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Optimization of honeycomb battery package based on space mapping algorithm

A new honeycomb battery package structure is designed in this study. It is a honeycomb structure but using a grid to reinforce its strength. It can not only satisfy the requirements of strength but also reduce the mass of battery package. To obtain the highly accurate finite element (FE) model, the material parameters of 18650 cylindrical Li-ion battery are identified by using optimization techniques based on flat compression test data. Due to the expensive cost of finite element evaluation, the space mapping (SM) algorithm is suggested to optimize the structure of the package. Compared with other space mapping algorithms, the coarse model of space mapping is based on a pseudo-plane-strain model. Moreover, to guarantee the reliability, the mean and variance values of battery stress are used to be the objective function. The final optimum shows that the maximum stress of inner battery is decreased from 32.1733MPa to 12.9642MPa, the mean of battery stress is decreased from 5.3231MPa to 3.5836MPa and the variance is decreased from 5.9944 to 4.3904. It means the magnitude of stress and the distribution of stress are improved significantly.


Introduction
The Li-ion battery is considered to be an ideal electric vehicle battery due to its higher energy density, higher power density and longer cycle life. However, the safety of the cell should be more valued. When the cell is abused, it might explode or get fire due to internal short circuit [2]. These abuses include mechanical abuse, overcharge, over-discharge and overheating. In this work, only mechanical abuse is concerned. In order to understand the mechanism of internal short circuit, experiments are conducted. However, these experiments are ruinous and unrepeatable. Therefore, FE method is commonly used to simulate and predict the behavior of battery. However, it is difficult to develop the constitutive model of Li-ion battery when build the FE models. Several In this study, the new battery pack structure is suggested. Compared with the typical honeycomb structure, the distinctive characteristic of this structure is that it uses a grid to reinforce the strength [1]. It is well known that typical honeycomb structure is an excellent function structure which has outstanding crashworthiness and energy-absorption property. Therefore it has been wildly used and investigated. Yamashita and Gotoh (2005)  formulations under different impact energy. However, the typical honeycomb structure is too soft to be used as battery package structure directly. It should be reinforced its strength by grid, and the largest difference from typical honeycomb is that the honeycomb is not applied as buffer structure of battery package. The batteries are placed into its hexagon cells, so that the special hexagon cell is utilized to protect internal battery.
In order to obtain better performance of battery package structure, optimization should be conducted. However, the computational cost of FE model is too high. It results in that typical direct optimization algorithm cannot be used in this  The Reanalysis-based space mapping method showed a significant improvement in the efficiency of expensive simulation-based problems. Due to above mentioned advantages, the SM algorithm seems to be suitable for the problems with highly computational cost. Therefore, the SM is applied to structure optimization of honeycomb package in this study.
The rest of paper is organized as follows. In section 2, the material parameters of

Identification of material parameters of Li-ion battery
and Al alloy 6061-T6

Inverse identification method
For the battery package design, it is important to use high accurate FE evaluation in the optimization procedure. The material parameters is the critical issue to the accuracy of simulation. To guarantee the accuracy of the FE evaluation, the hybrid numerical method is used to obtain the material parameters in this study.
The flow chart of the inverse identification method is shown in Fig. 1.

Flat compression test of 18650 battery
The Li-ion batteries that were studied in this work are commercial 18650 ternary polymer lithium batteries. Their capacity is smaller than pouch battery's, is only 2600mAh. It means an electric vehicle needs more batteries than using pouch battery. However, they are safer than pouch battery. The specifications of these 18650 batteries are detailed in Table 1.
The contribution of shell casing to total force was less than 1% in the flat compression test [5]. Therefore, whether there is or not shell casing is not important for the test. In order to measure the voltage of battery, the shell casing is retained, as shown in Fig. 2(a). The compression test was conducted in INSTRON 150kN Universal Test Machine, and digital image correlation (DIC) was used to measure local strain ( Fig. 2(c)). The UNI-T U322 Thermometer and UNI-T UT61C Multimeter were used to obtain temperature and voltage respectively. When test was started, the deformation rate was set as 1mm/min. When the internal short circuit happens, the test would be stopped. The response of internal short circuit is that the load and the voltage dropped rapidly and the temperature increase simultaneously [2].
The force, displacement, temperature and voltage were measured in the test.
After the test, the electrolyte of battery was all leaked and the shell casing was flawed. The battery after deformation is shown in Fig. 2(d). Figure. 2(e) shows the result data which obtained in the test. The data reveals the load dropped rapidly from 5.7×10 4 N when the displacement was about 6.3mm. Meantime, the temperature started increase significantly and the voltage dropped from 3.6V to 0.096V in a short time. These phenomena mean that internal short circuit has happened.  The simulation parameters are detailed in Table 2. The whole model is shown in Fig. 2(b). The contact force between rigid board and battery is exported as history output.

