Ups Uninterruptible Power Supply Matlab

Exploitation of material consolidation trade-offs in multi-tier

Complex Supply Networks

Vinod Kumar Chauhan∗1,2, Muhannad Alomari3, James Arney3, Ajith Kumar Parlikad1,

Abstract

While consolidation strategies form the backbone of many supply chain optimisation problems, ex- ploitation of multi-tier material relationships through consolidation remains an understudied area, despite being a prominent feature of industries that produce complex made-to-order products. In this paper, we propose an optimisation framework for exploiting multi-to-multi relationship between tiers of a supply chain. The resulting formulation is flexible such that quantity discounts, inventory holding, and transport costs can be included. The framework introduces a new trade-off between tiers, leading to cost reductions in one tier but increased costs in the other, which helps to reduce the overall procurement cost in the supply chain. A mixed integer linear programming model is developed and tested with a range of small to large-scale test problems from aerospace manufacturing. Our comparison to benchmark results shows that there is indeed a cost trade-off between two tiers, and that its reduction can be achieved using a holistic approach to reconfiguration. Costs are decreased when second tier fixed ordering costs and the number of machining options increase. Consolidation results in reduced inventory holding costs in all scenarios. Several secondary effects such as simplified supplier selection may also be observed.

Keywords: Supply chain management; multi-tier; supply network complexity; configuration; procure- ment cost optimisation; mixed integer programming; consolidation.Analytics, Artificial Intelligence

Introduction

Procurement of parts from suppliers is a key task in the supply chain management, greatly impacting its competitiveness and performance (Amid et al. (2006)). In many industries, procurement cost often forms the highest proportion of total cost of a product (Willard (2012)).

Consolidation has been a key underlying strategy in the context of procurement decisions. Consolidation, as the name suggests, is a process of combining related activities or materials to improve performance of a supply chain resulting from cooperation and coordination (Schulz and Blecken (2010); Chadha et al.

(2022)) and can help in reduction of costs, increase efficiency and improve performance (Vaillancourt (2016); Giampoldaki et al. (2023)). Material consolidation consists of purchasing, transportation and inventory activities (Brauner and Gebman (1993)), where a buyer or a set of buyers may choose to group items or orders to obtain quantity discounts (Monczka et al. (1993); Hagberg and Hulth´en (2022)). While this helps in increased efficiency and reduction of costs it can reduce flexibility of sourcing options, thus reducing supply chain resilience. Inventory consolidation considers relocation of warehouses to increase inventories in order to

Arxiv:2210.11479V3 [Cs.Ce] 19 Nov 2023

make use of reduced operational costs but may introduce increased transport cost (Wanke and Saliby (2009); Ralfs and Kiesm¨uller (2022)). Transportation consolidation merges small deliveries into single dispatch of economical load but increase uncertainty in delivery times (Trent and Monczka (1998); C¸etinkaya (2005); Torbali and Alpan (2023)).

Consolidation activities to date have been overwhelmingly studied within the span of single supply eche- lons (Stenius et al. (2018)). This is not surprising, as buyers often have control of their dyadic connections, gradually losing both visibility and influence beyond their immediate connections, making consolidation de- cisions not applicable beyond their immediate connections. There are, however, an increasing number of industrial contexts where a buyer may influence its wider supply chain, and there is willingness for coop- erative decision making for collective performance (Chauhan et al. (2023)). Examples include production of complex, made-to-order products, such as heavy machinery, turbines, aerospace products, and medical devices. Due to long-term supply relations involved in these sectors, a manufacturer may be involved in configuration of whole supply chain. In addition to whole supply chain configurability, longevity of relation- ships necessitates de-risking through multi-sourcing activities. This increased span of control, coupled with multi-sourcing offers a unique multi-to-multi relationship structure whereby products may be consolidated further upstream, affecting cost structures at different tiers.

In this paper, we highlight this understudied multi-tier consolidation problem presented by the above context and formulate it through a case study. We term this new consolidation opportunity as “multi-tier material consolidation problem”.

To contextualise the multi-tier consolidation problem, we consider following example from an aerospace industry (Fig. 1). Here, aircraft engines are produced, requiring different types of parts, which manufacturer outsources from a set of certified machining suppliers. These Tier 1 suppliers need different types of forged metal to manufacture final finished parts, which are themselves outsourced to Tier 2 forging suppliers. The forgings that could be used for manufacturing different parts is predetermined by the company.

