Materials and Design 156 (2018) 538–557
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Crashworthiness multi-objective optimization of the thin-walled tubes under probabilistic 3D oblique load Mohammad Arjomandi Rad, Abolfazl Khalkhali ⁎ Automotive Simulation and Optimal Design Research Laboratory, School of Automotive Engineering, Iran University of Science & Technology, Tehran, Iran
G R A P H I C A L
a r t i c l e
A B S T R A C T
i n f o
Article history: Received 7 March 2018 Received in revised form 14 June 2018 Accepted 6 July 2018 Available online 11 July 2018 Keywords: 3D oblique loading Thin-walled tube GEvoM Multi-objective optimization Reliability-based robust design optimization Multiple-criteria decision-making
a b s t r a c t In a real-world loading case (e.g., car crash accidents), energy-absorbing components are subject to oblique loads at various uncertain angles. This paper aims to investigate the behavior of such components under threedimensional (3D) oblique loads in deterministic and probabilistic loading conditions. In this way, some square tubes are tested experimentally, and results are utilized to validate numerical models. To apply the 3D oblique load, a special test setup is designed, constructed, and installed on a universal tensile testing machine. Hammersley method is employed to design sample points. ABAQUS software is used for the finite element modeling and analysis. GEvoM software is implemented for mapping design variables onto crashworthiness characteristics including energy absorption (EA) and peak crush force (PCF). Both deterministic and reliabilitybased robust design (RBRD) optimizations are performed, and their results are compared with each other. The primary outcome of this research is the effect of incidence angles on the energy-absorbing characteristics, as well as some remarkable trade-off design points obtained from various multiple-criteria decision-making (MCDM) methods. It was discovered that the obtained design points of probabilistic study, which satisfied the reliability constraint, were roughly 60% more robust than deterministic points. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction Crashworthiness, which is the ability of a structure to dissipate the energy of a crash event and prevent occupants from injury [1], is usually ⁎ Corresponding author. E-mail address:
[email protected] (A. Khalkhali).
https://doi.org/10.1016/j.matdes.2018.07.008 0264-1275/© 2018 Elsevier Ltd. All rights reserved.
achieved with by means of plastic deformation of the materials. Most of the studies in the respective literature analyze standard loading conditions such as straight axial or bending loads [2–4]. However, in a real crash event, energy-absorbing components are subject to oblique loads at various unpredictable angles. Nowadays, the demand for safety regulations obliges manufacturers to design these components to tolerate oblique loads [5].
M. Arjomandi Rad, A. Khalkhali / Materials and Design 156 (2018) 538–557
Crashworthiness of thin-walled tubes under oblique loading has raised interest among many authors. Han et al. [6] studied the crushing behavior of thin-walled square tubes under oblique loading using LSDYNA for the first time in 1999. Their numerical studies proved a critical load angle at which axial collapse mode changes to the bending mode. They also concluded that this critical load angle has almost no correlation with thickness, yet varies significantly with respect to the width and length of the tube. Reyes et al. (2002 & 2004) [7, 8] performed experimental and numerical studies on the crushing behavior of square aluminum tubes under oblique loads. It was concluded that the presence of the initial angle in crushing significantly lowers the peak load. Later, they completed their study [5], suggesting that deformation mode depends on thickness. Yang et al. (2012) [9] investigated the crushing behavior of tapered square tubes with two types of geometries (straight and tapered) and two kinds of cross-sections (single-cell and multi-cell). It was found that the optimum tube configurations were different for different load angles in general, either with or without constraint on the PCF level. Therefore, they concluded that it is necessary to include the load angle uncertainty effect in the optimization process; however, no such investigation was carried out. In 2013, Tarlochan et al. [10] numerically studied the crushing behavior of numerous crosssection shapes and found hexagonal profile to be a better option for energy absorption application. Later, some hexagonal profiles were employed for further investigation of tube's wall thickness, foam filling, and the effect of trigger mechanism. Despite their promising conclusions, this study solely relies on numerical simulations and no optimization was proposed for the suggested improvements. Yang and Qi (2014) [11] studied oblique impact behaviors of a class of thin-walled circular aluminum tubes (empty conical, empty round, foam-filled round, and foam-filled). It was shown that the tube's optimal shape differed from the change of crushing angles, and multiple load angles in the optimization process could result in improved robustness of the tube. Yet again, no such robustness study was performed. In 2015, Guangyao Li et al. [17] demonstrated that the functionally graded thickness tubes were preferable to other types, such as tapered tubes or straight uniform thickness tubes, in withstanding oblique impact loading. It was found that the energy absorption capacity of functionally graded thickness tubes, especially with gradient exponent 2.0, was well maintained as the impacting angle increased. The paper justly acknowledges the shortcomings of the functionally graded thickness tubes, but proposes no feasible solution for it.
539
Wang et al. (2017) [18] studied a single-cell tapered elliptical tube under oblique impact loading. A comparison also made between conventional configurations (straight, tapered, and multicell tapered) and cross-sections (square, rectangular, and circular). It was concluded that the single-cell tapered elliptical tube showed better crashworthiness performance at multiple loading angles. Yao et al. (2017) [19] investigated functionally graded thickness circular tubes under multiple loading angles through a surrogate model and non-dominated sorting genetic algorithm II. Their results showed that the thickness gradient index and the length-todiameter ratio had greater influence on the crashworthiness characteristics of circular tubes with functionally graded thickness under oblique impact load. Alkhatib et al. (2018) [20] numerically studied the effect of loading angles on crushing characteristics of corrugated tapered tubes (CTTs) at 7 different loading angles and the effect of the geometric parameters on the performance. They found that increasing the impact angles from 0° to 40° led to a reduction by 54% in EA and SEA. Foam-filled tubes under oblique impact loading are among the intensively investigated research topics [12–16]. Moreover, numerous alternative shapes, such as honeycomb filled crash boxes [21], octagonal thin-walled sandwich tube [22], conical tube clamped at both ends [23], expansion-splitting hybrid tubes [24], and hierarchical honeycombs [25], were investigated under oblique loading; however, we leave out their review since foam-filled tubes are out of the scope of this paper. All the aforementioned studies have defined oblique loads with a single angle in a 2D coordinate system, which impedes the method's accuracy, because, in a real crash event, the energy-absorbing structure crushes in a 3D direction. These studies also lack uncertainty analysis over their crushing angle; yet, we are certain that this is definitely the case in the real world. Fig. 1 depicts a tube which is being crushed on a horizontal plane. Angles between projections of sides a and b on a horizontal plane are denoted by α and β, respectively. Such 3D oblique loading offers highly accurate results in accordance with the actual crash circumstances. The safety performance and manufacturing costs are purposes of most energy absorbing design studies. Such a trade-off between cost and safety metrics, in general, was addressed by multi-objective optimization [9, 21, 22, and 26]. Optimization objectives of these studies are taken typically as energy absorption (EA) and peak crushing force (PCF) that are required to be maximized and minimized, respectively.
Fig. 1. Tube being crushed on a horizontal plane.
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Solid Square
Gripper
(b) Gripper
Specimen
(a) Exploded view
(c) Gripper assembled on a specimen
Fig. 2. Upper-end fixture configuration.
Performing such optimization and predicting the crashworthiness indices (EA and PCF) require meta-modeling techniques such as Response surface method (RSM) [9 and 21], Kriging algorithm [22], or Artificial
Neural Network like Radial Basis Function (RBF) [26]. These studies generally managed to find a Pareto frontier, i.e., a set of Pareto optimal solutions, which could be employed later in Multi-Criteria Decision-
(a) Fig. 3. Schematic figures of the fixture: a) Isometric view, b) two angles representing inclined plate.
(b)
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Table 2 Mechanical characteristics of the constitutive material of the specimens Density ρ (g/cm3)
Elasticity module E (GPa)
Ultimate tensile strength Su (MPa)
Yield stress Sy (MPa)
7.78
203.489
872
407
Fig. 5. Illustration of stress-strain behavior for the material of the specimens.
