Frontal Vehicular Crash Energy Management Using Analytical Model in Multiple Conditions

: When it comes to frontal vehicular crash development, matching the stiffness of the front-end structures reasonably, i.e., impact energy management, can effectively improve the safety of the vehicle. A multi-condition analytical model for a frontal vehicular crash is constructed by a three-dimensional decomposition theory. In the analytical model, the spring is used to express the equivalent stiffness of the local energy absorption space at the front-end structure. Then based on the analytical model, the dynamic responses and evaluation indexes of the vehicle in MPDB and SOB conditions are derived with the input of the crash pulse decomposition scheme. Comparing the actual vehicle crash data and the calculation results of the proposed solution method, the error is less than 15%, which veriﬁes validity of the modeling and the accuracy of the solution. Finally, based on the solution method in the MPDB and the SOB conditions, the sensitivities of the crash pulse decomposition scheme to evaluation indexes are analyzed to obtain qualitative rules which guide crash energy management. This research reveals the energy absorption principle of the front-end structure during the frontal impact process, and provides an effective optimization method to manage the multiple conditions of the vehicle crash energy such as the FRB (frontal rigid barrier), the MPDB (mobile progressive deformable barrier), and the SOB (small overlap barrier).


Introduction
The process of an automobile crash includes the following three parts: barriers, vehicles, and occupants, and is a complex dynamics system. According to the annual statistical report of road traffic accidents from the Chinese government, frontal impact of automobile accidents account for the highest proportion among all accident forms [1,2]. In a frontal crash, the crash energy is absorbed by the front-end structures of vehicle [3]. With the development of crash safety regulations, today's vehicle front-end structural design needs to meet the requirements of multiple conditions such as FRB (frontal rigid barrier), MPDB (mobile progressive deformable barrier), and SOB (small overlap barrier) [4,5]. This greatly increases the difficulty of the vehicle safety design. From the perspective of dynamics, revealing the crash energy dissipation mechanism and constructing a dynamic model of the front-end structure, which solves energy management in the early design stage, is of great significance for improving the safety of the vehicle [6].
The ultimate goal of vehicular frontal crash safety research is to protect the safety of its occupants. In the crash process, the initial kinetic energy of the occupants is dissipated through the following two ways: the deformation of the vehicle structure and the action of the restraint system [7]. According to statistics about existing vehicles, the structure absorbs more than 60% of the kinetic energy of the occupants and is the most important way to absorb energy [8,9]. Crash energy management is meant to control the dissipation of energy At present, it is mainly focused on the research of crash energy management methods of the vehicle front-end structure based on engineering experience in the FRB impact condition [27][28][29]. Considering the impact force transmission characteristics of FRB condition, the front-end structure of the vehicle is divided into space, and the crash pulse is decomposed according to the sub-space as the energy absorption target [12]. In addition to FRB condition, the vehicle also needs to absorb energy through the deformation of the front-end structure in other frontal impact conditions, such as the MPDB impact condition which mainly test crash compatibility and SOB impact conditions which mainly test the safety of the passenger compartment [5,30,31]. Therefore, managing the energy absorption of front-end structures to meet the performance requirements and evaluation indicators of multiple frontal crash conditions is an urgent problem in the vehicle design industry, and it is also the research purpose of this paper.
Based on the frontal multi-condition crash pulse design method studied in the previous stage, this paper proposes a multi-condition crash energy management method, that is, the decomposition method of crash pulse. The proposed method optimizes the energy absorption of the vehicle front-end structures considering the design requirements of three conditions (that is FRB, MPDB, and SOB). The method can help enterprises to carry out the forward design of frontal vehicular crashes when the empirical data are insufficient. The main contributions of this paper include the following three aspects: (1) constructing a three-dimensional analytical model of vehicle front-end structure to describe the energy absorption space, impact load path, and structural stiffness; (2) deriving the dynamic responses and evaluation indexes of vehicle and barrier in the MPDB and SOB conditions to realize the crash pulse decomposition considering multiple conditions based on the proposed analytical model; (3) analyze the sensitivity of the crash pulse decomposition scheme to the evaluation indexes to obtain the qualitative crash pulse energy management strategy based on the analytical model.
The paper is organized as follows: Section 2 introduces a mechanical analytical model of vehicle front-end structure. In Section 3, we derive the solution of analytical model in MPDB and SOB conditions. An empirical case and the simulation results of existing vehicles that have been running in China are applied to verify this constructed model and the solution method in Section 3. Analysis and discussion are conducted in Section 4. Finally, the conclusions of this paper are drawn in Section 5.