Figure 1: Two-tier supply chain of a manufacturing company The forging process involves manufacturing roughly shaped parts from melted alloys and machining refines those into final parts. The supply chain has N different forgings to manufacture M different parts, and creates a multi-to-multi relationship between forgings and parts.

That Is, One Forging Can Be Used

to manufacture many parts, and similarly, one part can be manufactured in multiple ways from different forgings. The total procurement cost of parts from Tier 1 and forgings from Tier 2 depend on ordering cost, unit cost and consequent transportation costs.

Since Parts Can Be Manufactured In Multiple Ways

from different forgings, requiring different machining costs, forgings can be consolidated into a smaller set, thereby, reducing the cost of forgings at the expense of increased machining time to manufacture parts from a limited set of forgings, and hence increased machining cost. Thus, there is a trade-off between the reduced cost of forgings at Tier 2 and increased machining cost at Tier 1. Additionally, since forging process takes longer compared to machining, the company also maintains a specific inventory of forgings. This leads the company to order additional forgings resulting in extra purchasing cost and cost of holding inventories.

Hence, objective of the multi-tier material consolidation problem in this example is to minimise overall procurement cost across the supply chain by consolidating forgings and striking an optimal balance between

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cost of forgings at second tier and machining cost at first tier. This problem can be visualised as a clustering problem, as presented in Fig. 2. Here, each forging is represented as a point in some space depicted as a circle (as shown in left panel). The problem is to find clusters whereby all forging in a cluster can be replaced with a single forging from the group. That is, the selected forging is used to manufacture all machined parts that were manufactured using different forgings in the cluster (as shown in middle-panel). Thus, we need to find minimum number of clusters, and hence minimum number of forgings in the consolidated set (as shown in right-panel), which balances the trade-off between cost of forgings and machining cost. Since ordering items in different quantities affects the suppliers so quantity discounts also need to be considered.

Figure 2: Illustration of the multi-tier consolidation problem: Different colours represent clusters where a square-enclosed forging is used to replace all other forgings in its cluster. The rest of this paper is structured as follows. In Section 2, we present related work on supply chain consolidation to situate context of our contribution. In Section 3, we characterise the multi-tier material consolidation problem, formulate it using an aerospace supply chain as a case study, discuss our solution approach and choices of modelling languages as well as solver libraries, and present our analysis. We then present concluding remarks, limitations and future scope of the study in Section 4.

Related Work

Scholars in supply chain management have proposed several consolidation strategies to improve cost against operational decision criteria. These can be broadly categorised as purchasing, shipment, inventory and part consolidation (Brauner and Gebman (1993)), as discussed below.

Purchasing Consolidation

Purchasing consolidation considers regrouping of items for purchase, which may involve grouping of multiple, related types of products to be purchased from same supplier in order to obtain contractual and logistics discounts (Monczka et al. (1993); Chauhan et al. (2023)), or pooling items to be purchased with other buyers (group buying) to increase economies-of-scale and obtain a reduction on unit cost of production and delivery (Vaillancourt (2017); Hu et al. (2022)). Both of these strategies may result in a loss of flexibility, due to the need to align production and deliveries with other product lines (in case of product grouping), or other buyers (in case of group buying) (Vaillancourt (2016)). Early deliveries may result in increased inventory costs (Guiffrida and Nagi (2006)), and over-reliance on a single supplier may increase risk and opportunism (Chopra and Sodhi (2014)).

A related but separate strand of consolidation literature considers multi-sourcing decisions determining number of suppliers supplying an item. It is generally presumed that single-sourcing, i.e., procurement from a single supplier, results in the cheapest unit cost (Silbermayr and Minner (2016)) while dual-sourcing and multi-sourcing avoid supplier monopoly of items and help reduce the risk of disruptions in a supply chain (Tomlin (2006)).

Researchers have proposed a number of analytical models to characterise these trade-offs. For example, Gaur et al. (2020) considered a real-world case study of an automotive-parts manufacturer to study impact of disruption on a closed-loop supply chain using sourcing policies. They developed a mixed-integer non-linear programming model for the problem and found that, under the risk of supply chain disruption, multi-sourcing generates more profit as compared to single sourcing.

Shipment, Volume And Order Consolidation

Shipment consolidation, also known as freight consolidation, transportation consolidation and terminal con- solidation, is a logistics strategy which refers to merging of small deliveries into a single dispatch of economical load (¨Ulk¨u (2012); Wagner et al. (2023)). This helps in increased efficiency and reduction of CO2 emissions, and delivery costs, e.g., Mu˜noz-Villamizar et al. (2021) investigated shipment consolidation by pooling and developed a mixed integer linear programming (MILP) model to study its effect on CO2 emission, distance and transport costs. However, the practice can cause uncertainty in delivery times leading to poor service for customers (Masters (1980)).

