truss, a 10-bar truss, and a simply supported RC beam. These models are related to the minimization of the best objective functions, which are total volume, weight, and cost for structures for models 1, 2, and 3, respectively. Two different optimizations (cases 1 and 2) were carried out for these model processes while an ANN estimation application (case 3) generates decision-making models.

Optimizations were operated for evaluation of combinations of HS parameters, which are FW and HMCR, and usage and comparison of different iteration and population numbers. ANN training and an estimation process for test models were performed by using optimum results of design variables together with the best (minimum) objectives belonging to presented structural models. All of these three cases’ parameter information is expressed in Table 2.

4.1 3-bar truss model

The first optimization problem was a 3-bar truss structure model [52], which can be seen in Fig. 1. As mentioned previously, the aim was minimization of the volume of the 3-bar truss. Design variables were section areas that are the same for A_1 and A_3 due to symmetric axis, except A_2. Also, a load symbolized as P was applied to the connection point between these bars, which is assumed as 2 kN for cases 1 and 2. Also, different P values (0.05–2.8 kN increasing by increments of 0.05 kN) are considered for case 3 (ANN estimation) to use as input, while outputs are handled as section areas of bars and minimum volume. 56 data points in total were used for input parameter as P. On the other hand, bar section areas were the design variables, which have limit values as 0 cm² for minimum and 1 cm² for maximum. Also, design constants are elasticity modulus of bars (E_s = 200000 kN/cm²) and distances between supports (L = 100 cm).

The objective function of this truss problem is related to the determination of the best volume as a minimum. The formulization was made for minimization of volume as expressed with {Min}_{\mathrm{f}\left(\mathrm{v}\right)} in Eq. (3). Also, constraints for optimum design were considered for controlling bar stresses. For this reason, stresses should not violate \sigma (2 kN/cm²), which is the allowable stress level. These constraints are expressed in Eqs. (4)–(6) for these three bars, independently.

(3){Min}_{\mathrm{f}\left(\mathrm{v}\right)}=\left({2\sqrt{2}A}_1+A_2\right)L,

(4)g_1=\frac{{\sqrt{2}A}_1+A_2}{{{\sqrt{2}(A}_1)}^2+{2A_{1}A}_2}P⩽\sigma ,

(5)g_2=\frac{A_2}{{{\sqrt{2}(A}_1)}^2+{2A_{1}A}_2}P⩽\sigma ,

(6)g_3=\frac{1}{A_1+{\sqrt{2}A}_2}P⩽\sigma .

In Fig. 2, the change of minimum volume of truss is shown according to the values of HMCR and FW parameters presented in Table 2. In Table 3, for case 1, the obtained optimum results and statistical evaluations of the 3-bar truss model and the detected best values of HS parameters were expressed.

Moreover, Fig. 3 shows the improvement in best volume as the result of iteration-population numbers’ alterations. The best iteration and population numbers, which can reach the minimum volume for the truss by using the constant FW and HMCR, are given in Table 4. It can be said that the correct determination of effective values for HMCR and FW (1st case) is more significant because the standard deviation of minimum volume in case 1 is bigger than that in case 2.

In addition, the error amount (according to HS results) of ANN estimations for training data was calculated as metrics containing the MAE, MSE, and RMSE and illustrated in Fig. 4. Various test models for truss structure were generated in case 3, and actual optimum results of A_1=A_3, A_2 and {Min}_{\mathrm{f}\left(\mathrm{v}\right)} for each model are observed by using the main estimation model with low error rates (Table 5).

4.2 10-bar truss model

The second optimization structure is a 10-bar truss model [53], which was handled to minimize total weight as presented in Fig. 5 for cases 1 and 2. Design variables are section areas for each truss member from A_1 to A_{10}, and applied loads are P₁ and P₂ with values of 150 and 50 kips, respectively. Values of P₁ and P₂ between 145 and 155 kips and 45–55 kips by increasingly as 0.2 and 0.1 kips, respectively, to use on the training of ANN through the usage of 5149 total data (case 3). The reason for this, P₁ and P₂ are assumed as input parameters, and cross-section areas of bars with minimum weight are handled as output parameters. Also, design properties belonging to the truss model are given in Table 6.

The objective function is related to the minimization of total weight, given in Eq. (7), where L_i is the length of the ith bar.

(7){Min}_{\mathrm{f}\left(\mathrm{w}\right)}=\rho \sum _{i=1}^{barnumber}{A}_i{L}_i.