Three-Dimensional Decomposition of Energy Absorption Space
In this section, the energy absorption space during a frontal impact can be determined as shown in Figure 1, where B represents the width of the whole vehicle; H represents the vertical distance from the intersection point of pillar A and the front finger beam to the chassis; D represents the longitudinal space L in the forward compartment of the vehicle minus the engine or motor occupancy E. The total energy absorption space can be decomposed into longitudinal, lateral, and vertical dimensions as follows, also considering the overlap rate of impact conditions, topological structure of the car body, and the layout position of the engine and motor.
Lateral decomposition: On both sides of the body symmetry plane, the energy absorption structure of the vehicle is almost exactly the same, and the contact area between the vehicle and the wall barrier in MPDB impact condition is also 50% of the width of the vehicle. Thus, the total energy absorption space can be divided horizontally into two equal regions. To ensure a better safety level of the vehicle in the condition of 25% small bias impact, the energy absorption structure should be set within 25% wide range on both sides of the vehicle. If the requirements of meeting various impact conditions are considered at the same time, four regions can be divided horizontally. The width of each region is 25% of the total car width.
Longitudinal decomposition: According to the impact force transfer path of the vehicle in the frontal impact, the total energy absorption space can be vertically divided into two, Sustainability 2022, 14, 16913 4 of 18 three, and four tiers longitudinally. Take a typical passenger vehicle as an example, which includes four tiers of the total energy-absorbing apace, the first tier is the engine cover; the second one is the front finger beam; the third one is the anti-impact beam, energy absorption box, and longitudinal beam; and the fourth one is the sub-frame structure. The energy absorption corresponds to the impact force. The energy absorption of each tier accounts for about 10%, 20%, 50%, and 20% of the total energy absorption, respectively [22]. The third tier includes the main energy absorption area of the vehicle. To improve the lightweight effect of vehicles, the engine hood is designed to be thinner and absorb less energy, so the energy absorption space is mainly divided into three tiers, that is, the first tier contains the engine hood, front finger beam and other structures. In addition, the sub-frames of some vehicles are removed in consideration of the economic and lightweight factors, so that the longitudinal space is two tiers. Lateral decomposition: On both sides of the body symmetry plane, the energy absorption structure of the vehicle is almost exactly the same, and the contact area between the vehicle and the wall barrier in MPDB impact condition is also 50% of the width of the vehicle. Thus, the total energy absorption space can be divided horizontally into two equal regions. To ensure a better safety level of the vehicle in the condition of 25% small bias impact, the energy absorption structure should be set within 25% wide range on both sides of the vehicle. If the requirements of meeting various impact conditions are considered at the same time, four regions can be divided horizontally. The width of each region is 25% of the total car width.
Longitudinal decomposition: According to the impact force transfer path of the vehicle in the frontal impact, the total energy absorption space can be vertically divided into two, three, and four tiers longitudinally. Take a typical passenger vehicle as an example, which includes four tiers of the total energy-absorbing apace, the first tier is the engine cover; the second one is the front finger beam; the third one is the anti-impact beam, energy absorption box, and longitudinal beam; and the fourth one is the sub-frame structure. The energy absorption corresponds to the impact force. The energy absorption of each tier accounts for about 10%, 20%, 50%, and 20% of the total energy absorption, respectively [22]. The third tier includes the main energy absorption area of the vehicle. To improve the lightweight effect of vehicles, the engine hood is designed to be thinner and absorb less energy, so the energy absorption space is mainly divided into three tiers, that is, the first tier contains the engine hood, front finger beam and other structures. In addition, the sub-frames of some vehicles are removed in consideration of the economic and lightweight factors, so that the longitudinal space is two tiers.
Vertical decomposition: The engine of the traditional automobile and the motor of a pure electric vehicle almost do not deform during the impact, which can be regarded as a rigid structure; therefore, according to the layout position of rigid components such as engine or motor, the total space of energy absorption can be divided longitudinally. For traditional gasoline cars and hybrid cars, rigid components such as engine need to be placed in the middle of the front end firewall considering the connection between the engine and drive shaft and maintenance problems. Thus, the two sections from the front end of the engine to the anti-impact beam and from the back end of the engine to the firewall are set as impact energy absorption spaces. For pure electric vehicles, the motor Vertical decomposition: The engine of the traditional automobile and the motor of a pure electric vehicle almost do not deform during the impact, which can be regarded as a rigid structure; therefore, according to the layout position of rigid components such as engine or motor, the total space of energy absorption can be divided longitudinally. For traditional gasoline cars and hybrid cars, rigid components such as engine need to be placed in the middle of the front end firewall considering the connection between the engine and drive shaft and maintenance problems. Thus, the two sections from the front end of the engine to the anti-impact beam and from the back end of the engine to the firewall are set as impact energy absorption spaces. For pure electric vehicles, the motor can be considered close to the firewall layout because there is no drive shaft on the vehicle. In this way, the hole from the anti-impact beam to the front end of the motor can be used as a section of energy absorption space.
In general, the three-dimensional decomposition of the total space of energy absorption at the front end of vehicle is to take into account the impact condition, the transmission path of impact force, and the arrangement of the engine and motor, etc., and conduct horizontal division, vertical stratification, and longitudinal segmentation successively. The decomposed fore cabin energy absorption space becomes the accumulation of the energy absorption subspace. As a sample, the total space of energy absorption at the front end of vehicle is decomposed 4 × 3 × 2 sections as shown in Figure 2.

