1 Introduction

Passenger safety is one of the most essential design elements in automotive industry (in particular, for busses and coaches) and gains more importance day by day. To protect driver and co-driver for a bus or coach in case of a collision, engineers study how to build crashworthy vehicles. Energy-absorbing elements are among the main structures used in vehicle design to achieve this goal. These structures absorb crash energy by transforming it into strain energy, while they deform. Due to huge occurrence ratio of frontal crashes among all, bumper systems are the most important and most studied energy-absorbing structures in vehicle design.

Crashboxes are the main energy-absorbing components of the structure, while the bumper beam has more rigid structure to transmit crash energy to the crashboxes in various crash scenarios with a less energy-absorbing capability. Therefore, substantial number of studies is performed on crashboxes with various shapes, such as cylindrical straight tubes [1,2,3], square extrusions [4,5,6] and multi-cell cross-sectioned tubes [7,8,9,10,11]. It is found that these structures are efficient energy absorbers that have capability to crush and fold stably. Many researchers studied on tapered circular tubes [12,13,14] and tapered rectangular tubes [15,16,17]. They showed the energy absorption advantages of tapered tubes under axial impact. In many studies honeycomb [18,19,20,21,22] and foam [18, 23,24,25,26] filling of these thin-walled tubes are investigated. These studies show that honeycomb structures and aluminum foams are significantly improved the crash performance of crashboxes.

There exist many studies on shape design of bumper beams to increase their crash performance [27,28,29,30,31,32,33,34]. These studies show that energy-absorbing capability of a bumper can be improved by using different shapes and optimizing them. Li et al. [35] studied optimization of foam-filled bumper beams and found that foam filling increases the energy absorption capability of a bumper. Jacob and Arunkumar [36] determined that foam and honeycomb incorporated bumpers absorb significantly more energy compared to a hollow steel bumper.

Finite element (FE) simulations are crucial design tools for engineers to design crashworthy vehicles. In order to obtain realistic results, high degrees of fidelity and robust simulations must be performed. However, this ability to design competitive products comes with a cost. The higher the fidelity of the FE model is, the higher the computational cost. A full FE crash simulation of a vehicle takes hours to complete. Also, many simulations must be performed in order to optimize a multi-variable design depending on the number of variables and the expected precision. Combination of these two may induce days of FE analysis and eventually leads to increased design time and cost.

However, there is an optimization concept called multi-fidelity modeling that gains popularity as a remedy for the computational burden [37,38,39,40,41,42,43,44]. This method essentially combines different fidelity FE model results to obtain high accuracy with significantly improved computational time. After creating a low-fidelity model complementary to the high-fidelity model, few high-fidelity analyses are performed at some design points and a function is used to calculate the offset between high-fidelity and low-fidelity results. Then, optimization runs are performed on low-fidelity model and predicted high-fidelity results are calculated using the offset function. This method helps to decrease computational cost by allowing designers to perform all optimization analyses using computationally inexpensive low-fidelity model with few costly high-fidelity simulations instead of running all simulations on high-fidelity FE model. It should be noted that the multi-fidelity approach used in this study is similar to the space mapping approach, which has been used in various applications including crashworthiness optimization [45], friction stir welding [46], fluid–structure interaction [47], airfoil shape optimization [48] and others [49, 50].

In this study, the effects of various geometrical parameters such as length of the crashboxes, wall angle and wall thickness of honeycomb on crash performance of a honeycomb filled bumper are investigated. A commercial finite element (FE) analysis software LS-DYNA [51] is used to simulate crash behavior of the bumper system under impact conditions of ECE R-29 tests [52]. The constructed finite element model is validated by using experimental results available in the literature. The optimum values of geometrical variables are obtained through multi-objective optimization by maximizing a composite objective function that provides a compromised value between crush force efficiency and specific energy absorption.

This paper is structured as follows. The next section provides the problem description for the optimization of honeycomb structure filled bumper system. Section 2 presents the details of finite element model and its validation. Section 4 explains multi-fidelity optimization concept. Section 5 describes construction of surrogate models and their accuracies. Section 6 discusses high-fidelity, low-fidelity and multi-fidelity optimization results. The paper is concluded with some remarks given in Sect. 7.

2 The bumper system

Description of the problem of interest, the crash metrics used and the formulation of the optimization problem are explained in the following sub-sections.

