Push Pull Inverter Matlab

Abstract

The Push, the Pull and the Push&Pull algorithms are well-known rumor spreading protocols. In all three, in the beginning one node of a graph is informed. In the Push setting, every round every informed node chooses a neighbor uniformly at random and, if it is not already informed anyway, informs it. In the Pull setting, each round each uninformed node chooses a neighbor uniformly at random and asks it for the rumor; if the asked neighbor is informed, now also the asking node is informed. Push&Pull is a combination of Push and Pull: In each round, each node picks a neighbor uniformly at random. If at least one of both knows the rumor, after this round, both know the rumor.

Clementi et al. have considered Push in settings where the underlying graph changes each round (). In one setting they investigated, in each round the underlying graph is a newly sampled Erdős-Rényi random graph G(n, p). They show that if p ≥1/n then with probability 1 −o(1) (as n →∞) the number of rounds needed until all nodes are informed is O(ln(n)). Doerr and Kostrygin introduced a general framework to analyze rumor spreading algorithms (); using this framework, for a > 0 and p = a/n they improved the results from in the described setting. In particular the expected number of rounds needed by Push was determined to be log2−e−a(n)+1/(1−e−a) ln(n)+O(1); also large deviation bounds were obtained. Using their framework, we investigate Pull and Push&Pull in that setting: We prove that the expected number of rounds needed by Pull to inform all nodes is log2−e−a(n) + 1/a ln(n) + O(1). Let γ := 2(1 −e−a) − (1 −e−a)2/a; we prove that the expected number of rounds needed by Push&Pull is log1+γ(n) + 1/a ln(n) + O(1); as a byproduct, we obtain large deviation bounds, too.

Introduction

The Push, the Pull and the Push&Pull algorithms are important and well-studied rumor spreading protocols [6, 10, 3, 5, 7, 8, 9, 2]. In all three, in the beginning one node of a graph is informed. In the Push setting, every round every informed node chooses a neighbor uni- formly at random and, if it is not already informed anyway, informs it. In the Pull setting, each round each uninformed node chooses a neighbor uniformly at random and asks it for the rumor; if the asked neighbor is informed, now also the asking node is informed. Push&Pull is a combination of Push and Pull: In each round, each node picks a neighbor uniformly at

1

random. If at least one of both knows the rumor, then, after this round, both know the rumor. Recently Clementi et al. have investigated Push on random evolving graphs (), i.e. in a set- ting where the underlying graph is not fixed but changes over time. One such setting treated in is the following: Each round the underlying graph is a newly (and independently of the previous graphs) sampled Erdős-Rényi random graph G(n, p). We are interested in large values for n, thus all asymptotic notation is with respect to n →∞if not explicitly stated differently. Among other results, in it is shown that if p ≥1/n then whp (with high probability, i.e. with probability 1−o(1)) the number of rounds needed by Push is O(ln(n)).

Let a > 0 and let n > a be a natural number. For p = a/n Doerr and Kostrygin have improved this bound (). They have shown that the expected number of rounds needed is log2−e−a(n) + 1/(1 −e−a) ln(n) + O(1); moreover, it is shown that constants α, A > 0 exist such that, if Tn (or short T) denotes the needed number of rounds, then for all r, n ∈N we have P[|T −E[T]| > r] ≤A exp(−αr). This was shown by applying a general framework developed in . This framework exploits that many rumor spreading algorithms are suf- ficiently characterized by the probability pk of a node to become informed in a round that starts with k informed nodes and a bound on the covariances between the indicator variables each indicating whether an uninformed node becomes informed in that round. By bounding pk and the mentioned covariances, the framework allows to obtain the expected number of rounds needed up to constant additive terms as well as large deviation bounds.

We use this framework to investigate Pull and Push&Pull in random evolving graphs. We show that the expected number of rounds needed by the Pull algorithm in the setting de- scribed above (i.e. each round a new G(n, p) is sampled independently of what happened before) is log2−e−a(n)+1/a ln(n)+O(1). Let γ = 2(1−e−a)−(1−e−a)2/a; then the expected number of rounds needed by Push&Pull is log1+γ(n) + 1/a ln(n) + O(1). As a byproduct, we also obtain large deviation bounds.

Particularly the results for Push&Pull are interesting. While both, Push and Pull, need loga- rithmic time for the last phase of the rumor spreading, when combining them in Push&Pull, Push becomes useless in the last phase which might be unexpected. Another interesting aspect is that Push&Pull in the investigated setting is an example where in the first phase when almost no nodes are informed Push and Pull get in each other’s way in the sense that even at the very beginning they inform significantly fewer nodes than the sum of the numbers of nodes they would have informed individually; in other words: even in the beginning many nodes are informed by Push as well as by Pull.

