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Structural optimisation for the collapse zone of a railway vehicle
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Structural optimisation for the collapse zone of a railway vehicle Chengxing Yang , Q.M. Li PII: DOI: Reference:
S0020-7403(19)31960-5 https://doi.org/10.1016/j.ijmecsci.2019.105201 MS 105201
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
31 May 2019 23 September 2019 29 September 2019
Please cite this article as: Chengxing Yang , Q.M. Li , Structural optimisation for the collapse zone of a railway vehicle, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105201
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Highlights Two stage optimal design with manufacture constrains was proposed to consider both static and crash loads for the collapse zone of a high-speed train; Dynamic weighting was used to balance the static and dynamic loading cases; The collapse zone designed in the paper demonstrated considerable improvements on energy absorption, crash force characteristics and weight saving.
1
Structural optimisation for the collapse zone of a railway vehicle Chengxing Yang, Q. M. Li a
*
Department of Mechanical, Aerospace and Civil Engineering, School of Engineering, The University of Manchester, Manchester M13 9PL, UK
ABSTRACT The collapse zone is a critical component of railway vehicle, which is designed with multiple functions, i.e., supporting neighbouring components and attached equipment in its normal operation and absorbing impact energy in a crash accident. However, there is a lack of systematic study on the design of the collapse zone based on the strategy of topological and size configurations. To make full use of the material, topology optimisation for conceptual design is conducted under concurrent static and crash loads. The importance of assigning dynamic weighting to each loading case has been proved to avoid the existence of a dominant case. The topological results show that the load transferring paths (i.e., the space where materials are distributed to) are in the longitudinal direction. In the detailed design, a revised model is created, which addresses structural and manufacturability concerns. To optimise the thickness assignment, the desirability approach is combined with the discrete optimisation method. The final design is obtained and compared with the currently-used collapse zone structure. The radar map of crashworthiness indicators shows that the overall crashing performance of the optimised structure is superior to the traditional one. Keywords: Collapse zone; Topology optimisation; Hybrid cellular automata (HCA); Multi-objective discrete optimisation; Crashworthiness
1. Introduction
*
Corresponding author.
E-mail address: [email protected] (Q.M. Li). 2
Structural crashworthiness of transportation vehicles (e.g., automotive, rail, ship, aerospace, etc.) has drawn increasing attention since it is highly related to occupant safety [1-4]. Considerable studies on impact energy absorber (IEA) have recently been performed for the passive safety of railway vehicles due to the rapid development of high-speed train [5-7] and new generation of subway [8,9]. IEA of rail vehicles can be classified into two categories, i.e., special energy-absorbing structure (SES) and bearing energy-absorbing structure (BES). The former is specially designed to act only when crash accidents happen while the latter has multiple roles of bearing static loads from connected neighbouring components in normal service condition and managing dynamic loads in crash accidents. For example, Peng et al. [10] introduced a duplex structure as a SES, which is composed of anti-climbing gears, square tubes with a diaphragm, an end plate, a supporting guide bar and energy-absorbing components. A five-cell thin-walled aluminium IEA for high-speed train was studied using hybrid optimisation approach [7]. With the consideration of structural bending strength, Yao et al. [11] optimised the cross-sectional shape of the guide of the structure developed in [12] to avoid the occurrence of instability. Xie et al. [5,13,14] designed two types of BES, which were composed of a thin-walled metal structure and aluminium honeycomb fillers. In [8,9,15], a gradual energy-absorbing structure in the collapse zone of a railway vehicle was proposed where the design started from an 1D collision analysis of two trains using a mass-spring model to calculate the impact energy absorbed by the collapse zone. Then, a 3D structure was designed based on numerical simulation and experimental validation. However, most of these studies on the collapse zone were based on predefined topological geometries, making it difficult to perform systematic investigations about the optimal material distribution to develop more efficient structures. Topology optimisation is an ideal method for conceptual design, which is able to derive an optimal topological structure by distributing materials to the most efficient load-path network. Due to this benefit, numerous studies have made contributions to the applications of topology optimisation. For railway vehicles, Aderiani et al. [16] investigated the structural optimisation of ER24PC locomotive carbody subjected to various static loads based on European standard EN12663. Kim et al. [17] performed the conceptual design for the leading-cab structure. The optimal structure was found with its mass reduced by 40% after topology optimisation. Mrzygł´od and Kuczek [18] dealt with a new uniform crashworthiness concept of carbody for a high-speed train. The optimisation problem was to minimise the mass of the survival space under stress constraint, i.e., the elastic limit of the material. Lee 3
et al. [19] investigated the structural-optimisation-based design process for the body of a railway vehicle made from extruded aluminium panels, in which topology optimisation is adopted for conceptual design while size optimisation is employed for detailed design. The mass of the passenger car was reduced by about 14% comparing with the mass of the existing passenger car. Nevertheless, these studies on railway vehicles only considered either static or crash loading cases, rather than both of them. There is a lack of systematic study on the design of the collapse zone considering both loading cases simultaneously in a practical application. As a type of BES, the collapse zone is expected to have satisfactory performance under multiple loading cases. In general, longitudinal dynamic compression and vertical static loading cases are the two most common loading types. However, the majority of the previous studies for the design of the collapse zone focused only on either the vertical static load [16] or the longitudinal dynamic compression load [15]. There are limited literatures studying topology optimisation under multiple loading cases [20]. To address this issue, topology optimisation under multiple loading cases will be conducted in LS-TaSC™ based on hybrid cellular automaton (HCA) algorithm [21]. HCA is a non-gradient approach that has been proved to be suitable for solving both linear problems such as structural stiffness and highly non-linear problems such as structural crashworthiness [22]. Patel [23] and Patel et al. [24] showed that a uniform distribution of the internal energy density (i.e., total elastic and plastic strain energy density) can lead to the optimised structures with favourable crashworthiness performance. Zeng and Duddeck [25] proposed an approach based on HCA for crashworthiness topology optimisation with a special focus on thin-walled structures. The thickness of each thin wall was treated as a design variable. Aulig et al. [22] proposed a preference-based scaled energy weighting HCA (SEW-HCA) approach to address the topology optimisation of both static and crash loading cases concurrently. The main idea is the use of weighting coefficient for each loading case. However, the weighting coefficients are constant and dependant on user’s preference. In this paper, the enhanced HCA algorithm with dynamic weightings proposed by Bandi et al. [26,27] will be employed to take both loading cases into consideration. The proposed collapse zone structure using optimisation methods will be based on a currently-used collapse zone structure in Lu et al. [15]. Size optimisation is an effective tool to find optimal configurations after conceptual design, which helps engineers to attain the optimal crashworthiness of IEA [14, 28-30]. For 4
example, Yin et al. [31-33] conducted size optimisation for W-beam guardrail, bio-inspired hierarchical honeycomb and functionally graded foam-filled structure. A comprehensive review was presented by Fang et al. [34], where the design optimisation methods for IEA have been systematically summarised. However, the traditional optimisation algorithms have been focusing on continuous variable problems. In real-life, many design variables can only be selected from a standard database such as thickness of commercially-available sheet materials, implying that the design space is restricted and discrete [29]. To address the optimal design in a discrete space, Lee et al. [35] proposed a discrete optimisation algorithm based on Taguchi method to select the best configurations, which was also adopted by Wu et al [36] and Sun et al. [29,37,38]. For the trade-off issue between conflicting objectives in discrete optimisation problem, Sun et al. [29] converted multiple indices into one unified cost function by coupling the grey relational analysis (GRA) with principal component analysis (PCA). In this study, the overall desirability function [39] is employed to combine multiple responses into a dimensionless measure of performance. The objectives of this research are 1) to perform the conceptual design of a real-world collapse zone based on a finite element (FE) model, using large-scale topology optimisation, and 2) to further improve the design in the subsequent detailed design stage, using size optimisation. It is expected that the outcome of this work will improve the effectiveness and efficiency of the design process for the collapse zone with considering the important performance metrics such as stiffness and crashworthiness in optimisation. Section 2 presents the optimisation problem statement for the entire design process of a collapse zone. Section 3 shows the topology optimisation task consisting of finite element modelling and topological results. In Section 4, detailed design is performed based on the combination of discrete optimisation method and the overall desirability approach. Finally, the paper is summarised and concluded in Section 5.