Fig. 4. The fixture installed on the universal tensile testing machine.
Making methods (MCDM) to find the best trade-off design point [10, 27–29]. Essentially, in a real crash event, energy absorption is affected by different uncertainties such as angles of incidence and/or geometrical dimensions caused by manufacturing deficiencies. Any uncertainty indicates risk or the possibility of failure which is aimed to be avoided. Therefore, reliability-based robust analysis is of great importance as a topic, which lacks sufficient respective studies in literature, hence its dire need for more investigation and studies in this regard. To the best knowledge of the authors, crashworthiness multi-objective optimization of tubes under probabilistic oblique loading has not been studied, which makes this paper an unprecedented in its own kind. The main objective of this paper is to optimize energy-absorbing characteristics of thin-walled tubes under probabilistic 3D oblique load, which gives a better simulation of the real-world loading such as
crash incidents. Therefore, an experimental test is implemented to study deformation and collapse of the tube through derivation of the load-displacement diagram. Later, finite-element (FE) models of diverse specimens are analyzed and validated using experimental tests in the first section. Output data of designed models in ABAQUS software are used to train GDMH-type neural network. Polynomials extracted from GEvoM software are adopted in the deterministic multi-objective optimization process. To improve robustness and reliability of design as well as taking unpredictability of incidence angles into account, probabilistic optimization is carried out and, then, compared with deterministic results. MCDM methods are used to attain trade-off points in both deterministic and probabilistic approaches as the primary findings of this study. Moreover, the oblique angles are studied briefly, and effects of each angle as well as their relation with each other are studied. 2. Experimental studies In this section, totally 12 experimental tests were carried out with quasi-static condition. Most of the papers exist in literature utilize quasi-static compression test to perform crashworthiness optimization on thin-wall structures. The dynamic plastic collapse of energy-
Table 1 Geometrical properties of the specimens and comparison between experimental and numerical results Ex no.
A1 A2 A3 A4 A5 A6 B1 B2 B3 B4 B5 B6
C (mm)
50 50 50 50 50 50 70 70 70 70 70 70
t (mm)
1.4 1.4 1.4 1.4 1.4 1.4 1.6 1.6 1.6 1.6 1.6 1.6
θ1 (deg)
0 0 0 0 15 30 0 30 15 15 15 0
θ2 (deg)
0 15 25 35 35 35 15 15 15 25 35 0
PCF (KN)
EA. (KJ)
Exp.
Num.
Error (%)
Exp.
Num.
Error (%)
95.1454 49.1816 28.9403 20.9294 18.8715 16.594 65.328 55.2955 60.118 51.1243 39.5117 154.89147
102.0228 47.2987 27.2523 22.6695 18.6154 15.7613 65.0202 61.2299 63.9873 51.3559 40.4104 161.5994
6.741 3.981 6.194 7.676 1.376 5.283 0.473 9.692 6.047 0.451 2.224 4.151
3.969 2.1337 1.341 1.2332 1.1765 1.0743 4.6084 3.3855 3.7338 2.7343 2.2238 4.953
4.2894 2.0792 1.2826 1.3098 1.1272 1.0408 4.4433 3.7097 4.1181 2.6929 2.153 5.2952
7.466 2.624 4.543 5.848 4.372 3.218 3.717 8.738 9.331 1.539 3.29 6.462
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20 mm
40 mm
60 mm
80 mm
100 mm
Fig. 6. Comparison of the experimental and numerical shape of deformation for specimen B3 in different displacements.
20 mm
40 mm
60 mm
80 mm
100 mm
Fig. 7. Comparison of the experimental and numerical shape of deformation for specimen B4 in different displacements.
M. Arjomandi Rad, A. Khalkhali / Materials and Design 156 (2018) 538–557
20 mm
40 mm
60 mm
80 mm
543
100 mm
Fig. 8. Comparison of the experimental and numerical shape of deformation for specimen A5 in different displacements.
absorbing structures is more difficult to understand than the quasistatic collapse, on account of three effects which may be described as the “strain-rate factor”, the “inertia factor” and the “mode shape change”. In this paper, for all the quasi static tests, SANTAM universal testing machine (STM 150 series) was used to attain loaddisplacement curves. The speed of the upper cross-head was set to 5 mm/min has been chosen for this study to ensure the quasi-static
condition as well as maintaining moderate runtime. Each test lasted for nearly 20 min. 2.1. Fixture preparation For applying 3D load on the specimen and accurately producing incidence angles, α and β (see Fig. 1), a unique fixture was designed and
Fig. 9. Load-displacement diagrams of experimental and numerical studies for specimens B3, B4 and A5.
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Fig. 10. The contour of the relation between θ1 and θ2 and corresponding collapse mode.
manufactured. This fixture has two ends: upper and lower ends. To ensure that appropriate constraints were applied to the specimens at the upper end, the fixture consists of two parts on this side: a gripper and a solid square. Fig. 2 represents this fixture as an exploded view at the upper end (a), assembled gripper with bolts (b), and assembled gripper on the specimen (c). As depicted in Fig. 2, the solid square enters the hollow space of the specimen from the upper end, and the gripper surrounds the upper end of the specimen. Using the bolts on the gripper, the specimen, solid square, and cover plate were tightly fastened together to fully lock the upper end of the specimens to the upper fixture during the loading. At the lower end, specimens were in contact with an inclined flat plate mounted on the lower end fixture. The position of the inclined plate is defined by two angles. θ1is inclined plate's rotational angle along the vertical axis, through the centroid of cross-head and, θ2 is the angle of the inclined plate relative to the horizon. The importance of taking these angles along each mentioned axis is discussed in the next sections. It is worth mentioning that α and β, which are the angles between sides a and b with their corresponding projections on the inclined plate, depends on θ1and θ2. By increasing θ1 from 0 up to 45°, the value of α decreases and the value of β increases. Hence, increasing the value of θ2 will result in an increase in the value of α and β. Relation of these angles with response functions is discussed in the next sections. As shown in Fig. 3, the lower end of fixture consists of the inclined plate mounted on a circular plate. This inclined plate material in the lower fixture is steel MO40. To prevent any damages to its surface and to minimize the effect of this damage on the friction between the plate and specimens during the tests, cementation (surface hardening by carburizing) has been performed on the surface of this part. Fig. 4 shows both ends of fixture installed on the universal tensile testing machine.
2.2. Specimen preparation The material of all specimens used here is of AISI Carbon Steel 1000 Series. The dimensions of the specimens and the crushing angles of θ1and θ2 in each test are presented in Table 1, where C and t are the cross-sectional width and thickness. The design of the variables in this table was carried out by Hammersley method, which is discussed in Section 4.1. All specimens have machined at both ends to acquire the height of 200 mm, exactly. To reduce the residual stresses before performing the tests, in addition to keeping specimen's integrity during the test, heat treatment was carried out for stress-relieving purposes. All the specimens were subjected to 900 °C for 10 min and, then, gradually adjusted back to the room temperature. Moreover, one standard specimen for the tensile test was manufactured by wire-cut technique, and the same heat treatment was performed equally on it before testing. The mechanical properties, as well as the stress-strain graph of this tensile test, are shown in Table 2 and Fig. 5, respectively.
2.3. Test results Overall, three different crushing modes, including progressive, mix, and bending modes, were attained. Herein, one example of each mode was presented by sequential pictures through Figs. 6 to 8. In these figures, the crushing modes of specimens B3, B4, and A5 are progressive, mix and bending, respectively. Additionally, a load-displacement diagram of these 3 specimens is illustrated in Fig. 9. It is clear in Fig. 9 that crushing peaks take the ascending trend after the first peak. These graphs are entirely different from the conventional loaddisplacement graph of the crushing behavior of a thin-walled tube under axial loading. Under conventional axial loading, as the crosshead displacement increases, the load reaches a peak value, which is
M. Arjomandi Rad, A. Khalkhali / Materials and Design 156 (2018) 538–557
545
Fig. 11. Flowchart of the performed process in this study.
followed by some fluctuations with lower peaks. In this study, there is a subsequent higher peak after the initial one. Notably, as the angles of the declined plate approach 0, the load-displacement graph becomes more similar to the axial loading condition. A comparison between the figures and load-displacement graphs indicates that, in the initial contact between specimens and declined plate, tube section has not entirely made contact with the declined plate; therefore, the load has not been applied equally on the tube interfaces. This condition leads to the local collapse of the tube, and the load that results in the first local collapse is equal to the first peak crushing force. This peak's value entirely depends on what percent cross-section gets into contact with the inclined plate, the value which depends highly on θ1and θ2. By increasing the cross-head displacement, when the whole tube section undergoes in contact with the declined plate, the remaining part of the tube shows the crushing behavior similar to axial loading conditions on a relatively flat support. The obtained values of EA and PCF for 12 experimental tests are reported in Table 1. Results of the experimental studies are used to validate finite-element model of numerical studies in the next section.