Three-Dimensional Decomposition of Crash Pulse
In the FRB impact condition, the wall is rigid and the overlap rate between the vehicle and the wall is 100%. This indicates that almost all the energy absorption structures in the front end of the vehicle are involved in deformation energy absorption during the impact, and the acceleration response of the vehicle, namely the crash pulse, is also the result of the comprehensive action of the energy absorption structure. Thus, we take the crash pulse of FRB condition as the overall energy absorption objective of the vehicle's front end structure, and gradually decompose it into each energy absorption subspace and the design target tion of the energy absorption subspace. As a sample, the total space of energy absorption at the front end of vehicle is decomposed 4 × 3 × 2 sections as shown in Figure 2.

Three-Dimensional Decomposition of Crash Pulse
In the FRB impact condition, the wall is rigid and the overlap rate between the vehicle and the wall is 100%. This indicates that almost all the energy absorption structures in the front end of the vehicle are involved in deformation energy absorption during the impact, and the acceleration response of the vehicle, namely the crash pulse, is also the result of the comprehensive action of the energy absorption structure. Thus, we take the crash pulse of FRB condition as the overall energy absorption objective of the vehicle's front end structure, and gradually decompose it into each energy absorption subspace and the design target of the energy absorption structure in space, so as to achieve the forward design of the front-end structure.
The horizontal coordinate of the crash pulse in the displacement domain is the deformation, which corresponds to the longitudinal direction of the energy absorption space, but there is no difference between the lateral direction and the vertical direction. Therefore, the three-dimensional decomposition method proposed in this section is mainly to decompose the crash pulse into each energy-absorbing subspace in accordance with a given proportion according to the three-dimensional decomposition scheme of the total energy-absorbing space. If the longitudinal part of the total energy absorbing space is divided into two parts, the crash pulse is divided into two parts at the deformation corresponding to the longitudinal subspace.
When the front energy absorption space of the vehicle is divided into N tiers vertically and M zones horizontally, the decomposition scheme Q of the crash pulse is obtained as shown in Figure 3. Note that the percentage of the absorbed energy of the sub-absorbent space in the total absorbed energy is represented by nm q (n = 1, 2,…, N; m = 1, 2,…, M).  The horizontal coordinate of the crash pulse in the displacement domain is the deformation, which corresponds to the longitudinal direction of the energy absorption space, but there is no difference between the lateral direction and the vertical direction. Therefore, the three-dimensional decomposition method proposed in this section is mainly to decompose the crash pulse into each energy-absorbing subspace in accordance with a given proportion according to the three-dimensional decomposition scheme of the total energy-absorbing space. If the longitudinal part of the total energy absorbing space is divided into two parts, the crash pulse is divided into two parts at the deformation corresponding to the longitudinal subspace.
When the front energy absorption space of the vehicle is divided into N tiers vertically and M zones horizontally, the decomposition scheme Q of the crash pulse is obtained as shown in Figure 3. Note that the percentage of the absorbed energy of the sub-absorbent space in the total absorbed energy is represented by q nm (n = 1, 2, . . . , N; m = 1, 2, . . . , M).
Sustainability 2022, 14, x FOR PEER REVIEW 5 of 20 tion of the energy absorption subspace. As a sample, the total space of energy absorption at the front end of vehicle is decomposed 4 × 3 × 2 sections as shown in Figure 2.

Three-Dimensional Decomposition of Crash Pulse
In the FRB impact condition, the wall is rigid and the overlap rate between the vehicle and the wall is 100%. This indicates that almost all the energy absorption structures in the front end of the vehicle are involved in deformation energy absorption during the impact, and the acceleration response of the vehicle, namely the crash pulse, is also the result of the comprehensive action of the energy absorption structure. Thus, we take the crash pulse of FRB condition as the overall energy absorption objective of the vehicle's front end structure, and gradually decompose it into each energy absorption subspace and the design target of the energy absorption structure in space, so as to achieve the forward design of the front-end structure.
The horizontal coordinate of the crash pulse in the displacement domain is the deformation, which corresponds to the longitudinal direction of the energy absorption space, but there is no difference between the lateral direction and the vertical direction. Therefore, the three-dimensional decomposition method proposed in this section is mainly to decompose the crash pulse into each energy-absorbing subspace in accordance with a given proportion according to the three-dimensional decomposition scheme of the total energy-absorbing space. If the longitudinal part of the total energy absorbing space is divided into two parts, the crash pulse is divided into two parts at the deformation corresponding to the longitudinal subspace.
When the front energy absorption space of the vehicle is divided into N tiers vertically and M zones horizontally, the decomposition scheme Q of the crash pulse is obtained as shown in Figure 3. Note that the percentage of the absorbed energy of the sub-absorbent space in the total absorbed energy is represented by nm q (n = 1, 2,…, N; m = 1, 2,…, M).  In Equations (1) and (2), Q represents the total absorbed energy; Q Ym represents the absorbed energy of mth region, m = 1, 2, . . . , M; Q Ym represents the absorbed energy of nth tier, n = 1, 2, . . . , N.