2.1 Problem description

The original study of this paper stems from the occupant safety systems for busses. The existing safety regulation for frontal impact for trucks is ECE R-29, and it is adapted for safety of driver and co-driver of busses [53]. The pendulum test according to ECE R-29 regulation is shown in Fig. 1. A bumper system having honeycomb structures inside the bumper beam is analyzed in this study. The generic model of the bumper system is shown in Fig. 2.

Fig. 1
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Pendulum test according to ECE R-29 regulation

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Fig. 2
figure 2

Bumper system

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All designs have the same bumper beam with length of 1112 mm, height of 129 mm, width of 65 mm and wall thickness of 1 mm. Two crashboxes in the model have the following dimensions, height of 120 mm, width of 80 mm, wall thickness of 1.6 mm and length of L (see Fig. 3).

Fig. 3
figure 3

Dimensions of the bumper system

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Bumper beam is filled with two pieces of honeycomb structures in front of each crashbox with length of 200 mm, height of 128 mm, width of 64 mm, cell edge length of 20 mm, angle of cell walls \(\theta\) and wall thickness of honeycomb t (see Fig. 4).

Fig. 4
figure 4

Dimensions of the honeycomb structures

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Examples of four different honeycomb structures with different cell wall angle used in FE models are shown in Fig. 5. In crash performance optimization for the bumper, three dimensions are chosen as design variables: (i) the length of the crashboxes L, (ii) the angle of cell walls \(\theta\) and (iii) the wall thickness of honeycomb t.

Fig. 5
figure 5

Honeycomb with various design with different \(\theta\)

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For the crash performance evaluation of the bumper system, the following design problem is considered. The bumper system is assembled at the front end of a generic heavy vehicle chassis, while the chassis is attached to a stiff fixture. Then, the whole assembly is placed in front of a rigid pendulum. Bumper system is impacted with the 1500-kg pendulum that has initial kinetic energy of 45 kJ in accordance with United Nations’ ECE R-29 safety requirements for heavy commercial vehicles [52].

2.2 Crash performance metrics

Total energy (\(E_t\)) absorbed by the structure is defined as work done by the crushing force P over the deformation distance d (the maximum deformation before elastic rebound).

$$\begin{aligned} {E_t}=\int \limits _{0}^{d}{P\,dx} \end{aligned}$$
(1)

Mean crush force (MCF) for a deformation is defined as the total energy absorbed by the structure divided by the deformation distance, d as:

$$\begin{aligned} \text {MCF}=\frac{E_t}{d} \end{aligned}$$
(2)

Crush force efficiency (CFE) is defined as the ratio of the mean crush force (MCF) to the peak crush force (PCF) to evaluate the efficiency of an energy absorber.

$$\begin{aligned} \text {CFE}=\frac{{\text {MCF}}}{{\text {PCF}}} \end{aligned}$$
(3)

where we used SAE-1000 filter in force–displacement curve while determining the PCF.

Specific energy absorption (SEA) is calculated as the total energy absorbed (\(E_t\)) divided by the mass of the absorber structure (m), and it is calculated as:

$$\begin{aligned} \text {SEA}=\frac{E_t}{m} \end{aligned}$$
(4)

2.3 Formulation of the optimization problem

In this study, crashworthiness of the bumper system is evaluated by using CFE and SEA, so that the bumper system is designed to maximize these two metrics. As noted earlier, three design variables are chosen: (1) length of the crashboxes, L, (2) wall angle of honeycomb cells, \(\theta\) and (3) wall thickness of honeycomb cells, t. Thus, the optimization problem can be stated as:

$$\begin{aligned} \begin{array}{lll} \text {Min}& -f \\ \text {S.t.}& 80\text { mm}\le L\le 160\text { mm} \\& 30^{\circ} \le \theta \le 75^{\circ} \\& 0.25\text { mm}\le t\le 0.75\text { mm} \end{array} \end{aligned}$$
(5)

Here f is a composite objective function that can provide a compromise between SEA and CFE. The composite objective function can be defined as:

$$\begin{aligned} f=w\frac{{\text {CFE}}}{{\text {CFE}}_0}+(1-w)\frac{{\text {SEA}}}{{\text {SEA}}_0} \end{aligned}$$
(6)

where w is the weight factor to determine the importance of the metrics relative to each other. For this study weight factor is selected as 0.5 because both metrics are decided to be equally important. CFE\(_0\) and SEA\(_0\) are the normalization constants for CFE and SEA. Normalization constants are taken as maximum values of CFE and SEA at training points (see Sect. 5.2).