The remainder of this paper is structured as follows: In section 2 needed preliminaries are considered; in particular this includes the framework introduced in . In section 3 the result for Pull is proven and in section 4 the result for Push&Pull is proven.

Preliminaries

We start with stating the framework from . Therefore we consider only homogeneous rumor spreading processes characterized as follows: We consider graphs with n nodes, in the beginning one node is informed, the other nodes are uninformed. Once a node is informed it remains informed. The process is partitioned into rounds, in each round each uninformed node can become informed. Whenever a round starts with k nodes, we assume that there is a pk (only depending on k) such that each uninformed node becomes informed in that round with probability pk; hence pk is called the success probability. A rumor spreading process as described is called homogeneous (). By suitably bounding the success probability and the covariance numbers defined as follows, bounds on the rumor spreading time (see Definition 2) can be obtained.

Definition 1 (Covariance numbers, ). For a given homogeneous process and k ∈{1, . . , n− 1} let ck be the smallest number such that whenever a round starts with k informed nodes for any two uninformed nodes x1, x2, the indicator random variables X1, X2 for the events that these nodes become informed in this round satisfy Cov[X1, X2] ≤ck.

Definition 2 (Rumor spreading times, ). Consider a homogeneous rumor spreading pro- cess. For all t = 0, 1, . . denote It the number of informed nodes at the end of the t-th round (I0 := 1). Let k ≤m ≤n. Let Tn(k, m) (or short T(k, m)) denote the time it takes to increase the number of informed nodes from k to m or more, that is, T(k, m) = min{t −s | Is = k and It ≥m}. We call T(1, n) the rumor spreading time of the process.

If the following exponential growth condition is fulfilled, then Theorem 4 states that there is an exponential growing phase, i.e. if few enough nodes are informed, then the number of informed nodes essentially increases by a constant factor each round and the rumor spreading time can be bounded respectively.

Definition 3 (Exponential growth conditions, ). Let γn be bounded between two positive constants. Let a, b, c ≥0 and 0 < f < 1. We say that a homogeneous rumor spreading process satisfies the upper (respectively lower) exponential growth conditions in [1, fn[ if for any n ∈N big enough the following properties are satisfied for any k < fn.

B

ln(n))).

• Ck ≤C K

n2. In the case of the upper exponential growth condition, we also require af < 1. Theorem 4 (). If a homogeneous rumor spreading process satisfies the upper (lower) exponential growth conditions in [1, fn[, then there are constants A, α > 0 such that

Log1+Γn(N) +

(−) r] ≤A exp(−αr) for all r, n ∈N. When the lower exponential growth conditions are satisfied, then also there is an f ′ ∈]f, 1[ such that with probability 1 −O(1/n) at most f ′n nodes are informed at the end of round T(1, fn).

3

If the following exponential shrinking condition is fulfilled, then Theorem 6 states that there is an exponential shrinking phase, i.e. if enough nodes are informed, then the number of unin- formed nodes essentially decreases by a constant factor each round and the rumor spreading time can be bounded respectively.

Definition 5 (Exponential shrinking conditions, ). Let ρn be bounded between two positive constants. Let 0 < g < 1, and a, c ∈R≥0. We say that a homogeneous rumor spreading process satisfies the upper (respectively lower) exponential shrinking conditions if for any n ∈N big enough, the following properties are satisfied for all u = n −k ≤gn.

U

For the upper exponential shrinking conditions, we also assume that e−ρn + ag < 1. Theorem 6 (). If a homogeneous rumor spreading process satisfies the upper (lower) exponential shrinking conditions, then there are A′, α′ > 0 such that

Ln(N) +

(−) r] ≤A′ exp(−α′r) for all r, n ∈N. Remark 7. It suffices to compute γn and ρn from Theorems 4 and 6 respectively up to additive O(1/ ln(n)) terms.

We will use the following two well-known facts in our proofs; Fact 9 is a simple consequence of Fact 8. Fact 8. Let (xn)n∈N be a sequence of real numbers such that for each n ∈N we have 0 <

Xn < 1. Then (1 + Xn/N)N = Exn + O(X2

n/n). Fact 9. Let a > 0; consider an Erdős-Rényi random graph G = G(n, a/n). Let x be a node. The probability that x is isolated is e−a + O(1/n).

Theorem 10 considers the number of rounds Push needs in the described setting. While we do not need the Theorem for the proof of our results, we state it for completeness. Theorem 10 (). Let a > 0 and let Tn be the time the push protocol needs to inform all n nodes when in each round a newly sampled Erdős-Rényi random graph G = G(n, a/n) is the

1 −E−A Ln(N) + O(1)

and there are constants A, α > 0 such that for all r, n ∈N P[|T −E[T]| ≥r] ≤A exp(−αr). It is observed that the obtained rumor spreading time is the same (up to constant terms) as if the underlying graph is a complete graph but message transmissions fail independently with probability e−a which, up to additive O(1/n) terms is the probability that a vertex is isolated.

