Simulation of the Effects of an Air Blast Wave - ELSA - Europa

Simulation of the Effectsof an Air Blast WaveMartin Larchera) t=2e-5 b) t=6e-5c) t=1e-4 d) t=1.4e-4PUBSY JRC41337 - 2007

The Institute for the Protection and Security of the Citizen provides researchbased, systemsorientedsupport to EU policies so as to protect the citizen against economic and technologicalrisk. The Institute maintains and develops its expertise and networks in information,communication, space and engineering technologies in support of its mission. The strongcrossfertilisation between its nuclear and non-nuclear activities strengthens the expertise it canbring to the benefit of customers in both domains.European CommissionJoint Research CentreInstitute for the Protection and Security of the CitizenContact informationAddress: Martin Larcher, T.P. 480, Joint Research Centre, I-21020 Ispra, ITALYE-mail: martin.larcher@jrc.itTel.: +390332789004Fax: +390332789049http://ipsc.jrc.ec.europa.euhttp://www.jrc.ec.europa.euLegal NoticeNeither the European Commission nor any person acting on behalf of the Commission isresponsible for the use which might be made of this publication.A great deal of additional information on the European Union is available on the Internet.It can be accessed through the Europa serverhttp://europa.eu/JRC 41337ISSN 1018-5593Luxembourg: Office for Official Publications of the European Communities© European Communities, 2007Reproduction is authorised provided the source is acknowledgedPrinted in Italy

Distribution ListLechner S.Anthoine A.Casadei F.Dyngeland T.Géradin M.Giannopoulos G.Gutierrez E.Halleux J.P.Larcher M.Paffumi E. (JRC Petten)Pegon P.Solomos G.DG TranExternal:Mr. Bung H. (CEA)Faucher V. (CEA)Galon P. (CEA)Kill N. (Samtech)Cheruet A. (Samtech)Potapov S. (EDF)S. LechnerThe information contained in this document may not be disseminated,copied or utilized without the written authorization of the Commission.The Commission reserves specifically its rights to apply for patents orto obtain other protection for the matter open to intellectual or industrialprotection.

CONTENTS1 Introduction ...................................................................................................................................52 Air Blast Waves.............................................................................................................................62.1 Introduction..........................................................................................................................62.1.1 Detonations ......................................................................................................................62.1.2 Air Blast Waves ...............................................................................................................72.2 Literature Data .....................................................................................................................82.2.1 Pressure-Time Distribution ..............................................................................................82.2.2 Maximum / Minimum Pressure .....................................................................................102.2.3 Impulse...........................................................................................................................112.2.4 Negative Phase...............................................................................................................112.2.5 Wave Form Parameter ...................................................................................................132.2.6 Shock Front Velocity .....................................................................................................162.2.7 Specific Heat Ratio ........................................................................................................163 Numerical Loading of a Structure with Air Blast Waves............................................................184 Investigations with Explosives as a Charge (Solid TNT) ...........................................................204.1 Modelling of the explosive ................................................................................................204.2 Behaviour in the Explosive ................................................................................................244.3 Cone with two Symmetry Axes .........................................................................................294.4 Cubic Charge with two Symmetry Axes............................................................................384.5 Spherical Charge with two Symmetry Axes ......................................................................424.6 Comparison between the Different Models .......................................................................464.6.1 Maximum Pressure.........................................................................................................464.6.2 Impulse...........................................................................................................................474.6.3 Arrival Time...................................................................................................................494.6.4 Positive Phase Duration .................................................................................................494.6.5 Comparison with results of other authors ......................................................................504.7 Influence of several parameters .........................................................................................504.7.1 Specific heat ratio (CON15) ..........................................................................................504.7.2 Values for γ, E 0, ρ .........................................................................................................514.7.3 Parameters for the explosive ..........................................................................................514.7.4 Burn mass fraction .........................................................................................................515 Bubble model...............................................................................................................................526 Control Volume...........................................................................................................................586.1 Flux between the CL3D and the fluid element ..................................................................586.2 Several models ...................................................................................................................587 Implementation of an Air Blast Loading Function .....................................................................627.1 Motivation..........................................................................................................................627.2 Used Function ....................................................................................................................627.3 Implementation ..................................................................................................................627.4 Verification with Examples................................................................................................638 Mesh generation for LS-DYNA ..................................................................................................649 References ...................................................................................................................................6510 Apendix..................................................................................................................................6710.1 EUROPLEXUS Code ........................................................................................................6710.2 Miscellaneous code ............................................................................................................7110.3 Sample input files...............................................................................................................744

