Lead Acid Battery Model Matlab

Faster Lead-Acid Battery Simulations from Porous-Electrode Theory:

Ii. Asymptotic Analysis

Valentin Sulzera,∗, S. Jon Chapmana,c, Colin P. Pleasea,c, David A. Howeyb,c, Charles W. Monroeb,c

Abstract

Electrochemical and equivalent-circuit modelling are the two most popular approaches to battery simulation, but the former is computationally expensive and the latter provides limited physical insight.

A Theoretical Middle Ground

would be useful to support battery management, on-line diagnostics, and cell design. We analyse a thermodynamically consistent, isothermal porous-electrode model of a discharging lead-acid battery. Asymptotic analysis of this full model produces three reduced-order models, which relate the electrical behaviour to microscopic material properties, but simulate discharge at speeds approaching an equivalent circuit.

A Lumped-Parameter Model, Which Neglects Spatial

property variations, proves accurate for C-rates below 0.1C, while a spatially resolved higher-order solution retains accuracy up to 5C. The problem of parameter estimation is addressed by fitting experimental data with the reduced- order models.

1. Introduction

The popular equivalent-circuit approach to battery mod- elling is efficient, but has limited physical detail and ex- trapolates poorly. Electrochemical models require far more computational power, but include detailed de- scriptions of physical mechanisms, which presumably en- hances predictive capability. Battery management could be improved if there existed easily-solved models with greater mechanistic detail. To that end, this paper puts forward several reduced-order models of lead-acid battery discharge, each derived from a mechanistic description based on an extension of Newman’s porous-electrode theory , which we developed in part I.

Several authors have simplified mechanistic lead-acid- battery models to improve their computational efficiency. Newman and Tiedemann recognise that spatial gradi- ents can be ignored at low current; they state a ‘lumped parameter model’ (LPM) that depends only on time, but do not show how it derives from a porous-electrode model.

Gandhi et al. propose a LPM to underpin an analytical current/voltage relation. Knauff simplifies a porous- electrode model by assuming, without justification, that current is linear in space, and acid molarity, quadratic.

We Deploy Perturbation Methods To Produce A Hi-

erarchy of increasingly complex models. After nondimen-

∗Corresponding Author

sionalization, a diffusional C-rate, Cd–the C-rate scaled with the diffusion time-scale—is found to control how sim- ply the full model can be approximated. Three reduced- order models are derived, validated against the full model, and applied to experiments for parameter estimation.

A leading-order expansion in the diffusional C-rate pro- duces a LPM of the Newman–Tiedemann type, found to be accurate for C-rates below 0.1C. The first-order expan- sion accounts for quasi-static spatial heterogeneity within the electrode sandwich. As well as improving the fit of the full model, this correction has a computationally efficient closed-form expression. Finally, the first-order solution is improved by accounting for diffusion transients. This com- posite model includes just one linear partial differential equation, but matches the full model well up to 5C.

2. Dimensionless Model

In part I, we proposed a general three-dimensional, thermodynamically consistent, isothermal porous-electrode model of a discharging lead-acid battery.

The Detailed

model was simplified slightly on the basis of dimensional analysis to allow solution in a one-dimensional setting. After nondimensionalization, we obtained the following dimensionless system governing the electrolyte concentra- tion c, porosity ε, current density i and potential Φ, elec- trode current density is and potential Φs, and interfacial Preprint submitted to Journal of the Electrochemical Society

1 −ℓp < X < 1. (2.1M)

Equation (2.1e) with the boundary conditions (2.1h) and

(2.1N)

where property values in the negative and positive elec- trode are designated with subscripts n and p, respectively. Typical values of the dimensionless parameters Cd, ιs, βsurf, γdl, ℓ, s, q0, εmax and ε∆are given in Table 1, while concentration-dependent functions D, κ, χ, j0 and U are given in Table A.1. The dimensionless applied current is icell(t) = Icircuit(t)/8Acs, where Icircuit(t) is the applied current in the external circuit and Acs is the electrode cross-sectional area. We define ¯i to be the maximum value of icell(t) with respect to time.

The key parameter is the diffusional C-rate, Cd, which is the C-rate as measured on the diffusion time-scale (or alternatively, the ratio of the applied current scale to the scale of the limiting current).