3. Numerical study The purpose of this section is to develop an appropriate numerical model to study the behavior of a thin-walled tube under 3D oblique loading and corresponding effects of the incident angles. Commercial software ABAQUS has been used or simulation purposes in the entire study. The shell element used in FE models is S4R, 4-node doubly curved, that uses six degrees of freedom at all nodes and is mainly suitable for large strain problems. To model bending deformations, five integration points are used through the thickness of the shell elements. Mechanical behavior of the material is isotropic plasticity through all of the simulations. True stress-strain values are calculated from engineering stressstrain curve obtained from tensile test (Fig. 5) and the post-yield material response was defined in ABAQUS using 27 points. Moreover, the upper cross-head is modeled as a rigid inclined plate, in which its entire degrees of freedom, except for displacement along the load direction, are fixed. The specimens are also completely tied to the upper-end of the rigid plate.
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Table 3 Numerical studies; geometrical properties of the specimens and their corresponding EA and PCF No.
a (mm)
b (mm)
t (mm)
θ1 (deg)
θ2 (deg)
PCF (N)
EA (J)
1 2 3 4 5 6 7 8 9 10 11 12 … 184
50 50 50 50 50 50 50 50 50 50 50 50 … 70
50 50 50 50 50 50 50 50 50 50 50 50 … 70
0.6 0.6 0.6 0.9 1.2 1.2 1.4 1.4 1.4 1.8 1.8 1.8 … 1.8
30 0 45 25 45 45 0 15 30 0 30 30 … 0
30 45 12 8 1 12 0 35 35 8 1 12 … 4
4955.98 4323.96 13,089.6 26,221.9 49,686.1 47,135.5 102,022.8 18,615.4 19,761.3 81,550.3 110,845.1 82,006.7 … 98,347.1
222.307 220.178 755.5 1806.3 3093.9 2335.4 4289.4 1127.2 1140.8 6521.1 6225.6 3735.4 … 8000.8
It should be noted that the loading rate, applied to the experimental quasi-static test, is 5 mm/min and, therefore, cannot be applied to the FE analysis, because it dramatically increases the computational cost. To address this challenge, different approaches have been employed previously in established studies, including load expediting, mass scaling, and smooth loading for efficient explicit quasi-static analysis [30–33]. In the present study, several considerations are made to make simulations feasible; the velocity of the upper rigid plate is raised to 0.1 m/s, the Amplitude option and SMOOTH STEP sub-option in ABAQUS/Explicit are used together to properly control the velocity for a valid quasi-static analysis with minimized dynamic loading effects. The lower rigid plate is configured according to incidence angles: θ1and θ2. Additionally, friction coefficient of 0.15 is considered for the contact between the tubes and the inclined plate as this value is in good correlation, compared to the test results. Equality between external work done and internal energy of specimens as well as negligible amount of kinetic energy all indicate the validity of the performed quasi-static conditions. Accordingly, Table 1 shows a comparison between numerical and experimental studies. As depicted in this table, all the results of numerical studies lie roughly within ±10% of the experimental tests, highlighting good reliability of the present numerical modeling. Moreover, Figs. 6–8 depict a comparison on the plastic folding mode of the specimens obtained in numerical and experimental studies. Fig. 9 shows diagrams of experimental and numerical studies for three specimens with distinct collapse modes. All of these comparisons indicate the acceptable accuracy of the numerical study; therefore, the numerical model can be applied to further investigations. 3.1. Effect of incidence angels on the collapse modes Data gathered through numerical analysis in this study reveal that θ1and θ2 are both critical factors in determining collapse mode, which has strong effect on EA and PCF. The relation between θ1 and θ2 for the given constant values of a, b, and t is depicted in Fig. 10. In this figure, 24 simulations (All with equal a, b, and t) are categorized based on their collapse mode on the plane of θ1 − θ2. It can be inferred that, for a certain value of θ2 such as 15°, the smaller θ1 value is, the more likely it is for the collapse mode to be a mix mode, while, in higher θ1 values, collapse mode is more likely to be bending. Similarly, for θ2 between 10 and 20°, changing θ1 can alter collapse mode and, consequently, PCF and EA. For different a, b and t values, this relation might be scaled a little, yet, the order will remain the same. The ability to control the collapse mode through adjusting θ1 is indeed a valuable inference for designing structures, implying the importance of 3D oblique loading method, presented by this study.
4. Multi-objective optimization In this section, deterministic and probabilistic multi-objective optimizations of the thin-walled tube under 3D oblique loading are presented. The deterministic study is used to find the best design variables with paramount EA, PCF, and Weight. Since incidence angles cannot be determined in real crash events, it is better to include them as a probabilistic parameter in the optimization problem. RBRDO studies help achieve design points with better performances, regardless of any oblique angles in a crash event. A flowchart of experimental, numerical, and optimization processes applied in this study is presented in Fig. 11 and discussed term by term in the next sections, respectively. 4.1. Designing sample points with Hammersley There exist many works on random sampling techniques that are mainly aimed at reducing the number of sample design points, in addition to maintaining reliability in the probabilistic distribution. QuasiMont Carlo method is a modification of crude Mont Carlo developed by Diwekar et al. [34]. This algorithm, which is known as Hammersley sequence sampling (HSS), has been widely acknowledged as a sampling technique with superior results [35]. HSS utilize Hammersley points, generating quasi-random numbers, to sample a unit hypercube in a uniform fashion. These points then are inverted by a joint cumulative probability distribution to deliver the set of samples for a given variable. Results provide good coverage of the distribution range and minimum set of sample points [36]. Let n Hammersley points be generated and denoted by mi in which i = 1, 2, …n through the inverse CDF of the uncertain parameter pi ¼ F −1 p ðmi Þ
ð1Þ
Clearly, any integer can be written in radix-R notation. Integer i expressed in radix-R notation can be written as follows: i¼
v X
i jR j
ð2Þ
j¼0
Table 4 The testing accuracy of meta-models Error criteria
Response functions EA
PCF
RMSE R2 MAPE
346.2 0.989 11.67
3230.45 0.997 7.718
M. Arjomandi Rad, A. Khalkhali / Materials and Design 156 (2018) 538–557
where R N 1 is an integer, ij is an integer between 0 and R-1 for each j = 0, 1, …v, and v = floor{logRi}. The inverse radix number of i, namely φR(i), is a distinctive section among 0 and 1 titled inverse radix number and can be assembled through reversing the order of the digits of n about the decimal point as follows: φR ðiÞ ¼
v X
ij
ð3Þ
jþ1 j¼0 R
The Hammersley points in a k-dimensional unit hypercube can be illustrated by the following sequence: h. iT mi ¼ 1− i ; φR1 ðiÞ; φR2 ðiÞ; …φRk−1 ðiÞ
4.2. GMDH-type neural network meta-modeling via GEvoM Based on [37,38], group method of data handling (GMDH) algorithm represents a given model as layers of neurons linked with each other in adjacent layers. These complex connections help map inputs onto outputs by quadratic polynomials. Identification problem aims to find a ^ for a specific function ^f such that it can be employed to predict output y input vector. In this regard, connections between input variables and output responses can be demonstrated by a discrete form of the Volterra functional series or Kolmogorov-Gabor polynomial [37]. n X i¼1
a i xi þ
n X n X i¼1 j¼1
aij xi x j þ
n X n X n X
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn ^ 2 i¼1 ðyi −yi Þ RMSE ¼ n
aijk xi x j xk þ ⋯
ð5Þ
i¼1 j¼1 k¼1
Mathematical statement consisting of only two variables (neurons) can be represented by partial quadratic polynomials: ^ ¼ G xi ; x j ¼ a0 þ a1 xi þ a2 x j þ a3 xi x j þ a4 x2i þ a5 x2j y
ð6Þ
Coefficient values of the polynomials and complexity of network are the main criteria for designing GMDH-type neural networks. To design connections in the network optimally and, also, embody polynomial coefficients effectively, Nariman-zadeh et al. [39,40] employed genetic algorithm and singular value decomposition to develop a software product, called GEvoM [41]. In fact, GEvoM is a lab-developed software that generates polynomials based on GMDH-type neural networks to model the relationship between input-output data pairs. In this study, in total, 184 patterns (see Table 3) obtained from the finite-element analysis are divided into 3 parts: 0 ≤ θ2 b 1, 1 ≤ θ2 b 15 and 15 ≤ θ2 b 45. Authors concluded that meta-models have minimum errors when the patterns are divided based on their θ2 value. Each of these parts again is divided into training and testing patterns. In order to attain highly accurate results, the order in which training patterns are presented to the network is randomized (shuffled) each time with 2-fold cross-validation [42]. With respect to evolutionary methods' parameters in sub-option of GEvoM software, population size, generation number, crossover probability, and mutation probability are taken as 60,700, 0.7, and 0.07, respectively. The algorithm is run several times, producing multiple generations until no further improvement is achieved in the network structure.