The optimization problem defined above is solved by using “ga” built-in function of MATLAB that uses genetic algorithm [54]. The population size is taken as 100, the elite count is taken 6, the crossover fraction is taken 80%, the maximum number of generations is taken 300, and remaining algorithm parameters are taken the default values in MATLAB.

3 Finite element modeling

Finite element method is used in this study to calculate crash behavior of the bumper system. In this section finite element (FE) model and validation of the FE model will be explained.

3.1 Validation of the FE Model of the Honeycomb Structure

The validation study is based on the experimental study by Zhang et al. [55], where a honeycomb structure is crushed with a rigid wall. The FE model prepared in this study is shown in Fig. 6 next to the experimental model of Zhang et al. [55].

Fig. 6
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Experimental setup of Zhang et al. [55] on the left, and FE validation model on the right

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Before the validation runs of the FE model, a mesh convergence study is performed on the base model. Mesh convergence decision is based on the mean crush force value. MCF, CFE and SEA comparison graphs of FE models having 0.5 mm, 1 mm, 1.5 mm, 1.75 mm, 2 mm and 2.5 mm mesh sizes are shown in Fig. 7. It is seen that the MCF, CFE and SEA values settle as mesh size decreases. From Fig. 7, the mesh size is determined as 0.5 mm to be used for all validation runs.

Fig. 7
figure 7

Mesh convergence study for the FE model of the honeycomb structure

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In this study, \(3 \times 3\) and \(5 \times 5\) honeycomb cell configurations are used for honeycombs as given in Zhang et al. [55]. For both \(3 \times 3\) and \(5 \times 5\) cell configurations, the height of honeycombs is taken as 100mm, honeycomb wall thickness is taken as 0.075 mm, and the central angle \(\alpha\) is taken as \(120^{\circ}\). Wall thickness of the honeycombs is doubled where two metal sheets are glued together because of the production method of honeycombs as shown in Fig. 8.

Fig. 8
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Contacts and double-layer surfaces of FE validation model

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For the contact algorithm, ”AUTOMATIC_SINGLE_SURFACE_CONTACT” is defined between the honeycomb and the fixed wall, ”AUTOMATIC_SURFACE _TO_SURFACE_CONTACT” between the honeycomb structure and the moving wall as shown in Fig. 6. Static and dynamic friction coefficients are taken as 0.3 and 0.2, respectively. Belytschko–Tsay 4-noded shell element type is used as element type. To simulate AA3003 H18 aluminum foil material in the reference experimental study, ”MAT_24_PIECEWISE_LINEAR_PLASTIC” material definition is used. This material model is widely used in crash simulations related to automotive industry [13, 51].

As a result of this validation study, load–displacement graphs for \(3 \times 3\) and \(5 \times 5\) cell configurations shown in Fig. 9 are obtained. Along with the load–displacement curves, energy absorption values of honeycomb structures are also calculated and compared with the reference experimental results. Note that Figure 7 of Zhang et al. [55] is digitized by using GetData Graph Digitizer, and the corresponding force and displacement values are extracted. Then, the area under force–displacement curve is obtained by using OriginPro graphical software to calculate the energy absorption. Energy absorption of FE analyses and experimental results are tabulated in Table 1. Comparison of experimental and numerical collapsed models with \(3 \times 3\) and \(5 \times 5\) cell configurations is shown in Fig. 10. It is seen that the FE results are in good agreement with the experimental results of Zhang et al. [55].

Table 1 Energy absorption comparison of FE analyses and experimental results of Zhang et al. [55]
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Fig. 9
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\(3 \times 3\) cell configuration (on the left), \(5 \times 5\) cell configuration (on the right). Load–displacement graph comparisons of FE analyses and experimental results of Zhang et al. [55]

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Fig. 10
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Comparison of a \(3 \times 3\), and b \(5 \times 5\) cell configuration experimental result of Zhang et al. [55] and numerical results

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3.2 Description of the FE model