In the Results section, we will take q0 to be unity (the battery starts from a fully charged state) unless explicitly stated.

Dimensionless Parameters, Relative To The C-

rate, C = Icircuit/Q.

Further Details And Interpretations

can be found in part I.

3. Solutions

We now derive three analytical, approximate solutions to the model system (2.1), and compare these to the nu- merical solution of the full model computed in part I, which we treat as ‘ground truth’.

To Do This, We Note

that the diffusional C-rate, Cd, is small for most practi- cal (low C-rate) applications, and perform an asymptotic analysis near the limit of small Cd.

3.1. Leading-Order Quasi-Static Solution

In this section, we will derive the quasi-static solution in the limit of small Cd, γdl and 1/ιs. Since γdl and 1/ιs are much smaller than one (Table 1), we only take the leading-order terms in their expansions. In contrast, Cd can sometimes be close to one, so we will consider both the leading order and first order in Cd.

(3.1)

in each electrode, and so Φs can be approximated as a function of time only. Applying the boundary condition (2.1h) and defining V (t) = Φs|x=1, we can now replace

Φs,N = 0,

Φs,p = V (t).

(3.2)

We use the integral condition (2.1n) so that we do not need to solve for is to find the voltage, V (t). Hence (2.1f) is only necessary if we want to find is having found j. We also take the leading order in γdl, so that the time derivatives in (2.1g) disappear.

In summary, we simplify the system (2.1) to the follow- ing equations for c(x, t), ε(x, t), j(x, t), Φ(x, t) and V (t):

(3.3G)

and initial conditions (2.1j). As shown in Table 1, the diffusional C-rate, Cd, is equal to 0.6C, where C is the C-rate. Most practical applications have a C-rate below 0.25C, so the diffusional C-rate is usually small. Hence we perform an asymptotic expansion in the limit Cd →0 and assume that we can expand all

(3.4)

where f = c, ε, Φ, j and V . Hence (3.3) becomes to leading

C(0)

.

C(0) = C0,

ε(0) = ε0.

(3.5H)

At first order, equating coefficients of Cd in (3.3) gives

C(1) = Ε(1) = 0

at t = 0.

(3.6L)

Leading-order quasi-static solution. We now seek the so- lution to the lowest order problem.

Integrating (3.5A)

with boundary conditions (3.5f), then integrating again, gives c(0) = c(0)(t). We then integrate (3.5c), use bound- ary conditions (3.5f), and integrate again, to find that

As Defined By (3.5D)

and (3.5e) are functions of time only; the boundary con-

P

= −icell/ℓp.

Sep ≡Εmax

sep ).

Hence To Leading Or-

der, the whole problem is quasi-static. To determine c(0), we need to consider the first-order problem (3.6a) for c(1). Integrating (3.6a) from x = 0 to x = 1 and using (3.7) and the boundary conditions (3.6j) gives a solvability con- dition that determines c(0).

We Can Combine This With

(3.5b), (3.5d) and (3.5e) to obtain a nonlinear differential-

(3.8E)

with initial conditions (3.5h). Integrate (3.8a-c) and re- arrange (3.8d,e) to find the final leading-order solution,

3.2. First-Order Quasi-Static Solution

We now solve the first-order system, (3.6), to find the O(Cd) correction to the voltage. We solve (3.6) as follows: (i) find c(1) using (3.6a), up to an arbitrary constant, k(t); (ii) find k using a solvability condition on c(2), the O(C2

D)

correction to c; (iii) find Φ(1) using (3.6c) up to an arbi- trary constant, An(t); (iv) find An using (3.6d); (v) find V (1) using (3.6e). Firstly, with known c(0) and ε(0), we can integrate (3.6a) with respect to x twice and use (3.6j) to find an explicit equation for c(1) (given in Appendix B).

Having found c(1), we integrate (3.6c), using (3.7) and

(3.10)

where An is an arbitrary constant. We can now integrate (3.6d) from x = 0 to x = ℓn and integrate (3.6e) from x = 1−ℓp to x = 1, using (3.6k) each

1−ℓp

· dx.