Here, Fig. 6 expresses where it was detected for the best weight values of this truss structure according to the usage of different parameter values, which are HMCR and FW.

Moreover, Table 7 shows the results of truss optimization performed with the usage of different HMCR and FW parameters for case 1. Some statistics for minimum weight and the best values of parameters are presented.

Figure 7 shows the change of weight values through case 2. In Table 8, the optimum design results for various iteration and population numbers are listed.

The results in Tables 7 and 8 indicate that the best result for objective function (minimum weight) was provided with a combination of HMCR and FW. For case 3, the ANN training data were ensured by using these parameters; all of the training errors from HS results were reflected in Fig. 8. As can be seen, section areas of bars can be detected with quite small errors in terms of each metric. For minimum weight, the RMSE value, which shows the successful convergence of the model to actual results, is at an acceptable level. Thus, estimations compared with HS results ensured the usage of optimum values for test models. Then, the errors were calculated and all of these results were expressed in Table 9 for A_1{-A}_{10} and {Min}_{\mathrm{f}\left(\mathrm{w}\right)}.

4.3 Simply supported RC beam model

The last model is a simply supported RC beam [54]. The optimization objective for this model is the minimization of total structure cost by determining the optimal section properties (beam height as 28 in < h < 40 in, and width as 5 in < b < 10 in) together with the best reinforcement area {(A}_{\mathrm{s}}). Two maximum bending moments due to dead and live load (M_DL and M_LL) occur at the beam. M_DL and M_LL have values as 1350 and 2700 in-kip, respectively. Beam length (L) is constant as 30 ft to be used in cases 1 and 2 (Fig. 9). But these bending moments and L were assumed as input parameters and utilized in different ranges for case 3, which is based on ANN training and estimation process. Training data ranges are between 15 and 20 ft, increasing in increments of 0.5 ft for L, and 1000–3000 in·kip increasing in increments of 200 in·kip for M_DL and M_LL. Thus, totally of 1331 data points were handled to train ANN.

The total cost minimization symbolized as {Min}_{\mathrm{f}\left(\mathrm{c}\right)} can be formulated with Eq. (8). Unit costs for steel reinforcement and concrete are $1.0 in⁻²·(linear ft)⁻¹ and $0.02 in⁻²·(linear ft)⁻¹, respectively. Here, Eq. (8) is comprised of the multiplication of unit costs and beam span length, respectively. For this reason, coefficients of A_{\mathrm{s}} and bh are expressed directly. Also, one of the design constraints is related to the required moment capacity/strength \left(M_{\mathrm{U}}\right) proposed by American Concrete Institute Building Code (ACI 318-14) [55] (Eq. (9)). The other constraint is expressed with Eq. (10).

(8){Min}_{\mathrm{f}\left(\mathrm{c}\right)}={29.4A}_{\mathrm{s}}+0.6bh,

(9)g_1=M_{\mathrm{u}}=0.9A_{\mathrm{s}}{f}_{\mathrm{y}}d\left(1-0.59\frac{A_{\mathrm{s}}{f}_{\mathrm{y}}}{f_{\mathrm{c}}db}\right)⩾1.4M_{\mathrm{D}\mathrm{L}}+1.7M_{\mathrm{L}\mathrm{L}},

(10)g_2=\frac{h}{b}-4⩽0,

where f_{\mathrm{y}} and f_{\mathrm{c}} are yield strength of steel reinforcement (50 ksi) and compressive strength of concrete (5 ksi), respectively. Also, d is the effective depth of the beam calculated as 0.8h.

In Table 10, objective function evaluations with optimum design variables were ensured for the minimization of total structure cost for both reinforcement steel and concrete. Then, the best parameter values were determined. Figure 10 shows the minimum cost results considering HMCR-FW differences.

In Table 11, beam section properties as optimal and minimum cost results can be seen in case 2. It can be understood from the results that the detection of the best iteration with population numbers is more considerable to reach the minimum cost with a smaller deviation.

Furthermore, Fig. 11 presents the whole cost values, which were determined according to the usage of numerous iteration and populations to reveal the best choice for minimum cost.

For beam structure design parameters with the best cost, error values between actual and estimations, which were determined based on the main ANN estimation model, can be seen in Fig. 12. The low error level shows that the ANN model is effective and precise in finding real optimum values directly for target outputs. Hence, in Table 12, ANN estimations for h, b, A_{\mathrm{s}}, and {Min}_{\mathrm{f}\left(\mathrm{c}\right)} can be seen for each beam design. Error-values are also presented to compare the real-optimum values with estimations.