The response of bumper systems under impact loading is predicted by using nonlinear, explicit finite element software LS-DYNA [51]. The finite element model is prepared as shown in Fig. 11. Bumper system is attached to a generic heavy vehicle chassis (see Sect. 2.1 for the detailed explanation of the bumper system). For the contact algorithm ”TIED_SURFACE_TO_SURFACE_CONTACT” is defined between endplates of crashboxes and chassis. Then, this assembly is attached to a fixture that is designed to hold the position of attached test sample and withstand crushing force of the pendulum. Again, ”TIED_SURFACE_TO_SURFACE _CONTACT” is defined between endplates of chassis and the front surface of fixture. Next, the whole unit is placed in front of a 1500-kg pendulum modeled according to ECE R-29 test standards [52]. Bottom surfaces of the fixture that touches the ground are fixed to maintain the position of the assembly during impact. All degrees of freedom of nodes that form the axis of pendulum are also fixed except rotation around y-axis, which allows pendulum to rotate (see Fig. 11). ”AUTOMATIC_SINGLE_SURFACE_CONTACT” is defined as containing all surfaces forming bumper system to prevent interference between the surfaces. For the contact algorithm ”AUTOMATIC_SURFACE_TO_SURFACE_CONTACT” is defined between pendulum and bumper system. Static and dynamic friction coefficients are taken as 0.3 and 0.2, respectively. The reader is referred to [56] for the details of the finite element analysis of the pendulum test setup.

Fig. 11
figure 11

Finite element setup

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Belytschko–Tsay 4-noded shell element type is used to generate elements. Materials of parts are determined as follows: ST44 steel for the fixture, DP1300 steel for the bumper beam, DP600 steel for the crashboxes, DP780 steel for the chassis and AA303 H18 aluminum for the honeycomb structures. To simulate these materials in FE model, ”MAT_24_PIECEWISE_LINEAR_ PLASTIC” material definition is used. ”MAT_20_RIGID” material definition is used for the pendulum.

Crash energy of the pendulum is provided via defining an initial angular velocity and inertia around y-axis. Inertia of the pendulum around y-axis is taken as \(I_{yy}=18.38\times {10}^6\) t\(\times {\text {mm}^2}\), and angular velocity around y-axis is taken as \(\omega =2.21\) rad/s to provide 45 kJ crash energy. These values are taken from a previous study performed by Guler et al. [56]. Note that the crash energy requirement of 45 kJ was increased to 55 kJ in the latest version of ECE R-29.

Two-step mesh convergence study is performed to determine the mesh sizes of the bumper system components. In the first step, FE models having 3, 4, 5 and 6 mm mesh size for the bumper beam and crashboxes are analyzed. Mean crush force comparison of FE models is shown in Fig. 12. It is seen that the mean crush force value settles as the mesh size decreases. From Fig. 12, the mesh size is determined as 4 mm for bumper beam and crashboxes.

Fig. 12
figure 12

Mean crush force comparison for different mesh sizes of bumper beam and crashboxes

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In the second step, FE models having 2.5, 3.0, 3.5, 4.0 and 5.0 mm mesh size for honeycomb structures are analyzed. Mean crush force comparison of FE models is shown in Fig. 13. Using the similar approach as before, the mesh size of honeycomb structures is determined as 3 mm (see Fig. 13).

Fig. 13
figure 13

Mean crush force comparison for different mesh sizes of honeycomb structures

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After determination of bumper beam, crashbox and honeycomb mesh sizes via a second mesh convergence study, the mesh sizes of the remaining parts are selected by intuition as they are subjected to much less deformation than bumper system. As shown in Fig. 14, mesh size of the chassis and the fixture is determined as 4 mm and 10 mm, respectively. Therefore, a base FE model is fully constructed to simulate crash performance of designed bumper system.

Fig. 14
figure 14

Mesh sizes of the high-fidelity finite element model

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As shown in Fig. 15, two different finite element (FE) models with different fidelities are used in this study. Previously constructed FE model is the complete model which consist of bumper system (that includes bumper beam, two crashboxes and two pieces of honeycomb structures), vehicle front rails and fixture that places the structure in front of the rigid pendulum. This complete model is named as high-fidelity (HF) model. The surfaces of fixture that touch to the ground are fixed in this FE model. A low-fidelity FE model is derived from the HF model. Fixture and heavy vehicle chassis are subtracted from the HF model to obtain low-fidelity (LF) model, which consist of only the bumper system. The bumper system is fixed from the end plates of the crashboxes. ”TIED_SURFACE_TO_ SURFACE_CONTACT” is deleted as corresponding parts do not exist, while all other FE model definitions remained identical to HF model. The comparison between the LF and HF models based on the force–displacement behavior for the baseline model is shown in Fig. 16. One FE simulation takes approximately 14 h for the HF model and 2.5 h for the LF model, with two Intel Xenon 3.1 GHz processors and 64 GB RAM.

