# Modeling Electric Vehicle’s Battery Module using Computational Homogenization Approach

Authors: Shantanu R. Shinde; Yihan Song; Elham Sahraei
Journal: International Journal of Energy Research
 (2023)

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## Abstract

Numerical simulations of heterogeneous structures like battery modules of electric vehicles are challenging due to the various length scales involved in it. Even with the latest computing technology, it is impossible to simulate the crash scene of the full vehicle resolving all length scales. Such hurdles have prevented manufacturers to understand the mechanical response of battery packs in vehicle crash scenarios. In this work, the problem of multiple length scales was solved using the RVE technique based on homogenization theory. An appropriate representative volume element was identified, and a 3D FE model was developed. Classical first-order boundary conditions were used in this research work. The RVE was subjected to several macroscopic deformations, and its response was obtained. The homogenized material properties were computed from the obtained responses, and a material model available in LS-Dyna’s material library was selected and calibrated to describe the nonlinear multiaxial behavior of the homogenized battery module at the macroscale. For validation, the USABC Crush and Drop Tests were simulated for the detailed and homogenized battery modules. The main output of this research is a robust and computationally efficient tool enabling satisfactory integration of a battery pack model to the vehicle for crash simulations, eliminating the need to simulate micro details at macro length scales. This approach significantly reduced the computational cost. For example, for a Drop Test simulation, the homogenized model reduced the simulation time from 40 hours (detailed model) to about 3 minutes, while maintaining a high precision (R2=0.9871) in predicting the load-displacement response. The system level modeling will enable the stakeholders to perform efficient optimization and safety evaluation for full-scale crashworthiness of electric vehicles.


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## Body

## 1. Introduction

Lithium-ion batteries are extensively used as a power source to propel electric vehicles (EVs) and Hybrid Electric Vehicles (HEV). The past decade has seen a rise in the sale of electric vehicles due to its various advantages over conventional internal combustion engine vehicles [1]. These high voltage battery packs are made up of modules and cells which are connected in series or parallel. A cell is the smallest form of the battery. A module consists of cells arranged in series or parallel along with other components as a single electrical and mechanical entity. Several modules further connected in series or parallel make up a battery pack. This is illustrated in Figure 1.Figure 1
Lithium-ion cell, module, and battery pack.In an event of a crash, these batteries may be subjected to various local forces and deformations which may lead to fire, electric shock (up to 400 volts), extremely high temperatures, thermal runaway, toxic fumes, lithium burns, and toxic runoffs [2]. Hence, it is of paramount importance to have a better understanding of the crash response of the battery pack that is integrated into the vehicle.In recent years, a great deal of research has been focused on the modeling of Lithium-ion batteries including electrochemical models [3–12], mechanical models [13–21], Battery Thermal Management System/thermal models [22–30], and multiphysics models [31–37] at scales varying from nanolevel to cell level. Such efforts had immense contribution in improving the safety of lithium-ion battery cells. However, very few studies have focused on mechanical modeling of lithium-ion batteries at module and pack level.Zhu et al. [38] conducted series of experimental studies on commercial vehicle battery modules. In this study, the battery modules were subjected to quasistatic and low speed crush loads with steel impactors having hemispherical and wedge shapes to study the effects of various parameters on the overall response of the battery module. Xia et al. [39] and Zhu et al. [40] conducted thorough studies on lithium-ion battery packs subjected to ground impact. For these studies, they established a detailed model for the battery pack with homogenized jellyrolls. Scurtu et al. [41] developed a detailed model of a battery pack to study the benefits of shock absorbers mounted on the external faces of the battery pack casing on reducing the crash impact propagated to the battery pack. Zhang et al. [42] developed a methodology to determine the state of health (SOH) of mechanically damaged lithium-ion batteries for evaluation of battery module safety. Machine learning and finite element model techniques were used to achieve the goals of the study. A detailed model of the battery module was used to evaluate the mechanical deformation of the battery cells when the module was subjected to impact loading. Xi et al. [43] conducted a study to investigate the mechanical response of the battery module subjected to static and dynamic loading. For this study, a detailed model of the battery module was developed.The main disadvantages of conducting such studies [38–43] are the extensive cost and time involved. This is due to the expensive set up for experimental testing at module and pack level as well as material nonlinearities, complex deformations, etc. at the component level and large number of elements required to model each component for detailed computational studies. Though in some cases it is inevitable to require an understanding of details of movements and deformations, the computational time can be substantially reduced by using multiscale analysis and submodeling techniques. The multiscale analysis starts by developing a computational homogenization framework, which is the subject of the current study, and then using submodeling for specific locations, which is the scope of the next publication of the current authors. Though it is possible to simulate the detailed model of battery modules at subsystem level, the same is not true for vehicle (system) level simulation with an integrated battery pack. The challenge in modeling large battery packs is in modeling hundreds or thousands of cells which exist in them. Even by using homogenized cell models, the finite element model of a battery pack itself can have ten billion elements, which would not be feasible to solve with the current computational power available to most researchers and designers. Hence, more recent studies have opted for homogenized models for modeling the battery modules and packs to reduce computational cost.Sahraei et al. [44] modeled the battery pack as a homogeneous crushable foam bulk material to simulate a Drop Test to predict the initiation and evolution of fracture of the battery pack casing when subjected to impact loading. Gilaki et al. [45] developed a framework for optimal design of lithium-ion battery packs. An isotropic foam material model from LS-Dyna’s material library was calibrated from response obtained from the RVE analysis to describe the behavior of the homogenized. Tang et al. [46] established an isotropic elastic-plastic constitutive law to describe the behavior of a homogenized battery module. A 2D simplification was done by taking the advantage of plane strain conditions. Ali et al. [47], Lai et al. [48], and Amodeo et al. [49] used an RVE model of the battery module to conduct their studies and to calibrate an isotropic foam model to describe behavior of the homogenized battery module. Zhu et al. [50, 51] conducted computational studies on lithium-ion battery modules subjected to mechanical deformation. For these studies, detailed and homogenized models were used. An isotropic model was calibrated to describe the homogenized behavior of the whole battery module. The averaged properties were obtained by uniaxial crushing of the detailed finite element model of the battery module. Pan et al. [52] studied the dynamic behavior of battery modules subjected to crushing loads using a combined finite element analysis- (FEA-) deep neural network (DNN) technique to design durable battery packs. In this work, the battery module was modelled as a homogeneous elastic structure with Young’s modulus and Poisson’s ratio of 500 MPa and 0.01, respectively, which were obtained from the literature.These approaches [44–52] use an isotropic material model at the macroscale. However, a close inspection of the battery module reveals that the structure is anisotropic in nature. Moreover, to simulate the scenarios like crash analysis of vehicles where rigid body rotation of the structure under investigation is possible, a globally defined material axis would lead to inaccurate results. This greatly limits the use of such models to simulation of only simple load cases. To the authors’ best knowledge, there are currently no studies addressing the stated issues. This means the constitutive behavior of battery modules as a bulk homogenized material that can be simulated with computationally efficient elements and integrated with full vehicle models for crashworthiness studies remains largely unknown. This motivates conductance of the current study to shed light on mechanical properties of battery modules and provide guidance on steps required to extract such properties computationally. The methods used in previous studies and the factors leading to its limitations are summarized in Table 1.Table 1
Summary of previous works.
StudyMethodFactors leading to limitationXia et al. [39]Detailed modelingComputational cost is too high for full electric vehicle simulationZhu et al. [40]Scurtu et al. [41]Zhang et al. [42]Xi et al. [43]Sahraei et al. [44]Homogenized modelIsotropic model, global material axis definition. Not a correct representation of the properties at pack level beyond simple loadsGilaki et al. [45]Tang et al. [46]Ali et al. [47]Lai et al. [48]Amodeo et al. [49]Zhu et al. [50, 51]Pan et al. [52]This work utilizes a computational homogenization technique for characterizing the averaged properties of battery modules. For this purpose, an appropriate representative volume element (RVE) is identified in a battery module, and displacement boundary conditions are applied. The averaged macroscopic properties are obtained by applying several types of macroscopic deformations. A nonlinear decoupled constitutive material model along with locally defined material axes is calibrated to model the anisotropic behavior of the battery module. Finally, the homogenized model is validated against the detailed model by comparing the results of standard mechanical abuse test protocols designed by the United States Advanced Battery Consortium (USABC) [53].