Construction of Analytical Model
After the three-dimensional decomposition of the total energy absorption space, each subspace corresponds to a decomposed crash pulse, or a sub-pulse. It is the same as the original pulse in shape, but the amplitude is different, which is the product of the original pulse and the proportion of energy absorbed by each subspace. The crash pulse in the displacement domain can be regarded as the equivalent specific stiffness of the front-end structure of the vehicle; similarly, the sub-pulse decomposed to each energy-absorbing subspace can be regarded as the equivalent specific stiffness of the space. Figure 4 represents the three-dimensional vehicle analytical model for frontal impact. In this model, the equivalent specific stiffness of each energy-absorbing subspace is expressed by spring stiffness k ij , rigid bodies such as engine, motor, and passenger cabin are expressed by a mass block.

Construction of Analytical Model
After the three-dimensional decomposition of the total energy absorption space, each subspace corresponds to a decomposed crash pulse, or a sub-pulse. It is the same as the original pulse in shape, but the amplitude is different, which is the product of the original pulse and the proportion of energy absorbed by each subspace. The crash pulse in the displacement domain can be regarded as the equivalent specific stiffness of the front-end structure of the vehicle; similarly, the sub-pulse decomposed to each energy-absorbing subspace can be regarded as the equivalent specific stiffness of the space. Figure 4 represents the three-dimensional vehicle analytical model for frontal impact. In this model, the equivalent specific stiffness of each energy-absorbing subspace is expressed by spring stiffness kij, rigid bodies such as engine, motor, and passenger cabin are expressed by a mass block. The spring stiffness kij can be calculated as follows: where, av represents the crash pulse; dv represents vehicle displacement; M represents vehicle mass; i is the number of lateral decomposition of the total space of energy absorption in the front compartment, and j is the number of vertical decomposition of the total space of energy absorption.

Solution Method for MPDB Condition
In the MPDB impact system, it mainly consists of the following two parts: barrier and vehicle. The impact process of the MPDB condition is defined as the process in which the vehicle and the barrier start to contact either of them with a deceleration of 0. It is assumed that the masses of the MPDB and vehicle are constant and the energy-absorbing The spring stiffness k ij can be calculated as follows: where, a v represents the crash pulse; d v represents vehicle displacement; M represents vehicle mass; i is the number of lateral decomposition of the total space of energy absorption in the front compartment, and j is the number of vertical decomposition of the total space of energy absorption.

Solution Method for MPDB Condition
In the MPDB impact system, it mainly consists of the following two parts: barrier and vehicle. The impact process of the MPDB condition is defined as the process in which the vehicle and the barrier start to contact either of them with a deceleration of 0. It is assumed that the masses of the MPDB and vehicle are constant and the energy-absorbing structures of the two are only plastically deformed without elastic deformation during the impact process. Due to the rotations of vehicle and barrier are all small during the impact process, structural deformation is the most important way to absorb energy [24]. The simplified model of MPDB condition can be constructed ignoring the energy converted to rotation as shown in Figure 5. structures of the two are only plastically deformed without elastic deformation during the impact process. Due to the rotations of vehicle and barrier are all small during the impact process, structural deformation is the most important way to absorb energy [24]. The simplified model of MPDB condition can be constructed ignoring the energy converted to rotation as shown in Figure 5. In this model, the barrier parameters obtained in the conceptual design stage include the mass and equivalent stiffness of the barrier, the general layout parameters of the vehicle, including the energy absorption space of the front-end structure and the mass of  In this model, the barrier parameters obtained in the conceptual design stage include the mass and equivalent stiffness of the barrier, the general layout parameters of the vehicle, including the energy absorption space of the front-end structure and the mass of the vehicle as the system parameters. Taking the crash pulse of vehicle FRB condition as input; the dynamic response of the vehicle and the barrier during the impact was solved, and the vehicle compatibility evaluation indexes are calculated.
In Figure 5, the vehicle and the barrier move in relative motion; M and M B are the masses of the vehicle and the barrier; d Bv is their relative displacement; d ij is the local deformation of the vehicle; d Bij is the local deformation of the barrier; k ij is the local stiffness of the vehicle, and k Bij is the local stiffness of the barrier.
During the impact, the local stiffness of the vehicle and the barrier, k ij and k Bij , are first connected in series to obtain the equivalent stiffness k eqij of each energy absorption subspace, and then the equivalent stiffness of each energy absorption subspace is connected in parallel to obtain the equivalent stiffness K eq of the vehicle and the barrier, as follows: where, M eq represents the equivalent mass of the model, and v eq represents the equivalent velocity. The motion response of the vehicle and the barrier is obtained as follows: where, d B and d v are respectively for the barrier and the deformation of the vehicle impact process; a B and a v are respectively for the barrier and the vehicle acceleration, v B and v v are the barrier and the vehicle speed, respectively; ω is the natural frequency of the impact system. In addition, three compatibility evaluation indexes of MPDB condition, i.e., the relative motion displacement of the vehicle and the barrier (d Bv ), the uniformity index of the barrier (SD), and the maximum deformation of the barrier (MD), as follows: where, m represents the number of forward compartment subspaces that absorb energy.