Fig. 15
figure 15

a High-fidelity model, b low-fidelity model

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Fig. 16
figure 16

Comparison of the low-fidelity (LF) and high-fidelity (HF) models based on the force–displacement behavior for the baseline model

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4 Multi-fidelity optimization concept

High-fidelity models simulate a crash scenario closest to the physical case in a finite element analysis. However, they are computationally expensive models to run especially where many analyses must be done (e.g., for an optimization study). Low-fidelity models decrease the computation time while sacrificing accuracy. To provide a remedy for this problem, multi-fidelity models can be used. Multi-fidelity optimization combines low-fidelity models with few high-fidelity model results to obtain an accurate response prediction while using computationally inexpensive low-fidelity models for optimization runs. In this paper, linear regression multi-fidelity surrogates are used for multi-fidelity optimization as in Zhang et al. [44]. In linear regression multi-fidelity surrogate approach, the low-fidelity finite element response is used as a base function with a scale factor as the regression coefficient. High-fidelity response behavior is expressed as a linear combination of low-fidelity response and a polynomial discrepancy function. Prediction of the response through multi-fidelity surrogate approach is expressed as:

$$\begin{aligned} {\hat{f}}_\mathrm{MF}(x)=\rho f_L(x)+\delta (x) \end{aligned}$$
(7)

where \(\rho\) is the low-fidelity scale factor, and \(\delta (x)\) is the discrepancy function. Because of limited high-fidelity samples, discrepancy function often determined as a constant or a low-order polynomial function. In our study, we used a first-order discrepancy function as

$$\begin{aligned} \delta (x)=c_0+\sum _{i=1}^{p} c_ix_i \end{aligned}$$
(8)

where \(x_i\) represents the input variables, \(c_i\) is the unknown coefficients of the variables, and p is the number of variables. In order to find the scale factor \(\rho\) and coefficients of discrepancy function \(\delta (x)\), least square estimation is used. In order to apply least-square method, errors between multi-fidelity surrogates and high-fidelity samples are calculated as

$$\begin{aligned} e^{(i)}={y_\mathrm{H}}^{(i)}-{\hat{f}}_\mathrm{MF}({x_\mathrm{H}}^{(i)}) \end{aligned}$$
(9)

where \(y_\mathrm{H}\) represents the responses of the high-fidelity model computed at the high-fidelity sampling points \(x_\mathrm{H}\). In vector form, these errors of high-fidelity samples can be written as

$$\begin{aligned} e=Y-XB \end{aligned}$$
(10)

where

$$\begin{aligned} X= & {} \begin{bmatrix} f_L({x_\mathrm{H}^{(1)}}) &{} c_1{x_\mathrm{H}^{(1)}} &{} \cdots &{} c_p{x_\mathrm{H}^{(1)}} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ f_L({x_\mathrm{H}^{(n_\mathrm{H})}}) &{} c_1{x_\mathrm{H}^{(n_\mathrm{H})}} &{} \cdots &{} c_p{x_\mathrm{H}^{(n_\mathrm{H})}} \end{bmatrix}\nonumber, \\ Y= & {} \begin{bmatrix} {y_\mathrm{H}}^{(1)} \\ \vdots \\ {y_\mathrm{H}}^{(n_\mathrm{H})} \end{bmatrix}\nonumber, \\ B= & {} \begin{bmatrix} \rho \\ L \\ \theta \\ t \end{bmatrix} \end{aligned}$$
(11)

where \(n_\mathrm{H}\) is the number of high-fidelity samples. Finally, the unknown coefficients of discrepancy function and the scale factor are obtained by minimizing the square sum of errors. Unknown coefficients can be expressed as

$$\begin{aligned} B={(X^T X)}^{-1} X^T Y \end{aligned}$$
(12)

After the calculation of unknown coefficients, multi-fidelity surrogate model for prediction of high-fidelity model response is generated as given in Eq. (7). In the process discrepancy function and scale factor are calculated using common datasets of high- and low-fidelity analyses, while low-fidelity surrogate model is constructed using whole dataset of low-fidelity analyses. The multi-fidelity model is generated using low-fidelity model data and high-fidelity samples in order to predict high-fidelity model response in design space.

5 Construction of surrogate models

5.1 Design of experiments

Selecting design of experiment (DoE) type is the first step to form a surrogate model. There are two main groups of DoE [13]: classic designs and space filling designs. Full factorial design (FFD), central composite design (CCD) and Box–Behnken design are the most common classic DoE designs. Latin hypercube sampling (LHS) designs, maximum entropy designs, orthogonal arrays, minimax and maximin designs are the most commonly used space filling DoE designs. In this study LHS is used to generate the training points. Detailed explanation of latin hypercube sampling can be found in Acar et al. [13].