2. Statement of the optimisation problem For a discretised design space, the optimisation problem for the whole process of conceptual design and detailed design can be mathematically stated as
5
(
( ) ( )⁄
){
(
( ) ( )
)
{
(1)
{
where the objective of the conceptual design is to maximise the target function
( ), which
could be the global stiffness matrix K, the total energy absorption EA or their combination. The objective of the detail design is to have combined effect of maximum specific energy absorption (SEA) and minimum initial peak crushing force (IPCF). The multi-objective problem can be treated as a single-objective problem through applying weighting factors based on practical considerations. x is the vector of elemental relative density xi for topology optimisation and xmin is introduced to avoid numerical problems. If xi approaches to 1, the i-th element will be kept; while it will be deleted if xi approaches 0. vectors of nodal force and displacement, which formulate
( )
and .
are the and
( ) are
respectively the initial mass and the topology-optimised mass of the design domain. A constraint
is imposed on the mass fraction. Me is the mass of the structure after size
optimisation and M1 is its upper bound obtained in Lu et al. [15]. ACF indicates the average crushing force and ACF1 is its lower bound obtained in Lu et al. [15]. design variable for size optimisation.
and
bounds for the size design variables.
and
represents the j-th
are respectively the lower and upper are the total number of design variables
for topology and size optimisation, respectively.
3. Topology optimisation in concept design 3.1 Finite element model Collapse zone is defined as an area in the front-end of the underframe of a rail vehicle, illustrated in Fig. 1. It supports other structural components and internal equipment, and thus, bearing static loads in normal operation. While in crash accidents, the collapse zone is expected to collapse in a controlled manner and absorb the impact energy to protect the survival space where occupants stay in. Attached areas used for the installation of other systems are fixed throughout the topology optimisation process, and therefore, they are excluded from the design domain. To save the computational time without losing accuracy, a quarter (1/4) of the collapse zone is modelled with symmetry boundary conditions (see Fig. 1). The design domain is separated into two parts, i.e., domain A and domain B, by fixed 6
beams. Excluding the elements of non-designable areas, the total number of finite elements used as design variables for the topology optimisation is nearly 100,000. The material is high-strength steel modelled by elastic-plastic constitutive model, which is referred to Yang et al. [28]. But there was a typo of tangent modulus in Ref. [28], whose unit is corrected to be MPa rather than GPa. The elastic-plastic material is modelled in Ls-Dyna® by using “*MAT_024_PIECEWISE_LINEAR_PLASTICITY”. The total mass of the design domains is 384.9 kg, excluding the non-designable areas.
Fig. 1. The collapse zone of a rail vehicle: (a) a typical rail vehicle showing the installation; (b) the framework; (c) the ½ model; (d) the ¼ model illustrating non-designable areas and design domains.
Current studies on topology optimisation for the collapse zone design mainly concentrate just on its stiffness for static load or energy absorption for crash load. But the topological results may change when considering both static and crash loads simultaneously, which will be dealt with in this study. The distributed weight Fw is simplified to two concentrated static forces (F1 and F2) acting on the centre of gravity of two design domains, shown in Fig. 2(a). Referring to the standard JIS E7106:2011 and Xu et al. [8], F1 and F2 are roughly calculated as 2945.33 N. Under this condition, one end of the supporting beam is fixed without motion. The translational motion in y-direction and the rotational motion in xand z-directions of the symmetry plane are also fixed to define the symmetry constraint. For the crash case shown in Fig. 2(b), the front-end of the collapse zone is impacted by the rigid wall with a mass of 13.825 tons [8] at an initial velocity of 20 m/s, while the rear-end was 7
restrained to the rigid base. Attention was paid to ensure that the static loading case remains in the elastic portion, while the crash loading case can go beyond plastic yield. The implicit and explicit solvers of Ls-Dyna® are used for the analyses of static and crash loads, respectively.
(a)
(b)
Fig. 2. Illustration of mechanical loads and constraints: (a) static loads in normal operation; (b) crash load in accident condition.
3.2 Optimisation method The solid isotropic material with penalization (SIMP) model [40,41] is a well-known interpolation scheme that heuristically relates the relative density to the elastic modulus of material. Patel [23] developed a continuous material model based on SIMP method by relating the relative density to the yield stress and the strain hardening modulus. Therefore, an elastic-plastic material model parameterised by the relative density is expressed as, ( )
(2)
( )
(3)
( )
(4)
( )
(5)
where ?? denotes the density of the material,
is the yield stress, ?? represents the
Young’s modulus and ??ℎ is the strain hardening modulus. They are variables in optimisation process with regard to x. The subscript 0 refers to the base material properties. p and q are penalisation exponents and both of them are equal to 1 in HCA algorithm [23,42]. The design variable x with 0
Based on the elastic-static assumption, topology optimisation for maximising stiffness is treated as minimising elastic strain energy (U), i.e., ( ) ( ) ( )⁄
( ) (6)
{ where U(x) is the elastic strain energy of the structure, represents the volume of i-th element.
( )
∑
( )
and
is the total number of design variables (around
100,000). The maximisation of the energy absorption (EA) can be formulated by ( ) ( )⁄
{
(
) (7)
where EA(x, t= td) is the internal energy of the structure at the final time step td in finite element analysis. The field variables (e.g., stress, strain, strain energy density (i.e., elastic strain energy density), internal energy density or functions of these quantities) of the loading cases can be combined by linear weighting factors when multiple loading cases are considered [23,26], i.e., ∑
(8)
where Si is the field variable of the i-th element, Sil is the field variable of the i-th element in l-th loading case, wl is the weighting factor of the l-th loading case and nl denotes the total number of loading cases. In a more specific expression, energy density of the i-th element in l-th static loading case;
( ) is the elastic strain (
) is the
internal energy density of the i-th element in l-th crash loading case at the final time step of t= . To link the objective functions in Eqs. (6) and (7) and the field variables in Eq. (8), formulations are given as follows: ( )
( )
∑
(
( )
( ))
(9)
9
( )
(
)
∑
(
( )
(
))
(10)
where I denotes the iteration step. In order to avoid the dominance by a single loading case, a dynamic weighting method is adopted. The goal is to achieve balanced performance for all loading cases, and thus, the average of the performance measure (PM), i.e., stiffness or energy absorption, for all loading cases is used as the target [26,27], ( )
where ( )
( )
∑
(11)
( )
is the performance measure of the l-th loading case at I-th iteration and
represents the average of the performance measure for all loading cases at I-th
iteration. From Eq. (11), the increment ( )
( )
( )
is computed as
( )
(12)
( )
where
( )
is a scale factor, which equals 0.1 in this study [42]. An upper limit of
is placed on
( )
to ensure convergence in a reasonable number of iterations. The weight
for the first loading case can be fixed as unity and the loading case weight ( variables can then be rewritten as (1,
, ⋯,
,
, ⋯,
)
) because it is the relative ratios of weights
that are important, rather than their absolute values [42]. Other details about HCA method can be found in Tover et al. [43], Patel [23], Bandi [26,27] and LS-TaSC™ [42].