Volume consolidation, i.e., a strategy where a buyer purchases most of its supply from one supplier, results in shipment consolidation and helps to reduce shipping costs. For example, Cai et al. (2010) studied volume consolidation and its effect on supply chain outcomes. Through an empirical study, they found that volume consolidation enhances buyer’s ability to learn from the supplier, and supplier performance. However, coordination costs negatively affect buyer satisfaction and supplier performance.

Order consolidation refers to consolidation of a customer’s orders at a delivery station so as to organise delivery in fewer trips. For example, Zhang et al. (2019) studied order consolidation for last-mile split- delivery in online retailing and developed an integer programming model to study the trade-off between splitting orders and consolidating shipments.

Inventory Consolidation

Inventory consolidation, also called facility location problem, is identification of optimal warehousing and distribution centre locations and capacities to stock up inventories with aim of meeting customer demand (Wanke and Saliby (2009); Seyedan et al. (2023)). Minimisation of inventory holding locations in a supply chain helps to reduce operational costs, however, leads to increased distance travelled to customers and thus increased CO2 and transport costs (Gabler and Meindl (2007)).

A classical and widely studied strategy is postponement, which refers to late differentiation of products to cater to fast changing demand (Zinn (2019)), resulting in cost savings (Geetha and Prabha (2021)). For a systematic review on postponement strategy refer to Ferreira et al. (2018); Zinn (2019) and for a review on consolidation effect and inventory portfolio analyses refer to Wanke (2009).

Part Consolidation Through Product Redesign

Part consolidation is an activity that aims to drive supply chain costs down through part redesign (?Kunov- janek et al. (2022)). Here, the assembled unit may be redesigned to contain fewer but more complex parts leading to a trade-off between increased manufacturing cost and reduced supply chain cost (Knofius et al.

(2019)), whilst manufacturing lead time may depend on process technology used. For example, Knofius et al. (2019) observed that part consolidation through Additive Manufacturing reduced lead times but resulted in increased total costs due to loss of flexibility.

Part consolidation is widely studied with different objectives. For example, Johnson and Kirchain (2009) applied a process-based cost model to quantify effects of parts consolidation and costs on material selection choices, and Crispo and Kim (2021) studied a multi-layered topology-based optimisation approach for part consolidation in Additive Manufacturing. Gan et al. (2021) explored concurrent design of product and supply chain and presented a trade-off between modularity of product and sourcing flexibility in supply chain. For a detailed review on part consolidation, refer to Sigmund and Maute (2013); Liu (2016); Gan and Grunow (2016).

From this brief literature review, it is clear that consolidation has been studied widely in supply chains at different levels and with different perspectives. Our work presents a unique perspective, different from the existing research, in its focus on the trade- off between two supply tiers. We consider the problem of minimising procurement cost by consolidating material through exploitation of multi-to-multi relationships in complex made-to-order products and show that consolidation in one tier results in cost savings at that tier but increased manufacturing and inventory

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costs in the next downstream tier, necessitating a trade-off formulation. While our work may appear similar to recent studies on part consolidation through redesign, it’s important to note that, in our case, there is no redesign activity involved. Instead, we focus on the exploitation of multiple potential relationships between materials and production across two tiers (please refer to Gan and Grunow (2016) for a review of trade-off in concurrent design of product and supply chain). Our proposed formulation is extended to incorporate a number of other considerations including shipment and purchasing consolidation so as to explore interaction between multiple consolidation strategies. We present this conceptualisation and problem formulation next.

Problem Formulation

Our context consists of a two-echelon supply chain, yielding a high-value complex assembled product with long-term supplier relationships and a large number of suppliers. Typical examples include precision engi- neered products such as aircraft engines, medical devices, wind turbines and heavy machinery. Here, forged metal alloys are precision machined to manufacture finished parts, which are then assembled into a final product. The manufacturer who assembles the final product has overall visibility and ability to configure the whole supply chain.

