Quasi Resonant Converter Matlab

September 26, 2013

Abstract A concise Matlab implementation of a stable parallelizable space-time Petrov-Galerkin discretization for parabolic evolution equations is given. Emphasis is on the reusability of spatial finite element codes.

Keywords heat equation · parabolic · space-time discretization · parallel ·

1 Introduction

The spectrum of numerical methods for parabolic evolution equations is extremely broad, which attests to the ubiquity and the relevance of such equations. With the aim of developing reliable massively parallel algorithms, e.g. for optimization problems constrained by parabolic evolution equations, several attempts have been made to go beyond time-stepping methods, see [3, Section 5.1] for a modest at- tempt of an overview. In this paper we give a concise Matlab implementation, partly motivated by , of a specific space-time Petrov-Galerkin discretization for parabolic evolution equations from , hoping to provide a basis for possible fur- ther developments. Spanning just a few lines of Matlab code, it is parallelizable and stable in the Petrov-Galerkin sense, which already distinguishes it from conven- tional methods for parabolic evolution equations. Stability implies quasi-optimality of the discrete solution in the natural solution spaces, and is an important property in the resolution of nonlinear problems. Moreover, the implementation is modu- lar with respect to the spatial discretization, admits time-dependent inputs and nonuniform temporal grids. Since the algorithm is based on an iterative solution of a single linear system, another significant advantage to conventional methods is the ability to terminate the iteration at a specified global accuracy.

The model parabolic evolution equation under consideration is presented in Section 2 and is restated in a variational formulation. The space-time Petrov- Galerkin discrete trial and test spaces that will be used to discretize the variational

Roman Andreev

formulation are introduced in Section 3. In order to obtain stable algorithms we develop in Section 4 a generalization of the usual variational framework for lin- ear operator equations, called the minimal residual Petrov-Galerkin discretization.

Choosing bases on the discrete trial and test spaces, it leads to a linear system of generalized Gauß normal equations along with a natural preconditioner. In that framework, certain norm-inducing operators play an important role. Specifically for the parabolic evolution problem, these are defined in Section 5. In Section 6 we detail the Kronecker product structure of the parabolic space-time operator and the norm-inducing operators when assembled using space-time tensor product bases, and comment on the data structures employed. In the solution process, the inverses of the matrix representations (of norm-inducing operators) are required.

Their Kronecker product structure is discussed in Section 7.1. The assembly pro- cedure for the space-time source is in Section 7.2. A generalization of the LSQR algorithm of Paige and Saunders for the iterative resolution of the linear sys- tem is given in Section 7.3. With those preparations, the Matlab implementation is presented in Section 8. Two numerical experiments are given in Section 9. We conclude and point out some limitations in Section 10.

2 Model problem and its space-time variational formulation Let D ⊂Rn, n ∈{1, 2, 3}, be a bounded connected domain with a polyhedral boundary Γ = ∂D. If n = 1 then D is an open bounded interval; if n = 2 then D is a polygon; if n = 3 then D is a polyhedron. We partition Γ into two disjoint subsets Γ0 and ΓN such that ¯Γ = ¯Γ0 ∪¯ΓN. On Γ0 we will impose homogeneous boundary conditions of Dirichlet type. For that reason, the Dirichlet boundary Γ0 is assumed to be of positive measure (with respect to the surface measure which will subsequently be denoted by σ), i.e., it contains at least one endpoint if n = 1; a curve of positive length if n = 2; or a surface of positive surface measure if n = 3.

Let J = (0, T), T > 0, denote the temporal interval. The model for parabolic evolution equations that we consider is the heat equa-

U(T, X) = H(X),

(t, x) ∈{0} × D.

(4)

Here, a, f, g and h are given scalar valued functions, while u is the unknown. Further, div and grad denote the divergence and the gradient operator with respect to the spatial variable x ∈D. The derivative in the direction of the outward normal at the Neumann part ΓN of the boundary is denoted by ∂u

∂N. The Precise Meaning Of

the heat equation will be fixed by means of a well-posed space-time variational for- mulation in the following. The time-independent operator “div(a(x) grad u(t, x))” could be replaced by the time-dependent one “div(A(t, x) grad u(t, x))”, where

A ∈L∞(J × D; Rn×N

sym ), without affecting most considerations below (the techni- cal reason why this is possible is given in [12, Lemma 4.4.1]). However, if A is not a finite sum of separable functions, the implementation becomes significantly

3

less transparent, and we therefore discard this case from the onset on; on the other hand, A being a finite sum of separable functions entails modifications of secondary relevance to this exposition.