3.3 Topological results The topology optimisation is carried out in LS-TaSC™ by integrating with Ls-Dyna®. To validate the importance of dynamic weighting method, the weighting factors in Eq. (8) are changed to consider the different loading conditions: the static loading case (Case I), the crash loading case (Case II), the multiple loading cases with equal constant weightings (Case III) and the multiple loading cases with dynamic weightings (Case IV). Case I:
,
10
Case II:
,
Case III:
,
Case IV:
,
(
)
For all four cases, the target mass fraction
is defined as 30%. Histories of mass
redistribution and weighting factors, and optimised structures of the final iteration are shown in Figs. 3-5, respectively. The mass redistribution curves illustrate the fraction of the total mass of the design domain that has been redistributed in each iteration. All four curves approach to zero showing the convergence of optimisation. Case I with sole static loads converges at I = 26, while other cases with crash load included converge at I 40. The main reason is that the explicit dynamic behaviour involving high nonlinearities consumes more computing time than that consumed for an implicit linear problem. The topological structure of Case I is very different from that derived from Case II, which proves the importance of the concurrent design of the collapse zone. In Case I, the upper and lower surfaces are kept for supporting static loads and the middle part is a truss-like frame. However, the middle part in Case II contains more materials to resist the longitudinal impact. It is interesting to point out that the result of Case III is the same as that of Case II, which reveals that the crash load is dominant and the static loads are neglected in this concurrent optimisation with equal constant weighting factors. However, the topological structure derived from Case IV shows the combined characteristics of both static and crash loads (see Fig. 5(d)). As plotted in Fig. 4, the weighting factor
of static loads varies during optimisation process. But it is higher
than unit to avoid the dominance of crash load, thus, promoting the importance of static loads.
11
Fig. 3. Histories of the mass redistribution, i.e., the fraction of the total mass of the design domain that has been redistributed during iteration, in Case I~IV.
Fig. 4. Histories of weighting factors,
12
and
, in Case IV.
(a)
(b)
(c)
(d) Fig. 5. Topological results with enlarged views of design domains: (a) Case I; (b) Case II; (c) Case III; (d) Case IV.
13
Manufacturability control is a critical parameter affecting the optimised result. The topological structures of the above four cases in Fig. 5 need additive manufacturing, which is effective but also costly for such large scale metal structure. There are a number of manufacturing processes used for mass production in automotive industry, in which extrusion is particularly favourable because of its low cost and fast processing time. This can be considered and implemented in topology optimisation as Case V, which is based on Case IV, but with further definition of extrusion constraint. The history of mass redistribution shown in Fig. 6 converges at I =28, which reveals that less computing time is needed with the definition of extrusion constraint. The weighting factor
increases during optimisation
process to increase the importance of static case (see Fig. 7). The topological structure shown in Fig. 8 presents much clearer load paths than the results in Fig. 5. Materials are mainly longitudinally-aligned although a few branches are formed, which are the most direct and efficient paths for force transmission. Obviously, this structure can be easily manufactured using traditional method rather than the costly additive manufacture. The total mass of the design spaces including design domain A and design domain B decreases from 384.9 kg to 118 kg with further remove of 69.34% materials. It should be noted that the HCA algorithm may not guarantee an overall maximum of energy absorption although the uniform internal energy density (IED), as the objective in HCA algorithm, can lead to a topological result with more uniform distribution of energy absorption in the material [34,44].
Fig. 6. History of the mass redistribution, i.e., the fraction of the total mass of the design domain that has been redistributed during iteration, in Case V.
14
Fig. 7. Histories of weighting factors,
and
, in Case V.
Fig. 8. Topological result with enlarged views of design domains in Case V.
4. Multi-objective discrete optimisation in detailed design 4.1 Finite element model A careful review of the topological structure in Case V revealed three design concerns: (i) disconnected structural components; (ii) rough side surfaces; (iii) very thin plates. Disconnected parts obviously cannot carry mechanical loads. Roughness of the surface causes stress concentration and crack initiation. Very thin elements are prone to fractures and buckling, and causes manufacture difficulty (e.g., welding) [45]. In order to address these concerns, each part of the optimal solution is carefully examined and a revised collapse zone is rebuilt, as shown in Fig. 9. The extruded solution is modelled using thin-walled tubes with prefabricated grooves because thin-walled structures can usually support loads more efficiently with minimum mass than bulky structures. The symmetric 15
half of the actual structure is shown in Fig. 10, which will be adopted for size optimisation in next sub-sections. The FE model is composed of an outer beam, inner beam, domain A and domain B with thicknesses marked as T1, T2, T3 and T4, respectively. The rigid wall moves towards the specimen at an initial velocity of 10 m/s and the impacting mass is 27.65 tons [28]. The automatic single-surface contact algorithm is chosen to simulate the self-contact of the whole FE model. Other detailed definitions of the FE model are the same as those in [15,28].
Fig. 9. Re-interpretation of the optimal result in Case V: (a) the topology-optimised structure; (b) the reinterpreted structure.
Fig. 10. Half of the reconstructed optimal collapse zone structure.
4.2 Optimisation method 16
The overall flowchart of discrete size optimisation algorithm [35] in Stage II is illustrated in Fig. 11.
Fig. 11. The overall flowchart of discrete size optimisation algorithm [35].
The detailed steps of the algorithm are further depicted as follows: (1) The 1st step: optimisation problem definition. [
( ) ( ) ( )
( )] (13)
{ where SEA means specific energy absorption, which is the energy absorbed per the mass of the structure. IPCF is the negative initial or first peak crushing force.
and
equal
901.05 kN and 98.34 kg, respectively, which are derived from [15]. The number of design variables
is 4 and the corresponding variable intervals are 1 mm, 2 mm, 3 mm, 4 mm, 5
mm, 6 mm, as tabulated in Table 1. 17
Table 1 The discrete values of design variables. Design variables
1
2
3
4
5
6
T1 (mm)
1.00
2.00
3.00
4.00
5.00
6.00
T2 (mm)
1.00
2.00
3.00
4.00
5.00
6.00
T3 (mm)
1.00
2.00
3.00
4.00
5.00
6.00
T4 (mm)
1.00
2.00
3.00
4.00
5.00
6.00
The desirability method [39] is adopted here to combine multiple responses into a dimensionless measure of performance. √
(14)
(
( )
(15)
( )
( where
)
,
) and
and IPCF, respectively.
(16) ,
and
represent the lower and upper bounds on SEA are weighting factors of SEA and IPCF, respectively. In
this study, equal importance is assigned to both indicators. (2) The 2nd step: design of experiments (DOE). The minimum orthogonal array is selected for the design of experiments according to the number of design variables and the number of design levels. In this research, the L9(34) orthogonal array (see Table 2) is chosen because there are four variables with three levels assigned to each variable. In each iteration, nine simulations are conducted. Table 2 The orthogonal array L9(34). Design variables Simulation number
Results (DR) A (T1)
B (T2)
C (T3)
D (T4)
1
1
1
1
1
DR1
2
1
2
2
2
DR2
3
1
3
3
3
DR3
18
4
2
1
2
3
DR4
5
2
2
3
1
DR5
6
2
3
1
2
DR6
7
3
1
3
2
DR7
8
3
2
1
3
DR8
9
3
3
2
1
DR9
(3) The 3rd step: selection of level values for design variables. Real discrete values of design variables are placed to the columns of the orthogonal array chosen in the second step. Levels of each variable should be sequenced according to their relative importance. In the initial design, an arbitrary discrete value can be selected, which is defined as the second level. In general, the discrete value with one step greater than the second level is assigned to the first level, while the discrete value of one step smaller is assigned to the third level. However, if the initial design has the smallest candidate values, it must be assigned to the first level; on the other hand, if the initial design has the largest candidate values, it must be assigned to the third level. Then, two successively increased or decreased discrete values are assigned to the second and third level or second and first level, respectively. Table 3 illustrates the arrangements of level values for various design variables. Table 3 The arrangement of level values for design variables. Levels Design variables 1 ( 𝐵 (𝐵 ( (
𝐵
3
4
5
6)
𝐵3
𝐵4
𝐵5
𝐵6 )
3
4
5
6)
3
4
5
𝐵
6)
(4) The 4th step: calculation of the penalised objective function.