1 IntroductionThis work is being conducted in the framework of the project RAILPROTECT, which deals withthe security and safety of rail transport against terrorist attacks. The bombing threat is onlyconsidered, and focus is placed on predicting the effects of explosions in railway and metro stationsand rolling stock and on assessing the vulnerability of such structures.The project is based on numerical simulations, which are carried out with the explicit FiniteElement Code EUROPLEXUS that is written for the calculation of fast dynamic fluid-structureinteractions. This program has been developed in a collaboration of the French Commissariat àl'Energie Atomique (CEA Saclay) and the Joint Research Centre of the European Commission (JRCIspra).As the aim of this project is to calculate the behaviour of structures under a loading produced by airblast waves, an indispensable starting point in this study is the ability to simulate the generation ofsuch waves from a given quantity of explosive, and to follow their propagation through 3D spacesas they finally impinge onto the structures under consideration.The results of such numerical tests of free air blasts are presented in this report and are compared toexperimental data available in the literature. In the absence of such data the EUROPLEXUS resultsare compared to the results of other codes, in particular to LS-DYNA, which is run in collaborationwith the University of Karlsruhe. This analysis is preceded by an exposition of some basic conceptson blast wave characteristics, explosives, and a description of the equation of state adopted hereinfor the modelling of the detonation of a solid explosive.5

2 Air Blast Waves2.1 Introduction2.1.1 DetonationsExplosions can be distinguished in detonations and deflagrations. The difference between detonationsand deflagrations is the velocity of the reaction zone in the explosive. Deflagrations have aslower reaction zone than the sound speed. Examples for deflagrations are the burning of gas-airmixturesand slow explosives like gun powder.Detonations have a faster reaction zone than the sound speed. The most common explosives reactwith detonations.To compare different explosives the TNT equivalent can be used. The TNT equivalent is a methodfor quantifying the energy released in the detonation of an explosive substance, by comparing it tothat of an equal quantity of TNT. It is known that 1 kg TNT releases the energy of 4.520x10 6 J.The TNT equivalent is available for standard explosives and for some of them it is summarized inTable 1.Explosive Mass Specificenergy [kJ/kg]TNTEquivalentTNT 4520 1Torpex 7540 1.667Semtex 1A 4980 1.102C4 6057 1.34Table 1: TNT equivalent for different explosivesThe effects of an explosion can be distinguished in three ranges:• Contact detonation: The explosive is in contact with the loaded material. The load-time functiondepends on the loaded material, which, in most cases, is destroyed. Occurrences are the blastingof concrete (demolition etc.) or terrorist attacks where the explosive is located directly on thestructure.• Near zone of the explosion: In most cases he material is also directly damaged like in thecontact zone.6

• Far zone. The blast wave resulting from the detonation dominates the effects on humans andstructures.The size of all these zones depends on the quantity of the explosive charge.Additional parameters for a detonation, depending on the size of the explosive, can be defined. Forexample, the radius in which debris from the explosion (not from the blast wave) are possible isgiven by Kinney [15] as13r = 45W(1)where, r is expressed in m and W is the TNT equivalent of the explosive in kg.2.1.2 Air Blast WavesThe pressure that arrives at a certain point depends on the distance and on the size of the explosive.ppmaxpmintp0t at d t nFigure 1: Pressure-time curve for a free air blast waveThe main characteristics of the development of this pressure wave are the following:- The arrival time t a of the shock wave to the point under consideration. This includes the timeof the detonation wave to propagate through the explosive charge.- The peak overpressure p max . The pressure attains its maximum very fast (extremely shortrise-time), and then starts decreasing until it reaches the reference pressure p o (in most casesthe normal atmospheric pressure).- The positive phase duration t d , which is the time for reaching the reference pressure. Afterthis point the pressure drops below the reference pressure until the maximum negative7