5 Results and discussion

5.1 Model 1: 3-bar truss

For model 1, optimum section areas of bars and minimum volume of truss structure were calculated for cases 1 and 2 by evaluating HMCR-FW and iteration-population combinations, respectively; thereafter, in case 3, the values of the same variables were observed for various designs through conducting ANN estimation, and automatic determinations were performed for test models.

When the results in Tables 3 and 4 are considered, the standard error of the best volume (263.8959 cm³), which is ensured for all solutions among the determined parameter combinations, is pretty small as 0.0031 cm³. Iteration numbers at extremely small levels are not effective in achieving target results as minimum volume. Owing to use of the ANN estimation model which provides predictions with low errors, the expected optimization data were acquired directly and rapidly for a test model comprising five designs that are not previously encountered/handled. Estimation for optimum section areas and minimum truss volume {(Min}_{\mathrm{f}\left(\mathrm{v}\right)}) was obtained from each test design. Errors of estimation results compared to real optimum outputs were calculated as shown in Table 4. It can be recognized from results that all error metrics, MAE, MSE, and RMSE, are extremely small for the predicted values of section areas and best volume. The target outputs for the 3-bar truss model can be determined without any additional optimization process, and with the use of the currently developed estimation model via ANN directly and quickly.

5.2 Model 2: 10-bar truss

Concerning model 2 and its design with weight minimization for the 10-bar truss structure, the procedures of each case were implemented from model 1. The minimum total weight can be calculated as 4680.8384 lb through the case based on observation of best HMCR-FW combinations, even if the standard deviation of the objective function is slightly bigger than the iteration-population evaluation case. Also, as shown in Fig. 6, the best weight cannot be provided when HMCR is bigger than 0.95 approximately, under any values of FW. The best parameters in terms of detection of minimum weight were selected for generating training data to be used via the ANN estimation model.

According to the results illustrated by both Fig. 8 and Table 9 where ANN estimations are for section areas together with truss minimum weight for each test model, it can be concluded that the developed model is effective and can accurately foresee parameters of different structural designs of the 10-bar truss.

5.3 Model 3: simply supported RC beam

As for model 3, optimum section sizes with reinforcement area and minimum cost of simply supported RC beam structure were obtained via a combination of HMCR-FW (case 1) and iteration-population numbers (case 2), respectively. Also, as per parameters of the best case, ANN estimations of optimal properties and the desired cost value for different test models were provided to observe directly and speedily.

As optimization results of both Tables 10 and 11 were evaluated, the minimum cost as the best result was $359.2080 based on both cases, but the smallest deviation for cost among total solutions has arisen as $0.04 when the case of iteration-population combinations was operated. Also, the observations for each case were shown in Figs. 10 and 11. For case 1, small fluctuations occur for minimum cost values until almost 0.8 for HMCR whether FW values are small or not. Cost results of case 2 were observed to be extremely stable for any iteration-population combination, except for small rates of iterations. Case 2 is more convenient for finding optimum properties with the desired cost in every respect.

In addition, values determined by ANN modeling were predicted based on actual optimum results of section sizes ( h and b), reinforcement area ( A_{\mathrm{s}}) with the best cost ( {Min}_{\mathrm{f}\left(\mathrm{c}\right)}) for five different test beam designs. Error amounts between estimations and actual values were also calculated for test designs as shown in Table 12. Errors, namely deviations of MAE, MSE, and RMSE for estimations of optimum results from real values are extremely low, especially for b and A_{\mathrm{s}}. It shows that differences and deviations between actual and estimations occur at a low rate.

6 Conclusions

As a summary of optimization processes for all structural models, the determination of the best and proper iteration and population numbers is more significant and effective in terms of reaching the best objective values with smaller errors and fewer standard deviations. The detection of these best parameters is also important for use during the generation of ANN training data.

The ANN modeling results show estimations with low errors and fewer deviation rates. It can be said that ANN technique is both intelligent and convenient for providing sensitive and time-efficient outcomes. It is used to make estimations of numerical values as in regression analysis. Therefore, ANN estimation and decision-making models can be utilized to determine actual optimization results for 3-bar and 10-bar truss structures and the RC beam model without applying any extra processes in a time-efficient way. Thus, usable, quick, and accurate tools were developed in this study for determining various numerical and optimized data for different structural models using ANN.

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Higher Education Press 2021.