19
2
3
3
4
𝐵3
𝐵4
3
4
3
4
To convert a constraint problem to a non-constraint problem, a penalised objective is formulated as, ( ) ( )
∑
[
]
(18)
( ) is the penalty function,
where and
(17)
is the maximum violation of the l-th constraint,
is the scale factor of the l-th constraint. The scale factor should be carefully selected
to avoid neglecting the effect of objective functions and causing infeasible design. In this research, the scale factor is set to a value so that the order of penalty function is one-order larger than that of the original objective function [35]. Specifically, the penalty functions for the average crushing force and structural mass are formulated as, ( )
[
( )
]
[
( )
]
(19)
where r1 and r2 are respectively given as 0.1 and 2 in this study. The optimisation problem in Eq. (13) can be finally rewritten as, ( )
3
{
4
[ [ [ [
( ) ] ] ] ]
(20)
(5) The 5th step: analysis of means (ANOM) and selection of optimal design. Analysis of means (ANOM) is calculated using the results (
~
) of nine
rows obtained from experiments or simulations in Table 2, which is conducted to determine the optimal configurations. Taking the first level of design variable A (A1) as an example, the main effect of A1 is calculated as follow, (
)
3
(
3)
(21)
It should be noted that results derived from the optimal configurations should be compared with the best combination from the matrix experiments. The best level for each design variable is selected as the new level of that variable in the next iteration. (6) The 6th step: convergence criteria. 20
For the successive Taguchi method employed, two convergence criteria are used for the termination, i.e., (i) the iteration number, at which
converges, is larger than
five; (ii) the iteration number reaches the maximum value pre-defined by the designer [38]. Otherwise, the design process goes back to the 3rd step if none of the convergence criteria mentioned above is satisfied. In this new circle, the second levels as design variables are replaced by the former optimal levels.
4.3 Optimisation results To enlarge the searching space in discrete design space, the present optimisation is started from the intermediate value of each design variable. Table 4 shows the model parameters and DRnew at the first iteration. ANOM is employed to determine the optimal levels. As shown in Table 5, by selecting the highest average level as the optimal level of each design variable, the best configurations are T1=4 mm, T2=3 mm, T3=3 mm and T4=4 mm. A new simulation for the model with the best configurations is conducted and DRnew is obtained as 0.72, which is larger than the combinations in Table 4. Therefore, the optimal configurations are selected as the new middle levels for the next iteration. Table 4 The discrete values of design variables in the first iteration and corresponding results. Design variables Simulation number
1 2 3 4 5 6 7 8 9
Results ( A (T1)
B (T2)
C (T3)
D (T4)
2
2
2
2
0.03
2
3
3
3
0.60
2
4
4
4
0.06
3
2
3
4
0.69
3
3
4
2
0.00
3
4
2
3
0.58
4
2
4
3
0.06
4
3
2
4
0.64
4
4
3
2
0.58
Table 5 ANOM of the first iteration and the best level of the first iteration. 21
)
Levels Design variables
A (T1) B (T2) C (T3) D (T4)
Best levels
Best results
0.426
3
4 mm
0.413
0.406
2
3 mm
0.417
0.623
0.040
2
3 mm
0.205
0.412
0.463
3
4 mm
1
2
3
0.232
0.422
0.260
The iteration history of the optimisation is shown in Fig. 12. It can be seen that the design objective of the new level was not enhanced since the 6th iteration, and the optimisation procedure was thus terminated at the 11th iteration. An optimal design was reached with DRnew = 49.24 at the 6th iteration.
Fig. 12. Evolution history of the penalised objective, DRnew, in the discrete size optimisation.
Table 6 summarises the crashing performances of the optimised collapse zone, the referenced collapse zone [15] and the initial design without grooves, in which the optimal design is obtained with T1=5 mm, T2=5 mm, T3=2 mm and T4=4 mm (at iteration 6). It should be noted that only 80% of crushing distance is calculated for EA, SEA and ACF to have a fair comparison. Comparing with the currently-used collapse zone in [15], which has the same design space in this paper, the optimised collapse zone enhances SEA, ACF and EA by 32.12%, 25.75% and 25.75%, respectively. Meanwhile, the IPCF and Me are reduced by 34.07% and 4.83%, respectively. Comparing with the initial design (T1=3 mm, T2=3 mm, T3=3 mm and T4=3 mm), the optimised structure increases SEA by 58.04%. Meanwhile, the average crushing force is enhanced by 101.99%. Note that the ACF is positively related to the 22
energy absorption. Therefore, energy absorption is improved substantially, although the mass of optimal design also increases by 27.89%. Table 6 Comparison between the optimal collapse zone, the referenced collapse zone [15] and the initial design without grooves. EA 3
(mm)
(mm)
(mm)
4
(mm)
Lu et al. [15]
SEA
IPCF
ACF
Me
(J/g)
(kN)
(kN)
(kg)
(kJ)
7.44
1137.50
901.05
98.34
731.65
3
3
3
3
6.22
1510
560.95
73.18
455.18
Optimal design
5
5
2
4
9.83
750
1133.05
93.59
919.99
Increasing ratio
-
-
-
-
32.12%
-34.07%
25.75%
-4.83%
25.75%
-
-
-
-
58.04%
-50.33%
101.99%
27.89%
102.12
Initial design
(compared to the results in [15]) Increasing ratio
%
(compared to initial design)
To illustrate the improvement of crashworthiness performance after optimisation, the impact force versus crush displacement histories of these three structures are plotted in Fig. 13. In addition, the radar maps for each crashworthiness indicators are also presented in Fig. 14. It can be clearly seen that the collapse zone in [15] is entirely surrounded by the optimised design in the radar map. Therefore, it is concluded that the overall crashing performance of the optimised structure has been improved.
23
Fig. 13. The impact force-crush displacement curves of the initial and optimal design and the collapse zone in Lu et al. [15].
Fig. 14. Radar map of crashworthiness indicators for the optimal design and the collapse zone in Lu et al. [15].
It should be noted that the stiffness is not an objective function in detailed design because the detailed design concentrates on the crashing performance of the collapse zone. It only changes the thickness of components rather than the loading paths and the results show that the optimisation process leads to the increase of thickness. Nevertheless, the stiffness of the whole carbody (including the collapse zone) should be checked in a detailed design in order to satisfy related requirements.
5. Conclusions This paper presented an efficient design process for the collapse zone of railway vehicle from conceptual design to detailed design using optimisation methods. In the conceptual design, the importance of assigning dynamic weightings to concurrent static and crash loading cases has been proved since the crash load may be dominant in topology optimisation process if constant weighting factors are assigned. For detailed design, this study addressed a multi-objective discrete optimisation by adopting the desirability approach, thus, the optimal thickness distribution has been determined. The initial design domains (1/4 model) had the mass of 384.9 kg; topology optimisation in conceptual design reduced mass of the design domain to 118 kg; and the detailed design based on size optimisation further reduced the mass to 46.795 kg. Comparing with the currently used 24
collapse zone in Lu et al. [15], the optimised structure in this study enhanced the SEA, EA and ACF by 32.12%, 25.75% and 25.75%, respectively, while reduced the IPCF and Me by 34.07% and 4.83% respectively. Further, compared to the traditional design, the optimised structure derived from topology and size optimisations is more objective, reliable and economic.
Acknowledgments Authors would like to acknowledge the support of the China Scholarship Council (CSC) and The University of Manchester for the funding of the Ph.D work of C.X. Yang. (No. 201706370205). We would like to thank Professor Ping Xu from Central South University for suggestions and Imtiaz Gandikota from Livermore Software Technology Corporation for the technical support of LS-TaSC™. Conflict of interests We confirm there is no conflict of interests in this research and the submission of this paper to the International Journal of Mechanical Sciences.