pressure p min . The duration of the negative phase is denoted as t n .- The incident overpressure impulse, which is the integral of the overpressure curve over thepositive phase t d .The idealised (free air blast) form of the pressure wave of Figure 1 can be greatly altered by themorphology of the medium encountered along its propagation. For instance, peak pressure can beincreased up to 8 times if the wave is reflected on a rigid obstacle. The effects of the reflectiondepend on the geometry, the size and the angle of incidence. By setting γ = 1.4 (ratio of specificheats of air), it can be shown that the reflected overpressure p r ispr2 p⎡7p+ 4p0 max=max ⎢ ⎥7 p0 + pmax⎣⎤⎦(2)All parameters of the pressure time curve are normally written in terms of a scaled distanceZ = d3W(3)where W is the mass of the explosive charge and d the distance to the centre of the charge.2.2 Literature Data2.2.1 Pressure-Time DistributionThere are available in the literature several pressure-time-curves for different kinds of explosions.The effects of nuclear explosions here should be disregarded.The pressure at a known point can be described by the modified Friedlander equation (from Baker[2]) and depends on the time t from the arrival of the pressure wave at this point ( t = t0−t a)⎛ t ⎞pt () = p0 + pmax⎜1−⎟⎝ td⎠bt−td(4)The other parameters involved are the atmospheric pressure p 0 , the maximum overpressure p max andthe duration of the positive pressure t d . The parameter b describes the decay of the curve. It can becalculated with a known minimum pressure after the positive phase. Alternatively, the parameter bcan be calculated with the knowledge of the impulse. This will be done in chapter 2.2.5.All parameters for the pressure-time curve can be taken from different diagrams and equations(Baker [2], Kinney [15], Kingery [14], see e.g. Figure 2).8

Figure 2: Model of Kingery [14] with scaled distances9

2.2.2 Maximum / Minimum PressureKingery [14] developed in 1984 curves for the description of the different air blast parameters byusing a rich body of experimental data, which had been properly homogenised. The parameters arepresented in double logarithmic diagrams with the scaled distance Z as abscissa, but are alsoavailable as polynomial equations. These diagrams and equations enjoy the greatest overallacceptance and are widely used as reference by most researchers. The parameters are alsoimplemented in different computer programs that can be used for the calculation of air blast wavevalues. e.g. they are implemented in ConWep – a program developed from the US-Army thatcalculates conventional weapons effects. The same curves are also used for an easy air blast loadmodel (*LOAD_BLAST) in LS-DYNA. Also in [14] curves are provided for reflection effects(surface burst of hemispherical charges) and free air conditions (spherical charge).Another equation has been proposed by Kinney [15], in which the overpressure-distance relation forchemical explosions can be written asp=2⎡ ⎛ Z ⎞ ⎤808⎢1+ ⎜ ⎟ ⎥⎢⎣⎝4.5⎠ ⎥⎦maxp0Z2Z2Z2⎛ ⎞ ⎛ ⎞ ⎛ ⎞1+ ⎜ ⎟ 1+ ⎜ ⎟ 1+⎜ ⎟⎝0.048 ⎠ ⎝0.32 ⎠ ⎝1.35⎠(5)Figure 3 shows the small differences between the two models.Figure 3: Difference of the model of Kingery and the model of Kinney with 1kg TNT10

2.2.3 ImpulseThe impulse of the air blast wave has a big influence on the response of the structures. The impulseis defined here as the area under the pressure time curve with the unit of pressure*sec. The impulsecan be calculated with (Kinney [15])0.067 1 + ( Z / 0.23)I =2 34Z 1 + ( Z /1.55)4(6)Another possibility is the polynomial equation of Kingery [14]. The comparison of the impulseresulting from both equations shows that the equation of Kinney simplifies the curve of the impulsebetween a scaled distance of 0.5 and 1.5 m/kg 1/3 .800600impulse [Pa sec]400200KinneyKingery00 0.5 1 1.5 2 2.5 3scaled distance [m/kg 1/3 ]Figure 4: Different equations for the impulse (Kinney [15] and Kingery [14])2.2.4 Negative PhaseDetonations produce an overpressure peak, and afterwards the pressure decreases and drops belowthe reference pressure (generally the atmospheric pressure). The influence of the so-called negativephase depends on the scaled distance. For scaled distances Z larger than 20 and especially for Zlarger than 50 the influence of the negative phase can not always be neglected. The size of thepositive impulse and of the negative impulse is then nearly the same. If the structure can react11