Inverse identification of material parameters of battery
The material parameters of the cell are obtained by inverse identification method. The inverse identification method is an optimization method essentially.
where is the nearest displacement from the in the experimental displacement data. represents the design variables where is elasticity modulus, is Poisson ratio, is yield stress in harden curve which shown in Table 3. The Genetic algorithm (GA) algorithm is used to obtain the inverse identification result.
The inverse identification result is shown in Table 4. Figure. 3(a) shows the force-displacement curves of experiments and simulation. Figure. 3(b) shows the stress contours of battery. The R-square of the two curves is 0.9. Therefore, the inverse identification parameters can be accepted. Table 3 The harden curve of material of battery  Table 4 The material parameters of battery

Uniaxial tensile test of Al alloy 6061-T6
The material of the battery package is Al alloy 6061-T6. In order to obtain the material parameters of Al alloy 6061-T6, the uniaxial tensile test was conducted.
The size of tensile test coupon is obtained according to GB/T 228-2010, it is shown in Fig. 4(a). The uniaxial tensile test was conducted in INSTRON 150kN Universal Test Machine. To obtain the strain of tensile test coupon during the test, the extensometer was used. The tensile rate was 2mm/min. Figure. 4

(b)
shows the fractured tensile test coupon. The strain-stress curve obtained from test is shown in Fig. 4(c).

Validation of material parameters of Al alloy 6061-T6
According to the above illustration, the material parameters can be obtained from the strain-stress curve. The material parameters of Al alloy 6061-T6 are b c d summarized in Table 5. In order to validate the material parameters, the FE model of tensile sample has been built, as shown in Fig. 4(b). The strain-stress curve of elements of FE model is output to validate the accuracy of parameters.
The validation result is shown in Fig. 4(d). The result indicates the material parameters of Al alloy are accurate. Table 5 The material parameters of Al alloy

Quasi-static compression test of battery package
In order to validate the modeling method of battery package, the quasi-static compression test was conducted to compare with FE model. The structure of the test battery package which without battery is shown in Fig. 5(a). Figure. 5 The deformed sample is shown in Fig. 5(d).

The validation of modeling method
The material parameters of the whole package had been obtained. The FE model of battery package is constructed in ABAQUS. The FE model of the battery has been introduced in section 2. In order to improve the efficiency of simulation, the FE model is simplified and different from the entity. The thickness of honeycomb core is 0.9mm which is much less than the height of package 65mm. It could be modeled by shell element. Therefore, the FE model divided into two parts. One is the honeycomb core which is modeled by 4-nodes shell element (S4RSW).
Another is the reinforce grid which is modeled by solid element (C3D8R). These two parts are tied together by using tie constraint. Figure. 5 Fig. 5(d). The deformed shape of test and simulation is similar by comparing the two pictures in Fig. 5(d).

Fine FE model of battery package
The structure of optimized battery package is shown in Fig. 6(a). The FE model of package also consisted of two parts which are honeycomb core and grid. The honeycomb core is modeled by using 4-node shell elements (S4RSW), its thickness is 0.9mm. The grid is modeled by using solid elements (C3D8R). Figure. 6(b) shows the FE model of the whole battery package. The simulation parameters are detailed in Table 6. Figure. 7(a) shows the stress contours of the package in 0s, 0.018s, 0.036s, 0.054s, 0.072s, respectively. These contours show the stress of model is increasing. In order to only show the integral stress contours of battery, the grid and honeycomb are excluded from the simulation output and only the battery are output as shown in Fig. 7(b). The maximum stress of battery was 32.1733MPa which is still in safe loading range.  Table 6 The

Coarse FE model of battery package
The fine FE model of battery package can simulate the stress-strain behaviors of the battery package accurately, however, the computational cost is high. It needs 2.5 hours to finish a simulation when use 8 CPUs. Therefore, the typical optimization method cannot be available for optimizing the battery package.
However, SM algorithms are doing well in this kind of complex and expensive problems. Therefore, the SM algorithms be used in this study. In SM algorithms, an easier and cheaper FE model is needed which called coarse FE model. As shown in Fig. 6(c), the stress of battery was almost changeless in the axial direction of battery. Therefore, the fine FE model can be simplified as a pseudoplane-strain model. The coarse FE model is shown in Fig. 6(d). All the nodes of battery and package were constrained their movement along the Z axis and their rotation along the X and Y axis. The simulation parameters of the coarse FE model were the same as the fine model. Figure.