## 2. Multiscale Computational Homogenization Approach

Modeling of structures like an EV’s Batteries involves multiple scales which is shown schematically in Figure2. The smallest scale includes the components inside a cell. The jellyroll/electrode stacks inside battery cells consist of a cathode (positive electrode), an anode (a negative electrode), and a separator. The anode is made up of a copper current collector coated with graphite. The cathode is an aluminum foil coated with lithium metal oxide, such as LiCoO2. The separator is a thin polymeric porous membrane that acts as an insulator between the two electrodes. In case of a cylindrical cell, the jellyroll is made by layering down a separator sheet followed by a negative electrode, then another separator, and finally a positive electrode. This multilayer structure forms the RVE of a cell. Later, this sandwich of sheets is rolled up and inserted into the metal shell casing. Then, numerous cells can be put together and held with supporting structures and connections to form a battery module for an EV.Figure 2
Multiscale modeling of LIB.The two methods of modeling such structures are detailed modeling and homogenized modeling. In detailed modeling, all the components of the battery pack are discretized separately, and constitutive models are defined for each component of the system. However, the complexities arising due to this approach are so immense that it becomes impossible to simulate the whole systems, which motivates the use of the homogenized modeling approaches. Amongst the several homogenization methods, computational homogenization is found to be the most efficient method due to its ability to deal with complex structures with strong nonlinear behavior at the subscale. In this method, the effective properties are extracted by performing finite element analysis on representative volume elements (RVE). The computational homogenization method in which the deformation gradient is the link between the scales is called the first-order computational homogenization. In this approach, special types of boundary conditions are imposed for solving the problem at the subscale, and then effective properties are calculated using averaging theorems. In the second-order computational homogenization, higher-order gradients are incorporated to increase the accuracy of the simulation. However, this method is more complicated and computationally expensive to implement. In this work, the first-order computational homogenization method is used to obtain the effective properties of the battery module.


### 2.1. Problem Formulation

In this research, the Lagrangian formulation is used for the analysis. For a material pointX, in the undeformed configuration, the position x at any time t is defined by the deformation mapping:
(1)x=φX,t.The homogenization problem for a continuum at the macroscopic scale can be formulated as [54]
(2)∇PM=0inΩ0,(3)s.t.PMN=T^onδΩ0,N(4)φM=φ^onδΩ0,D,where PM is the macroscopic first Piola-Kirchoff stress tensor, N is the unit normal vector to the domain in the reference configuration, and T^ and φ^ are the prescribed traction and deformation mapping, respectively. Also Ω0 refers to the full domain, δΩ0,N is the Neumann boundary, and δΩ0,D is the Dirichlet boundary of the domain.


### 2.2. Micro- to Macrotransition

For micro-to-macro transition, the Hill-Mandel principle states that the virtual work densities on both scales are equal; for finite strain problem, this is satisfied as [54]:
(5)PM:FM=1V∫Pm:FmdV,where FM is the macroscopic deformation gradient tensor and Pm,Fm are the microscopic first Piola-Kirchoff stress and deformation gradient tensors, respectively. Here, micro and macro do not refer to length scales, but the smaller and bigger scale of the problem under consideration. It is proven that by imposing classical first-order boundary conditions at a microscale the above energetic equivalence (Equation (5)) is satisfied.After solving the microscopic boundary value problem (BVP), the macroscopic properties can be calculated using Equation (6). Therefore, average quantities can be evaluated as
(6)gM=1V∫gmdV,where g=materialpropertystress,strain,anddensity.


### 2.3. RVE Boundary Conditions

To impose the macroscopic stress or strain constraints, different types of boundary conditions can be used on the RVE. The first-order boundary conditions that satisfy the energetic equivalence stated in Equation (5) are displacement, uniform traction, and periodic boundary conditions. In this work, strain-driven displacement boundary conditions are used, which enforce Dirichlet boundary conditions [54],
(7)xm=FMXmonΩ0,D,where xm,Xm are the position vectors in deformed and undeformed configurations at microlevel, respectively.After imposing the macroscopic deformation gradientFM on the RVE, the macroscopic stress tensor PM can be calculated as
(8)PM=1V∫PmdV.In this research, the first-order computational homogenization method is used to obtain the average properties of a battery module.


## 2.1. Problem Formulation

In this research, the Lagrangian formulation is used for the analysis. For a material pointX, in the undeformed configuration, the position x at any time t is defined by the deformation mapping:
(1)x=φX,t.The homogenization problem for a continuum at the macroscopic scale can be formulated as [54]
(2)∇PM=0inΩ0,(3)s.t.PMN=T^onδΩ0,N(4)φM=φ^onδΩ0,D,where PM is the macroscopic first Piola-Kirchoff stress tensor, N is the unit normal vector to the domain in the reference configuration, and T^ and φ^ are the prescribed traction and deformation mapping, respectively. Also Ω0 refers to the full domain, δΩ0,N is the Neumann boundary, and δΩ0,D is the Dirichlet boundary of the domain.