Using this DoE technique, 21 training points are generated within the bounds of variables (see Table 2). Generated points are listed in Table 3. Then, FE simulations of both FE models are performed to obtain CFE and SEA responses of FE models at the training points (see Table 4).

Table 2 Lower and upper bounds of the design variables
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Table 3 Design of experiments
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Table 4 CFE and SEA responses of finite element models at training points
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After realization of the FE simulations at the training points, CFE and SEA responses of the high-fidelity model and the low-fidelity model are calculated. These CFE and SEA calculations are tabulated in Table 4. Using corresponding CFE of SEA response of FE models, total of four surrogate models are constructed for HF and LF models.

5.2 Accuracy of surrogate models

Response surface models are used to predict actual response of FE model at any point in the design space. Quadratic response surface models are used in this study. Quadratic response surface model can be expressed as [13]

$$\begin{aligned} {\hat{y}}(x) = b_0 + \sum _{i=1}^{L} b_ix_i + \sum _{i=1}^{L} b_{ii}{x_i}^2 + \sum _{i=1}^{L-1} \sum _{j=1}^{L} b_{ij}x_ix_j \end{aligned}$$
(13)

where \({\hat{y}}(x)\) is the response prediction, L is the size of input vector x and \(b_0\), \(b_i\), \(b_{ii}\), \(b_{ij}\) are the response surface parameters to be determined using linear regression.

In order to construct multi-fidelity surrogate models, all data of LF model and data of HF model at randomly selected 10 training points are used (training points number 1, 4, 5, 6, 7, 10, 11, 13, 14, 18 given in Tables 3 and 4 ). After determination of low-fidelity scale factor (\(\rho\)) and the discrepancy function (\(\delta (x\))), multi-fidelity surrogate model is constructed as it is explained in Sect. 4. Therefore, along with the multi-fidelity surrogates, a total of six surrogate models are constructed in order to predict SEA and CFE responses.

Accuracy of surrogate models is evaluated using the root-mean-square error, RMSE, which can be calculated from:

$$\begin{aligned} {\text {RMSE}} = \sqrt{\frac{1}{N} \sum _{k=1}^{N} (y_k-{\hat{y}}_k)^2 } \end{aligned}$$
(14)

where \(y_i\) is the actual response and \({\hat{y}}_i\) is the surface model response at the \(i^{th}\) training point. RMSE value can be normalized with the mean value (see Eq. 15) or the range (see Eq. 16) of the actual responses at the training points as follows:

$$\begin{aligned} {\text {RMSE}}_\mathrm{nor1} = \frac{{\text {RMSE}}}{\sum _{k=1}^{N} y_k} \end{aligned}$$
(15)
$$\begin{aligned} {\text {RMSE}}_\mathrm{nor2} = \frac{{\text {RMSE}}}{y_\mathrm{max}-y_\mathrm{min}} \end{aligned}$$
(16)

where \(y_\mathrm{max}\) and \(y_\mathrm{min}\) are the maximum and minimum values of the responses evaluated at the training points, respectively.

Table 5 provides the RMSE and RMSE\(_\mathrm{nor}\) values for the surrogate models constructed for the CFE and SEA prediction. For the surrogate models, it is seen that the RMSE\(_\mathrm{nor1}\) values range between 4.9 and 7.6%, RMSE\(_\mathrm{nor2}\) values range between 12.7 and 17.6%, and these error values are acceptable for response prediction of a crash as a highly nonlinear phenomenon. It is also noticed that the surrogate models constructed for SEA prediction are more accurate than the ones constructed for CFE prediction.