3.3. Composite Solution

The quasi-static solution developed in the ‘Leading- order quasi-static solution’ and ‘First-order quasi-static solution’ sections is valid when the current varies slowly, but fails to capture transient behaviour when the current changes more rapidly, such as a jump. To capture such transients, we could rescale time with τ = (t −t∗)/Cd, where t∗is the time of the jump in the current, define C(τ) = c(t) (and likewise for other variables) and expand in powers of Cd. We give the details of such an approach in Appendix C.

Such a transient solution is valid at short times after a jump time t∗, but breaks down at times long after the jump time. To obtain a solution that is valid both at short times after a jump in current and at long times, without having to repeatedly ‘reset’ the transient solution, we use a ‘composite’ solution, which we now develop here.

We consider the lowest order and first order correction for the concentration by taking ˜c = c(0) + Cd c(1). We then

(3.13)

where c(0) and ε(0) are given by the quasi-static problem (3.9) and j(0) is given by (3.7).

4

times, Cd ∂c(1)/∂t is a higher-order term and we retrieve the quasi-static problem (3.6a), while for short times, re- scaling τ = (t −t∗)/Cd, c(0) is constant and we have the transient problem (C.3a) for c(1).

Hence (3.13) Is Valid

uniformly at both short times and long times. The composite solution then consists of solving (3.13)

(3.14)

and finally finding V (1) through (3.10) and (3.11) with c(1) given by (3.14).

4. Results

In the Solutions section,we derived four systems that are approximately equivalent to the full dimensionless sys-

2. Leading-Order Quasi-Static (Loqs) – (3.9)

3. First-order quasi-static (FOQS) – (B.1), (3.10) and

4. Composite – (3.13), (3.10) And (3.11)

The code used to solve the models and generate the results below is available publicly on GitHub . Note that to obtain either the first-order quasi-static solution or the composite solution, we must first solve the leading-order quasi-static problem.

4.1. Reduced-Order Solutions

We now compare results from the four models.

We

treat the full numerical model as ‘ground truth’, and in- vestigate the speed and accuracy of the three other models compared to the numerical model.

The most important output from the model is the volt- age, since this is the variable that we can compare to experimental data (treating current as a known input).

In Figure 1, we compare the voltage during a complete constant-current discharge at a range of C-rates. The dis- charge is deemed to be finished either when the concentra- tion reaches zero anywhere in the cell, or when the voltage reaches a cut-offvoltage of 10.5V.

We observe that all three reduced-order solutions agree well with the numerical solution at very low C-rates (Fig- ure 1a).

As We Increase The C-Rate (Figures1B-D), Only

the first-order solutions (FOQS and composite) agree with the numerical solution; further, a discrepancy appears be- tween the FOQS solution and the numerical solution at early times.

Finally, For Very High C-Rates (Figures 1E-

f) the composite solution still agrees very well with the numerical solution, but the FOQS solution does not, and terminates early, for reasons that we explain below.

To explain the behaviour observed in the voltages, we investigate internal variables, such as the concentration at various states of charge (Figure 2). At a very low C-rate of 0.1C (Figure 2a), the concentration remains almost uni- form throughout the discharge; hence the LOQS solution, which does not take into account any spatial variations, provides a good fit to the numerical solution. At a higher C-rate of 0.5C (Figure 2b), the concentration in the nu- merical solution is no longer spatially homogeneous; this is non-uniformity is captured well by the FOQS and com- posite solutions, but not by the LOQS solution. However, even with the FOQS and composite solutions, there is a discrepancy in the concentration profiles in the positive electrode (Figures 2b,c, right-hand side of the spatial do- main). This is because the solutions from the asymptotic methods assume a uniform interfacial current density, but in the numerical solution the interfacial current density is non-uniform.

Finally, at high C-rates (2C, Figure 2c), there is a dif- fusion transient at the start of the discharge; this is only captured by the composite solution, and not the FOQS solution. This initial diffusion transient also explains the discrepancy between the FOQS and numerical solutions at early times in Figure 1d. In addition to this, we can now see that the FOQS solution terminates early in Figure 1f because the concentration quickly reaches zero.

In Figure 3, we show the relative errors of the voltage obtained from reduced-order models compared to the volt- age obtained from the numerical model. Then, in Table 2, we compare the time taken to solve the various models. We see that the composite solution, leading-order quasi-static and first-order quasi-static solutions are roughly one, two and three orders of magnitude faster than the full numeri- cal solution respectively. Coupled with the errors shown in Figure 3, the speeds shown in Table 2 suggest that in order to solve the model accurately and as quickly as possible, we should use the LOQS model for C-rates below 0.1C, the FOQS model for C-rates of 0.1-1C, and the composite model for C-rates above 1C.