Solution for Small Overlap Condition
In the small overlap condition, the vehicle decelerates along the longitudinal direction and rotates around its contact point with the rigid barrier after contacting with the barrier. If one of the following two point occurs, the impact process of small overlap condition is considered to be over: (1) when the speed of the vehicle decreases to 0, the structures of the vehicle no longer deform longitudinally; (2) if the vehicle displacement in the Y direction is ≥25% of the vehicle width, the vehicle is detached with the barrier without structural deformation. Therefore, the vehicle motion responses in the impact process of the small overlap condition can be considered from two aspects, i.e., the deceleration motion and rotation motion of the vehicle.
Based on the theory of impact mechanics, the simplified model of small overlap condition can be obtained as shown in Figure 6. In this model, O is the contact point between the vehicle and the barrier, and also the rotation center of the vehicle. v 0 is the impact speed, M is the mass of the vehicle, and k ij is the vehicle stiffness. The motion response of the vehicle in this condition can be considered from two aspects, i.e., the deceleration motion and the rotation motion of the vehicle. The 25% overlap area between the vehicle and the barrier is mainly the energy-absorbing area during the impact, and its equivalent stiffness (k25%) is calculated as follows: The vehicle makes a single-degree-of-freedom free vibration in the X-axis direction. The vibration equation is as follows: where, x(t) is the displacement of the vehicle, and ( ) t x  is the acceleration response of the vehicle. (17) In this condition, . Thus, the motion response of the vehicle is obtained as follows: The motion response of the vehicle in this condition can be considered from two aspects, i.e., the deceleration motion and the rotation motion of the vehicle. The 25% overlap area between the vehicle and the barrier is mainly the energy-absorbing area during the impact, and its equivalent stiffness (k 25% ) is calculated as follows: The vehicle makes a single-degree-of-freedom free vibration in the X-axis direction. The vibration equation is as follows: where, x(t) is the displacement of the vehicle, and ..
x(t) is the acceleration response of the vehicle.
x = v 0 . Thus, the motion response of the vehicle is obtained as follows: x(t) = v 0 cos(ωt) a v = ..
The force analysis of the vehicle under the condition of the small overlap condition is shown in Figure 7. The vehicle will rotate around point O under the action of impact reaction F, and the distance between the vehicle's center of mass and the center of rotation is the radius of rotation r. When the vehicle is decelerating, its longitudinal displacement is x v , and the longitudinal distance between the center of mass and the center of rotation is 1/2L − x v . When the vehicle is rotating, the transverse displacement of the center of mass is y, and the transverse distance from the center of mass to the center of rotation is 25% B + y. The radius of rotation (r) and the rotation angle (θ) of the vehicle are calculated as follows: The torque of the vehicle (Me), the angular acceleration of the vehicle rotation (β), and the lateral displacement of the vehicle (y) is obtained as follows: where, F′ represents the component force of the impact reaction F perpendicular to the direction of the radius of rotation; J represents the moment of inertia of the vehicle.
The maximum longitudinal displacement of the vehicle is the structural deformation of the vehicle in the process of impact. The intrusion of the crew compartment (D25%max) can be calculated as follows:

1) Occupant load criterion (OLC)
The OLC is calculated by the velocity-time curve of the barrier as Figure 8. A smaller OLC is preferable [19]. At the time t1 in the Figure 8, the virtual occupant makes free movement relative to the barrier, and the displacement is S1 = 0.065 m; from the time t1 to t2, the relative displacement between virtual occupant and barrier is S2 = 0.235 m. The radius of rotation (r) and the rotation angle (θ) of the vehicle are calculated as follows: The torque of the vehicle (Me), the angular acceleration of the vehicle rotation (β), and the lateral displacement of the vehicle (y) is obtained as follows: where, F represents the component force of the impact reaction F perpendicular to the direction of the radius of rotation; J represents the moment of inertia of the vehicle. The maximum longitudinal displacement of the vehicle is the structural deformation of the vehicle in the process of impact. The intrusion of the crew compartment (D 25%max ) can be calculated as follows:

Evaluation Indexes (1) Occupant load criterion (OLC)
The OLC is calculated by the velocity-time curve of the barrier as Figure 8. A smaller OLC is preferable [19]. At the time t 1 in the Figure 8, the virtual occupant makes free movement relative to the barrier, and the displacement is S 1 = 0.065 m; from the time t 1 to t 2 , the relative displacement between virtual occupant and barrier is S 2 = 0.235 m. The SD of barrier intrusion is obtained by the homogeneity of footprint based on scans of barrier. A smaller SD is preferable. The barrier deformation uniformity factor h is calculated by SD as follows:

2) Maximum deformation (MD)
If there is an intrusion depth of 0.63 m in an area larger than 40 mm × 40 mm on the barrier, the barrier is considered to be bottoming out. At the time, deduction of two points is MD = 2, otherwise MD = 0.