To motivate the space-time variational formulation of the heat equation, we formally test the equation with a function v1 on J × D and integrate in space and time; the initial condition is tested with v2 on D and integrated in space.

Integration by parts is performed in space only, and the two resulting conditions (one “∀v1”, the other “∀v2”) are added together. The solution u will be sought in the space X, and the test functions are combined to v = (v1, v2) ∈Y := Y1 × Y2.

The spaces X and Y will be specified presently. We write (·, ·)D for the L2(D) and the [L2(D)]d scalar product, while (·, ·)ΓN is the scalar product on L2(ΓN) for the boundary measure σ introduced above. Generally, we omit the dependence of the integrands on the temporal variable t. In this way we obtain the continuous

(5)

where the system bilinear form B encoding the heat equation is

(6)

and the load functional b supplying the source term, as well as boundary and

J

(g, v1)ΓN dt + (h, v2)D.

(7)

Integrating by parts also in time, as in e.g. , leads to an alternative space- time variational formulation, which, however, will not be discussed below.

Where H1

Γ0(D) denotes the Sobolev space of functions in H1(D) with vanishing

Γ0(D)]′ Its Dual. We Identify H With

its dual H′ by means of the Riesz isomorphism on H. Then the duality pairing on V × V ′ (or V ′ × V ) is the continuous extension of the H-scalar product on V × V . In this way we obtain a so-called Gelfand triple of separable Hilbert spaces

(9)

with continuous and dense embeddings. In order for B : X × Y →R to be a continuous bilinear form and for b : Y →R to be a continuous linear functional,

(10)

For the definition of the Bochner spaces L2(J;V ), H1(J; V ′), and the like, we refer to e.g. or [14, Chapter 1]. Then (6)–(7) are well-defined for all (u, v) ∈X × Y

With

0 < ess inf a ≤ess sup a < ∞.

(12)

For the remainder of the article we assume f ∈L2(J; L2(D)). The spaces X and Y are themselves Banach spaces for the norms ∥·∥X and ∥·∥Y that are given by

H,

v = (v1, v2) ∈Y.

(14)

We recall from e.g. [11, Section 5.9.2] or [14, Chapter 1] that any (representant of any) u ∈X admits a modification on a negligible subset of J such that the resulting function coincides with a unique continuous H-valued function defined on the closed interval ¯J; moreover, the C0 norm of the latter is controlled by the X norm of the former. In other words, the following embedding is continuous

∥U(0)∥H ≤C∥U∥X

∀u ∈X.

(16)

In this way, the initial value u(0) of any u ∈X is well-defined in H. With the above assumptions, the space-time variational problem (5) has a unique solution u ∈X and the solution depends continuously on the functional b ∈Y ′, see [16, Theorem 5.1] and the references therein. Hence, the solution u also depends continuously on the input data f, g and h that define the load functional b in (7).

3 Space-time tensor product discrete trial and test spaces The continuous space-time variational formulation (5) will be discretized using finite-dimensional discrete trial and test spaces Xh ⊂X and Yh ⊂Y built up from finite-dimensional “univariate” temporal subspaces E ⊂H1(J), F ⊂L2(J), and spatial subspaces Vh ⊂V . These spaces assume the space-time tensor product

And

Yh := (F ⊗Vh) × Vh.

(17)

A key feature of this discretization and the implementation given below is the modularity with respect to the spatial subspaces Vh. To specify the temporal subspaces E and F we need to introduce some ter- minology. A temporal mesh T is a finite set of points in ¯J = [0, T] containing 0 and T. The connected components of J \ T are called the elements of T . Let max ∆T denote the maximal “time-step”, i.e., the maximal length of an element of T . For a temporal mesh T let T ⋆denote the temporal mesh obtained from T by a uniform refinement (each element is split into two smaller elements of equal length). Concerning E on the trial side and F on the test side, we will restrict ourselves to two types of pairs that differ in the choice of F.

