Neutral Point Clamped Npc Inverter Matlab

Failure Analysis and Reliability Evaluation of Modulation Techniques for Neutral Point Clamped Inverters—A Usage Model Approach

Masoud Farhadi, Mehdi Abapour, Mehran Sabahi

Please cite this article as: Farhadi Masoud, Abapour Mehdi, Sabahi Mehran, Failure Analysis and Reliability Evaluation of Modulation Techniques for Neutral Point Clamped

Modulation

Techniques for Neutral Point Clamped Inverters—A Usage Model

Approach

Masoud Farhadi 1, Mehdi Abapour 2*, Mehran Sabahi 3

Tabriz, Iran

Tabriz, Iran

Tabriz, Iran

Abstract: Up to now, many modulation techniques have been proposed for neutral point clamped (NPC) inverters. In this paper, for the first time, a general methodology is applied to calculate and compare the failure analysis and reliability of NPC inverter with most commonly used control strategies. Also, the mean time to failure (MTTF) of NPC inverter is derived for different control strategies. It is demonstrated that the key feature of control strategies in determining the reliability of inverter is their loss distribution among the switches. The failure rate of components that is relevant to this study and junction temperature calculation is developed, then conduction losses and switching losses of switches for different control strategies are calculated. Finally, the most reliable control strategy is identified. Experimental results obtained have promptly justified the theoretical analysis and outlined procedure.

Λfet

Failure rate of MOSFET (FIT = 10−9 failure/hour).

Λd

Failure rate of diode (FIT).

Λc

Failure rate of capacitor (FIT).

Λb

Base failure rate (FIT).

Πt

Temperature factor.

Πa

Application factor.

Πq

Quality factor.

Πe

Environment factor.

Πs

Electrical stress factor.

Πc

Contact construction factor.

Πcp

Capacitance factor.

Πv

Voltage stress factor.

Πsr

Series resistance factor.

Λnpc

Failure rate of inverter (FIT).

L

Lifetime under use condition.

L0

Lifetime under testing condition.

V

Voltage under use condition (V).

V0

Voltage under testing condition (V).

Ea

Activation energy (J).

Kb

Boltzmann’s constant (8.62 ×10−5 eV/K).

A

Constant describing the voltage and temperature dependency of Ea.

Ξ

Stress variable under operation.

Ξ0

Stress variable under test.

Pr

Rated power of MOSFETs (W).

Vs

Voltage Stress Ratio.

Tj

Junction temperature (°C).

Ta

Ambient temperature (°C).

Tc

Case temperature (°C).

Th

Heat sink temperature (°C).

Rth,Ca

Thermal resistance between the case and ambient (°C/W).

Rjc

Thermal resistance between the junction and case (°C/W).

Rth,Ch

Thermal resistance between the case and heat sink (°C/W).

Rth,Ha

Thermal resistance between the heat sink and ambient (°C/W).

Zjc

Thermal impedances between the junction and case (°C/W). PSW,MOSFET Switching losses of MOSFET (W).

Psw,D

Switching losses of diode (W).

Ts

Sampling period (Sec).

Eon

Turn on energy losses (J).

Eoff

Turn off energy losses (J).

Erec

Reverse recovery process energy (J).

E(M,Θ)

Commutation energy losses (J).

Il(M,Θ)

Load current (A).

Imax

Maximum collector current (A).

M

Modulation index.

Φ

Current lagging angle to voltage (Deg).

Vce

Collector to emitter voltage (V).

Vcen

Rated collector to emitter voltage (V).

Vceo

Threshold collector to emitter voltage (V).

Ic

Collector current (A).

Icn

Rated collector current (A).

Rs

Collector to emitter resistance (Ω).

Vf

Diode forward voltage (V).

Vfn

Rated diode forward voltage (V).

Vfo

Diode threshold voltage (J).

Rd

Diode resistance (Ω).

Pcond

Conduction losses (W).

Econd

Conduction energy (J).

Α

Command voltage vector angle (Deg).

Λd,F

Failure rate of freewheeling diode (FIT).

Λd,C

Failure rate of clamping diode (FIT). 1.

Introduction

Nowadays, continuous development of semiconductor switches has led to widespread application of power electronic systems. Usually, these systems have a large number of power semiconductor switches. In addition, most of power electronic converters are equipped with electrolytic capacitors. But the semiconductor switches and electrolytic capacitors are the most fragile components -. Also, cost reduction pressure from global competition dictates minimum reliability-oriented design margin. For these reasons, reliability is the number one challenge for power electronic systems. So, quantitative evaluation of reliability for power electronic systems being a significant concern, can be used as a criterion to compare different topologies and control strategies.