We will see that this also holds for Pull.

Interestingly It Does Not Hold For

Push&Pull we provide an explanation in Remark 15.

Pull In Random Evolving Graphs

Theorem 11. Let a > 0 and assume that each round a newly sampled Erdős-Rényi random graph G(n, a/n) is the underlying graph. Then for the rumor spreading time of Pull, Tn, we

A Ln(N) + O(1)

and there are constants A, α > 0 such that for all r, n ∈N P[|Tn −E[Tn]| ≥r] ≤A exp(−αr). Proof. We want to apply the framework from .

We Can Assume That At The Start Of

each round, the edges of the random graph are not yet sampled.

Before The G(N, P) Is

sampled, each uninformed node has the same probability of getting informed, hence the rumor spreading algorithm is homogeneous. First we consider the covariance numbers. To do this, consider two uninformed nodes x and y and let X and Y denote the random indicator variables indicating whether x or y respectively get informed in this round. Note that, as the edges are not yet sampled, there is some positive correlation between X and Y , because if we condition on the event that the uninformed node x becomes informed, then it is slightly less likely that x and the uninformed node y are neighbors which increases the probability that y has a higher fraction of informed neighbors and therefore y pulls the information more likely. However, the framework from allows for some positive correlation. We will bound the covariance accordingly. Let X := “X = 1”, Y := “Y = 1” and let E(G) denote the edge set of the random graph for the current round; define E := “{x, y} ∈E(G)”.

Cov(X, Y ) = P[X ∩Y] −P[X]P[Y] = P[X]P[Y | X] −P[X]P[Y] = P[X](P[Y | X] −P[Y]).

P[X](P[Y | X] −P[Y]) ≤P[X]O(1/N) ≤K

nO(1/n). Therefore the covariance conditions are fulfilled for the exponential growing and shrinking phases.

Now we have to estimate the probability pk for an uninformed node to become informed in a round starting with k informed nodes. If an uninformed node has a neighbor, i.e. if it is not isolated, then with probability k/(n−1) it becomes informed. However, if it is isolated, which according to Fact 9 is the case with probability e−a + O(1/n), the node does not become informed in this round deterministically. Thus pk = (1 −e−a + O(1/n))k/n. Hence both,

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upper and lower, exponential growth conditions are fulfilled for arbitrary 0 < f < 1 with γn = 1 −e−a + O(1/n). Recall that according to Remark 7, the O(1/n) term is negligible.

E[Tn(1, Fn)] = Log2−E−A(N) + O(1)

and that there are A1, α1 > 0 such that for all r, n ∈N

≤Log2−E−A(N) +

−r] ≤A1 exp(−α1r). Next, for the exponential shrinking conditions, we consider 1 −pn−u. We have

N −1(1 −E−A + O(1/N)) = E−A + (1 −E−A)U

n + O(1/n). The upper and lower exponential shrinking conditions are fulfilled with ρn = a + O(1/n) (because e−a + O(1/n) = e−a+O(1/n)) for an arbitrary 0 < g < 1. Note that according to Remark 7, the term O(1/n) is negligible. Theorem 6 therefore yields

A Ln(N) + O(1)

and that there are A2, α2 > 0 such that for all r, n ∈N

A Ln(N) +

−r] ≤A2 exp(−α2r). Thus, considering the exponential growth phase and the exponential shrinking phase to- gether, we obtain the claim.

Push&Pull In Random Evolving Graphs

Theorem 12. Let a > 0 and let γ := 2(1 −e−a) −(1 −e−a)2/a. Assume that each round a newly sampled Erdős-Rényi random graph G(n, a/n) is the underlying graph. Then for the

A Ln(N) + O(1)

and there are constants A, α > 0 such that for all r, n ∈N P[|Tn −E[Tn]| ≥r] ≤A exp(−αr). Before we prove Theorem 12 we introduce some notation. Consider an uninformed node y at the beginning of a round that starts with k ∈N informed nodes, let µ := k/n; we will refer to this round as the current round. Let PHy denote the event that y is pushed by an informed node in the current round. Analogously let PLy denote the event that y pulls the rumour in the current round from an informed node. Further set PPy = PHy ∪PLy, i.e. PPy denotes the event that y is pushed or pulls the rumour in the current round. For

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.

Figure: System Model & Simulation Flow for Ansys Airbag Injury Simulation

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).

Figure: System Model & Simulation Flow for Ansys Airbag Injury Simulation

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.

Figure: System Model & Simulation Flow for Ansys Airbag Injury Simulation

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).

Figure: System Model & Simulation Flow for Ansys Airbag Injury Simulation

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.

Figure: System Model & Simulation Flow for Ansys Airbag Injury Simulation

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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