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28
GRAPHICAL ABSTRACT
29
Structural optimisation for the collapse zone of a railway vehicle Chengxing Yang , Q.M. Li PII: DOI: Reference:
S0020-7403(19)31960-5 https://doi.org/10.1016/j.ijmecsci.2019.105201 MS 105201
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
31 May 2019 23 September 2019 29 September 2019
Please cite this article as: Chengxing Yang , Q.M. Li , Structural optimisation for the collapse zone of a railway vehicle, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105201
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Highlights Two stage optimal design with manufacture constrains was proposed to consider both static and crash loads for the collapse zone of a high-speed train; Dynamic weighting was used to balance the static and dynamic loading cases; The collapse zone designed in the paper demonstrated considerable improvements on energy absorption, crash force characteristics and weight saving.
1
Structural optimisation for the collapse zone of a railway vehicle Chengxing Yang, Q. M. Li a
*
Department of Mechanical, Aerospace and Civil Engineering, School of Engineering, The University of Manchester, Manchester M13 9PL, UK
ABSTRACT The collapse zone is a critical component of railway vehicle, which is designed with multiple functions, i.e., supporting neighbouring components and attached equipment in its normal operation and absorbing impact energy in a crash accident. However, there is a lack of systematic study on the design of the collapse zone based on the strategy of topological and size configurations. To make full use of the material, topology optimisation for conceptual design is conducted under concurrent static and crash loads. The importance of assigning dynamic weighting to each loading case has been proved to avoid the existence of a dominant case. The topological results show that the load transferring paths (i.e., the space where materials are distributed to) are in the longitudinal direction. In the detailed design, a revised model is created, which addresses structural and manufacturability concerns. To optimise the thickness assignment, the desirability approach is combined with the discrete optimisation method. The final design is obtained and compared with the currently-used collapse zone structure. The radar map of crashworthiness indicators shows that the overall crashing performance of the optimised structure is superior to the traditional one. Keywords: Collapse zone; Topology optimisation; Hybrid cellular automata (HCA); Multi-objective discrete optimisation; Crashworthiness
1. Introduction
*
Corresponding author.
E-mail address: [email protected] (Q.M. Li). 2
Structural crashworthiness of transportation vehicles (e.g., automotive, rail, ship, aerospace, etc.) has drawn increasing attention since it is highly related to occupant safety [1-4]. Considerable studies on impact energy absorber (IEA) have recently been performed for the passive safety of railway vehicles due to the rapid development of high-speed train [5-7] and new generation of subway [8,9]. IEA of rail vehicles can be classified into two categories, i.e., special energy-absorbing structure (SES) and bearing energy-absorbing structure (BES). The former is specially designed to act only when crash accidents happen while the latter has multiple roles of bearing static loads from connected neighbouring components in normal service condition and managing dynamic loads in crash accidents. For example, Peng et al. [10] introduced a duplex structure as a SES, which is composed of anti-climbing gears, square tubes with a diaphragm, an end plate, a supporting guide bar and energy-absorbing components. A five-cell thin-walled aluminium IEA for high-speed train was studied using hybrid optimisation approach [7]. With the consideration of structural bending strength, Yao et al. [11] optimised the cross-sectional shape of the guide of the structure developed in [12] to avoid the occurrence of instability. Xie et al. [5,13,14] designed two types of BES, which were composed of a thin-walled metal structure and aluminium honeycomb fillers. In [8,9,15], a gradual energy-absorbing structure in the collapse zone of a railway vehicle was proposed where the design started from an 1D collision analysis of two trains using a mass-spring model to calculate the impact energy absorbed by the collapse zone. Then, a 3D structure was designed based on numerical simulation and experimental validation. However, most of these studies on the collapse zone were based on predefined topological geometries, making it difficult to perform systematic investigations about the optimal material distribution to develop more efficient structures. Topology optimisation is an ideal method for conceptual design, which is able to derive an optimal topological structure by distributing materials to the most efficient load-path network. Due to this benefit, numerous studies have made contributions to the applications of topology optimisation. For railway vehicles, Aderiani et al. [16] investigated the structural optimisation of ER24PC locomotive carbody subjected to various static loads based on European standard EN12663. Kim et al. [17] performed the conceptual design for the leading-cab structure. The optimal structure was found with its mass reduced by 40% after topology optimisation. Mrzygł´od and Kuczek [18] dealt with a new uniform crashworthiness concept of carbody for a high-speed train. The optimisation problem was to minimise the mass of the survival space under stress constraint, i.e., the elastic limit of the material. Lee 3
et al. [19] investigated the structural-optimisation-based design process for the body of a railway vehicle made from extruded aluminium panels, in which topology optimisation is adopted for conceptual design while size optimisation is employed for detailed design. The mass of the passenger car was reduced by about 14% comparing with the mass of the existing passenger car. Nevertheless, these studies on railway vehicles only considered either static or crash loading cases, rather than both of them. There is a lack of systematic study on the design of the collapse zone considering both loading cases simultaneously in a practical application. As a type of BES, the collapse zone is expected to have satisfactory performance under multiple loading cases. In general, longitudinal dynamic compression and vertical static loading cases are the two most common loading types. However, the majority of the previous studies for the design of the collapse zone focused only on either the vertical static load [16] or the longitudinal dynamic compression load [15]. There are limited literatures studying topology optimisation under multiple loading cases [20]. To address this issue, topology optimisation under multiple loading cases will be conducted in LS-TaSC™ based on hybrid cellular automaton (HCA) algorithm [21]. HCA is a non-gradient approach that has been proved to be suitable for solving both linear problems such as structural stiffness and highly non-linear problems such as structural crashworthiness [22]. Patel [23] and Patel et al. [24] showed that a uniform distribution of the internal energy density (i.e., total elastic and plastic strain energy density) can lead to the optimised structures with favourable crashworthiness performance. Zeng and Duddeck [25] proposed an approach based on HCA for crashworthiness topology optimisation with a special focus on thin-walled structures. The thickness of each thin wall was treated as a design variable. Aulig et al. [22] proposed a preference-based scaled energy weighting HCA (SEW-HCA) approach to address the topology optimisation of both static and crash loading cases concurrently. The main idea is the use of weighting coefficient for each loading case. However, the weighting coefficients are constant and dependant on user’s preference. In this paper, the enhanced HCA algorithm with dynamic weightings proposed by Bandi et al. [26,27] will be employed to take both loading cases into consideration. The proposed collapse zone structure using optimisation methods will be based on a currently-used collapse zone structure in Lu et al. [15]. Size optimisation is an effective tool to find optimal configurations after conceptual design, which helps engineers to attain the optimal crashworthiness of IEA [14, 28-30]. For 4
example, Yin et al. [31-33] conducted size optimisation for W-beam guardrail, bio-inspired hierarchical honeycomb and functionally graded foam-filled structure. A comprehensive review was presented by Fang et al. [34], where the design optimisation methods for IEA have been systematically summarised. However, the traditional optimisation algorithms have been focusing on continuous variable problems. In real-life, many design variables can only be selected from a standard database such as thickness of commercially-available sheet materials, implying that the design space is restricted and discrete [29]. To address the optimal design in a discrete space, Lee et al. [35] proposed a discrete optimisation algorithm based on Taguchi method to select the best configurations, which was also adopted by Wu et al [36] and Sun et al. [29,37,38]. For the trade-off issue between conflicting objectives in discrete optimisation problem, Sun et al. [29] converted multiple indices into one unified cost function by coupling the grey relational analysis (GRA) with principal component analysis (PCA). In this study, the overall desirability function [39] is employed to combine multiple responses into a dimensionless measure of performance. The objectives of this research are 1) to perform the conceptual design of a real-world collapse zone based on a finite element (FE) model, using large-scale topology optimisation, and 2) to further improve the design in the subsequent detailed design stage, using size optimisation. It is expected that the outcome of this work will improve the effectiveness and efficiency of the design process for the collapse zone with considering the important performance metrics such as stiffness and crashworthiness in optimisation. Section 2 presents the optimisation problem statement for the entire design process of a collapse zone. Section 3 shows the topology optimisation task consisting of finite element modelling and topological results. In Section 4, detailed design is performed based on the combination of discrete optimisation method and the overall desirability approach. Finally, the paper is summarised and concluded in Section 5.