The past decade has witnessed an increasingly growing research interest in various aspects of reliability for power electronic systems, with focused specifically on inverters –. In , Chiodo et al. presented some crucial properties to evaluate reliability of the power electronic systems. During the last few decades, many recommendations are proposed to improve reliability, such as “fault-tolerant design”, “condition monitoring”, and “active thermal management”, to meet current and future industry needs.

Mirafzal presented an instructive survey of existing fault-tolerance techniques for three-phase, two-level, and multilevel inverters in . More comprehensive fault-tolerant techniques regarding power electronic converters in case of power semiconductor device failures, are reviewed by . For condition monitoring (CM), a review paper was presented by , which described the current state of the art in CM research

5

for power electronics. In , it is proposed to use the active thermal management to reduce the switching losses or to move them to less stressed devices. That can increase the reliability of power electronic modules. In , the authors present a global reliability comparison between two-level and three- level/five-level inverter topologies in single and three-phase operations. Harb et al. has proposed a new methodology for calculating the reliability of the photovoltaic module-integrated inverter (PV-MII) based on a stress factor approach . Various fault-tolerant configurations have been proposed in the literature for power electronics converters –. But, no reliability evaluation or comparisons of different control strategies have been presented in previous articles. For the first time to our knowledge, a general methodology is applied that permits us to compare different control strategies from the reliability point of view. Though the methodology presented here is general, results associated with a three-phase three-level neutral point clamped (NPC) inverter are presented and discussed here.

NPC inverters are the most widely used topology of multilevel inverters in MV high-power applications on the market and play an increasingly important role in electric motor speed control, utility interfaces with renewable energy resources, induction heating, flexible AC transmission systems (FACTS) and uninterruptible power supplies (UPS). Over the last decades, many modulation schemes are proposed to improve the performance of NPC inverters, which can generally be classified into two categories: pulse width modulation (PWM), and space vector modulation. In this paper, three common control methods: sinusoidal PWM (SPWM), third harmonic injection PWM (THIPWM), and space vector PWM (SVPWM) are compared to determine the most reliable modulation technique for NPC inverter. It is demonstrated that the effect of control strategy on reliability will be determined by its effect on the junction temperature of switches. The junction temperature is a function of three parameters, namely, ambient temperature, power loss of the semiconductor switch and the thermal resistance of the heat sink.

Clearly, control strategy effects on the junction temperature only through the power loss. So this paper presents a comprehensive analysis on conduction and switching loss for different modulation strategies.

6

2.

Basic Operation Of The Three-Level Npc Inverter

In NPC topology, to produce n different levels for output phase voltage, (n-1) capacitors (with DC voltages), 2(n-1) switches and (n-1)(n-2) clamping diodes are needed in each leg. Fig. 1 depicts the three- phase three-level NPC inverter topology. Switches (1, 3) and (2, 4) on each leg are a complementary switching pair, which means that when a switch is on, to avoid DC link short circuit, the other switch must be off and vice versa. Table 1 shows the three switching states of this topology and corresponding output voltage levels. Also, their corresponding equivalent circuit is highlighted in Fig. 2. To obtain the equation of power losses we need to calculate the effective duty cycle of each switching state. Table 2 shows the duty cycle of each state based on modulating function (MF). The modulating function is explained for different control strategies as a function of modulation index and current to voltage lagging angle, in Section 6.

Fig. 1. Three-phase three-level topology of a diode clamped inverter. Table 1 The switching state of NPC inverter and corresponding output voltage levels

7

Fig. 2. Corresponding equivalent circuit for switching states.

Dtn

3.

Failure Rate Of Components

In order to analysis the effect of modulation schemes on the reliability of the components, this section is devoted to the calculation of the failure rate of the components that is relevant to this study. Currently, failure rates provided by the Military Handbook for Reliability Prediction of Electronic Equipment, MIL-HDBK-217 F , are used most often for the reliability modeling . It covers the broad range of component types, and widely accepted for military and commercial electronic systems. So, in this paper MIL-HDBK-217 F will be used for failure rate calculations. However, the aim of this paper is to prepare a framework for reliability comparison of control strategies, and any available data source can be adopted in the outlined procedure. The field experience confirms that power switches and capacitors are the most vulnerable components. Moreover, magnetic components and control system are much more reliable , , , and . Therefore, only power switches and capacitors are considered in this paper and other electronic systems (e.g. gate drivers, control) are not taken into account. In addition, the type of the components must be specified in this regard. In this paper the switches are considered to be power MOSFETs. But, a similar discussion can be extended to other types of power semiconductor

8

switches. The failure rates of the MOSFET, diode, and capacitor are summarized in Table 3. These failure rates are expressed as a function of various stress factors.











As shown in Table 3, except the voltage stress factor, other factors are equal in each control strategy for the failure rate of capacitors. Also, the junction temperature is used as a common input for the failure rate calculation of the power switch. So, in the next three Sections, we will discuss about these factors in detail.