3) Standard deviation (SD)
The SD of barrier intrusion is obtained by the homogeneity of footprint based on scans of barrier. A smaller SD is preferable. The barrier deformation uniformity factor h is calculated by SD as follows: ( )

4) The penalty of Mcompat (PM)
The calculation rules for PM reference to [31] are shown in the following:

Verification of MPDB Condition
The main MATLAB algorithm of the solution method is established as shown in the Supplementary Information. The data about three quality grade vehicles in FRB and MPDB tests [30] are used to verify the accuracy and reliability of the solution method. In this condition, research of the influence of pulse parameters and vehicle quality on the evaluation index, including OLC, MD, SD, and PM, three kinds of vehicles, i.e., V1, V2, and V3, with the masses of 1700 kg, 1400 kg, and 1100 kg, respectively, are applied. The crash pulses of V1, V2, and V3 are shown in Figure 9. The calculation rules for PM reference to [31] are shown in the following:

Verification of MPDB Condition
The main MATLAB algorithm of the solution method is established as shown in the Supplementary Information. The data about three quality grade vehicles in FRB and MPDB tests [30] are used to verify the accuracy and reliability of the solution method. In this condition, research of the influence of pulse parameters and vehicle quality on the evaluation index, including OLC, MD, SD, and PM, three kinds of vehicles, i.e., V 1 , V 2 , and V 3 , with the masses of 1700 kg, 1400 kg, and 1100 kg, respectively, are applied. The crash pulses of V 1 , V 2 , and V 3 are shown in Figure 9. The stiffness decomposition schemes of the front structures of the three vehicles are all transverse 50% + 25% zone and vertical three floors, denoted as "4 × 3" decomposition, as shown in Figure 10. Decomposition plans (Q) of V1, V2, and V3 are obtained as shown in Equation (27).  The stiffness decomposition schemes of the front structures of the three vehicles are all transverse 50% + 25% zone and vertical three floors, denoted as "4 × 3" decomposition, as shown in Figure 10. Decomposition plans (Q) of V 1 , V 2 , and V 3 are obtained as shown in Equation (27). The stiffness decomposition schemes of the front structures of the three vehicles are all transverse 50% + 25% zone and vertical three floors, denoted as "4 × 3" decomposition, as shown in Figure 10. Decomposition plans (Q) of V1, V2, and V3 are obtained as shown in Equation (27).
The values of OLC, MD, SD and PM are obtained in Table 1. The errors between the calculation results and the existing data are all about 10%, which indicate that the accuracy of the solution method is acceptable.

Verification of Small Overlap Condition
The finite element calculation results of a vehicle in small overlap condition as the basic data are used to verify the accuracy of the analytical model by the maximum intrusion amount of the upper part of the passenger compartment. The vehicle mass is 1365 kg, and the energy absorption space in the front compartment is 0.68 m. The maximum  Table 1. The errors between the calculation results and the existing data are all about 10%, which indicate that the accuracy of the solution method is acceptable.

Verification of Small Overlap Condition
The finite element calculation results of a vehicle in small overlap condition as the basic data are used to verify the accuracy of the analytical model by the maximum intrusion amount of the upper part of the passenger compartment. The vehicle mass is 1365 kg, and the energy absorption space in the front compartment is 0.68 m. The maximum intrusion value of the four intrusion measurement points on the upper part of the passenger compartment is taken as 34.2 cm in the simulation results of the vehicle as shown in Figure 11 [32].  It is known that the energy absorption decomposition scheme of the front-end structure of the car body is shown in Equation (28). Take the crash pulse of the vehicle as shown in Figure 12 into the analytical model together with the decomposition scheme in Equation (28) to calculate the maximum longitudinal displacement of the vehicle as It is known that the energy absorption decomposition scheme of the front-end structure of the car body is shown in Equation (28). Take the crash pulse of the vehicle as shown in Figure 12 into the analytical model together with the decomposition scheme in Equation (28)  It is known that the energy absorption decomposition scheme of the front-end structure of the car body is shown in Equation (28). Take the crash pulse of the vehicle as shown in Figure 12 into the analytical model together with the decomposition scheme in Equation (28)

Impact Analysis of Different Waveform Decomposition Schemes
In this section, the effect of waveform decomposition scheme on three evaluation indexes of compatibility, i.e., OLC, MD, and SD, is studied. Note that the 4 × 3 decomposition method is adopted in this research. Considering the lateral symmetry of the car body, the variables of the decomposition method are the proportion of energy absorbed by six variables, and the sum of the six variables is 50%, as shown in Equation (28).