Table 5 Accuracy of surrogate models
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6 Optimization results

The constructed surrogate models are used for optimization of the bumper system to achieve the maximum value of composite objective function (f) in Eq. 6. Finite element analyses are performed at these optimum points, and the predicted values of CFE and SEA are compared with the FEA results. The percent error between the predicted values and FEA responses is calculated from

$$\begin{aligned} {\text {Err}}_\mathrm{CFE}= & {} \frac{|{\text {CFE}}_\mathrm{pred}-{\text {CFE}}_\mathrm{fea}|}{{\text {CFE}}_\mathrm{fea}} \times 100 \end{aligned}$$
(17)
$$\begin{aligned} {\text {Err}}_\mathrm{SEA}= & {} \frac{|{\text {SEA}}_\mathrm{pred}-{\text {SEA}}_\mathrm{fea}|}{{\text {SEA}}_\mathrm{fea}} \times 100 \end{aligned}$$
(18)

6.1 High-fidelity optimization results

Using surrogate models for CFE and SEA response prediction of high-fidelity model, optimum values of the design variables are found (Table 6, columns 2, 3 and 4). CFE and SEA responses of the optimum design are predicted using surrogate models (Table 6, columns 5 and 6). Predicted values of composite objective function f are calculated using predicted response values (Table 6, column 7). Then, the actual CFE and SEA response values are calculated using finite element simulations (Table 6, columns 8 and 9). Next, the actual value of f is calculated using actual response values (Table 6, column 10). Finally, the error in CFE and SEA predictions is calculated using Eqs. (17) and (18) (see Table 6, column 11 and 12).

Table 6 High-fidelity optimization results
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High-fidelity optimization study yields an optimum design that has actual f value of 0.9399. Also CFE and SEA responses of that design are predicted with an error of 2.4% and 3.3%. In order to obtain these results for high-fidelity model, 21 finite element analyses are performed at the training points and a verification run of the optimum point is performed. Therefore, considering that each high-fidelity analysis takes 14 h of computation time, the total cost of high-fidelity optimization is \(22\times 14=308\) h of computation time. Note that the computational cost of surrogate model construction and optimization is far smaller than that of a single FE run.

6.2 Low-fidelity optimization results

Optimization of low-fidelity model is performed similar to high-fidelity optimization. Optimum values of the design variables (columns 2, 3 and 4), predicted values of CFE, SEA and f (columns 5, 6 and 7), finite element response values of CFE, SEA and f (columns 8, 9 and 10) and errors of CFE and SEA are given in the first row of Table 7 for the low-fidelity model.

Low-fidelity optimization yields on optimum design that has f value of 0.9165, and prediction errors of CFE and SEA are 4.0% and 2.8%, respectively. The composite objective function values of the high-fidelity model and low-fidelity model are very close to each other. Even though these predicted values are very close to each other for different optimum designs, this result is misleading because the actual performance of different designs should be evaluated by using the high-fidelity model.

The second row of Table 7 shows the high-fidelity response values of optimum point obtained from low-fidelity optimization. The actual f value of the LF optimum is computed to be 0.8475 (smaller than its LF model predicted value of 0.9165). Also, the errors in CFE and SEA predictions are computed as 5.3% and 27.2%, respectively. Note that the LF and HF models present a good correlation in terms of the force–displacement behavior for the baseline model as shown earlier in Fig. 16. However, for the LF optimum design, the behavior of the low-fidelity and high-fidelity models differed a lot in terms of SEA. As the design space of the optimization problem is wide, this inconsistency is probable.

Table 7 Low-fidelity optimization results
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Computational cost of low-fidelity optimization is much smaller than that of the high-fidelity optimization. In addition to 21 training, one low-fidelity verification is performed. Considering 2.5 h of computation time for LF run \(22\times 2.5=55\) h of computational time is spent for LF optimization.

6.3 Multi-fidelity optimization results

In the following, the multi-fidelity surrogate model for prediction of high-fidelity response generated using the linear regression multi-fidelity approach is discussed as explained in Sect. 4. Responses of LF model at 21 training points and response of HF model at randomly selected 10 training points (number 1, 4, 5, 6, 7, 10, 11, 13, 14, 18 shown in Table 3 and Table 4) are used. After generation of multi-fidelity model, the same process as high-fidelity optimization is followed. Optimum values of design variables (columns 2, 3 and 4), predicted values of CFE, SEA and f (columns 5, 6 and 7), finite element response values of CFE, SEA and f (columns 8, 9 and 10) and errors of CFE and SEA are given in Table 8 for the multi-fidelity optimum.

Table 8 Multi-fidelity optimization results
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The value of composite objective function f is calculated as 0.9238 for high-fidelity FE response of multi-fidelity optimum design. CFE and SEA responses of that design are predicted with an error of 4.5% and 3.1%, respectively. Crash behavior of multi-fidelity optimum bumper design is shown in Fig. 17. Input data to generate multi-fidelity surrogate model required 21 LF model and 10 HF model analyses. In addition, verification of optimum design required an additional HF analysis. Therefore, \(21\times 2.5+11\times 14=206.5\) h of computation time is spent.