ð7Þ
Pn
^ 2 i¼1 ðyi −yi Þ n ∑i¼1 ðyi −yi Þ2
ð8Þ
n ^ 100 X yi −yi n i¼1 yi
ð9Þ
R2 ¼ 1−
MAPE ¼
where R, R1, …, Rk−1 show the first k − 1 prime numbers and i = 1, 2, …n. Totally, 184 finite-element models are designed and simulated using finite-element model, which are fully discussed in Section 3. Table 3 shows some of these points (a, b, t, θ1and θ2) and their corresponding EA and PCF values. In this table, a and b have a range of 50 to 70 mm, and t is designed to have one of the values of 0.6 to 1.8 mm. In addition, θ1and θ2 are considered in the range of 0 to 45. Some of the numerical simulations with their corresponding dimensions are presented in Table 3.
y ¼ a0 þ
The accuracy of the meta-models can be evaluated for testing data based on the coefficient of determination (R2), the root mean square error (RMSE) or the mean absolute percentage error (MAPE), which can be given by Eqs. (7) to (9).
ð4Þ
n
547
^i is the model prediction value, yi where yi is the experimental value, y is arithmetic mean variables, and n is the number of testing data, which is 184 in this study. Table 3 shows calculated error results of meta-models based on mean of each fold over testing data. RMSE is an average-error measure which indicates the overall interpolator performance [43]. Often, RMSE is preferred because it is on the same scale as the data. This criterion uses the difference between measured values and estimated response to report model's accuracy [44]. R2 (Coefficient of determination) of the multiple regression is like simple regression. It can be used for fitted means obtained by any estimation method. The value is always 0 b R2b1 and, in this way, 0.9 or above is considered as excellent precision, 0.8 or above good and, in some cases, 0.6 or above is considered satisfactory [45]. MAPE measures relative performance [46], and it has reported to be the most efficient error reporting method by most textbooks [47]. If value of MAPE is b10%, it is taken as excellent accuracy, between 10 and 20% as good, between 20 and 50% as acceptable, and over 50% as inaccurate [45]. Using the interpretations discussed and information of Table 4, it is concluded that extracted ANN models have minimum errors and, hence, are valid; therefore, they can be adopted for further studies in the optimization section. The GMDH polynomial functions of SEA and PCF for the tube under 3D oblique load in 3 angle span based on θ2 are presented in Appendix 1. 4.3. Deterministic multi-objective optimization As mentioned in the previous section (see Fig. 10), oblique angles θ1and θ2 have huge impact on collapse mode, energy absorption, and peak crushing force. Appling such suitable, yet small, angles could considerably reduce the PCF, whereas the amount of EA does not differ significantly. Hence, in this section, the optimization problem is defined with the aim of finding the best oblique angles to increase the EA and reduce the PCF and Weight at the same time. On the other hand, geometrical dimensions of the tube have noticeable effect on crashworthiness characteristics of the tube. In this regard, a, b, t, θ1, and θ2 are considered as optimization design variables. To attain the best crashworthiness and lightweight criteria, a three-objective optimization problem is formulated in order to minimize PCF and Weight and maximize EA as below: 8 Maximize f 1 ¼ EAðEnergy AbsorptionÞ > > > > Minimize f 2 ¼ PCF ðPeak Crushing ForceÞ > > > > Minimize 8f 3 ¼ W ðWeight Þ > > > > 35000 b PCF b 80000 > > > > < > > > > > > 50 mm b a b 70 mm < > > > > Subject to: 50 mm b b b 70 mm > > > > > > 0:6 mm b t b 1:8 mm > > > > > > > > > > > > 0 b θ1 b 45 deg : : 0 b θ2 b 45 deg
ð10Þ
548
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rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i E ðX−μ Þ2 Z þ∞ ðx−μ ðxÞÞ2 f X ðxÞdx≅ ¼
σ ðX Þ ¼
−∞
3
45
45
a
b
t
0.1
0.1
0.01
8 < Minimize fμ ½ f ðx; d; pÞ; ϑ f ðx; d; pÞg xðLÞ ≤x ≤xðU Þ : ðLÞ ðU Þ d ≤d≤d
In Eq. (10), a constraint on PCF between 35,000 N and 80,000 N is imposed to ensure attaining the global best point. These numbers are a minimum requirement to maintain sufficient strength in the structure and a maximum requirement to ensure the safety of downstream components. Any PCF which is lower or higher than this range will result in unstable structure and failure in adjacent components, respectively. Additionally, variables a and b are bounded between 50 and 70 and t is between 0.6 and 1.8, as these are the most common dimensions used for manufacturing energy absorbers in the automotive industry [48]. Given that the cross-sections of the tubes are taken as rectangular in shape, the design points with θ1 between 45 and 90 are equivalent to the points with θ1 between 0 and 45, because sides a and b rotate together. Due to this reason, θ1 is considered between 0 and 45. θ2 is also limited to 45°, because the main effect of loading over 45° will not be axial anymore. In this paper, a modified NSGA-II algorithm called ε-elimination diversity proposed by Nariman-zadeh et al. [49] is used. This developed NSGA-II method has been successfully used in previous research studies [27, 50]. 4.4. Reliability-based robust design optimization (RBRDO) A successful model for uncertainties in randomness (stochasticity) is probability density function (PDF) represented by fX(x) or alternatively cumulative distribution function (CFD) shown by FX(x). Z
b a
f X ðxÞdx Z
F X ðxÞ ¼ PrðX ≤xÞ ¼
x −∞
ð11Þ
f X ðuÞdu
ð12Þ
where Pr(.) refers to the probability that an event (X ≤ x) will occur, and X refers to random variable with CFD of FX. Additionally, statistical moments such as mean value (also known as expected value) denoted by μ(X) and Standard deviation (SD) denoted by σ(X) are defined as below: Z μ ðX Þ ¼ E½X ¼
þ∞ −∞
x f X ðxÞdx≅
N 1X x N i¼1 i
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi −μ ðxÞÞ2
ð14Þ
A robust design approach, which was initially proposed by a Taguchi [51], requires diminishing variability in objective function performance generated by fluctuations in uncertain variables. Therefore, the typical robust optimization problem can be formulated as follows: A robust design approach, which was initially proposed by a Taguchi [51], requires diminishing variability in objective function performance generated by fluctuations in uncertain variables. Therefore, the typical robust optimization problem can be formulated as follows:
Fig. 12. Applied Gaussian probabilistic distributions.