## 2.2. Micro- to Macrotransition

For micro-to-macro transition, the Hill-Mandel principle states that the virtual work densities on both scales are equal; for finite strain problem, this is satisfied as [54]:
(5)PM:FM=1V∫Pm:FmdV,where FM is the macroscopic deformation gradient tensor and Pm,Fm are the microscopic first Piola-Kirchoff stress and deformation gradient tensors, respectively. Here, micro and macro do not refer to length scales, but the smaller and bigger scale of the problem under consideration. It is proven that by imposing classical first-order boundary conditions at a microscale the above energetic equivalence (Equation (5)) is satisfied.After solving the microscopic boundary value problem (BVP), the macroscopic properties can be calculated using Equation (6). Therefore, average quantities can be evaluated as
(6)gM=1V∫gmdV,where g=materialpropertystress,strain,anddensity.


## 2.3. RVE Boundary Conditions

To impose the macroscopic stress or strain constraints, different types of boundary conditions can be used on the RVE. The first-order boundary conditions that satisfy the energetic equivalence stated in Equation (5) are displacement, uniform traction, and periodic boundary conditions. In this work, strain-driven displacement boundary conditions are used, which enforce Dirichlet boundary conditions [54],
(7)xm=FMXmonΩ0,D,where xm,Xm are the position vectors in deformed and undeformed configurations at microlevel, respectively.After imposing the macroscopic deformation gradientFM on the RVE, the macroscopic stress tensor PM can be calculated as
(8)PM=1V∫PmdV.In this research, the first-order computational homogenization method is used to obtain the average properties of a battery module.


## 3. Battery Pack of an Electric City Bus

### 3.1. Components of the Battery Pack

In this section, the battery pack of an electric city bus is being investigated and its components are explained. The architecture of the battery pack consists of 6 modules. Each module consists of 1008 cylindrical cells, arranged as shown in Figure3. The other structural components of the pack are foam supports to secure the cells in position, the cooling system, and sockets. In this research work, all the 3D meshes were generated using Altair Hypermesh, the preprocessing was done using LS-PREPOST, and the calculations were performed using LS-Dyna.Figure 3
Battery pack of city bus.The cylindrical cells used in the battery pack are of 18650 form factor. These cells, from the structural point of view, consist of three major components, jellyroll, shell casing, and endcaps. These cells are stacked together using sockets. These sockets facilitate electric connects and smooth stacking of the cells and damps the vibration up to a certain extent. Since the performance of the lithium-ion battery pack is greatly affected by the temperature of the cells, it is very important to have a cooling system to regulate the temperature during charging and discharging. The battery cooling system of the electric bus consists of cooling channels that wind through the battery pack and carries a liquid coolant. The cells are further secured in place with the help of polyurethane foam. This foam component acts as a cushion for the lithium-ion cells.