successfully to the positive pressure but is more sensitive to a negative pressure, failure of parts ofthe structure can result from this negative pressure phase (see Krauthammer [16]). However, inseveral cases the negative phase is neglected e.g. in the air blast function of the CONWEP-Code.Smith [22] presents the following equation to calculate the value of the negative pressurepmin0.35 105 Pa for Z 1.6= > (7)ZThe duration time of the negative pressuretnpmincan be calculated with1/3= 0.00125 W [sec](8)Another possibility to get these parameters is a diagram (see Figure 5) in Krauthammer [16]. Byusing this diagram the limitation of equation (7) can be overcome by assumingppminmin0.35 105 Pa for Z 3.5=Z>=

Figure 5: Different parameters for the negative phase (see Krauthammer [16])2.2.5 Wave Form ParameterThe decay or form parameter b in the Friedlander equation (4) describes the decay of the pressuretimecurve. The Friedlander equation has the parameters p max , t d and b. p max and t d can be readilyfound as explained before. There are several possibilities to calculate the decay parameter b byusing another known value of the pressure-time curve:1. Using the minimal pressure in the negative phase. Then, as it will be shown, the impulse ofthe positive phase is not accurate.2. Using the impulse of the positive phase. Then, as it will be shown, the minimal pressure inthe negative phase is not accurate. An additional equation for the negative phase should beused to avoid a smaller underpressure than the atmospheric pressure.Kinney [14] and Baker [3] calculate the parameter b by using the impulse of the positive phase.They use different equations for the pressure, for the impulse and for the duration of the positivephase. Therefore, the results for the parameter b differ (see Figure 6).Both methods, described above, for the calculation of the decay parameter b should be used here tosee the difference between the results. The Friedlander equation is too complex to solve13

analytically, and a program written in C++ can be used for the approximation. The listing is shownin the appendix.At the first step the negative pressure with the values from Kingery are used. The results for b differfrom the function of Kinney [13] and Baker [3] (see Figure 6). The comparison of the resultedimpulses (see Figure 7) shows that the parameter b calculated with the minimal pressure in thenegative phase gets a too small positive impulse and should not be used.108KinneyBakerb [-]64Parameter b by using pminParameter b by using theimpulse of the postive phase200 5 10 15 20 25 30 35 40Z [m/kg 1/3 ]Figure 6: Decay parameter b – different methods14

250200Impulse from KingeryImpulse with b from KinneyImpulse with b from BakerImpulse with b by using pminImpulse [Pa sec]1501005000 2 4 6 8 10 12 14 16 18 20Scaled distance Z [m/kg 1/3 ]Figure 7: Decay parameter b – resulting impulse in comparison with the impulse from KingeryTherefore, the parameter b is next calculated using the impulse of the positive phase. Then, theresulting curve of b is similar to the curves of Kinney and Baker. The exponential trend line givenby Excel has the following equationb1.1975= 5.2777⋅ Z −(11)The pressure time curve that is built with this b doesn’t fulfil the minimal pressure in the negativephase. Sometimes the pressure p is smaller than the atmospheric pressure. This results in animpossible state of the air. Therefore, the approximation of the negative phase is done with abilinear curve shown in equation (12) and Figure 8 by using the values of the negative phase shownin section 2.2.4.⎛tt ⎞ dp= p0 + pmax⎜1 − ⎟ for t < td⎝ td⎠2 pntnp= p0− ( t− td) for t > td ∧ t < td+t22 pntnp = p0− ( t + t − t) for t > t + ∧ t < t + tt2p= p for t > t + t0nnbt−d n d d ndn(12)15

Figure 8: Pressure-time curve for a free air blast wave – approximation of the negative phase2.2.6 Shock Front VelocityThe arrival time of the shock front at different points can be used to calculate the velocity of theshock front. With the knowledge of this velocity the pressure can be obtained with the Rankine-Hugoniot relationship.Kingery [14] calculates also the shock front velocity depending on the pressure as⎛ γ + 1 pu = c0⎜1+⎝ 2γpmax0⎞⎟⎠1/2(13)The parameter γ (ratio of specific heats of air) depends also on the overpressure and can be takenfrom a table in [13]; c 0is the sound velocity in air (331 m/sec);p0is the atmospheric pressure (101.3 kPa).2.2.7 Specific Heat RatioThe specific heat ratio γ is defined asccpmaxis the peak overpressure andpγ = (14)with c p being the specific heat at constant pressure and c v the specific heat at constant volume. Bothv16

the specific heat ratio and the speed of sound depend on the temperature, the pressure, the humidity,and the CO 2 concentration. Kingery [14] defines the variation of the specific heat ratio with a rangeof 1.402 to 1.176.17