Fig. 17
figure 17

Crash behavior of the multi-fidelity optimum bumper system

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6.4 Comparison of optimization results

Comparing the optimum designs obtained from all FE models, it is seen that the optimum design obtained by using different levels of fidelity is quite different. It is seen that the length of crashboxes (L) tends to take values close to the lower limit for the HF optimum and takes higher value for the LF optimum. The multi-fidelity optimum value of L is the same as that of the LF model. The wall angle of honeycombs (\(\theta\)) takes its upper limit value for HF and MF models, and for the LF model, it takes its lower limit. The wall thickness of honeycombs (t) takes values close to middle of design range for HF and LF optimum designs, but the value of t is close to upper limit for the MF optimum design.

Comparing the composite objective function values presented in Tables 6, 7 and 8, it is seen that optimization with HF model provides the optimum design with better performance and smaller error compared to the optimum design obtained through multi-fidelity and low-fidelity models. The composite objective function value of the optimum design obtained through HF model (f=0.9399) is 11% larger than that of the optimum design obtained through LF model (f=0.8475), and only 2% larger than that of the optimum design obtained through MF model (f=0.9238).

Force–displacement responses of all three optimum points obtained through high-fidelity FEA model are shown in Fig. 18. It is seen that the force value of HF optimum design is higher than the LF optimum design at any displacement. Therefore, difference between total energy absorption of HF optimum design and LF optimum design is easily observable at the graph. The behavior of the MF optimum design is different compared to the other two. Average force value of MF optimum design calculated from numerical data lies between the LF and HF optimum designs.

Fig. 18
figure 18

Force–displacement responses of the high-fidelity model at optimum points

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Finally, we provide a comparison in terms of computational cost. To complete 21 finite element simulations for training points and the verification run, 308 h of computation time is spent for high-fidelity optimization. Similarly, 22 low-fidelity FE simulations are completed in 55 h for the optimization with LF model. Finally, for 21 LF model analyses and 11 HF model analyses including the verification run required for multi-fidelity optimization, 206.5 h of computation time is spent.

7 Concluding remarks

In this study, surrogate-based multi-fidelity design optimization of a honeycomb filled bumper system was performed to achieve maximum crash force efficiency and specific energy absorption. Two different fidelity finite element models (HF and LF) were considered. The length of the crashboxes, the wall angle of honeycomb structures and the wall thickness of honeycomb structures were taken as design variables. From the results obtained in this study, the following conclusions were drawn:

  • Accuracy of response surfaces was evaluated using RMSE, normalized with the mean (\(RMSE_\mathrm{nor1}\)) and also with the range (\(RMSE_\mathrm{nor2}\)) of the responses evaluated at the training points. It was seen that \(RMSE_\mathrm{nor1}\) changed between 4.9 and 7.6%, \(RMSE_\mathrm{nor2}\) changed between 12.7 and 17.6%, and these error values were found to be acceptable for crash (a nonlinear phenomenon).

  • It was also noticed that the surrogate models constructed for SEA prediction are more accurate than the ones constructed for CFE prediction.

  • The optimum designs obtained by using different models were quite different. Length (L) of MF optimum model was the same as for the LF model, and cell wall angle (\(\theta\)) value was the same as for the HF model. Honeycomb wall thickness (t) values of the HF and LF optimum designs were close to each other, whereas it was close to the upper limit for MF optimum design.

  • Optimization with HF model provided the optimum design with better performance and smaller error compared to the optimum design obtained with low-fidelity and multi-fidelity optimizations. The composite objective function value of the optimum design obtained through HF model was 2% larger than that of the optimum design obtained through multi-fidelity optimization and 11% larger than that of the optimum design obtained through LF model.

  • Computational time spent to complete high-fidelity optimization was 308 h, whereas it was 206.5 h for multi-fidelity optimization (33% reduction) and 55 h for LF optimization (82% reduction). Multi-fidelity optimization provides a close result to high-fidelity optimization with 33% saving from computation cost.

Lastly, some limitations of the current study can be listed as follows:

  • In this study, the HF samples were selected from the pool of LF samples in a random manner. It should be noted that with different set of HF samples the results would be different, and finding a judicious selection procedure of the HF samples is subject of a future study.

  • In this study, the peak force values were not penalized in optimization. It should be noted that a high peak force relates to high accelerations, which should be avoided in a crashworthy design. This exercise is subject of a future study.