2. Statement of the optimisation problem For a discretised design space, the optimisation problem for the whole process of conceptual design and detailed design can be mathematically stated as
5
(
( ) ( )⁄
){
(
( ) ( )
)
{
(1)
{
where the objective of the conceptual design is to maximise the target function
( ), which
could be the global stiffness matrix K, the total energy absorption EA or their combination. The objective of the detail design is to have combined effect of maximum specific energy absorption (SEA) and minimum initial peak crushing force (IPCF). The multi-objective problem can be treated as a single-objective problem through applying weighting factors based on practical considerations. x is the vector of elemental relative density xi for topology optimisation and xmin is introduced to avoid numerical problems. If xi approaches to 1, the i-th element will be kept; while it will be deleted if xi approaches 0. vectors of nodal force and displacement, which formulate
( )
and .
are the and
( ) are
respectively the initial mass and the topology-optimised mass of the design domain. A constraint
is imposed on the mass fraction. Me is the mass of the structure after size
optimisation and M1 is its upper bound obtained in Lu et al. [15]. ACF indicates the average crushing force and ACF1 is its lower bound obtained in Lu et al. [15]. design variable for size optimisation.
and
bounds for the size design variables.
and
represents the j-th
are respectively the lower and upper are the total number of design variables
for topology and size optimisation, respectively.
3. Topology optimisation in concept design 3.1 Finite element model Collapse zone is defined as an area in the front-end of the underframe of a rail vehicle, illustrated in Fig. 1. It supports other structural components and internal equipment, and thus, bearing static loads in normal operation. While in crash accidents, the collapse zone is expected to collapse in a controlled manner and absorb the impact energy to protect the survival space where occupants stay in. Attached areas used for the installation of other systems are fixed throughout the topology optimisation process, and therefore, they are excluded from the design domain. To save the computational time without losing accuracy, a quarter (1/4) of the collapse zone is modelled with symmetry boundary conditions (see Fig. 1). The design domain is separated into two parts, i.e., domain A and domain B, by fixed 6
beams. Excluding the elements of non-designable areas, the total number of finite elements used as design variables for the topology optimisation is nearly 100,000. The material is high-strength steel modelled by elastic-plastic constitutive model, which is referred to Yang et al. [28]. But there was a typo of tangent modulus in Ref. [28], whose unit is corrected to be MPa rather than GPa. The elastic-plastic material is modelled in Ls-Dyna® by using “*MAT_024_PIECEWISE_LINEAR_PLASTICITY”. The total mass of the design domains is 384.9 kg, excluding the non-designable areas.
Fig. 1. The collapse zone of a rail vehicle: (a) a typical rail vehicle showing the installation; (b) the framework; (c) the ½ model; (d) the ¼ model illustrating non-designable areas and design domains.
Current studies on topology optimisation for the collapse zone design mainly concentrate just on its stiffness for static load or energy absorption for crash load. But the topological results may change when considering both static and crash loads simultaneously, which will be dealt with in this study. The distributed weight Fw is simplified to two concentrated static forces (F1 and F2) acting on the centre of gravity of two design domains, shown in Fig. 2(a). Referring to the standard JIS E7106:2011 and Xu et al. [8], F1 and F2 are roughly calculated as 2945.33 N. Under this condition, one end of the supporting beam is fixed without motion. The translational motion in y-direction and the rotational motion in xand z-directions of the symmetry plane are also fixed to define the symmetry constraint. For the crash case shown in Fig. 2(b), the front-end of the collapse zone is impacted by the rigid wall with a mass of 13.825 tons [8] at an initial velocity of 20 m/s, while the rear-end was 7
restrained to the rigid base. Attention was paid to ensure that the static loading case remains in the elastic portion, while the crash loading case can go beyond plastic yield. The implicit and explicit solvers of Ls-Dyna® are used for the analyses of static and crash loads, respectively.
(a)
(b)
Fig. 2. Illustration of mechanical loads and constraints: (a) static loads in normal operation; (b) crash load in accident condition.
3.2 Optimisation method The solid isotropic material with penalization (SIMP) model [40,41] is a well-known interpolation scheme that heuristically relates the relative density to the elastic modulus of material. Patel [23] developed a continuous material model based on SIMP method by relating the relative density to the yield stress and the strain hardening modulus. Therefore, an elastic-plastic material model parameterised by the relative density is expressed as, ( )
(2)
( )
(3)
( )
(4)
( )
(5)
where ?? denotes the density of the material,
is the yield stress, ?? represents the
Young’s modulus and ??ℎ is the strain hardening modulus. They are variables in optimisation process with regard to x. The subscript 0 refers to the base material properties. p and q are penalisation exponents and both of them are equal to 1 in HCA algorithm [23,42]. The design variable x with 0
Based on the elastic-static assumption, topology optimisation for maximising stiffness is treated as minimising elastic strain energy (U), i.e., ( ) ( ) ( )⁄
( ) (6)
{ where U(x) is the elastic strain energy of the structure, represents the volume of i-th element.
( )
∑
( )
and
is the total number of design variables (around
100,000). The maximisation of the energy absorption (EA) can be formulated by ( ) ( )⁄
{
(
) (7)
where EA(x, t= td) is the internal energy of the structure at the final time step td in finite element analysis. The field variables (e.g., stress, strain, strain energy density (i.e., elastic strain energy density), internal energy density or functions of these quantities) of the loading cases can be combined by linear weighting factors when multiple loading cases are considered [23,26], i.e., ∑
(8)
where Si is the field variable of the i-th element, Sil is the field variable of the i-th element in l-th loading case, wl is the weighting factor of the l-th loading case and nl denotes the total number of loading cases. In a more specific expression, energy density of the i-th element in l-th static loading case;
( ) is the elastic strain (
) is the
internal energy density of the i-th element in l-th crash loading case at the final time step of t= . To link the objective functions in Eqs. (6) and (7) and the field variables in Eq. (8), formulations are given as follows: ( )
( )
∑
(
( )
( ))
(9)
9
( )
(
)
∑
(
( )
(
))
(10)
where I denotes the iteration step. In order to avoid the dominance by a single loading case, a dynamic weighting method is adopted. The goal is to achieve balanced performance for all loading cases, and thus, the average of the performance measure (PM), i.e., stiffness or energy absorption, for all loading cases is used as the target [26,27], ( )
where ( )
( )
∑
(11)
( )
is the performance measure of the l-th loading case at I-th iteration and
represents the average of the performance measure for all loading cases at I-th
iteration. From Eq. (11), the increment ( )
( )
( )
is computed as
( )
(12)
( )
where
( )
is a scale factor, which equals 0.1 in this study [42]. An upper limit of
is placed on
( )
to ensure convergence in a reasonable number of iterations. The weight
for the first loading case can be fixed as unity and the loading case weight ( variables can then be rewritten as (1,
, ⋯,
,
, ⋯,
)
) because it is the relative ratios of weights
that are important, rather than their absolute values [42]. Other details about HCA method can be found in Tover et al. [43], Patel [23], Bandi [26,27] and LS-TaSC™ [42].