As discussed earlier (Fig. 1), Tier 2 involves manufacturing roughly shaped parts from melted alloys, called as forgings, and Tier 1 involves machining that refines forgings into final finished parts. A single forging can be used to manufacture multiple, different parts, and similarly, one type of part can be manufactured from a variety of different forgings. However, manufacturing of a part from different forgings results in different subsequent machining costs. A myopically ideal scenario from a machining cost reduction perspective would be to have a single forged part for single machined part, as the forging brings the part to as close a shape as possible to the machined part. However, a one-to-one relationship would increase transportation costs, and result in over-reliance on the forger. Additionally, quantities per forging type would decrease at Tier 2, preventing quantity discounts. Furthermore, as forging process takes longer compared to machining process, a certain inventory of forgings must be maintained to ensure continuity of production.

This Leads The

company to order extra forgings resulting in extra purchasing cost and cost of holding inventories. On the other hand, consolidating forgings such that multiple machined parts can be manufactured from a single type of forging means that machining costs in Tier 1 increase, along with lead times, resulting in a trade-off between costs of the two tiers. Another myopic scenario here would include one forging creating multiple machined parts, with minimum transportation costs and maximum economies-of-scale at Tier 2, but much increased machining costs and lead time in Tier 1. To benefit from economies-of-scale, both the tiers consider discounts based on quantity of items ordered, as pictorially presented in Fig. 3. For a given order of parts, along with inventories, discounts on parts can be pre-computed to simplify the modelling, as information required to calculate discounts on parts is given.

Our Objective is to optimise the overall procurement cost across the supply chain for a given requirement of parts, including inventories, by consolidating Tier 2 forgings to build Tier 1 parts, which balances the trade-off between cost of the tiers; under the constraint that there should be at least one way to manufacture each part from consolidated set of forgings.

In the development of model for the problem, we assumed the following points. (a) Demand is constant and a priori known for all the parts. (b) There is at least one way to manufacture all the parts from a given set of forgings.

(c) There can be multiple ways to manufacture a part from different forgings but each way needs only one type of forging to manufacture the part. Given the multi-to-multi relationship between forgings and parts, and multiple manufacturing ways for a part, the forging consolidation problem can become complex due to the need to consider different forging combinations for each part. To simplify our problem, we make this assumption that can be

5

achieved by increasing number of parts. For instance, if part P1 requires two distinct forgings, say F1 and F2, for its production, we replace P1 with two separate parts, P1A and P1B. P1A is produced using F1, and P1B is manufactured using F2. This adjustment is possible because, typically, each part relies on a single forging for its production.

(d) Consolidation does not result in additional costs within the supply chain, including any costs associated with reconfiguring production for increased quantities of certain forgings. Furthermore, consolidation does not have any adverse effects on the supply chain relationships with suppliers.

Consolidated forgings represent a subset of all forgings, requiring no additional machinery for produc- tion. Therefore, we anticipate minimal additional costs. Additionally, during the order assignment to suppliers stage (for details, please refer to Chauhan et al. (2023)), which follows consolidation, each supplier is assigned certain minimum orders. As a result, we expect that consolidation will not significantly impact the supply chain relationships.

Mathematical notations for the development of the model are defined in Table 1.

Objective Function:

The procurement cost of parts in the supply chain consists of the sum of forging cost, machining cost and associated inventory holding costs. So, the optimisation problem for procurement cost optimisation through forging consolidation is given as:

(1)

where CM, CF and CI are machining costs, forging costs and inventory holding costs, respectively. The costs of machining CM to manufacture a given order of parts is the sum of fixed cost and variable cost, which depend on number of units ordered, unit cost of part and unit transportation cost. The machining cost also depends on the forging used to manufacture a part. There can be multiple ways to manufacture a part from different forgings. A forging with minimum machining cost to manufacture a part is selected from consolidated set of forgings. So, the cost of machining can be calculated as given below:

(2)

where CMFi, Di, Mi and vi are fixed cost, discount level, ordered quantity and variable cost for part i, respectively. Di can be pre-computed using discount levels Di

D (As Defined In

Fig. 3) because both are available before solving the problem.

Discount Levels

Figure 3: Quantity intervals for calculating discount levels for forging k. Quantity intervals and discounts for parts can be represented in a similar way. Similarly, the cost of forging is sum of fixed cost and variable cost which depend on number of units ordered, unit cost of forging and unit transportation cost of forging. But number of forgings depend on number of parts ordered, and requirement of forging k is calculated using P

I Lik × Mi × Xik, Where Lik

is number of forgings k required to manufacture one unit of part i and xik is an indicator variable which indicates if forging k is used to manufacture part i. So, assuming CFFk, CFUk and CFTk denote fixed cost, unit cost and unit transportation cost, respectively, associated with ordering forging k then forging cost CF is calculated as below.