4.

Failure Rate Of Dc-Link Capacitors

DC link capacitors are widely used to balance the instantaneous power difference between the input source and output load, and minimize voltage variation in the dc link. Three types of capacitors are

9

generally available for dc-link applications, which are the aluminum electrolytic capacitors, metalized polypropylene film capacitors and high capacitance multi-layer ceramic capacitors -. Among these types, aluminum electrolytic capacitors due to low cost per joule are commonly used in DC- link application. So, in this paper dry electrolytic aluminum type is considered for capacitors. This choice is purely for illustrative purposes. Failure-rate models for other technologies are available in and can be incorporated into the analysis for comparing different technologies.

The failure rate of capacitor can be calculated using many lifetime models. Arrhenius equation-based models are most widely employed to analyze the reliability of capacitors. These models are generally

(1)

The Ea and n as a function of the capacitor type were obtained in , . For aluminum electrolytic capacitors, equation (1) can be simplified as follows :

(2)

In , a generic lifetime model of electrolytic capacitors is proposed based on the primary wear-out

10

In this paper, to integrate analytical tools for assessing converter reliability, the MIL-HDBK-217 F will be used to calculate the failure rate of capacitors. As previously mentioned, except the voltage stress factor, other factors are equal in each control strategy for failure rate of capacitors. To calculate the voltage stress, the voltage of DC link capacitors for different modulation techniques are obtained (see Fig. 3). Based on these results, it is clear that the failure rates of C1, C2 in SVPWM modulation are different. This is because of imbalance voltages of DC-link capacitors, which causes the sum of the failure rates of C1 and C2 to be higher in this method compared to the other two methods. Although some improved SVPWM methods are proposed to reduce the voltage imbalance, but these methods have a stronger impact on imbalance losses in power switches . So, usually these methods decrease the reliability. It should be noted that beside the voltage balancing, voltage ripple reduction has an equal importance from the power electronic designers perspective and more research efforts are expected to tackle these issues to achieve more reliable inverters.

180

0.00E+00 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01 7.00E-01

B

Fig. 3. Capacitors voltage comparison with different control strategies

B C2 Voltage

5.

Junction Temperature Calculation

As it was mentioned before, the temperature of the switch is the only factor that affects the failure rate of the switches in different control strategies. Consequently, in this section the thermal modeling of the switches is provided. The thermal models used for a single power switch and a power switch module are shown in Fig. 4, in which the thermal impedance between the junction and case usually is modeled as a multi-layers foster RC network in the manufacturer datasheets, (see Fig. 5c) , . Regardless of the thermal capacitance Cth, which describe dynamic changes, junction temperature based on the thermal equivalent model shown in Fig. 4, is calculated as follows:

(4)

Similar to (4), the case temperature of switch can be expressed in terms of its power loss and

(5)

Using (4) and (5), the junction temperature can be expressed as

12

So having the ambient temperature, power losses and thermal resistance we can calculate the junction temperature of power switches. Note that, if a heat sink is used, the thermal resistance between the case and ambient will be the thermal resistance of the heat sink, which is much less than the thermal resistance between the case and ambient when no heat sink is used .

It should be noted that junction temperature is not very sensitive to ambient temperature. Also, for a specific application with known power loss, the heat sink will be predetermined. Therefore, power loss change is the only factor that leads to a temperature change. The next section is devoted to the power loss calculation for different control strategies.

C

Fig. 4. The used thermal model and thermal equivalent for Zth,jC

A The Used Thermal Model For Single Power Switch

b The used thermal model for power switch with freewheeling diode module

13

6.

Evaluation Of Power Switches Losses

Power switch losses consist of conduction losses and switching losses. Up to now, many papers have reported conduction and switching loss calculation, but data sheet information based method for switching losses calculation is well known and widely accepted in both scientific and industrial applications. In this method, characteristic curves which are presented in the datasheets of each power semiconductor, are approximated by exponential equations using curve-fitting tools. This method provides an accurate power loss prediction for many different types of circuits. On the other hand, power loss prediction based on pulse by pulse calculation is used most often for the conduction loss calculation. It can be seen that the losses are dependent on the circuit parameters. In addition to this, it is easy to use in order to compare conduction loss of control techniques.

6.1. Switching Losses

Switching loss consists of the energy losses during turn-on and turn-off instants in one reference period. Turn-on losses are caused by the forward recovery process. As for fast diodes, this share of the losses can be neglected. However, the switching energy at turn-off can’t be neglected. The switching losses for power switches and diodes can be derived as:

(8)

Where the commutation energy loss as a function of load current is described as follows:

(9)

Where (A, B) and (C, D) are turn-on and turn-off curve fitting constants for power switch, respectively. For a typical MOSFET (IRF740), the turn-on energy losses, turn-off energy losses, and

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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