Pr½a≤x ≤b ¼
1 N−1
ð13Þ
ð15Þ
In this formulation, x is the probabilistic design variables vector (called control factors), d is the deterministic design variable vector, and p is the uncertain parameters vector, which are not counted in design variables (called noise factors). Additionally, μ is the mean value, f(x, d, p) is the cost function, and ϑ represents any indices for dispersion measure such as standard deviation or variance. In addition, a designer can perfectly change control factors, fitting the desired function. Noise factors cannot be controlled by the designer and depends on crash circumstances. Inequality constraints are defined as reliability metrics in the reliability-based design approach. To define probabilistic constraint, deterministic constraint of gi(x, d, p) ≤ gi is considered, where gi is the limiting value of the ith constraint. Gi ðx; d; pÞ ¼ g^i −g i ðx; d; pÞ
ð16Þ
The typical probability constraint is then represented as follows: P if ¼ P ½Gi ðx; d; pÞ≤0≤εi
x ¼ 1; 2; …; m
ð17Þ
where m indicates the number of inequality constraints, Pif shows failure probability for the ith reliability measure (ideal value = 0), and εi is the maximum and acceptable probability of failure. By dispersing a specific probabilistic distribution (of x and p), a set of N solutions can be attained. For any sample, constraint gi should be measured and inspected for violation. If r (of N) cases violate gi constraints, Pif (probability of failure) can be computed by r/N. In the reliability-based robust multi-objective optimization presented in this paper, there are various contradictory objectives that should be minimized simultaneously. This methodology can be formulated as follows: 8 Minimize fμ ½ f i ðx; d; pÞ; ϑ f i ðx; d; pÞg > > > > i ¼ 1; 2; 3; …k > 8 > > > P Gi ðx; d; pÞ≤ε j > < > > > < j ¼ 1; 2; 3; …m > > > Subject to: > > > ðLÞ ðU Þ > > > > > x ðLÞ ≤x ≤x ðU Þ > : : d ≤d≤d
ð18Þ
As mentioned in the previous sections, in the real crash events, axial loading is highly unlikely. In a real crash event, α and β that are the angles between sides a and b with their projections on the inclined plate are not under control; as a result, it is better to take them as uncertain probabilistic parameters, called noise factors. In this way, oblique angles θ1and θ2 are counted as two parameters having Gaussian probabilistic distribution with mean and standard deviation equal to 0 and 15, respectively. Such Gaussian probabilistic distribution is depicted in Fig. 12.
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Fig. 13. Obtained non-dominated optimum design points in 3-Dimension.
Solving such a probabilistic problem will result in design points with better performances, regardless of oblique angles in a crash event. Moreover, a, b, and t are considered as probabilistic design variables due to manufacturing tolerances. In this way, Gaussian probabilistic distribution with standard deviation of 0.1 for a and b as well as 0.01 for t (see Fig. 12) is considered.
To obtain optimum values for EA and PCF, the mean rates of EA and PCF are considered to be maximized and minimized, respectively. As a robust design requirement, standard deviation (σ) of EA and PCF is also included in objective functions optimization and, therefore, needs to be minimized. Additionally, weight is also included as a lightweight criterion. In order to obtain 90% reliable design, inequality constraints
Fig. 14. Obtained non-dominated design points in the plane of PCF-EA.
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(a)
(b)
(c)
(d)
Fig. 15. Relationships between design variables of non-dominated results.
Constrained
and probability of failure are defined. 5-objective optimization of this problem is formulated as below:
Un-Constrained
8 > > > > > > > > > > > > > > > > > >
> > > > > > > Where probability of failure is P PCF b 0:1 > > > > < > > > > Subject to: > > 50mm≤a ≤70mm > > > > > > > > 50mm≤b≤70mm > > > > : : 0:6 mm≤t ≤1:8 mm
6000
EA (J)
Minimize Minimize Maximize Minimize Minimize
5000 4000 3000
ð19Þ
The process of multi-objective optimization is carried out by 1000 Monte Carlo simulation and HSS distribution. In this section, parameters of modified NSGA-II algorithm, such as the population size, the crossover probability, and the mutation probability, are considered as 100, 0.7, and 0.07, respectively.
2000 1000 0 0
10
20
30
40
50
2 (degree) Fig. 16. Comparison between the results of constrained and unconstrained optimization problems in the plane of EA- θ2.
4.5. Multi-criteria decision making (MCDM) methods Balancing of multiple factors by considering different choices becomes a multiple-criteria decision-making (MCDM) choice, especially when there exist a number of such standards with conflict with an
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Table 5 Deterministic values of objective functions and their associated design variables of the different design points Point
Type
a(mm)
b(mm)
t(mm)
θ1(deg)
θ2(deg)
PCF(N)
EA(J)
W(kg)
A B C D E F
Max EA Min PCF Min W TOPSIS VIKOR SAW
63.3 50.2 52 50 50.2 50.2
57.1 51 50.4 50.1 52.4 52.1
1.8 1.1 0.8 1.3 1.4 1.3
11 21 0 11 19 19
10 7 0 8 7 8
79,817.46 35,160.71 39,571.44 45,225.85 53,620.35 45,915.64
6290.92 2806.49 1815.03 3634.42 4213.66 3664.75
0.66 0.34 0.251 0.39 0.43 0.4
extent [52]. To make a balance between objective functions in our optimization problems, three different MCDM methods are utilized to suggest a trade-off choice (point) out of Pareto-optimal points. The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), VIKOR, and Simple Additive Weighting (SAW) are methods, which are briefly discussed and implemented in this study. TOPSIS, originally developed by Hwang and Yoon in 1981 [53], faced many modifications in later years. Basically, TOPSIS requires an alternative that has the minimum distance from the positive ideal solution (PIS) as well as maximum distance from the negative ideal
solution (NIS) [54]. TOPSIS process used in this study is presented in Appendix 2. The basic VIKOR method [55] focuses on selecting and ranking a set of alternatives and providing comparisons between conflicting criteria; this can help decision-makers reach a final decision [56]. The procedure is presented in Appendix 2. Simple Additive Weighting, also called scoring method, is one of the simplest MCDM methods. The idea is to attain a weighted sum of each alternative ranked by performance under all attributes [57,58]. The process is presented in Appendix 2.