Trust region space mapping
The TRSM might be the most widely used SM algorithm. Its characteristic is setting a trust region for design parameter of fine model. The trust region of every cycle is used to judge whether the algorithm is convergent. Then, the TRSM will be introduced in detail.
Assuming is design parameter of fine model and is the objective function of fine model. Similarly, the symbol and represents the design parameter and the objective function of coarse model, respectively. The optimization problem of fine model is expressed as The constraint function of fine model is which is expressed as where and are the lower and upper bounds of constraint, respectively, and are the lower and upper bounds of design parameter . The optimal solution of fine model is expressed as The optimal solution of coarse model is expressed as (6) where and represent the design space of fine model and coarse model, respectively. The algorithms want to establish the mapping relation between the two design space, that is . The method to establish the relation is minimize the residual. The expression is (7) where is residual, and is norm. According to the expression, it follows (8) Therefore, the Equation.
(4) can be transformed into (9) Now, the complex and expensive optimization problem is transformed into easy and cheap problem. The specific algorithm steps are presented in Table 7. In every iteration, is accepted if it satisfies (10) The update rule of is (11) where .
The update rule of is (12) The algorithm converges if it satisfied any one of these conditions: (13) where both and are constant coefficient, is the maximum number of cycles

optimization of battery package
In the optimization problem, the design variables are the lengthways thickness of grid ,crosswise thickness of grid and thickness of honeycomb core .
The range of these design variables are listed in Table. 8. Because the stress of battery is increasing with time, the stress state of battery in the final time is concerned. In order to improve the magnitude and distribution of stress of batteries, the optimized target is suggested to minimize the mean and variance of stress of all the batteries which in the final time. It is a multi-objective problem.
The objective function is (14) where is the stress of all batteries in 0.072s. GA was used to be the The optimum solution of fine model is listed in Table 9. The initial mean and initial variance of stress of batteries in 0.072s was 5.3231MPa and 5.9944, respectively. The optimized mean and optimized variance of stress was 3.5836MPa and 4.3904, respectively. Figure. 8 shows the stress contours of the optimized fine model in 0s, 0.018s, 0.036s, 0.054s, 0.072s, respectively. The stress contours only output the stress of battery. The maximum stress of batteries was 12.9642MPa in 0.072s. It dropped by 59.71% compared to maximum stress of initial which is 32.1733MPa. The optimization consequence revealed the magnitude and distribution of stress of batteries has been improved significantly.  Table 9 The optimum solution of fine model

Conclusions
A new honeycomb battery package structure is designed in this study. It is different from traditional honeycomb structure but using a grid to reinforce its strength. In order to develop the FE model of the package, the material parameters of battery and Al alloy have to be obtained. Therefore, the flat compression test of 18650 battery and the uniaxial tensile test of Al alloy 6061-T6 were conducted. And then according to the data that obtained from flat compression test of battery, the material parameters of battery were identified by inverse identification method. In order to validate the modeling method of battery package, the quasi-static compression test of battery package was conducted. The verification results revealed that the deformation shape of the test package sample and the deformation shape of FE model is similar. Therefore, the FE model of battery package is accurate. In order to obtain the optimized structure of package, TRSM algorithm was used to optimize the structure.
Compared with other SM algorithms, the coarse model of SM is based on a pseudo-plane-strain model. Moreover, to guarantee the reliability, the mean and variance values of battery stress are used to be the objective function. Optimized consequence shows the maximum stress of inner battery was decreased from 32.1733MPa to 12.9642MPa, the mean of battery was decreased from 5.3231MPa to 3.5836MPa and the variance was decreased from 5.9944 to 4.3904.
The optimization effect is quite remarkable. Generally, the highlights of this work can be summarized as follows:  A new structure of battery package is designed. It is a kind of honeycomb structure. In order to reinforce its strength, an Al alloy grid is used to be the framework.
 The material parameters of cylindrical battery are obtained by inverse identification method instead of replacing by material parameters of pouch battery.
 The TRSM algorithm is used to optimize the structure of battery package.
Due to the expensive cost of FE evaluation, the typical optimization method cannot be available for optimizing the battery package. Therefore, the TRSM algorithm can be used to optimize the structure and obtain an ideal result in an acceptable time.