3.3 Topological results The topology optimisation is carried out in LS-TaSC™ by integrating with Ls-Dyna®. To validate the importance of dynamic weighting method, the weighting factors in Eq. (8) are changed to consider the different loading conditions: the static loading case (Case I), the crash loading case (Case II), the multiple loading cases with equal constant weightings (Case III) and the multiple loading cases with dynamic weightings (Case IV). Case I:
,
10
Case II:
,
Case III:
,
Case IV:
,
(
)
For all four cases, the target mass fraction
is defined as 30%. Histories of mass
redistribution and weighting factors, and optimised structures of the final iteration are shown in Figs. 3-5, respectively. The mass redistribution curves illustrate the fraction of the total mass of the design domain that has been redistributed in each iteration. All four curves approach to zero showing the convergence of optimisation. Case I with sole static loads converges at I = 26, while other cases with crash load included converge at I 40. The main reason is that the explicit dynamic behaviour involving high nonlinearities consumes more computing time than that consumed for an implicit linear problem. The topological structure of Case I is very different from that derived from Case II, which proves the importance of the concurrent design of the collapse zone. In Case I, the upper and lower surfaces are kept for supporting static loads and the middle part is a truss-like frame. However, the middle part in Case II contains more materials to resist the longitudinal impact. It is interesting to point out that the result of Case III is the same as that of Case II, which reveals that the crash load is dominant and the static loads are neglected in this concurrent optimisation with equal constant weighting factors. However, the topological structure derived from Case IV shows the combined characteristics of both static and crash loads (see Fig. 5(d)). As plotted in Fig. 4, the weighting factor
of static loads varies during optimisation process. But it is higher
than unit to avoid the dominance of crash load, thus, promoting the importance of static loads.
11
Fig. 3. Histories of the mass redistribution, i.e., the fraction of the total mass of the design domain that has been redistributed during iteration, in Case I~IV.
Fig. 4. Histories of weighting factors,
12
and
, in Case IV.
(a)
(b)
(c)
(d) Fig. 5. Topological results with enlarged views of design domains: (a) Case I; (b) Case II; (c) Case III; (d) Case IV.
13
Manufacturability control is a critical parameter affecting the optimised result. The topological structures of the above four cases in Fig. 5 need additive manufacturing, which is effective but also costly for such large scale metal structure. There are a number of manufacturing processes used for mass production in automotive industry, in which extrusion is particularly favourable because of its low cost and fast processing time. This can be considered and implemented in topology optimisation as Case V, which is based on Case IV, but with further definition of extrusion constraint. The history of mass redistribution shown in Fig. 6 converges at I =28, which reveals that less computing time is needed with the definition of extrusion constraint. The weighting factor
increases during optimisation
process to increase the importance of static case (see Fig. 7). The topological structure shown in Fig. 8 presents much clearer load paths than the results in Fig. 5. Materials are mainly longitudinally-aligned although a few branches are formed, which are the most direct and efficient paths for force transmission. Obviously, this structure can be easily manufactured using traditional method rather than the costly additive manufacture. The total mass of the design spaces including design domain A and design domain B decreases from 384.9 kg to 118 kg with further remove of 69.34% materials. It should be noted that the HCA algorithm may not guarantee an overall maximum of energy absorption although the uniform internal energy density (IED), as the objective in HCA algorithm, can lead to a topological result with more uniform distribution of energy absorption in the material [34,44].
Fig. 6. History of the mass redistribution, i.e., the fraction of the total mass of the design domain that has been redistributed during iteration, in Case V.
14
Fig. 7. Histories of weighting factors,
and
, in Case V.
Fig. 8. Topological result with enlarged views of design domains in Case V.
4. Multi-objective discrete optimisation in detailed design 4.1 Finite element model A careful review of the topological structure in Case V revealed three design concerns: (i) disconnected structural components; (ii) rough side surfaces; (iii) very thin plates. Disconnected parts obviously cannot carry mechanical loads. Roughness of the surface causes stress concentration and crack initiation. Very thin elements are prone to fractures and buckling, and causes manufacture difficulty (e.g., welding) [45]. In order to address these concerns, each part of the optimal solution is carefully examined and a revised collapse zone is rebuilt, as shown in Fig. 9. The extruded solution is modelled using thin-walled tubes with prefabricated grooves because thin-walled structures can usually support loads more efficiently with minimum mass than bulky structures. The symmetric 15
half of the actual structure is shown in Fig. 10, which will be adopted for size optimisation in next sub-sections. The FE model is composed of an outer beam, inner beam, domain A and domain B with thicknesses marked as T1, T2, T3 and T4, respectively. The rigid wall moves towards the specimen at an initial velocity of 10 m/s and the impacting mass is 27.65 tons [28]. The automatic single-surface contact algorithm is chosen to simulate the self-contact of the whole FE model. Other detailed definitions of the FE model are the same as those in [15,28].
Fig. 9. Re-interpretation of the optimal result in Case V: (a) the topology-optimised structure; (b) the reinterpreted structure.
Fig. 10. Half of the reconstructed optimal collapse zone structure.
4.2 Optimisation method 16
The overall flowchart of discrete size optimisation algorithm [35] in Stage II is illustrated in Fig. 11.
Fig. 11. The overall flowchart of discrete size optimisation algorithm [35].
The detailed steps of the algorithm are further depicted as follows: (1) The 1st step: optimisation problem definition. [
( ) ( ) ( )
( )] (13)
{ where SEA means specific energy absorption, which is the energy absorbed per the mass of the structure. IPCF is the negative initial or first peak crushing force.
and
equal
901.05 kN and 98.34 kg, respectively, which are derived from [15]. The number of design variables
is 4 and the corresponding variable intervals are 1 mm, 2 mm, 3 mm, 4 mm, 5
mm, 6 mm, as tabulated in Table 1. 17
Table 1 The discrete values of design variables. Design variables
1
2
3
4
5
6
T1 (mm)
1.00
2.00
3.00
4.00
5.00
6.00
T2 (mm)
1.00
2.00
3.00
4.00
5.00
6.00
T3 (mm)
1.00
2.00
3.00
4.00
5.00
6.00
T4 (mm)
1.00
2.00
3.00
4.00
5.00
6.00
The desirability method [39] is adopted here to combine multiple responses into a dimensionless measure of performance. √
(14)
(
( )
(15)
( )
( where
)
,
) and
and IPCF, respectively.
(16) ,
and
represent the lower and upper bounds on SEA are weighting factors of SEA and IPCF, respectively. In
this study, equal importance is assigned to both indicators. (2) The 2nd step: design of experiments (DOE). The minimum orthogonal array is selected for the design of experiments according to the number of design variables and the number of design levels. In this research, the L9(34) orthogonal array (see Table 2) is chosen because there are four variables with three levels assigned to each variable. In each iteration, nine simulations are conducted. Table 2 The orthogonal array L9(34). Design variables Simulation number
Results (DR) A (T1)
B (T2)
C (T3)
D (T4)
1
1
1
1
1
DR1
2
1
2
2
2
DR2
3
1
3
3
3
DR3
18
4
2
1
2
3
DR4
5
2
2
3
1
DR5
6
2
3
1
2
DR6
7
3
1
3
2
DR7
8
3
2
1
3
DR8
9
3
3
2
1
DR9
(3) The 3rd step: selection of level values for design variables. Real discrete values of design variables are placed to the columns of the orthogonal array chosen in the second step. Levels of each variable should be sequenced according to their relative importance. In the initial design, an arbitrary discrete value can be selected, which is defined as the second level. In general, the discrete value with one step greater than the second level is assigned to the first level, while the discrete value of one step smaller is assigned to the third level. However, if the initial design has the smallest candidate values, it must be assigned to the first level; on the other hand, if the initial design has the largest candidate values, it must be assigned to the third level. Then, two successively increased or decreased discrete values are assigned to the second and third level or second and first level, respectively. Table 3 illustrates the arrangements of level values for various design variables. Table 3 The arrangement of level values for design variables. Levels Design variables 1 ( 𝐵 (𝐵 ( (
𝐵
3
4
5
6)
𝐵3
𝐵4
𝐵5
𝐵6 )
3
4
5
6)
3
4
5
𝐵
6)
(4) The 4th step: calculation of the penalised objective function.