(a)
(b)
(d)
(c) Fig. 17. Re-fronted optimum points in different planes
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Table 6 Probabilistic values of objective functions and their associated design variables of the different design points Point
Type
t(mm)
b(mm)
a(mm)
μ(PCF) N
σ(PCF) (N)
μ(EA) (J)
W(kg)
σ(EA) (J)
G H I J K L M N
Min μ(PCF) Min σ(PCF) Max μ(EA) Min σ(EA) Min W TOPSIS VIKOR SAW
1.1 1.1 1.5 1.1 1.2 1.1 1.3 1.1
50.7 61.3 60.7 64.2 50.9 57 51.6 61.3
69 69.5 66.2 69.3 50.2 70 69.5 69.5
37,429.24 38,353.51 64,644.55 38,279.49 41,885.37 38,302.72 50,319.95 38,353.51
8204.95 7529.94 12,293.43 7665.36 9717.95 7654.09 10,317.82 7529.95
2590.38 2613.06 4384.35 2610.64 2784.82 2614.56 3454.52 2613.06
0.40 0.44 0.58 0.45 0.37 0.42 0.48 0.44
358.91 313.27 763.10 311.35 740.84 319.75 481.44 313.27
Table 7 Values of probabilistic metrics for the points obtained from deterministic approaches Points
Type
A B C D E F
Max EA Min PCF Min W TOPSIS VIKOR SAW
Design variables
Deterministic
Probabilistic
Probability of failure
a (mm)
b (mm)
t (mm)
PCF(N)
EA(J)
W(kg)
μ(PCF) (N)
σ(PCF) (N)
μ(EA) (J)
σ(EA) (J)
1.8 1.1 0.8 1.3 1.4 1.3
57.1 51 50.4 50.1 52.4 52.1
63.3 50.2 52 50 50.2 50.2
0.66 0.34 0.251 0.39 0.43 0.4
6290.92 2806.49 1815.03 3634.42 4213.66 3664.75
79,817.46 35,160.71 39,571.44 45,225.85 53,620.35 45,915.64
49.52 20.29 96.44 11.78 10.16 11.68
1599.45 729.31 443.53 1019.67 1198.12 1008.23
5298.83 2293.21 1428.22 2927.33 3144.35 2938.17
19,486.70 8457.59 5054.96 11,581.48 13,528.67 11,546.11
4.6. Optimization results and discussion In this section, deterministic and probabilistic optimization results are discussed elaborately, and some important designing keynotes will be highlighted. Deterministic optimization using modified NSGA-II is
80,573.33 34,722.90 20,977.41 45,153.47 49,886.40 45,570.37
run 10 times, and all the 1512 points are added up together and, then, re-fronted. All 275 non-dominated points in the first front are taken as a set of ultimate optimum points. Fig. 13 shows those 1512 obtained points together with non-dominated optimum design points in a three-Dimensional view. Moreover, single-objective optimum points,
(a)
(b)
Fig. 18. Probability density function of obtained deterministic and probabilistic optimum points for EA and PCF
Table 8 Comparison between GMDH and FE results for the optimal design points Points
L
M
N
θ1
30 15 0 15 30 0 15 0 30
θ2
0 30 15 0 15 30 30 15 0
Peak crushing force (KN)
Energy absorption (KJ)
GMDH
FE
Error%
GMDH
FE
Error%
59.88 20.594 33.878 83.083 45.64 20.538 20.998 32.188 60.294
57.464 18.886 34.452 84.566 42.922 22.275 20.652 35.711 58.785
4.2 9.04 1.67 1.75 6.33 7.8 1.68 9.87 2.57
2.931 0.975 1.613 3.361 2.11 1.08 1.005 1.753 2.953
2.712 0.907 1.638 3.669 2.036 1.024 0.966 1.837 2.739
8.08 7.5 1.53 8.39 3.63 5.47 4.04 4.57 7.81
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namely A, B, and C, indicate maximum energy, minimum peak crushing force, and minimum weight, respectively, which are depicted in this figure. Fig. 14 depicts the obtained optimum points of the three-objective deterministic optimization problem on the plane of PCF-EA. It is interesting that these optimum points can be categorized into two linear groups according to their weights. Designers could choose a weight to be 0.251 to 0.38 kg or 0.34 to 0.71 kg and, then, use the linear relation between PCF and EA to acquire the best choice by simply knowing one of them. Single-objective optimum points are ideal to choose if radical requirements are set over only one of the objective functions. Otherwise, multi-criteria decision-making methods can be employed to choose trade-off optimum points among all 275 non-dominated optimum points. Consequently, points D, E, and F are found using TOPSIS, VIKOR, and SAW methods, respectively. In order to investigate optimization results and fully understand feasible space in optimization procedure, design variables of all the obtained optimum points have been plotted against each other. Fig. 15(a) depicts variables a and b against each other. In the case that θ1 equals zero, the angle between side a with its projection on the inclined plate (α) is equal to θ2, and the angle of side b with its projection on the inclined plate (β) is zero. As mentioned earlier, with the increase of θ1, β angle enlarges and α angle shrinks. Generally, between 0 and 45, α angle is larger than β angle, while θ1increases, the difference between α and β decreases up to the point in which θ1 is equal to 45°, in which they become equal. In Fig. 15(a), the values of side a are larger than those of side b. Given that the angle of side a (α) in θ1is between 0 and 45 and is larger than the angle of side b (β), it can be concluded that the side of the tube with a larger angle with the inclined plate is more important in oblique loading and should have a larger value. In Fig. 15(a), values smaller than 60 mm are the governing case of variable a, occurring when variable b lies between 50 and 53 mm. Therefore, when a is between 60 and 70 mm, variable b has more diverse values between 50 and 70 mm. This finding could be used to divide all points into two categories: yellow squares (a b 60) and blue circles (a ≥ 60), which will be addressed onward by sets “s” and “c”, respectively. Furthermore, Fig. 15(b) illustrates the relation between θ1 and θ2 values, and reveals that optimum points tend to occur in the range of 0 to 40 for θ1 and 0 to 12 for θ2 value. Note that the range of 8 to 12, which is the changing zone from progressive collapse mode to mixed mode, is located in the optimum obtained range of θ2 (0−12). Comparing this figure with Fig. 10 proves that, for a given dimension of a tube, there exist a range of angles close to the changing mode from progressive collapse mode to the mixed mode, which is likely to result in optimum points. It can be inferred from Fig. 15(c) and (d) that set “c” occurs when θ1 and θ2 both have higher values, and it is in this case that absorbed energy is satisfying. Note that, in these cases, tubes' sides on the inclined plate also have higher values. In order to investigate the effect of PCF constraints, the optimization problem (Eq. (19)) was solved twice: with and without considering the constraint. Fig. 16 shows the result of these two different solutions in two ways: the red dot for the first answer and the green dot for the second answer. It can be inferred that points with better EA values, which occur in θ2 = 0, are omitted by the constraint. An interesting inference is that, for achieving optimum EA and PCF, a little angle in the crushing incident is indispensable. Table 5 shows the corresponding design variables and objective functions for points “A to F”. Based on the table, TOPSIS and SAW methods produce similar results. In comparison with point D suggested by TOPSIS, point E suggested by VIKOR has 18.5% more PCF and 15.9% more EA. Note that PCF of point E is in the range of the defined constraints; thus, due to suppressing the value of energy absorption as well as neglecting minor weight differences, this point may be suggested as a perfect design point. Without a doubt, important optimal
553
design facts between these three objective functions cannot be revealed without multi-objective optimization. For solving the reliability-based robust design optimization problem, modified NSGA-II with ε-elimination diversity is also employed. In this way, this algorithm is run 10 times, and total 1064 design points are added up together and, then, re-fronted. All 224 non-dominated points in the first front are taken as a set of ultimate optimum points. All the obtained optimum design points together with 4 single-objective optimum points (G, H, I and J) and 3 MCDM trade-off choices (L, M, N) suggested by TOPSIS, VIKOR, SAW methods are depicted on 4 different planes in Fig. 17. The tube's thickness (t) at the optimum points is between 1.1 and 1.5, which divides the Fig. 17(a, b, and c) into five segments. The value of t is also an important parameter in the probability of failure. Points with a value of t N1.5 as well as values smaller than 1.1 have a probability failure rate of over 10%. Fig. 17(a) reveals a noteworthy linear relation between μ(EA) and μ(PCF) which is depicted by a corresponding trend line and a mathematical formulation. Designers could use this formula to choose one of the afore-mentioned objectives by simply knowing the other value. A comparison of Fig. 17(a, b, and c) shows that, in a specific t with increasing μ(EA), there comes an increase in μ(PCF) and weight. On the other hand, σ(EA) decreases. Fig. 17(d) shows that decreasing σ(EA) follows a decrease in σ(PCF). Therefore, points H and J which represent minimum σ(EA) and σ(PCF) almost coincide with each other. 5 points out of 9 design choices in Fig. 17(d) are congregated in an area with minimum σ(EA) and σ(PCF). Values of design variables and objective functions for the obtained single-objective optimum points and MCDM trade-off choices are illustrated in Table 6. Results of the table reveal that design points G, H, J, L, N have close values in mean EA, mean PCF and in dispersion (SD) of EA and PCF. Moreover, one could note that all the mentioned design points have the thickness of 1.1 mm. To investigate reliability and robustness of the deterministic points and find their probabilistic metrics, the suggested deterministic points (A, B, C, D, E, and F) are examined by the probabilistic algorithm, and the results are depicted in Table 7. In this algorithm, probability density functions of θ1 and θ2 are considered according to Fig. 12. Furthermore, mean value of the Gaussian function (μ) is considered equal to those deterministic values reported in Table 5. Results indicate that, considering uncertainties, the deterministic points (A, B, C, D, E, and F) failed with probability rates of 49.52, 20.29, 96.44, 11.78, 10.16, and 11.68%, respectively. These numbers indicate high risk of choosing deterministic points, especially single-objective ones. In Table 7, the comparison between μ(PCF) values of probabilistic MCDM points and PCF values of deterministic MCDM points shows how close these values are in the two studies. Nevertheless, the same evaluation over EA values reveals an intolerable decrease in μ(EA) of probabilistic MCDM points as compared to the EA of MCDM deterministic points. Clearly, deterministic points suffer from poor performances when θ1 and θ2 are considered probabilistic. Therefore, it can be concluded that carrying out a probabilistic study is indeed essential to acquire a reliable design. Fig. 18(a) depicts the probability density function of PCF for the trade-off optimum design points of deterministic study (D, E, and F) as compared to trade-off optimum design points of the probabilistic study (L, M, and N). A noticeable fact about this figure is the robustness improved within the probabilistic study in all the probabilistic points. As expected, on the plane of PCF-PDF, compared to deterministic points, probabilistic points have better σ(PCF) and μ(PCF) results. For instance, a comparison of the suggested points by TOPSIS method in deterministic and probabilistic approaches (points D and L) shows 15% and 34% improvements in μ(PCF) and σ(PCF), respectively. This inference indicates that without using probabilistic analysis, optimization could present wrong points, or lead to at least poor performance points.