### 3.2. Finite Element Model of the RVE

The RVE is assumed to include the smallest repeating unit in the system. In the current battery module, it includes three full cells, two half cells, and surrounding components, such as sockets, foam cushion, and cooling channel (see Figure4(a)). The RVE BVP is solved using LS-Dyna’s Explicit Solver. In this section, the finite element models of the parts are discussed in detail. There is extensive literature on constitutive modeling of various materials used to manufacture the components of the battery pack. In this work, material properties from the literature explained below are used to define the properties of the parts in the RVE. Song et al. [55] determined the averaged mechanical properties of the jellyroll of an 18650 cell in transverse and axial directions. The specifications and properties are listed in Tables 2 and 3, respectively. An anisotropic foam type material model in LS-Dyna (MAT_126_MODIFIED_HONEYCOMB) was used to model the computationally homogenized jellyroll. The authors used a cylindrical coordinate system to model a single cell in the above study.Figure 4
(a) RVE, (b) material axis definition:AOPT=0 [56], and (c) jellyroll mesh.
(a)(b)(c)Table 2
Specifications.
Nominal capacity2600 mAhCharge methodConstant current and constant voltage (4.2 V)Nominal voltage3.7 VMax. charge current2600 mAMax. discharge current5200 mADischarge temperature−20°C−+60°CEnergy density565 Wh/L, 215 Wh kg-1Charging time3 h (standard) or 2.5 h (rapid)Table 3
Properties for the homogenized jellyroll [55].
PropertyValueDensity2.5E−06 kg/mm3Young’s modulus (uncompacted)500 MPaShear modulus (uncompacted)250 MPaSince the jellyroll exhibits an anisotropic behavior, a proper definition of material axes is crucial. While it is enough to use a cylindrical coordinate system for a single cell simulation, such coordinate system will not be valid for collection of cells in a RVE, where each cell has its own cylindrical coordinates. In the present research, a modification was performed on the material coordinate system to make it local for each cell. Since the battery module is subjected to complex deformations, global material axes would yield inaccurate results. Hence, a locally orthotropic system with material axes determined by element nodes is used (in LS-Dyna keyword,AOPT=0). Figure 4(b) shows the establishment of material axes by the means of element nodes. The direction “a” is from node 1 to node 2 of the element. The direction “b” is orthogonal to direction “a” and is in the plane formed by nodes 1, 2, and 4. Direction “c” is the direction of “a×b.” To get accurate results, it is necessary to have the elements of the jellyroll to be arranged concentrically. However, it is not possible to have a uniform mesh size, which is required when failure criteria are to be defined. Since much of the deformation happens at the outer layers of the jellyroll, it is found that having a concentric mesh up to a certain extent (see Figure 4(c)) yields accurate results. Here, vector “A” is radial, vector “B” represents tangential, and vector “C” (perpendicular to plane of the paper) represents the axial direction. The input curves for this anisotropic model are presented in Figure 5.Figure 5
Input curves: (a) jellyroll shear, (b) axial direction, and (c) radial and tangential direction. According to LS Dyna convention for the input curves of MAT Honeycomb, compression is shown in the positive quarter.
(a)(b)(c)The shell casing and the endcaps are made up of stainless steel and modeled with very fast fully integrated shell elements (Elform Type 16). A linear-elastic-plastic material model (MAT_PIECEWISE_ LINEAR_PLASTICITY) is used to define its material properties (see Table4). For the purposes of the present study and to satisfy requirements of computational homogenization in conservation of mass, no material failure was considered for any of the components in the RVE. A study of fractures or failures will be covered under submodeling work which will be subject of future publication of the authors, where all components will be modeled in details in a local area of the battery pack, subjected to extreme deformations conditions that warrant failure prediction.Table 4
Properties for shell casing and caps.
PropertyValueDensity7.8E−6 kg/mm3Young’s modulus207 GPaPoison’s ratio0.3Yield strength253 MPaTangent modulus1.035 GPaTo validate the cell model, it was simulated under four different load cases and the results were compared to the experimental data. The results of this study (see Figure6) show that the present model of the cell with the modifications of (i) jellyroll coordinate system, (ii) isotropic material used for the shell casing of the cell, and (iii) no failure criteria for either jellyroll or casing components yields good results up to 6 mm of displacement in each loading case, which is the extent required for the current research work. The rest of components of the RVE were simulated per well studied methods reported in the literature. The cooling channel is made up of Aluminum 6063 alloy and modeled with first-order 8-node linear hex fully integrated solid elements (Elform Type 2). A linear-elastic-plastic material model (MAT_PIECEWISE_ LINEAR_PLASTICITY) is used to define its material properties [57] (see Table 5).Figure 6
Cell validation: test vs. simulation: (a) rod compression, (b) 3-point bending, (c) hemispherical compression, and (d) axial compression.
(a)(b)(c)(d)Table 5
Properties for Aluminum 6063 [57].
PropertyValueDensity2.7E−6 kg/mm3Young’s modulus68.9 GPaPoison’s ratio0.3Yield strength214 MPaTangent modulus35 MPaThe polyurethane foam supports are modeled with first-order 6-node linear wedge fully integrated solid elements (Elform Type 2). A low-density foam material model (∗MAT_LOW_DENSITY_VISCOUS_FOAM) is used to define its elasto-viscoplastic behavior [58] (see Table 6). The input curve for this foam is shown in Figure 7. Additional stiffness is added to the densification region to avoid numerical instability (negative volume issue).Table 6
Polyurethane foam [58].
PropertyValueDensity2.700E−08 kg/mm3Young’s modulus0.1000000 GPaTensile cutoff strength0.0075 MPaFigure 7
Input curve for the foam support (compressive properties).The sockets are made up of polycarbonate material and modeled with first-order 4-node tetrahedral fully integrated solid elements (Elform Type 2). A linear-elastic-plastic material model (MAT_PIECEWISE_LINEAR_PLASTICITY) is used to define its material behavior [59] (see Table 7).Table 7
Properties for polycarbonate plastic [59].
PropertyValueDensity1.200E−06 kg/mm3Young’s modulus1.5300000 GPaPoison’s ratio0.38Yield strength63 MPaTangent modulus35 MPaThe interfaces between the cooling channel and the foam material and between the cell holders are bonded with an adhesive. Since the type of adhesive is not known, it is assumed to be perfectly bonded, and the nodes were merged to simulate such behavior. Contact between other components and self-contacts are modeled using a single surface algorithm (∗CONTACT_AUTOMATIC_SINGLE_SURFACE). The rigid walls were modelled using RIGIDWALL option in Ls-Dyna. ∗RIGIDWALL_PLANAR was used for stationary rigid walls, and ∗RIGIDWALL_GEOMETRIC_FLAT_MOTION was used for moving rigid walls. The displacement boundary conditions are prescribed by coupling the global displacements of boundary nodes when the RVE is subjected to a uniaxial state of stress. When the RVE is subjected to shear loading, the boundary nodes are subjected to constrained global rotation. These boundary conditions ensure that straight lines deform to straight lines, planes deform to planes, and cube or rectangular parallelepiped deform to parallelepiped.The macroscopic strain in the case of uniaxial stress state is prescribed by using two rigid walls with a coefficient of friction equal to zero to allow sliding along the plane without inducing stress in other directions. The lower rigid wall is set stationary, and the prescribed motion is applied on the upper wall. For inducing shear stress, motion is prescribed directly on the boundary nodes, as shown in Figure8. The average mesh size of all the components is 1 mm.Figure 8
Implementation of boundary conditions: (a) normal stress and (b) shear stress.Once the finite element analysis is done, macroscopic properties are obtained from the Finite Element response. The macroscopic First-Piola-Kirchoff stress is computed in terms of reaction forces as(9)PM=1VRVE∗fxx∗Xfyx∗Xfzx∗Xfxy∗Yfyy∗Yfzy∗Yfxz∗Zfyz∗Zfzz∗Z,where fmm is the reaction contact force in mth direction when the motion of rigid wall is along m-axis, fnm is the reaction force on the defined node-set perpendicular to mth direction (shearing plane) when the motion of node-set is along the n-axis, VRVE is the volume of the RVE, and X,Y,Z is the dimensions of RVE in X,Y,Z directions, respectively.


## 3.1. Components of the Battery Pack

In this section, the battery pack of an electric city bus is being investigated and its components are explained. The architecture of the battery pack consists of 6 modules. Each module consists of 1008 cylindrical cells, arranged as shown in Figure3. The other structural components of the pack are foam supports to secure the cells in position, the cooling system, and sockets. In this research work, all the 3D meshes were generated using Altair Hypermesh, the preprocessing was done using LS-PREPOST, and the calculations were performed using LS-Dyna.Figure 3
Battery pack of city bus.The cylindrical cells used in the battery pack are of 18650 form factor. These cells, from the structural point of view, consist of three major components, jellyroll, shell casing, and endcaps. These cells are stacked together using sockets. These sockets facilitate electric connects and smooth stacking of the cells and damps the vibration up to a certain extent. Since the performance of the lithium-ion battery pack is greatly affected by the temperature of the cells, it is very important to have a cooling system to regulate the temperature during charging and discharging. The battery cooling system of the electric bus consists of cooling channels that wind through the battery pack and carries a liquid coolant. The cells are further secured in place with the help of polyurethane foam. This foam component acts as a cushion for the lithium-ion cells.