The time taken for the leading-order quasi-static model is independent of grid size, while the time taken for the other models scales linearly with grid size. Note that we can expect to obtain a faster numerical solution by using a different spatial discretisation scheme than Finite Vol- umes, such as Chebyshev orthogonal collocation , and the relative speed-up of the composite solution by using the same discretisation would be similar.

4.2. Voltage Breakdown

As well as obtaining a faster solution to the model, the composite solution allows us to identify the individual overpotentials that contribute to the total drop in voltage from full charge. We write the total dimensional voltage

(F) 5C

Figure 1: Comparing voltages for a constant-current discharge using the parameters from literature (Tables 1 and A.2), for a range of C-rates.

(4.2)

where VU,i, i = n, p are the open-circuit voltages; Vk,i, i = n, p are the kinetic overpotentials, accounting for losses due to the reactions at the electrode-electrolyte interfaces; Vc is the concentration overpotential, accounting for losses due to concentration gradients; and Vo is the Ohmic over- potential in the electrolyte, accounting for losses due to the electric resistance of the electrolyte. Equation 4.2 would usually include a term to account for Ohmic losses in the solid electrodes, but in our reduced-order models this term is zero since ιs is large (c.f. equation 3.1).

, (4.3F)

Together with the quasi-static formulas (3.9a) for c(0) and (B.1) for c(1), equations (4.2) and (4.3a) give an exact formula for the voltage that is valid for most operating C-rates (below 0.5C). For higher C-rates, we must solve (3.13) and use (3.14), instead of (B.1), to find c(1).

In Figure 4, we show the relative contribution from

Deployment In Out-Of-Position Situations

D. Bendjaballah1, A. Bouchoucha1, M. L. Sahli1,2* and J-C. Gelin2

Abstract

Side-impact collisions represent the second greatest cause of fatality in motor vehicle accidents. Side-impact airbags have been installed in recent model year vehicle due to its effectiveness in reducing passengers’ injuries and fatality rates. In meeting these requirements, simulations of folding and deploying airbags are very useful and are widely used. The paper presents a simulation method for the deploying airbags using three materials in different working conditions. Finite element analysis is primarily used to evaluate this concept. In these simulations, the gas flow is described by the conservation laws of mass, momentum, and energy. The numerical results indicate that the FE method in this paper is capable of capturing airbag deploying process accurately.

Keywords: Airbag simulations, Out-of-position, Crash, Modeling, Out-of-position

Background

The passive safety of cars has become a very high prior- ity issue for the automotive industry. Today, there are not only one or two airbags in a car; certain models have ten times more than that. With the increasing usage of airbags, the number of accidents where the airbag itself can cause an injury to the occupant also increases

(Augenstein Et Al. 2003; Gabauer And Gabler 2010;

Audrey et al. 2011). As is well known, safety belts are also now devices designed to provide protection to the users of vehicles during crash events, minimizing the loads necessary to adapt their movement to the move- ment of the car (Freesmeier and Butler 1999; Schmitt et al. 1997). In general, the seat belt is designed to restrain the occupant in the vehicle and prevent the

Occupant From Having Harsh Contacts With Interior

surfaces of the vehicles. The airbag acts to cushion any impact with vehicle structure and has positive internal pressure, which can exert distributed restraining forces over the head and face. As a safety component of auto- mobile, an airbag decreases occupants’ injury likelihood effectively in case of an accident (Ruff et al. 2007). These safety elements can reduce the death rates on the roads, and its protection effects have been widely approved (Crandall et al. 2001; Teru and Ishikawa 2003). With computational tools such as finite element methods designed for dynamic contact problems, crashworthiness simulations can now be used with reliable accuracy to evaluate occupant protection in various collision condi- tions with safety metric/parameters such as acceleration, head injury criteria, intrusion distance, intrusion vel- ocity, and neck forces (neck injury risk or whiplash).

Thus, new types of airbag products are being developed to handle different collision scenarios.