5

Type 1 temporal subspaces. Given a temporal mesh TE, we define E as the standard space of continuous piecewise affine functions, and F as the space of piecewise constant functions on TF := TE.

Type 2 temporal subspaces. Given a temporal mesh TE, the space E is de- fined as above. Let another temporal mesh, TF be obtained from TE by a succession of uniform refinements, i.e., TF = [T 7→T ⋆]n(TE) for some positive n ∈N. Then F is defined as the space of piecewise constant functions on TF .

In the first case the discrete variational formulation

(18)

is an example of continuous Galerkin time-stepping schemes . In the second case the dimension of Yh is larger than that of Xh, and the above discrete varia- tional formulation is meaningless. A generalization based on residual minimization is therefore introduced in Section 4. Concerning the stability of the resulting min- imal residual Petrov-Galerkin method there is a fundamental difference between Type 1 and Type 2 temporal subspaces. This is the subject of the following two propositions that summarize the relevant main results from [3, Section 5.2.3]. Note carefully that the present concept of stability, namely the validity of the discrete

(19)

(its role is discussed in Section 4) uniformly in the choice of the temporal dis- cretization, is different from e.g. A-stability for time-stepping methods. The following measure of self-duality for the spatial subspace Vh ⊂V will be

H∥V ′∥Χh∥V

.

(20)

Note that κh is bounded (independently of Vh) and necessarily positive for a finite-dimensional Vh. Proposition 1 Let {0}̸ = Vh ⊂V be a finite-dimensional subspace. Let E ⊂ H1(J) and F ⊂L2(J) be of Type 2. Then there exists a constant γ⋆

0 > 0 Indepen-

dent of Vh, E and F, such that the discrete inf-sup condition (19) holds for the

Γh ≥Γ⋆

0κh.

We Remark That Γ⋆

0 in (21), as a function of the number of refinements between TE and TF , is monotonically increasing and saturates (exponentially quickly). Type 1 temporal subspaces, on the other hand, do not lead to unconditional stability of the form (21). To formalize this, we define the CFL number

∥Χh∥V

∥χh∥V ′ .

Roman Andreev

Proposition 2 Let {0}̸ = Vh ⊂V be a finite-dimensional subspace. Let E ⊂ H1(J) and F ⊂L2(J) be of Type 1. Then there exists a constant γ0 > 0 indepen- dent of Vh, E and F, such that the discrete inf-sup condition (19) holds for the

Γh ≥Γ0Κh Min{1, Cfl−1

h }.

(23)

In general, the dependence on the CFL number cannot be improved.

4 Minimal Residual Petrov-Galerkin Discretization

In this section we consider an abstract continuous bilinear form B : X × Y →R, where X and Y are Hilbert spaces with norms ∥·∥X and ∥·∥Y . Let

∥W∥X∥V∥Y

denote the norm of the bilinear form B. Further, let b be a linear continuous func- tional of Y . For the remainder of the section, two finite-dimensional subspaces Xh ⊂X and Yh ⊂Y are fixed. We aim at relaxing the discrete variational formu- lation (18) to admit the case dim Xh < dim Yh. To guarantee well-posedness, the discrete inf-sup condition of B on Xh × Yh will be essential (cf. Proposition 1):

∥Wh∥X∥Vh∥Y

> 0.

(24)

We introduce norms |||·|||X and |||·|||Y on Xh and Yh that are induced by (positive definite) linear continuous operators M : Xh →X′ and N : Yh →Y ′ as follows:

Y := (Nvh)(Vh),

vh ∈Yh.

(26)

The operators M and N are moreover assumed to be symmetric, i.e., (Muh)(wh) = (Mwh)(uh) for all wh, uh ∈Xh, and similarly for N. The Gram matrices of the operators M and N, defined below, will essentially act as preconditioners for the discrete system, and should therefore be easy to invert approximately, cf. Section 7.1. Let 0 < dM ≤DM < ∞and 0 < dN ≤DN < ∞be constants such that

(27)

on Xh and Yh, respectively. We emphasize that the operators M and N, the induced norms, and hence the constants in (27) may depend on h, but in this section, the pair Xh × Yh is fixed to lighten the notation.