## 3.2. Finite Element Model of the RVE

The RVE is assumed to include the smallest repeating unit in the system. In the current battery module, it includes three full cells, two half cells, and surrounding components, such as sockets, foam cushion, and cooling channel (see Figure4(a)). The RVE BVP is solved using LS-Dyna’s Explicit Solver. In this section, the finite element models of the parts are discussed in detail. There is extensive literature on constitutive modeling of various materials used to manufacture the components of the battery pack. In this work, material properties from the literature explained below are used to define the properties of the parts in the RVE. Song et al. [55] determined the averaged mechanical properties of the jellyroll of an 18650 cell in transverse and axial directions. The specifications and properties are listed in Tables 2 and 3, respectively. An anisotropic foam type material model in LS-Dyna (MAT_126_MODIFIED_HONEYCOMB) was used to model the computationally homogenized jellyroll. The authors used a cylindrical coordinate system to model a single cell in the above study.Figure 4
(a) RVE, (b) material axis definition:AOPT=0 [56], and (c) jellyroll mesh.
(a)(b)(c)Table 2
Specifications.
Nominal capacity2600 mAhCharge methodConstant current and constant voltage (4.2 V)Nominal voltage3.7 VMax. charge current2600 mAMax. discharge current5200 mADischarge temperature−20°C−+60°CEnergy density565 Wh/L, 215 Wh kg-1Charging time3 h (standard) or 2.5 h (rapid)Table 3
Properties for the homogenized jellyroll [55].
PropertyValueDensity2.5E−06 kg/mm3Young’s modulus (uncompacted)500 MPaShear modulus (uncompacted)250 MPaSince the jellyroll exhibits an anisotropic behavior, a proper definition of material axes is crucial. While it is enough to use a cylindrical coordinate system for a single cell simulation, such coordinate system will not be valid for collection of cells in a RVE, where each cell has its own cylindrical coordinates. In the present research, a modification was performed on the material coordinate system to make it local for each cell. Since the battery module is subjected to complex deformations, global material axes would yield inaccurate results. Hence, a locally orthotropic system with material axes determined by element nodes is used (in LS-Dyna keyword,AOPT=0). Figure 4(b) shows the establishment of material axes by the means of element nodes. The direction “a” is from node 1 to node 2 of the element. The direction “b” is orthogonal to direction “a” and is in the plane formed by nodes 1, 2, and 4. Direction “c” is the direction of “a×b.” To get accurate results, it is necessary to have the elements of the jellyroll to be arranged concentrically. However, it is not possible to have a uniform mesh size, which is required when failure criteria are to be defined. Since much of the deformation happens at the outer layers of the jellyroll, it is found that having a concentric mesh up to a certain extent (see Figure 4(c)) yields accurate results. Here, vector “A” is radial, vector “B” represents tangential, and vector “C” (perpendicular to plane of the paper) represents the axial direction. The input curves for this anisotropic model are presented in Figure 5.Figure 5
Input curves: (a) jellyroll shear, (b) axial direction, and (c) radial and tangential direction. According to LS Dyna convention for the input curves of MAT Honeycomb, compression is shown in the positive quarter.
(a)(b)(c)The shell casing and the endcaps are made up of stainless steel and modeled with very fast fully integrated shell elements (Elform Type 16). A linear-elastic-plastic material model (MAT_PIECEWISE_ LINEAR_PLASTICITY) is used to define its material properties (see Table4). For the purposes of the present study and to satisfy requirements of computational homogenization in conservation of mass, no material failure was considered for any of the components in the RVE. A study of fractures or failures will be covered under submodeling work which will be subject of future publication of the authors, where all components will be modeled in details in a local area of the battery pack, subjected to extreme deformations conditions that warrant failure prediction.Table 4
Properties for shell casing and caps.
PropertyValueDensity7.8E−6 kg/mm3Young’s modulus207 GPaPoison’s ratio0.3Yield strength253 MPaTangent modulus1.035 GPaTo validate the cell model, it was simulated under four different load cases and the results were compared to the experimental data. The results of this study (see Figure6) show that the present model of the cell with the modifications of (i) jellyroll coordinate system, (ii) isotropic material used for the shell casing of the cell, and (iii) no failure criteria for either jellyroll or casing components yields good results up to 6 mm of displacement in each loading case, which is the extent required for the current research work. The rest of components of the RVE were simulated per well studied methods reported in the literature. The cooling channel is made up of Aluminum 6063 alloy and modeled with first-order 8-node linear hex fully integrated solid elements (Elform Type 2). A linear-elastic-plastic material model (MAT_PIECEWISE_ LINEAR_PLASTICITY) is used to define its material properties [57] (see Table 5).Figure 6
Cell validation: test vs. simulation: (a) rod compression, (b) 3-point bending, (c) hemispherical compression, and (d) axial compression.
(a)(b)(c)(d)Table 5
Properties for Aluminum 6063 [57].
PropertyValueDensity2.7E−6 kg/mm3Young’s modulus68.9 GPaPoison’s ratio0.3Yield strength214 MPaTangent modulus35 MPaThe polyurethane foam supports are modeled with first-order 6-node linear wedge fully integrated solid elements (Elform Type 2). A low-density foam material model (∗MAT_LOW_DENSITY_VISCOUS_FOAM) is used to define its elasto-viscoplastic behavior [58] (see Table 6). The input curve for this foam is shown in Figure 7. Additional stiffness is added to the densification region to avoid numerical instability (negative volume issue).Table 6
Polyurethane foam [58].
PropertyValueDensity2.700E−08 kg/mm3Young’s modulus0.1000000 GPaTensile cutoff strength0.0075 MPaFigure 7
Input curve for the foam support (compressive properties).The sockets are made up of polycarbonate material and modeled with first-order 4-node tetrahedral fully integrated solid elements (Elform Type 2). A linear-elastic-plastic material model (MAT_PIECEWISE_LINEAR_PLASTICITY) is used to define its material behavior [59] (see Table 7).Table 7
Properties for polycarbonate plastic [59].
PropertyValueDensity1.200E−06 kg/mm3Young’s modulus1.5300000 GPaPoison’s ratio0.38Yield strength63 MPaTangent modulus35 MPaThe interfaces between the cooling channel and the foam material and between the cell holders are bonded with an adhesive. Since the type of adhesive is not known, it is assumed to be perfectly bonded, and the nodes were merged to simulate such behavior. Contact between other components and self-contacts are modeled using a single surface algorithm (∗CONTACT_AUTOMATIC_SINGLE_SURFACE). The rigid walls were modelled using RIGIDWALL option in Ls-Dyna. ∗RIGIDWALL_PLANAR was used for stationary rigid walls, and ∗RIGIDWALL_GEOMETRIC_FLAT_MOTION was used for moving rigid walls. The displacement boundary conditions are prescribed by coupling the global displacements of boundary nodes when the RVE is subjected to a uniaxial state of stress. When the RVE is subjected to shear loading, the boundary nodes are subjected to constrained global rotation. These boundary conditions ensure that straight lines deform to straight lines, planes deform to planes, and cube or rectangular parallelepiped deform to parallelepiped.The macroscopic strain in the case of uniaxial stress state is prescribed by using two rigid walls with a coefficient of friction equal to zero to allow sliding along the plane without inducing stress in other directions. The lower rigid wall is set stationary, and the prescribed motion is applied on the upper wall. For inducing shear stress, motion is prescribed directly on the boundary nodes, as shown in Figure8. The average mesh size of all the components is 1 mm.Figure 8
Implementation of boundary conditions: (a) normal stress and (b) shear stress.Once the finite element analysis is done, macroscopic properties are obtained from the Finite Element response. The macroscopic First-Piola-Kirchoff stress is computed in terms of reaction forces as(9)PM=1VRVE∗fxx∗Xfyx∗Xfzx∗Xfxy∗Yfyy∗Yfzy∗Yfxz∗Zfyz∗Zfzz∗Z,where fmm is the reaction contact force in mth direction when the motion of rigid wall is along m-axis, fnm is the reaction force on the defined node-set perpendicular to mth direction (shearing plane) when the motion of node-set is along the n-axis, VRVE is the volume of the RVE, and X,Y,Z is the dimensions of RVE in X,Y,Z directions, respectively.