Become Standard Equipment On Most New Passenger

vehicles (Braver and Kyrychenko 2004; Teng et al. 2007; Yoganandan et al. 2007). The airbag cushion is com- posed of a woven fabric which is rapidly inflated during a car crash. The airbag dissipates the passenger’s kinetic energy thereby reducing injury through biaxial stretching of the fabric bag and escaping gas through vents. There- fore, the performance of the airbag is greatly influenced by the mechanical properties of the fabric. Generally, air bags are designed to deploy in a crash that is equivalent to a vehicle crashing into a solid wall at 8 to 14 mph.

Air bags most often deploy when a vehicle collides with another vehicle or with a solid object like a tree. There are various types of airbags: frontal, side-impact, and curtain airbags. In general, the passenger side airbags are usually larger than the driver airbags (see Fig. 1).

Besançon, France

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Bendjaballah et al. International Journal of Mechanical

Doi 10.1186/S40712-016-0070-2

Extensive studies have shown that the airbag deploy- ment in load cases consists of two occupant loading phases: a punch-out effect where the airbag bursts out of its container with the airbag and airbag module cover accelerating towards the occupant and a second loading phase during which the airbag is taking on its deployed shape and volume (membrane-loading effect). Bankdak et al. (2002) developed an experimental airbag test system to study airbag-occupant interactions during close proximity deployment. The results provided insight for simulating the effect of inflation energy and mass flow on target response. Bedard et al. (2002) found that while left-side (driver-side) impacts accounted for only 13.5% of all crashes, the fatality rate among these

Crashes Was 68.3% In Comparison To Front Impact

(48.3%), right-side impact (31.3%), and rear impact (38.4%). These studies underscore the importance of oc- cupant safety during side-impact collisions. In the last years, the current market requested to reduce the time and cost airbag development. In order to achieve this result, virtual simulations play an important role since they allow to minimize the number of experimental tests (Pei et al. 2013; Cao et al. 2014). Several simulation models of airbag were established (Wang et al. 2007). It is feasible to optimize the parameters of airbag deploy- ment using simulation technology. Experimental and numerical studies have quantified injury risks to close- proximity occupants from deploying side airbags. These studies have focused on the prevention of the most ad- verse effects of airbag deployment (Duma et al. 2003).

Other studies have proposed airbag characteristics to minimize particular biomechanical responses (Haland and Pipkorn 1996). In a more recent study, Marklund and Nilsson (2003) compared deformation patterns with experimental data as well as the computational costs associated with three different airbag deployment simu- lation methods; they concluded that the SPH method is relatively inexpensive and produces incremental deform- ation patterns that compare most closely to the experi- mental results. The process of inflation of an airbag is one of the determining factors in saving lives. The duration from the initial impact of the crash to the full inflation of an airbag is about 40 ms, and during this time, the airbag goes from being in a folded state to a fully inflated state, with a high internal pressure. After achieving this state, the airbag begins to deflate, thus providing a nice cushion for the body impacting it.

Ideally, the person in the crash should come into contact with the airbag at this time. In the present study, a large volume passenger side airbag model is developed to handle different collision scenarios. The main aim is evaluate the performance of deploying of passenger side airbag using finite element methods (FEM).

Materials

The tensile specimens were made in different airbags (P: Peugeot, R: Renault, and VW: Volkswagen) with a length of 200 mm long and a width of 40 mm. Table 1 shows the mechanical properties of the airbag.

Tensile Tests

To determine the mechanical properties of the material of airbag used in the test pieces, tensile tests were performed on Lloyd EZ20 universal testing machine in Constantine. These tests were conducted using rect- angular samples. The axial force and axial displacement acquired during a test are converted into stress and the strain in order to be used for the fabric material model.

The continuous recording of the stress-strain data was performed during both the load and unload phases. A minimum of five samples were made in order to check the repeatability of the measurements. All the data was collected by using a PC-based data acquisition system and analyzed by commercial software. The picture frame test device that is made for this study is shown in Fig. 2.