Impact Analysis of Different Waveform Decomposition Schemes
In this section, the effect of waveform decomposition scheme on three evaluation indexes of compatibility, i.e., OLC, MD, and SD, is studied. Note that the 4 × 3 decomposition method is adopted in this research. Considering the lateral symmetry of the car body, the variables of the decomposition method are the proportion of energy absorbed by six variables, and the sum of the six variables is 50%, as shown in Equation (28).
Three variation schemes for six variables are proposed, as follows: Alternative 1: setting any one of the six variables to increase from 0 to 50%, and the other variables are equal.
Alternative 2: setting any two of the six variables are increased from 0 to 50%, and the other variables are equal.
Alternative 3: setting any three of the six variables and increase them from 0 to 50%, and the other variables are equal.
The decomposition difference of each scheme is calculated by Equation (29). In addition, the design and decomposition difference (W) of the three alternatives are shown in Table 2. In this section, the decomposition difference and the stiffness-change position of the three schemes are adopted as the criteria to measure the uniformity of the barrier. The greater W is, the poorer the uniformity of the decomposition scheme. At the same W, with the more varied locations, the uniformity of the decomposition scheme is better. With W as the horizontal coordinate, the values of OLC, MD, and SD calculated by the three schemes are drawn, as shown in Figure 13, respectively.
When W is 0, the local stiffnesses of vehicle are evenly distributed. As the W increases, the uniformity becomes worse. Through the above analysis and the information in Figure 13, it is found that when W is 0, the OLC is the largest, the MD is the smallest, and the SD equals 0. In the Figure 13, as the W increases, the decomposition scheme becomes more and more uneven, the OLC value gradually decreases, and the MD and SD value gradually increases. Corresponding to the same value of W, the positional relationship of local stiffness changes in the three stiffness decomposition schemes is: Alternative 3 > Alternative 2 > Alternative 1.
When W is the same, the OLC values of three alternatives are basically the same, which shows that OLC is more sensitive to W, but has nothing to do with the position of the stiffness distribution. When the W is the same, the increase in the stiffness-change position helps to reduce the MD, and increase SD. It shows that MD and SD are more sensitive to the change position of local stiffness and W. In general, the worse the uniformity of the vehicle stiffness distribution, the smaller the OLC, and the larger the MD and SD. In order to ensure that the barrier is not penetrated and to control the deformation uniformity index SD, it is necessary to reasonably allocate the stiffness of the front-end structure.

Discussion about Stiffness Decomposition of Vehicle
The study of automobile crash safety is very important to improve road traffic safety [33][34][35]. In the reference [24], an analytical model integrating barrier, vehicle, and occupant is established to obtain not only the system dynamic responses but also the two evaluation indexes in MPDB (OLC and MD), known as BVO model, as shown in Figure 14. The BVO model system refers to simplified models with the elements of masses and springs. It is assumed that vehicle and barrier have constant masses during the impact process. The spring stiffnesses are used to represent the deformation energy absorption process of the structures. Since the overlap between vehicle and barrier are ignored in the modeling, the rotation responses of the BVO system and the local deformation index of barrier SD cannot be calculated [31,36]. When W is 0, the local stiffnesses of vehicle are evenly distributed. As the W increases, the uniformity becomes worse. Through the above analysis and the information in Figure 13, it is found that when W is 0, the OLC is the largest, the MD is the smallest, and the SD equals 0. In the Figure 13, as the W increases, the decomposition scheme becomes more and more uneven, the OLC value gradually decreases, and the MD and SD value gradually increases. Corresponding to the same value of W, the positional relationship of local stiffness changes in the three stiffness decomposition schemes is: Alternative 3 > Alternative 2 > Alternative 1. When W is the same, the OLC values of three alternatives are basically the same, which shows that OLC is more sensitive to W, but has nothing to do with the position of the stiffness distribution. When the W is the same, the increase in the stiffness-change position helps to reduce the MD, and increase SD. It shows that MD and SD are more sensitive to the change position of local stiffness and W. In general, the worse the uniformity of the vehicle stiffness distribution, the smaller the OLC, and the larger the MD and SD. In order to ensure that the barrier is not penetrated and to control the deformation uniformity index SD, it is necessary to reasonably allocate the stiffness of the front-end structure.