## 4. Homogenized Response under Various Loading Conditions

In this section, the average response of the RVE subjected to various macroscopic deformations is discussed. The first loading scenario was uniaxial compression alongx-axis, Figure 9(a). In this case, the lithium-ion cells are compressed along the transverse direction. The force-displacement curve shows an initial linear behavior, with a local peak at 38.8 kN followed by a small region of constant force at 34.3 kN (see Figure 10(a)). This flat region is when the cooling channel starts to buckle. In the later stage, when the gaps between components close, the load increases rapidly to a maximum of 196 kN at 22.5 mm displacement. It should be noted that in presenting the force-displacement curves here, the positive axis is showing compression; this is consistent with LS Dyna’s convention for input curves of the honeycomb material. Under uniaxial tension in the X direction, the sockets, cooling channels, and foams will be tolerating the force. However, the cells will move along with these components without much contribution to the tensile strength of the RVE, because individual cells can move without any contact with the surrounding ones. Therefore, the strength of the RVE in tension is much lower than that in compression.Figure 9
Deformed configuration: (a) normal compression (xx), (b) normal compression (yy), (c) normal compression (zz), (d) shear loading (xy), (e) shear loading (zy), and (f) shear loading (zx).
(a)(b)(c)(d)(e)(f)Figure 10
Force vs. displacement: (a) normal compression (xx), (b) normal compression (yy), (c) normal compression (zz), (d) shear loading (xy), (e) shear loading (zy), and (f) shear loading (zx).
(a)(b)(c)(d)(e)(f)The second case simulated was uniaxial compression in theY direction (Figure 9(b)). In this type of loading, the cell is compressed in axial direction. Here again, the force-displacement curve shows a linear behavior in the beginning, until reaching a force of 27.6 kN (Figure 10(b)). After this stage, the force drops with an increase in displacement, indicating the initiation of buckling of the shell casings of the cells and the cooling channel. Once all the components are compacted, a similar rapid increase in force is observed, indicating densification of the RVE. The maximum force in this simulation is 88.4 kN at the maximum displacement of 25 mm. The tension in this direction mostly involves only the foam and the cooling channel. With minimal tension, the cells move out of the sockets, and these components seize to contribute to the tensile strength. This loading quickly causes localization of deformation in cooling channel, and the maximum force asymptotes to 10 kN.The third scenario was uniaxial compression along theZ direction (Figure 9(c)). In this case, the cooling channel and the foam are compressed along their thickness, unlike the X and Y directions, in which these components are under in-plane loading and buckle (see Figure 10(c)). The lithium-ion cells are compressed transversely too, so there is no buckling in the shell casings of the cells either. In this case, the linear region is very small and gentle. The initial slope of the load-displacement curve is also smaller than that for the other two directions. The maximum force is reached at 23.7 mm displacement and is 189 kN. Under tensile loading in this direction, the cooling channel layers start to separate, and the displacement localizes in the narrow cross-section in the two ends of cooling channel where the two layers are connected. The maximum tensile force in this direction is at 4.5 kN, which is much smaller than those of the two other loading directions.Since, in a crash scenario, the battery module may also experience a substantial amount of shear deformation along with compression, it is important to incorporate shear behavior in the macroscopic model to increase the accuracy of the full-scale simulation. Three shear loading scenarios were also studied (see Figures9(d)–9(f) for the deformed configuration and Figures 10(d)–10(f) for the force displacement curves). The results show that the strength in shear for the YZ and ZX directions is much lower when compared to compression, with the maximum shear force being less than 10 kN. The main reason for the low shear strength in the RVE is that in these cases, the contributing parts are the sockets, cooling channel, and foam. The cells just dislodge from their socket connection and have minimal contribution to the strength of the RVE under shear loading. In the case of XY direction, we see the comparatively high force due to the additional shearing of the metallic cooling channel.Poisson’s effect was also studied. It was observed that in pure compression, the change in transverse dimension is negligible compared to that in the axial change in dimension. Figure11(a) shows the initial and deformed configuration when subjected to pure compression, and Figure 11(b) shows the variation of Poisson’s ratio vs. strain. This shows that Poisson’s effect is negligible.Figure 11
(a) Change in transverse dimension (pure compression); (b) Poisson’s ratio vs. strain.
(a)(b)