3 Numerical Loading of a Structure with Air Blast WavesThere are several ways of numerical modelling in order to load a structure with an air blast wave.These methods differ in the number of used elements and with them in the calculation time.• Model with the mechanical modelling of the explosive (JWL-equation (15)). A fine mesh isessential to get realistic results. The size of the element in the range around the explosive shouldbe approximately 1 mm. These calculations are very expensive. To reduce the computation timepartitioning can be used. This method reduces the calculation time for models with a largevariation of element sizes.• The method proposed by Clutter [9] is also a solid TNT model and uses only one element forthe explosive. This is possible by different not specified methods in combination with theBecker-Kistiakowsky-Wilson EOS for the explosive.• 1D to 3D. This modelling is proposed in [4] and is also a solid TNT model. A 1D calculation isused until the wave reaches a surface. Then the values of the density, energy, velocity andpressure are mapped into the 3D mesh. Rose [20] maps the 1D model to 2D when the wavearrives the first surface and maps the 2D model to 3D when the wave arrives a second surfacewith another direction. EUROPLEXUS allows the implementation of this method. The methodshould also be validated with a calculation of the first model. The model is a mixture of the firstand the third model. The calculation time should be larger than for the second model.• Model with a compressed bubble. The pressure-time function resulting from a compressedbubble can not easily match the curve of an air blast wave. The size of the compression can becalibrated with the maximum pressure or the impulse. The calculation time is smaller than forthe first model.• Control volume. A volume around the explosive is loaded by a pressure-time curve. Thispressure time curve can be calculated with a model based on the modelling of the explosive.Alternatively, the well known curves from Kingery [14] can be used. This method should bevalidated through comparisons with calculations of the first model. The computation time is inthe range of the second model.• Load-time function. This is only usable for an estimation of the behaviour of a structure loadedby an air blast wave. The structure is loaded by a load-time function built with the pressure-timefunction e.g. from Kingery [14]. This function is implemented in EUROPLEXUS (see chapter7). The calculation is relatively inexpensive. Alternatively, the pressure-time function can be18

determined with a fluid pre-calculation with fixed boundaries for the structure. The structure isthen loaded by the pressures resulting from this fluid calculation.The choice among these methods depends on the scope of the analysis and on further investigationsabout their advantages and shortcomings. Figure 9 shows different models for the simulation of anair blast wave.Solid TNTCompressed bubbleLoad on a control volumeLoad-time functionFigure 9: Several models for air blast wave simulations19

4 Investigations with Explosives as a Charge (Solid TNT)The aim of the RAILPROTECT project is to contribute to alleviating the vulnerability of Europe'spassenger transport infrastructures. The effects of a terrorist attack should be simulated numerically,and for a numerical investigation the knowledge of the loading of the structures is necessary. Thereare different approaches and possibilities for the calculation of a detonation inside buildings, asdiscussed in chapter 3.A detonation releases a large amount of energy in a very short time. This results in an air shockwavewhich is spread outwards from the charge. Then, the air blast wave reaches the structure,which, depending on the size of the charge and on the distance, will respond to this wave loading.A calculation of the behaviour of the air blast wave requires the knowledge of the behaviour of theexplosive and of the air around the explosive. The results of the numerical investigation can becompared for the validity of the calculations with existing experimental-analytical data. As will beshown in this chapter, the experimental-analytical results of Kingery [14] will constitute the basisfor these comparisons.4.1 Modelling of the explosiveThe explosive for the numerical investigation can be built up e.g. with the Jones–Wilkins–Lee(JWL)-equation. This equation of state (EOS) is widely used because of its simplicity and due to thefact that most high explosives are well modelled by this equation. According to it, the value ofpressure is given as⎛ ω ⎞−RV⎛ ω ⎞1 −R2VEpEOS= A⎜1− ⎟e + B⎜1− ⎟e+ ω⎝ RV1 ⎠ ⎝ RV2 ⎠ V(15)In this equation A, B, R1, R2 and ω are the model parameters, V is the ratio ρ sol /ρ, where ρ=currentdensity and ρ sol =density of solid explosive, and E is the internal energy per unit volume of theexplosive. It is noted that E=ρ sol e int , where e int is the current internal energy per unit mass. Theparameters of this equation for most explosives are shown in Dobratz [10]. Different authors useslightly differing parameter values for this equation, as shown in Table 2. Note that the equationwill be reduced only to its last term if the solid explosive is exhausted and the resulting gases fullyexpanded. The last term of equation (15) is the EOS of an ideal gas that can be used e.g. for the air.pEOSE= ω (16)V20