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Fig. 18(b) depicts the probability density function of EA for the same trade-off optimum design points. As anticipated, probabilistic design points have better σ(EA) on this plane. Higher μ(EA) for point M (probabilistic VIKOR) and its lower σ(EA) in comparison with the deterministic points make this design point an interesting candidate with superior energy absorption capability in the probabilistic design. Specifically, in comparison with point E (deterministic VIKOR), point M has 9% and 60% better μ(EA) and σ(EA), respectively. In order to validate RBRDO results in a post-numerical study, the suggested design points in the probabilistic section are analyzed by FEM using ABAQUS. Table 8 depicts such re-evaluation and corresponding errors of such analysis with GMDH results. In this re-evaluation, 3 different θ1 and θ2 are taken within their distribution range. It can be asserted that all errors are within an expected range, and the optimization procedure has a valid outcome. 5. Conclusion It was demonstrated that the proposed 3D oblique loading was feasible and effective for acquiring a highly realistic approach to crash events. In this way, crushing behavior of thin-walled tubes on the inclined plate with two spatial angles was investigated experimentally and numerically. The comparison of the results obtained from experimental tests with numerical simulations validated the accuracy of FE modeling. The data gathered from numerical results were used to train and test GMDH-type neural network, and the extracted polynomial functions were used to map inputs onto outputs. Deterministic and probabilistic multi-objective optimizations were successfully applied with the goal of optimizing crushing characteristics as well as improving reliability and robustness. Three different multi-criteria decision-making methods were utilized to suggest a trade-off choice (point) out of Pareto-optimal points, namely TOPSIS, VIKOR, and SAW. The obtained reliable and robust points were compared in their values of ABAQUS and GMDH to exhibit the accuracy and validity of the approach. Results of this paper can be concluded as follows: • Experimental and Numerical analyses proved that θ1 and θ2, which are the tube's incidence angles, could be used to control tube's collapse mode as well as crashworthiness characteristics. • The deterministic study showed that optimum points for a given dimension of a tube occurred in a range of angles close to changing the mode from progressive to mixed collapse mode. • In oblique loading, it can be concluded that the side of the tube with a larger angle of the inclined plate is more important and has higher impact on energy absorption. • The presence of a small oblique angle highly decreased PCF value, while EA remained without severe touch. • Although MCDM deterministic points were more reliable than singleobjective deterministic points, they all failed during the analysis with the probabilistic condition. • The optimum points obtained by implementing MCDM methods in an RBRDO study resulted in a much better mean of PCF and standard deviation of PCF than the deterministic results. • The comparison of the suggested points by TOPSIS method in deterministic and probabilistic approaches (points D and L) showed 15% and 34% improvements in the mean of PCF and standard deviation of PCF, respectively. • The comparison of the suggested points by VIKOR method in deterministic and probabilistic approaches (points M and E) showed 9% and 60% improvements in the mean of EA and standard deviation of EA, respectively. • Using the meta-model-based stochastic analysis, engineers can secure the best crushing angle for highly reliable and robust energy absorbers that can balance cost and safety in close correlation to real crash events.
Appendix 1 The quadratic polynomial functions of SEA and PCF for the tube under 3D oblique load in 3 angle span for θ2 are as below: Mathematical equations according to the neural network of EA for 0 ≤ θ2 b 1 Y12 ¼ 16987:106 þ 2394:311 a−2657:712 b−15:937 a2 2 þ 26:828 b −10:318 a b; 2
Y23 ¼ −1672:675 þ 46:051 b þ 2129:925 t−0:161 b þ 2394:333 t2 −39:484 b t;
Y13 ¼ 5809:464−214:1804 a þ 2266:2604 t þ 2:024 a2 þ 2355:705 t2 −38:816 a t; Y1244 ¼ 5:278−0:659 Y12 þ 228:348 θ1 þ 0:00039 Y12 2 −6:337 θ1 2 −0:0085 Y12 θ1 ; Y2313 ¼ 0:00028 þ 0:8717 Y23 þ 0:143 Y13 −0:00217 Y23 2 −0:00218 Y13 2 þ 0:00436 Y23 Y13 ; EAð0−1Þ ¼ 0:00026 þ 0:011 Y1244 þ 0:971 Y2313 þ 2:246e−05 Y1244 2 þ 2:601e−05 Y2313 2 −4:633e−05 Y1244 Y2313 ; Mathematical equations according to the neural network of PCF for 0 ≤ θ2 b 1 Y15 ¼ −1095454:202 þ 41718:339 a−363:798 a2 ; Y12 ¼ −60078:786 þ 50118:15 a−43003:351 b−251:051 a2 2 þ 556:102 b −389:205 a b; Y23 ¼ 45135:705−1448:945 b−3071:003 t þ 5:45 b þ 19115:78 t2 þ 1093:378 b t;
2
Y45 ¼ 92114:339−1255:588 θ1 þ 20:149 θ1 2 Y1512 ¼ 2:785e−05 þ 0:873 Y15 þ 0:097 Y12 −1:213e−05 Y15 2 þ 7:606e−07 Y12 2 þ 1:163e−05 Y15 Y12 ; Y2345 ¼ −1:0095e−05 þ 0:884 Y23 −0:432 Y45 þ 1:426e−06 Y23 2 þ 6:789e−06 Y45 2 −1:834e−06 Y23 Y45 ; PCFð0−1Þ ¼ 1:13e−06 þ 0:037 Y1512 þ 0:976 Y2345 −1:76e−06 Y1512 2 −1:93e−06 Y2345 2 þ 3:834e−06 Y1512 Y2345 ; Mathematical equations according to the neural network of EA for 1 ≤ θ2 ≤ 15 Y34 ¼ −732:072 þ 1965:715 t þ 14:369 θ1 þ 805:747 t2 −0:199 θ1 2 −2:373 t θ1 ; Y15 ¼ 9661:972−196:854 a−445:361 θ2 þ 1:767 a2 þ 0:032 θ2 2 þ 5:451 a θ2 ; Y35 ¼ −1391:28 þ 2589:91 t þ 238:529 θ2 þ 998:308 t2 −10:7679 θ2 2 −113:419 t θ2 ; Y3433 ¼ 0:0016 þ 0:943 Y34 þ 0:00079 t−2:7409e−05 Y34:2 þ 0:00024 t2 þ 0:11466 Y34 t;
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Y1535 ¼ 0:00018 þ 0:2592 Y15 þ 0:5392 Y35 −7:0441e−05 Y15 2 −1:4567e−05 Y35 2 þ 0:00014 Y15 Y35 ;