## 5. Constitutive Material Model for Homogenized Battery Module

After obtaining the mechanical response of the RVE, a material model from LS-Dyna’s material library was selected to describe the nonlinear multiaxial behavior of the homogenized battery module at the macro scale. The battery module is assumed to be a continuum at this scale. Material model MAT_126 (Modified Honeycomb) from LS-DYNA’s material library was selected for this purpose. The motivation for selecting this model lies in the behavior observed in Section4. This material model is majorly used to model honeycomb and foam materials with anisotropic behavior. A nonlinear elastoplastic material behavior can be defined separately for all normal and shear stresses.The mathematical formulation of MAT_126 has two deformation phases, which are independent of each other. These are uncompacted and compacted phases. In the uncompacted phase, the stress and strain are uncoupled in all the directions and each element is treated as six independent 1-D elements (3-Normal and 3 Shear). In the second phase, i.e., compacted phase, the material behavior is assumed to be elastic-perfectly plastic.The detailed mathematical formulation for this constitutive model is given in LS-Dyna’s user’s manual Volume II [56]. The software has several different approaches for the implementation of this material model.The moduli (elastic and shear) in the uncompacted phase vary from initial to fully compacted value as(10)Eii=Eiiuncp+βEcp−Eiiuncp,(11)Gij=Gijuncp+βEcp21+μ−Gijuncp,where Ecp is the elastic modulus (compression) in fully compacted phase, Eiiuncp,Gijuncp is the elastic and shear moduli in uncompacted phase, and μ is Poisson’s ratio (compacted phase).
(12)β=maxmin1−VR1−VRf,1,0,where VR is the relative volume and VRf is the relative volume at which the honeycomb becomes fully compacted.For element type 0, the stress tensor components remain uncoupled throughout the deformation. So, in this case,Ecp is ignored. Eiiuncp and Gijuncp are used.The stress in the uncompacted phase is updated using the input stress-strain curve as(13)σiin+1trial=σiin+EiiΔεii,(14)σijn+1trial=σijn+2GijΔεij,where Δεij are the strain increment.Then, it was made sure that these stress components do not exceed the value defined in the load curves.If(15)σijn+1trial>λσijεij,Then,(16)σijn+1=σijεijλσijn+1trialσijn+1trial.The values forσijεij are defined in the load curves. λ is a parameter which is defined as a function of strain-rate. If the strain-rate dependence is not defined, the value of λis 1.For element formulation 1, in the fully compacted phase, the material is assumed to be elastic perfectly plastic. The stress is updated.(17)sijn+1trial=sijn+2GijΔεijdevn+1/2.Δεijdev is the deviatoric strain increment.After this, the value of yield stress defined in the material card is compared to the effective trial stress.(18)sefftrial=32∗sijtrial∗sijtrial1/2to find if the yield strength in compacted material is exceeded by effective trial stress. If this happens, the stress is scaled back to the yield surface.
(19)sijn+1=σysefftrialsijtrial.After this, the pressurep is found as
(20)pn+1=pn−KΔεkkn+1/2,whereK is the bulk modulus.Finally, the Cauchy stress is calculated as(21)σijn+1=sijn+1−pn+1δij.The material axes of the homogenized battery module are defined using a locally defined material axes. The definition of such axes is explained in Section3.2. During the process of deformation, if the homogenized structure undergoes substantial rigid body rotations, which is commonly seen in crash test, the response obtained with global coordinate systems will be inaccurate. Hence, the inaccuracies due to rigid body rotations in such simulations are eliminated by using locally defined material axes.Ecp is a property of fully compacted honeycomb, and in the present study, it is considered equal to Young’s modulus of aluminum, when element formulation 1 is used. The input curves for MAT_126 with element formulation 0 should use engineering strain, and logarithmic strain is used for element formulation 1 [56]. For this research, both Elform 0 and 1 were used, and it was concluded that no significant difference is observed in the simulations with using either of these element formulations. Therefore, for simplicity, the research continued by using ELform=0 with hourglass control 6. ∗DAMPING_FREQUENCY_RANGE keyword was used to damp the vibrations in the homogenized model. The range was set to 0.05 to 3 Hz with damping in fraction of critical CDAMP=0.05. The density at the macrolevel was calculated by dividing the mass of the detailed model by the volume of the bulk homogenized model. To develop the input curves for the material card, first, the force-displacement curves were converted to stress strain relations, using Equation (9). The elastic modulus in the uncompacted configuration corresponds to the slope of the initial part of the curve. In the next step, these stress-strain curves are modified as per LS-Dyna user manual recommendations. For the compression curve, the initial part of the curve is removed, and the curve is extended to the zero-strain value which corresponds to the yield stress. The corresponding tensile stress starts from this stress value at zero strain and increases linearly with the slope equivalent to the tangent modulus along that direction. For the shear load case, the initial linear part is removed, and the curve is simply extended to the zero-strain value. The homogenized input curves in all directions are shown in Figure 12. It should be noted that formatting of the stress-strain curves as explained will result in slight elevation of the strength of the cell in uniaxial tensile loading (see Figure 13). However, the difference is minimal and acceptable to ensure stability of the model. Although it was found that the effect of the slope is not significant for this kind of applications, it should be ensured that the extrapolated stress value in tension should not become negative at any point, or else numerical instability might occur.Figure 12
Input stress-strain curves: (a) normal compression (xx), (b) normal compression (yy), (c) normal compression (zz), (d) shear loading (xy), (e) shear loading (zy), and (f) shear loading (zx).
(a)(b)(c)(d)(e)(f)Figure 13
Stress-strain curves: original vs. input.


## 6. Validation

Two different load cases were taken to ensure the versatility of the homogenized model. Two models were used, one where the interior of battery module was modeled with homogenized elements and another where the interior was modeled with all the details of cells, sockets, foams, and cooling channels. The load cases studied were actual battery abuse test procedures defined by the United States Advanced Battery Consortium (USABC). These tests were designed by USABC to simulate actual accidental abusive conditions and potential internally initiated failures that may be experienced in batteries [53]. The LS-Dyna MPP solver was used to simulate these test scenarios. These simulations were run on Temple University’s Owlnest cluster with hardware specifications listed in Table 8.Table 8
HPC specs.
DetailsModelDell PowerEdge R830Processor4x Intel Xeon E5-4655 v4No. of nodes1No. of cores32RAM1536 GB


### 6.1. USABC Crush Test

In this test, the battery module is crushed with a corrugated platen, as shown in Figure14. According to USABC guideline, the compression should continue to 50% of thickness, unless the crushing force reaches a limit of 1000 times the weight of the battery module. In this case, the latter was reached first.Figure 14
Crush platen [53].A mesh convergence study was performed to find the effect of mesh size on the desired results. For this purpose, the USABC Crush test was simulated with the homogenized battery module with mesh size varying from 3 mm to 10 mm. The results (Figure15) show that the effect of mesh size is not very significant. Owing to the computational cost and accuracy, 7 mm mesh size was chosen.Figure 15
Mesh dependence study.Figure16 shows the deformed shape of the detailed and homogenized battery module. The simulation in this case results in significant computational saving, where the detailed model takes about 74 hours to run, and the homogenized model takes about 11 minutes. The force-displacement shows very close prediction (see Figure 17(a)). The coefficient of determination (R2 value) for this loading case was at 0.9699.Figure 16
Deformation: detailed vs. homogenized: USABC Crush Test.Figure 17
Force vs. displacement: detailed vs. homogenized: (a) USABC Crush Test and (b) USABC Drop Test.
(a)(b)


### 6.2. USABC Drop Test

The second validation study was simulation of USABC Drop experiment. In this case, the battery module is dropped from a height of 10 m on a cylindrical steel object placed centrally. The cylinder has a diameter of 300 mm. This test can also be carried out in horizontal arrangement with equivalent impact velocity. Figure18 shows the deformed shape of the detailed and homogenized battery modules. From Figure 17(b), it is evident that the force-displacement shows a good correlation in this case as well, with a coefficient of determination (R2 value) at 0.9871. The computational savings in this case are also significant, from 40 hours in the case of detailed model to less than 4 minutes for the homogenized model.Figure 18
Deformation: detailed vs. homogenized: USABC Drop Test.