Instead of the usual discrete variational formulation we now introduce the discrete (functional) residual minimization problem

(29)

is the (functional) residual. The following can be shown .

7

Theorem 1 Let the discrete inf-sup condition (24) hold. Then there exists a

(30)

holds. Moreover, uh satisfies the quasi-optimality estimate

(31)

for any u ∈X such that B(u, v) = b(v) for all v ∈Y . Proof (Sketch) Invoking the open mapping theorem, one can show that the the mapping b 7→uh is well-defined, linear and continuous, and its norm is dominated

Dn

dN . Thus the composition u 7→Bu 7→uh is a continuous projection with norm not exceeding Ch given in (31). An application of [18, Lemma 5] finishes the argument.

Let us describe a computable algebraic equivalent of the somewhat nonstan- dard variational definition (28) of the discrete solution. To that end let Φ ⊂Xh and Ψ ⊂Yh be bases for the respective discrete spaces. Having fixed the pair Xh × Yh, the possible dependence on h is again omitted. The algebraic representants of the system bilinear form B, of the load functional b and of the norm-inducing operators N and M, are defined with respect to the chosen basis in the usual way,

(32)

or in componentwise notation Bψφ = B(φ, ψ), bψ = b(ψ), Nψ′ψ = (Nψ)(ψ′), Mφ′φ = (Mφ)(φ′) for φ, φ′ ∈Φ and ψ, ψ′ ∈Ψ. The basis functions are used to index the components of matrices and vectors. Similarly, RΦ will denote vectors of real numbers indexed by φ ∈Φ. The matrix B is injective if and only if the discrete inf-sup condition (24) holds; further, N and M are symmetric positive definite matrices due to the analogous properties (27) of the operators N and M.

√

wTMw, w ∈RΦ, defines a norm, and we use similar notation for other matrices. With these definitions, the discrete functional residual minimization (28) can be seen to be equivalent to the discrete algebraic residual minimization

U := Arg Min

w∈RΦ ∥Bw −b∥N−1.

(33)

A vector u is a stationary point of (33) if and only if it satisfies the first order optimality conditions, namely the generalized Gauß normal equations BTN−1Bu = BTN−1b.

(34)

If the discrete inf-sup condition (24) holds, the matrix BTN−1B is symmetric positive definite; then, the Gauß normal equations (34), and therefore also the discrete minimization problem (33), have a unique solution. Finally, the matrices M and BTN−1B are spectrally equivalent with the bounds [3, Section 4.1]

Γhdmdn∥W∥M ≤∥W∥Btn−1B ≤∥B∥Dmdn∥W∥M

∀w ∈RΦ.

(35)

Therefore, M is a preconditioner for the Gauß normal equations (34). By the estimate (35), the quality of this preconditioner is controlled by the discrete inf- sup constant γh in (24) and the norm equivalence constants in (27), and does not depend on the choice of the basis.

5 Parabolic Space-Time Preconditioners

In Section 4 we admitted general norm-inducing operators M and N on the fixed pair of finite-dimensional discrete trial and test spaces Xh × Yh. For the space- time variational formulation (5) several practical choice are available [3, Chapter 6]. Here, to simplify the exposition, we will only use the canonical choice of the

H,

v = (v1, v2) ∈Y.

(37)

These definitions extend to the off-diagonal by the imposed symmetry of M and

V ′ := (Φ, A−1Ιϕ)D,

ϕ ∈H = L2(D).

(39)

Here, A is the operator A : V →V ′, σ 7→(a grad σ, grad ·)D, where a is the heat conduction coefficient from (1) satisfying the bounds (12), and ιϕ ∈V ′ is the functional on V defined by ιϕ := (ϕ, ·)D, ϕ ∈H. For β ∈V ′, the definition

Extends To ∥Β∥2

V ′ := β(A−1β). 6 Kronecker product structure of the discretized operators

6.1 The System Bilinear Form

Recall from Section 2 the definition of the system bilinear form

(A Grad U, Grad V1)Ddt + (U(0,·), V2)D

for the space-time variational formulation of the model parabolic evolution equa- tion, where u ∈X = L2(J;V ) ∩H1(J; V ′) and v = (v1, v2) ∈Y = L2(J;V ) × H. As described in Section 3, we consider two types of discrete trial and test spaces.