Y1535 ¼ 1:945e−05 þ 0:258 Y15 þ 0:706 Y35 −6:763e−06 Y15 2 −2:055e−06 Y35 2 þ 1:056e−05 Y15 Y35 ;
EAð1−15Þ ¼ 0:00036 þ 0:6623 Y3433 þ 0:2908 Y1535 −0:000577 Y3433 2 −0:00049 Y1535 2 þ 0:0011 Y3433 Y1535 ;
Y2311 ¼ −8:0429e−05−0:674 Y23 −0:0039 a−1:551e−06 Y23 2 −0:192 a2 þ 0:029 Y23 a;
Mathematical equations according to the neural network of PCF for 1 ≤ θ2 ≤ 15
PCFð15−45Þ ¼ 2:636e−05 þ 0:77 Y1535 þ 0:07 Y2311 þ 3:716e−06 Y1535 2 þ 5:25e−06 Y2311 2 −5:86e−06 Y1535 Y2311 ;
Y35 ¼ −7172:194 þ 26821:52 t−223:344 θ2 þ 22154:645 t2 þ 75:429 θ2 2 −1807:171 t θ2 ; Y14 ¼ 83852:458−1574:542 a−82:981 θ1 þ 16:7606 a2 þ 5:767 θ1 2 −0:886 a θ1 ; Y45 ¼ 77433:277−240:551 θ1 −6652:485 θ2 þ 7:571 θ1 2 þ 295:682 θ2 2 þ 9:945 θ1 θ2 ;
Appendix 2 1- TOPSIS Step 1: Start with creating an evaluation matrix consisting of m alternatives and n criteria, the final matrix should be in this form (xij)m×n xij Step 2: using the normalization method r ij ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi Pm ffi and evaluating
2
i¼1
Y23 ¼ −80280:545 þ 2709:663 b þ 8603:492 t−23:178 b þ 20033:313 t2 þ 106:3756 b t;
x2ij
matrix R = (rij)m×n Step 3: Calculate the weighted normalized decision matrix tij = rij × Wj
wj . Where wj is also normalized weight as w j ¼ Pn
Y3514 ¼ 4:167e−06 þ 0:478 Y35 þ 0:1077 Y14 þ 6:2326e−07 Y35 2 −1:55e−06 Y14 2 þ 8:616e−06 Y35 Y14 ;
j¼1
Wj
in which, Wj is
the original weight given to indicator. Step 4: Determine the worst alternative Aw and the best alternative Ab.
Y4523 ¼ 4:6903e−06 þ 0:126 Y45 þ 0:228 Y23 −2:3496e−06 Y45 2 −3:937e−07 Y23 2 þ 1:5157e−05 Y45 Y23 ;
Aw ¼
PCFð1−15Þ ¼ 1:135e−05 þ 0:408 Y3514 þ 0:589 Y4523 −2:85e−05 Y3514 2 −3:44e−05 Y4523 2 þ 6:3e−05 Y3514 Y4523 ;
Mathematical equations according to the neural network of EA for 15 b θ2 ≤ 45 Y15 ¼ 11034:865−334:763 a−45:023 θ2 þ 3:68 a2 þ 1:928 θ2 2 −1:924 a θ2 ;
Ab ¼
max t ij ∀ J − ; min t ij ∀ J þ ≡ t wj
min t ij ∀ J − ; max t ij ∀ J þ ≡ t bj
J+ is associated with criteria having a positive impact and J− associated with the criteria having a negative impact. Step 5: Calculate the distance between the target alternative and the worst and best condition
diw
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 2 u n ¼t t ij −t wj
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 2 u n dib ¼ t t ij −t bj
j¼1
2
Y23 ¼ −5072:297 þ 158:367 b þ 526:535 t−1:324 b þ 378:952 t2 þ 11:568 b t;
j¼1
where diw and dib are norm distances from the target alternative i to the worst and best conditions, respectively.
Y35 ¼ 77:7661 þ 2955:148 t−66:334 θ2 þ 423:672 t2 þ 1:55 θ2 2 −66:625 t θ2 ; Y1523 ¼ 0:000147 þ 0:1411 Y15 þ 0:107 Y23 −4:442e−05 Y15 2 −1:702e−05 Y23 2 þ 0:00049 Y15 Y23 ; Y3511 ¼ 0:0988−0:476 Y35 þ 2:947 a−3:336e−05 Y35 2 −0:04 a2 þ 0:0252 Y35 a; EAð15−45Þ ¼ 0:00022 þ 0:1092 Y1523 þ 0:875 Y3511 þ 0:00029 Y1523 2 þ 0:000177 Y3511 2 −0:00047 Y1523 Y3511 ; Mathematical equations according to the neural network of PCF for 15 b θ2 ≤ 45 2
Y15 ¼ 187631:2123−4522:347 a−2300:19 θ2 þ 43:09 a þ 25:605 θ2 2 −1:674 a θ2 ;
iw ,0 Step 6: Calculate the similarity to the worst condition: Siw ¼ ðdiwdþd Þ ib
b Siw b 1 If and only if Siw = 1 the alternative solution has the best condition; if and only if Siw = 0 the alternative solution has the worst condition. Step 7: Rank the alternatives according to Siw 2- VIKOR − Step 1: Determine the best ∅+ i and the worst ∅i values of all criterion functions, i = 1, 2, …, n; − ∅+ i = max (∅ij, j = 1, …, J), ∅i = min (∅ij, j = 1, …, J), if the i-th function is benefit; − ∅+ i = min (∅ij, j = 1, …, J), ∅i = max (∅ij, j = 1, …, J), if the i-th function is cost. Step 2: Compute the values Sj and Rj, j = 1,2,…,J, by the relations: P wi ð∅þi −∅ij Þ , weighted and normalized Manhattan distance; S j ¼ ni¼1 ∅ þ −∅− i
R j ¼ max i
i
wi ð∅þ i −∅ij Þ − ∅þ i −∅i
, weighted and normalized Chebyshev
distance; where wi are the weights of criteria, expressing the decision maker's (DM) preference as the relative importance of the criteria. Step 3: Compute the values Qj,
2
Y35 ¼ −7269:39 þ 50258:234 t−352:686 θ2 þ 8468:19 t þ 15:114 θ2 2 −1265:604 t θ2 ;
2
Y23 ¼ −130934:228 þ 4033:656 b þ 26611:362 t−31:724 b þ 7206:96 t2 −134:241 b t;
Qj ¼ ϑ
S j –S− R j −R− þ ð 1−ϑ Þ Rþ −R− Sþ −S−
556
M. Arjomandi Rad, A. Khalkhali / Materials and Design 156 (2018) 538–557
where S+ = max (Sj), S− = min (Sj), R+ = max (Rj), R− = min (Rj); and ϑ is introduced as a weight for the strategy of maximum group utility, whereas 1 − ϑ is the weight of the individual regret. These strategies could be compromised by ϑ = 0.5 Step 4: Sorting by the values S, R and Q, to rank the alternatives. Step 5: If Alternative A1is the best ranked by the measure Q (minimum) and following two conditions are satisfied it is considered best point: 1 where, A2is Condition 1: “Acceptable Advantage”: Q ðA2 Þ–QðA1 Þ≥ J−1 the alternative with second position in the ranking list by Q Condition 2: “Acceptable Stability in decision making”: The alternative A1 must also be the best ranked by S or/and R. If only the condition 2 is not satisfied, Alternatives A1and A2 will be proposed. If the condition 1 is not satisfied, Alternatives A1, A2, …, Am 1 will be proposed. Am is determined by the relation Q ðAm Þ–Q ðA1 Þb J−1 for maximum m. 3- SAW Step 1: Let A = (a1, a2, …, an) be a set on alternatives. Let C = (c1, c2, .…, cn) be a set of criteria. Construct the decision matrix (dij)m×n Where dij is the rating of alternative Ai with respect to criterion Ci. Step 2: Step 2: Construct the normalized decision matrix. For positive d
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(cost) rij ¼
min
dij dij
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