From this asymptotic form it can also be concluded that ω=γ-1 (γ=ratio of specific heats).Parameter Description ref.[6] AUTODYN ref.[21]- manualparameters usedfor the airA (Pa) 3.738e11 3.7377e11 3.712e11B (Pa) 3.747e9 3.7471e9 3.21e9R1 4.15 4.15 4.15R2 0.90 0.90 0.95ρ sol (kg/m3) density 1630 1630 1630 1.3e int (J/kg) current internal 3.68e6 3.68e6 4.29e6 2.1978E5energy per unit massγ specific heat ratio 1.35 1.35 1.30 1.30v det (m/sec) detonation speed 6930 6930 6930Table 2: Parameters for the JWL equation for TNTFor the air the same EOS will be used without a detonation and different starting density andinternal energy. By ignoring the explosion, the last part of the JWL-equation will prevail andtherefore, an ideal gas will be used.Different FE-Codes smear the detonation front over different time steps. This procedure is calledburn fraction and its motivation is to control the release of the chemical energy for the simulation.The effects of the combustion on the pressure can be considered with this formula( ( 1 2))p = p min 1, max F,F(17)The burn mass functions F 1 and F 2 are computed by (see LS-DYNA and [18])EOS( )⎧2t−t1 d⋅Ae,max⎪if t > t1F1= ⎨ 3νe⎪⎩ 0 if t ≤ t1(18)1−VF2= (19)− V1CJwhere t 1 is the ignition time of the observed element (calculated with the detonation velocity d),A e,max is the maximum surface area and ν e the volume of the element. V is the actual specificvolume and V CJ the specific volume at the Chapman-Jouguet-pressure. The Chapman-Jouguet21

pressure is reached if the sonic velocity of the reaction gases reaches the detonation velocity. Thevolume at the Chapman-Jouguet-point isVCJ= − P(20)CJ12ρ0dThe term F 1 of the burn mass function intends to spread the burn front over several elements. Thesecond term should control the releasing of the energy. Interestingly, MSC-Dytran uses only theterm (19), whereas ABAQUS uses only one burn mass function( − )⎧ t t1d⎪ if t > t1Fb = ⎨ BS ⋅le⎪⎩ 0 if t ≤ t1(21)In this formula B S is the constant that controls the width of the burn wave (set to a value of 2.5) andl e is the element length. This function is very similar to (18).A 3D model of explosive is used to show the influence of the different burn mass fractions (Figure12, explosive4), whose different components can be compared in Figure 10 and Figure 11. Theinfluence of the term (21) is visible. The slope of the pressure peak decreases with F b , and thearrival of the pressure peak is later. The calculations with EUROPLEXUS do not show an influenceof the second term (F 2 ). Both curves are almost identical.LS-DYNA uses the functions (18) and (19) for calculations with theMAT_HIGH_EXPLOSIVE_BURN model. The input syntax allows calculations with bothfunctions (beta=0) or only with the function (19) (beta=1). The model in LS-DYNA was built withhexahedral with an element size of 0.001 m. The results of a 3D model show that the burn massfraction in LS-DYNA reduces also the slope of the pressure peak. Nevertheless, the differencebetween the calculations with beta=0 and beta=1 is negligible. The differences of the calculationsbetween EUROPLEXUS and LS-DYNA are the smaller peak in LS-DYNA and the higher pressurevalues behind the peak (here for a distance less than 0.04 m). The impulses are nearly the same (LS-DYNA 2.07 10 8 Pa sec, EUROPLEXUS 2.00 10 8 Pa sec).Therefore, the burn mass fraction is for the future work implemented only with the equation (21).22