(40)

where E ⊂H1(J) is the space of continuous piecewise affine functions on a tem- poral mesh, F ⊂L2(J) is the space of piecewise constant functions on the same mesh (Type 1) or on its n-fold uniform refinement (Type 2), and Vh ⊂V is a finite-dimensional subspace.

For the remainder of the section we fix the spatial discretization Vh ⊂V with a basis Σ ⊂Vh, and the temporal mesh TE = {0 = t0 < t1 < . . <

K′ = T} Be Either Te Or Any N-

fold uniform refinement of TE. Let E be as above, and let F denote the space

9

of piecewise constant functions with respect to the temporal mesh TF ⊇TE, which possibly refines TE. As basis for E we take the usual hat functions Θ := {θk : k = 0, . . , K} ⊂E defined by θk(t˜k) = δk˜k, where δk˜k denotes the Kronecker delta. In particular, the only function that does not vanish at t = 0 is θ0. As basis for F we take the indicator functions Ξ := {ξk := χ(t′

K−1, T′

k) of the temporal mesh TF ⊇TE. These univariate bases are first combined to the collections Φ ⊂X and Ψ1 ⊂Y1 as

Φ := {Θ ⊗Σ : Θ ∈Θ, Σ ∈Σ},

Ψ1 := {ξ ⊗σ : ξ ∈Ξ, σ ∈Σ}.

(42)

for the discrete trial and test spaces Xh and Yh. Discretizing the bilinear form B using these tensor product bases Φ and Ψ as described in abstract terms in Section

(44)

with the prime denoting the derivative with respect to t, and b) the usual “spatial FEM” mass and stiffness matrices Mx, Ax ∈RΣ×Σ are given by

D

a(x) grad ˜σ(x) · grad σ(x)dx.

(45)

Let us comment on the assembly of the temporal FEM matrices. First, if

2|I|, 0} Depending

on whether ξ and θ are both nonzero on the temporal element I of TE having length |I|, and on the sign of θ′ there. Therefore, assume now that TF is obtained from TE by a succession of uniform refinements (Type 2). Let T ⋆

E Denote The First Uniform

refinement of TE, and let E⋆be the space of continuous piecewise affine function

As Above, But With T ⋆

E in place of TE. Consider now the embedding operator

Se

t .

(46)

Moreover, denoting by tθ ∈T the node for which θ(tθ) = 1, and similarly for

T ]Θ⋆Θtθ

∀θ⋆∈Θ⋆.

6.2 The Norm-Inducing Operators

With the norms on V and V ′ taken to be (38)–(39), the discretized operators M

J

˜ξ(t)ξ(t)dt.

6.3 Data Structures

The Kronecker product structure of these matrices suggests regarding a vector w ∈RΣ×Θ as a rectangular array with #Σ rows and #Θ columns. Let Vec(w) denote the “vectorization” of such an array, i.e., its columns are collected one after another into one column vector Vec(w) of length #(Σ × Θ). Now, if T ∈RΘ×Θ

And X ∈Rς×Σ Are Matrices Then

(T ⊗X) Vec(w) = Vec(XwTT).

(52)

In the implementation we will exclusively use the representation as rectangular arrays. Moreover, load vectors derived from load functionals d ∈Y ′ will be stored as pairs d = (d1, d2) with d1 ∈RΣ×Ξ (rectangular array with #Σ rows and #Ξ columns) and d2 ∈RΣ (column vector of length #Σ) in the form of a Matlab structure {d1, d2}. To these, a formula analogous to (52) applies. In particular, we never store the operators B, M and N (or its inverses) as matrices.

7 Implementational Aspects

7.1 Inverses of the space-time parabolic preconditioners Consider the matrix representations M and N of the space-time parabolic pre- conditioners given in (48) and (50) in Section 6. In order to solve the generalized Gauß normal equations (34) with M as preconditioner, we need to (approximately) compute the inverses M−1 and N−1.

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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