Fig. 1 a Frontal and side airbags. b Oblique view of facet occupant model in sitting posture following airbag deployment (Lim et al. 2014)

0.150

Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

Page 2 Of 9

Figure 3 shows the stress-strain relationship of the airbag sample under axial tensile loads. The results are showing a linear increase in extension with the increas- ing stresses. This is an expected output and it confirms with the theoretical behavior of a sample subjected to tensile stress. The rupture strain values for different airbags (R/P/VW) were 0.322, 0.441, and 0.472, respect- ively. The measured elastic parameters (i.e., Young’s modulus E and initial yield strength) and Poisson’s ratio are summarized in Table 2. The tensile tests of the woven fabrics can show differences on mechanical prop- erties because woven fabrics can resist in-plane shear loads once the yarn lock-up angle has been reached. The differences of material property on material direction can affect the shape of fully deployed bag (see Fig. 3b).

Theoretical Background

Numerical simulations of airbags use very complex and techniques such as an orthotropic model to identify the mechanical behaviors during the airbag inflation and the fluid mechanics (gas flow) to describe the inflator gas flow (pressure gradient) and improve the representation of the pressures within the airbag. To model the airbag as an orthotropic model, three material constants have to be provided. Assuming a plane stress condition, the

Ð1Þ

where σ is the normal stress and τ is the shear stress, the subscript refers to the principal material directions, i.e., the fill and warp directions. Also, ε and γ are the strain components. The material elastic constants Qij are

Ð2Þ

where E1 and E2 are the Young’s modulus in the fill and wrap directions and G12 is the shear modulus of the fabric material. νij is the Poisson ratio of the material.

The gas exerts a pressure load on the airbag causing it to expand. This expansion puts the airbag under tensile stress lowering the expansion rate. In this study, heat conduction and heat transfer is not taken into account.

Fig. 2 A photograph of Lloyd EZ20 universal testing Fig. 3 Stress versus strain using Lloyd EZ20 machine for a three different airbags at 0° and 90° and b VW airbag test specimens at

Different Angles

Table 2 Physical and mechanical properties of the airbag

Page 3 Of 9

In the deployment of an airbag, an inflator supplies high velocity gas into an airbag causing it to expand rapidly. The gas inside the airbag is assumed to be ideal, to be of constant entropy, and to satisfy the equation of state:

Ð3Þ

Here p, ρ, and e are respectively the pressure, density, and specific internal energy, and γ is the ratio of the heat capacities of the gas. The gas flow is described by the conservation laws for mass, momentum, and energy that

Ð4Þ

here, V is a volume, A is the boundary of this volume,

N Is The Normal Vector Along The Surface A, And U

denotes the velocity vector in the volume. Applying Bernoulli’s equation in the case of an ideal gas with

Ð5Þ

Here, the subscript ex denotes quantities at the throat of the tube. Furthermore u, p, and ρ denote the quan- tities inside that part of the tube that is supplying mass.

Materials And Boundary Conditions

The airbag system mainly consists of three parts: the airbag itself, the inflator unit, and the crash sensor or diagnostic unit. Thus, to study the behavior of the airbag using FE simulations, we need to have an FE model of the airbag in the folded position. A FE model of the airbag was used to simulate the test condition as shown in Fig. 5. LS-DYNA® material model FABRIC (MAT_34) is used to simulate the airbag material. It is a variation of the layered orthotropic material model. Additionally, in the LS-DYNA® material model, fabric leakage can be accounted for. However, for this CAB material, the leak- age is almost negligible and therefore no leakage is specified. The mechanical properties can be determined from the physical test. Typical material properties for airbag fabrics are taken as given in Chawla et al. (2004a) (Table 3). These properties are used to simulate inflation process of airbag (see Table 1). The car dashboard is modeled as the rectangular thin plate using a MAT_RI-

Gid Material, And The Degrees Of Freedom Are Con-

strained in all the directions. The similar properties of thermoplastic polymer are assigned for contact purposes. The porosity of the fabric is assumed zero. The nitro- gen gas is taken for inflating the airbag. Properties of nitrogen gas and initial bag conditions are shown in Table 4. The example on which we perform the study is a typical passenger side airbag. The geometric de- tails have been measured from a commercially avail- able airbag. The initial state of the airbag is a closed rectangular whose sides are to be finished to 482 × 635 mm2 and is shown in Fig. 4.

Table 3 Material properties of airbag and rigid plate used in FE

–

Table 4 Initial values used for FE simulation of the swelling of

3.33 × 10−4

Fig. 4 The initial airbag geometry in the form of a rectangular Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

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