Discussion about Stiffness Decomposition of Vehicle
The study of automobile crash safety is very important to improve road traffic safety [33][34][35]. In the reference [24], an analytical model integrating barrier, vehicle, and occupant is established to obtain not only the system dynamic responses but also the two evaluation indexes in MPDB (OLC and MD), known as BVO model, as shown in Figure  14. The BVO model system refers to simplified models with the elements of masses and springs. It is assumed that vehicle and barrier have constant masses during the impact process. The spring stiffnesses are used to represent the deformation energy absorption process of the structures. Since the overlap between vehicle and barrier are ignored in the modeling, the rotation responses of the BVO system and the local deformation index of barrier SD cannot be calculated [31,36]. In order to solve the dynamic response of the vehicle in the overlap impact conditions, the equivalent stiffness of the BVO model (KB and K) are decomposed into the local stiffness in this paper, as shown in Figure 5. The three-dimensional analytical model of vehicle front end structure after stiffness decomposition is shown in Figure 4. The previous impact mechanics model can solve the one-dimensional dynamic response. The In order to solve the dynamic response of the vehicle in the overlap impact conditions, the equivalent stiffness of the BVO model (K B and K) are decomposed into the local stiffness in this paper, as shown in Figure 5. The three-dimensional analytical model of vehicle front end structure after stiffness decomposition is shown in Figure 4. The previous impact mechanics model can solve the one-dimensional dynamic response. The analytical model and solution method proposed in this paper provide new calculation ideas for the threedimensional dynamic response of the impact system.
The vehicle data as the verification of the analytical model solved in MPDB test, are the same with data to verify the accuracy of BVO model in the reference [24]. The results of the BVO model are calculated by the vehicle mass and the crash pulse in the Section 3.3.2, as shown in Table 3. Comparing with the results calculated by the BVO model in Table 3, the OLC values calculated by the analytical model are smaller and the MD values are larger. This may be because that the local stiffnesses obtained after decomposing the vehicle stiffness are connected in parallel when the vehicle parameters are the same. The local stiffnesses of vehicle and barrier are first connected in series and then in parallel as the overall equivalent stiffness between the two to calculate the impact responses under the MPDB condition. Therefore, if the decomposition is uniform, the equivalent stiffness between barrier and vehicle is constant, that is, the OLC and accelerations of the system calculated by the BVO model and proposed analytical model are the same, and the deformation standard deviation SD of the barrier is 0. If it is not uniformly decomposed, the equivalent stiffness between vehicle and barrier is decreased, so that the OLC is decreased; and a part of local stiffnesses of vehicle are increased to cause the local deformation of the barrier to increase, i.e., MD is increased; at the same time, the SD value is larger than 0.
In engineering application, the main methods to improve vehicle compatibility are the following: reducing the quality of the entire vehicle, evenly distributing structural stiffness, and designing a reasonable crash pulse [12,13]. In the process of safety development, the quality level of the vehicle should be determined first, then the BVO model should be used to optimize the crash pulse, and finally the proposed model in this paper should be used to decompose the stiffness of front-end structure [7,37].
The effect of vehicle stiffness decomposition on the dynamic responses of small offset impact conditions is not analyzed in this paper. Theoretically, the greater the stiffness of the 25% area (QY1 or QY4 in the Figure 10), the smaller the deformation of the vehicle front-end structure and the smaller intrusion into the passenger compartment. Therefore, in the engineering design, the sum of the stiffness of the three positions in the 25% area should be increased as much as possible to improve the safety of the passenger compartment. That is to say, adding an energy-absorbing structure or increasing the stiffnesses of the existing structures in the 25% area can effectively improve the safety of the vehicle in small offset impact condition.

Comparison with Existing Research
The energy management method distributes the total collision energy to the various energy absorption structures of the auto body. According to the difference of the decomposition method in time and space order, a variety of energy management methods are proposed. From 2006 to 2011, Qi proposed the magic cube approach which decomposes crashworthiness target into sub-structure design targets, and designs the load of front longitudinal beam by the dynamic topology optimisation method [20,21,38]. In 2011, the book of 'Fundamentals of Automobile Body Structure Design' divided the energy absorption space of the front-end structure according to the transmission path of the impact force [22]. From 2011 to 2013, Zhang et al. effectively decomposed total collision energy into sub-structure performance targets based on energy management technology [25,39]. From 2013 to 2016, Qiu presented an average allocation between ride-down energy and restraint energy to decrease occupant injury; in his book 'Automobile Crash Safety Engineering', the concepts of 'longitudinal energy management' and 'lateral energy management' were proposed [7,40].
The existing researches basically use qualitative methods to decompose the total energy of a frontal vehicular crash into the front-end structures [31,37]. The existing methods rely on empirical data or the subjective judgment of engineers and experts. Compared with the existing methods, the most important feature of this paper is to use an analytical method to optimize the crash energies of the vehicle front-end structures. This analytical method greatly reduces the empirical dependence of the frontal vehicular crash. In addition, the existing studies are based on empirical conclusions of FRB. The multi-conditions crash energy optimization method proposed in this paper is mainly based on the FRB, and takes into account the design requirements of MPDB and SOB.

Conclusions
This paper studies the modeling method of vehicle front-end structures and the solution method of system dynamic responses in multiple impact conditions to manage the crash energy. The energy absorption space of vehicle front-end is decomposed in three dimensions, and expressed by the spring stiffness to construct the analytical model of the front-end structure. Based on the three-dimensional analytical model of the front-end structure, the dynamic responses and evaluation indexes of the MPDB and SOB operating system are solved by the principle of engineering vibration with input of crash pulse decomposition scheme. Subsequently, the sensitivity of the crash pulse decomposition scheme to the evaluation indicators of each operating condition is analyzed based on the solution method in the multi-condition. Several concluding remarks are drawn as follows: (1) comparing the experimental data of the MPDB test and the calculation results of constructed analytical model, the errors of evaluation indexes, i.e., OLC, SD and PM, are all less than 15%, and judgments about the barrier bottoming out are all in accordance; (2) comparing the simulation data of the SOB test and the calculation results of constructed analytical model, the errors of the maximum intrusion into passenger compartment is −5.88%; (3) as the W increases, the decomposition scheme becomes more and more uneven, the OLC value gradually decreases, the MD and SD value gradually increases; (4) OLC is more sensitive to W; MD and SD are more sensitive to the change position of local stiffness and W; (5) the greater the stiffness of the 25% area, the smaller the deformation of the vehicle front-end structure and the intrusion into the passenger compartment.
To sum up, the proposed method reveals the energy absorption principle of the frontend structure during the frontal impact process. The calculation results provide references for vehicle crash energy management in three conditions. In the case of insufficient enterprise experience data, the optimization method in this paper can be used to carry out the forward design of frontal vehicular crash safety.