## 6.1. USABC Crush Test

In this test, the battery module is crushed with a corrugated platen, as shown in Figure14. According to USABC guideline, the compression should continue to 50% of thickness, unless the crushing force reaches a limit of 1000 times the weight of the battery module. In this case, the latter was reached first.Figure 14
Crush platen [53].A mesh convergence study was performed to find the effect of mesh size on the desired results. For this purpose, the USABC Crush test was simulated with the homogenized battery module with mesh size varying from 3 mm to 10 mm. The results (Figure15) show that the effect of mesh size is not very significant. Owing to the computational cost and accuracy, 7 mm mesh size was chosen.Figure 15
Mesh dependence study.Figure16 shows the deformed shape of the detailed and homogenized battery module. The simulation in this case results in significant computational saving, where the detailed model takes about 74 hours to run, and the homogenized model takes about 11 minutes. The force-displacement shows very close prediction (see Figure 17(a)). The coefficient of determination (R2 value) for this loading case was at 0.9699.Figure 16
Deformation: detailed vs. homogenized: USABC Crush Test.Figure 17
Force vs. displacement: detailed vs. homogenized: (a) USABC Crush Test and (b) USABC Drop Test.
(a)(b)


## 6.2. USABC Drop Test

The second validation study was simulation of USABC Drop experiment. In this case, the battery module is dropped from a height of 10 m on a cylindrical steel object placed centrally. The cylinder has a diameter of 300 mm. This test can also be carried out in horizontal arrangement with equivalent impact velocity. Figure18 shows the deformed shape of the detailed and homogenized battery modules. From Figure 17(b), it is evident that the force-displacement shows a good correlation in this case as well, with a coefficient of determination (R2 value) at 0.9871. The computational savings in this case are also significant, from 40 hours in the case of detailed model to less than 4 minutes for the homogenized model.Figure 18
Deformation: detailed vs. homogenized: USABC Drop Test.


## 7. Discussion

The validation results demonstrate a very good agreement between the homogenized and detailed models in both the case studies. This was achieved with significant savings in computational time (as was observed in Table9). The loading conditions included complex deformation loadings with static initial conditions, as well as dynamic drop with an initial velocity. These facts make it quite evident that the proposed homogenization approach is a very efficient tool for simulating the crash behavior of the battery pack when integrating it into a full vehicle model in various crash scenarios. During a crash, the battery pack experiences a variety of forces and deformations in different locations. The calculated stress and strain values in the elements of the homogenized model from the vehicle level simulation then can be applied back to the detailed battery pack with cells and other components, to evaluate the possibility of failure in the module or at the cell level in a full circle multiscale evaluation of the crash safety. The above procedure is subject to the next publication of the authors but beyond the scope of the current manuscript.Table 9
Computational cost of detailed vs. homogenized mode.
DetailedHomogenizedNo. of phases61Nodes11,137,684118,432Elements10,070,848105,430Average element size1 mm7 mmCrush Test Simulation Time (HH : MM : SS)74 : 10 : 160 : 11 : 14Drop Test Simulation Time (HH : MM : SS)40 : 07 : 550 : 03 : 51The developed homogenized models serve two major purposes. First, it acts as a surrogate to the complex construction of the battery pack in the full-scale simulations. It can be used to simulate the crash tests per the standards prescribed by FMVSS for safety evaluation of electric vehicles and crashworthiness and occupant protection analysis. It is fascinating to note that around 100 prototype vehicles are tested to ensure crashworthiness and compliance with the international and country-specific regulations of a vehicle which can cost a manufacturer around $400,000-$750,000 per prototype at the initial stage, which is why manufacturers are pushing for the need to simulate crash events using computational models [60]. If an attempt is made to simulate the full-scale crash test with the detailed model of just one battery module, it would take months to simulate 1 millisecond of the full vehicle crash with the most powerful HPC system available to most scientists and researchers. It should be noted that there are in total 12 battery modules installed on the electric bus subject of the present study.Second, the approach presented here facilitates the design changes required to optimize the weight, cost, and safety of the vehicle by efficiently designing the protective structures around the battery packs. Full vehicle simulations provide predictions about how the battery pack is affected during the crash and allow the designer to investigate if the damage is under safe limits. If not, changes can be made in the design to minimize the load and deformation transferred to the battery pack. This practice reduces overdesigning of the protective structures, due to the current practice used by many manufactures where the battery pack is considered a black box with mass and inertia and supporting structure is overly augmented to ensure zero deformation to this box. The present status of overdesigned structures in many EVs on the roads today reduces vehicle range and increases the cost due to bulk mass around the battery pack. This may be proved to not have any protective effects on the safety of the battery pack while increasing the mass and acceleration pulse transferred to the occupants, only because of a lack of proper understanding about the deformation and failure inside the formerly black box pack structures.While it is reasonable to assume that the model would yield accurate results in all the loading scenarios prescribed by crash safety regulatory bodies, a limitation could be lack of accuracy if the deformation is overly localized. The extent of localization and error that would jeopardize the predictions of this model has not been studied in the current work and will be subject of future investigations. This type of error is always a concern in multi-scale/homogenization studies. This study did not include strain-rate dependence and failure parameters, which also falls in the scope of on-going and future works.


## 8. Conclusion

This work mainly focuses on the evaluation of homogenized properties for the battery modules of electric vehicles for crashworthiness analyses. Results from these studies show that the homogenized model is versatile, robust, and computationally efficient when compared to the detailed model (R2values=0.9699and0.9871). This method significantly reduces the computational cost, ultimately leading to efficient optimization and safety evaluation for full-scale crashworthiness analysis of electric vehicles.



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*Source: 9210078-2023-02-13.xml*