Lik

number of forgings k needed to manufacture one unit of part i

Cmtik

per unit transportation cost of part i manufactured from forging k

Zk

1 if forging k is selected in solution, i.e., consolidated set contains forging k, otherwise 0

Vi

continuous variable to calculate per unit variable cost of part i

Xik

indicator variable; 1 if forging k is used to manufacture part i otherwise 0

Udk

indicator variable; 1 if forging k is purchased at discount level d otherwise 0

Where Dk

d and udk are discount level and corresponding indicator variable. The inventory cost CI results from the need to keep an inventory of forgings to manufacture Pi parts. It is sum of cost for purchasing inventory, which is like CF except fixed cost, and cost of holding inventory.

So, assuming CFHk be unit holding cost for forging k, CI is given below.



.

Constraints:

The problem requires that there should be at least one way to manufacture each part, which can be added as given below.

7

where vi is per unit variable cost for part i and E is a very small number (included to avoid strict inequalities), and this inequality ensures that part i is being manufactured. To ensure vi takes minimum value, i.e., part i is being machined from the cheapest forgings, we further add following constraints.

(6)

where CMUik and CMTik are per unit machining cost and transportation cost for manufacturing part i from forging k, M is a very large number and xik is an auxiliary indicator variable which helps to find minimum cost forging for the part. First and second part ensure that machining cost is equal to the minimum of different ways to manufacture the part, and third part of (6) ensures that non-zero minimum value is selected.

Constraints related to discounts for forgings, i.e., economies-of-scale for forgings are given below, where d = 0, 1, 2, ..., |Dk| −1 are discount levels for forging k, udk is an indicator variable that indicates if forging k

Gets Discount Level D, And Qk

d represents quantity intervals to calculate discounts, as explained in Fig. 3. The quantity discounts are calculated on forgings required to meet order and inventory requirements, as given below.

D Udk = 1,

∀k.

D ≤P

i Lik × (Mi + Pi) × xik −E.

(8)

For extreme cases of d = 0 and d = |Dk| −1, and ∀k,

(9)

Equation (7) ensures that only one discount level is applicable, and inequalities (8) and (9) force that

Discount Level Dk

d is applicable, i.e., udk = 1 when required quantity of forging k is in semi-closed interval

I

. The objective function and constraints have two non-linear terms as zk × xik × udk and xik × udk which can be simplified into linear terms to make the problem easier to solve. Hence, the following new variables are introduced for linearisation as yikd = zk × xik × udk and wikd = xik × udk, and corresponding constraints are given below.

≥Xik + Udk −1,

∀i, k, d.

(11)

So, simplifying constraint (8) and (9) using linearisation variables, we get following updated constraints.

Optimisation Problem:

The multi-tier material consolidation optimisation problem can be obtained by substituting values of CM, CF and CI into objective function (1), and simplifying the non-linear terms using the linearisation variables, yielding an MILP formulation as given below.



.

(14)

Subject to constraints (5), (6), (7), (10), (11), (12) and (13), where variable zk and auxiliary variables xik, udk, yikd and wikd are binary.

Solution Approach

In this section, we discuss the methodology used to solve the consolidation problem, which is based on efficient problem formulation and the use of exact methods to obtain an optimal solution, as discussed below.

Model Development

For the multi-tier material consolidation problem, as discussed in the previous subsection, we developed an MILP model using the following mathematical techniques for simplification.

Linearisation:

The consolidation problem results in a cubic integer programming problem due to the interaction of two tiers and the exploitation of economies-of-scale.

Since Non-Linear Problems Are More

complex than MILP, we simplified the model using linearization, a process that converts non-linear terms into linear terms by introducing additional auxiliary variables. For example, x1 and x2 are two binary variables and we want to linearise non-linear term x1 × x2 then we introduce a new binary variable z = x1 × x2 such

(16)

where (15) ensures that z is 0 when any of x1 or x2 is 0 and (16) ensures that z is 1 when both x1 and x2 are 1. Similarly, for n binary variables x1, x2, ..., xn, non-linear term Qn

Z = Qn

i=1 xi as given below.

I

xi −(n −1).

(18)

Now suppose, v is a continuous variable and u is a binary variable then to linearise v × u, we introduce

S ≥V −(1 −U) × M,

s ≥0.

(19)

Here, the first and last inequalities ensure that s is 0 when u is 0. Second inequality ensures that s is upper bounded by v and third inequality ensures that s is lower bounded by v when u is 1, i.e., s = v.

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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