19
2
3
3
4
𝐵3
𝐵4
3
4
3
4
To convert a constraint problem to a non-constraint problem, a penalised objective is formulated as, ( ) ( )
∑
[
]
(18)
( ) is the penalty function,
where and
(17)
is the maximum violation of the l-th constraint,
is the scale factor of the l-th constraint. The scale factor should be carefully selected
to avoid neglecting the effect of objective functions and causing infeasible design. In this research, the scale factor is set to a value so that the order of penalty function is one-order larger than that of the original objective function [35]. Specifically, the penalty functions for the average crushing force and structural mass are formulated as, ( )
[
( )
]
[
( )
]
(19)
where r1 and r2 are respectively given as 0.1 and 2 in this study. The optimisation problem in Eq. (13) can be finally rewritten as, ( )
3
{
4
[ [ [ [
( ) ] ] ] ]
(20)
(5) The 5th step: analysis of means (ANOM) and selection of optimal design. Analysis of means (ANOM) is calculated using the results (
~
) of nine
rows obtained from experiments or simulations in Table 2, which is conducted to determine the optimal configurations. Taking the first level of design variable A (A1) as an example, the main effect of A1 is calculated as follow, (
)
3
(
3)
(21)
It should be noted that results derived from the optimal configurations should be compared with the best combination from the matrix experiments. The best level for each design variable is selected as the new level of that variable in the next iteration. (6) The 6th step: convergence criteria. 20
For the successive Taguchi method employed, two convergence criteria are used for the termination, i.e., (i) the iteration number, at which
converges, is larger than
five; (ii) the iteration number reaches the maximum value pre-defined by the designer [38]. Otherwise, the design process goes back to the 3rd step if none of the convergence criteria mentioned above is satisfied. In this new circle, the second levels as design variables are replaced by the former optimal levels.
4.3 Optimisation results To enlarge the searching space in discrete design space, the present optimisation is started from the intermediate value of each design variable. Table 4 shows the model parameters and DRnew at the first iteration. ANOM is employed to determine the optimal levels. As shown in Table 5, by selecting the highest average level as the optimal level of each design variable, the best configurations are T1=4 mm, T2=3 mm, T3=3 mm and T4=4 mm. A new simulation for the model with the best configurations is conducted and DRnew is obtained as 0.72, which is larger than the combinations in Table 4. Therefore, the optimal configurations are selected as the new middle levels for the next iteration. Table 4 The discrete values of design variables in the first iteration and corresponding results. Design variables Simulation number
1 2 3 4 5 6 7 8 9
Results ( A (T1)
B (T2)
C (T3)
D (T4)
2
2
2
2
0.03
2
3
3
3
0.60
2
4
4
4
0.06
3
2
3
4
0.69
3
3
4
2
0.00
3
4
2
3
0.58
4
2
4
3
0.06
4
3
2
4
0.64
4
4
3
2
0.58
Table 5 ANOM of the first iteration and the best level of the first iteration. 21
)
Levels Design variables
A (T1) B (T2) C (T3) D (T4)
Best levels
Best results
0.426
3
4 mm
0.413
0.406
2
3 mm
0.417
0.623
0.040
2
3 mm
0.205
0.412
0.463
3
4 mm
1
2
3
0.232
0.422
0.260
The iteration history of the optimisation is shown in Fig. 12. It can be seen that the design objective of the new level was not enhanced since the 6th iteration, and the optimisation procedure was thus terminated at the 11th iteration. An optimal design was reached with DRnew = 49.24 at the 6th iteration.
Fig. 12. Evolution history of the penalised objective, DRnew, in the discrete size optimisation.
Table 6 summarises the crashing performances of the optimised collapse zone, the referenced collapse zone [15] and the initial design without grooves, in which the optimal design is obtained with T1=5 mm, T2=5 mm, T3=2 mm and T4=4 mm (at iteration 6). It should be noted that only 80% of crushing distance is calculated for EA, SEA and ACF to have a fair comparison. Comparing with the currently-used collapse zone in [15], which has the same design space in this paper, the optimised collapse zone enhances SEA, ACF and EA by 32.12%, 25.75% and 25.75%, respectively. Meanwhile, the IPCF and Me are reduced by 34.07% and 4.83%, respectively. Comparing with the initial design (T1=3 mm, T2=3 mm, T3=3 mm and T4=3 mm), the optimised structure increases SEA by 58.04%. Meanwhile, the average crushing force is enhanced by 101.99%. Note that the ACF is positively related to the 22
energy absorption. Therefore, energy absorption is improved substantially, although the mass of optimal design also increases by 27.89%. Table 6 Comparison between the optimal collapse zone, the referenced collapse zone [15] and the initial design without grooves. EA 3
(mm)
(mm)
(mm)
4
(mm)
Lu et al. [15]
SEA
IPCF
ACF
Me
(J/g)
(kN)
(kN)
(kg)
(kJ)
7.44
1137.50
901.05
98.34
731.65
3
3
3
3
6.22
1510
560.95
73.18
455.18
Optimal design
5
5
2
4
9.83
750
1133.05
93.59
919.99
Increasing ratio
-
-
-
-
32.12%
-34.07%
25.75%
-4.83%
25.75%
-
-
-
-
58.04%
-50.33%
101.99%
27.89%
102.12
Initial design
(compared to the results in [15]) Increasing ratio
%
(compared to initial design)
To illustrate the improvement of crashworthiness performance after optimisation, the impact force versus crush displacement histories of these three structures are plotted in Fig. 13. In addition, the radar maps for each crashworthiness indicators are also presented in Fig. 14. It can be clearly seen that the collapse zone in [15] is entirely surrounded by the optimised design in the radar map. Therefore, it is concluded that the overall crashing performance of the optimised structure has been improved.
23
Fig. 13. The impact force-crush displacement curves of the initial and optimal design and the collapse zone in Lu et al. [15].
Fig. 14. Radar map of crashworthiness indicators for the optimal design and the collapse zone in Lu et al. [15].
It should be noted that the stiffness is not an objective function in detailed design because the detailed design concentrates on the crashing performance of the collapse zone. It only changes the thickness of components rather than the loading paths and the results show that the optimisation process leads to the increase of thickness. Nevertheless, the stiffness of the whole carbody (including the collapse zone) should be checked in a detailed design in order to satisfy related requirements.
5. Conclusions This paper presented an efficient design process for the collapse zone of railway vehicle from conceptual design to detailed design using optimisation methods. In the conceptual design, the importance of assigning dynamic weightings to concurrent static and crash loading cases has been proved since the crash load may be dominant in topology optimisation process if constant weighting factors are assigned. For detailed design, this study addressed a multi-objective discrete optimisation by adopting the desirability approach, thus, the optimal thickness distribution has been determined. The initial design domains (1/4 model) had the mass of 384.9 kg; topology optimisation in conceptual design reduced mass of the design domain to 118 kg; and the detailed design based on size optimisation further reduced the mass to 46.795 kg. Comparing with the currently used 24
collapse zone in Lu et al. [15], the optimised structure in this study enhanced the SEA, EA and ACF by 32.12%, 25.75% and 25.75%, respectively, while reduced the IPCF and Me by 34.07% and 4.83% respectively. Further, compared to the traditional design, the optimised structure derived from topology and size optimisations is more objective, reliable and economic.
Acknowledgments Authors would like to acknowledge the support of the China Scholarship Council (CSC) and The University of Manchester for the funding of the Ph.D work of C.X. Yang. (No. 201706370205). We would like to thank Professor Ping Xu from Central South University for suggestions and Imtiaz Gandikota from Livermore Software Technology Corporation for the technical support of LS-TaSC™. Conflict of interests We confirm there is no conflict of interests in this research and the submission of this paper to the International Journal of Mechanical Sciences.
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GRAPHICAL ABSTRACT
29