2.0E+101.6E+10without burn mass fractionburn mass fraction Fbburn mass fraction Fb and F2LS-DYNA beta=0LS-DYNA beta=1Pressure [Pa]1.2E+108.0E+094.0E+090.0E+000 0.01 0.02 0.03 0.04 0.05 0.06Distance [m]Figure 10: Influence of the burn mass fraction in EUROPLEXUS (t=7 10 -6 sec)2.0E+101.6E+10EUROPLEXUS bmf=0EUROPLEXUS bmf=2.5LS-DYNA beta=0LS-DYNA beta=1Pressure [Pa]1.2E+108.0E+094.0E+090.0E+000.04 0.05 0.06 0.07 0.08 0.09 0.1Distance [m]Figure 11: Influence of the burn mass fraction in EUROPLEXUS (t=1.4 10 -5 sec)Another EOS for explosive is the “Ignition and Growth Reactive Model” based on Tarver et al.[23]. This model can also be used for the burning of propellant (deflagrations).23

4.2 Behaviour in the ExplosiveThe blast behaviour in the air is affected by the development of the pressure in the explosive.Therefore, it should be investigated, whether the numerical simulation can sufficiently represent thebehaviour of the explosive.The numerical model for the explosive calculates the pressure with the JWL-equation (15). Anaccurate model for the development of the detonation front is used (See [1]). The detonation startsat the initiation point, and the detonation front is moved with the given velocity of the detonation.An element detonates if the detonation front reaches this element. From this time the JWL-equationwill be used for this element.A spherical TNT charge of volume of 8000 cm 3 , i.e. with a radius of 0.124 m is considered. Tocontrol the behaviour of the pressure in the explosive a conical model is used (similar to the modelof chapter 4.3, see Figure 12).lengthxopeningd pyrad ex,ind ex,endFigure 12: 3D simplified model for the behaviour in the explosiveThe models use hexahedrons and a pyramid for the top of the model. The FSR-condition is used forall surfaces of the model. The difference between the meshes, listed in Table 3, is the refinement.24

Case opening d pyra d ex,in d ex,end Number of elementsexplosive1 Eulerian 2e-2 0.001 0.001 0.01 33explosive2 Eulerian 1e-2 0.0005 0.0005 0.005 65explosive3 Eulerian 4e-3 0.0005 0.0002 0.002 159explosive4 Eulerian 2e-3 0.0002 0.0001 0.001 318explosive5 Eulerian 2e-3 0.0002 0.0001 0.0005 499explosive6 Eulerian 2e-3 0.0002 0.0001 0.0002 859explosive7 ALE 2e-3 0.0002 0.0001 0.0005 499Table 3: Comparison of different models for the explosiveThe pressures are here analysed in intervals of 1µsec. The procedure to get the values in space(SCOURBE command in EUROPLEXUS) uses the averaged elemental pressure values for thenodal values. This results in a half value of the pressure at the node between an element inside andan element outside of the detonation zone (see Figure 13). The vertical lines in this figurecorrespond to the distances of the elements that are detonated at a certain time step (can be get fromthe listing). The detonation front can be identified by the steep increasing of the pressure.Therefore, it is important to consider the value of the last element of the detonation front and not thevalue of the last node.Figure 13: Distribution of pressures over distance (explosive1)The maximum pressure of the detonation is increasing with the distance from the initial detonationpoint. The Chapman-Jouguet-pressure is the maximum experimental resulted pressure. The25

parameters of the JWL-equation should represent this limitation (see Shin [21]). Shin shows theinfluence of the discretisation with a 1D model, where a finer mesh reproduces larger pressures.The results with the models explosive1 to explosive4 show the same dependency of the elementsizes (Figure 14). The Chapman-Jouguet-pressure seems to be the limit in the convergence study.2.40E+102.00E+10Pressure [Pa]1.60E+101.20E+108.00E+09t=7e-6t=1.4e-6Shin t=7e-6Shin t=1.4e-5Chapman-Jouguet-pressure4.00E+090.00E+000 200 400 600 800 1000 1200 1400 1600 1800 2000Number of elementsFigure 14: Maximum pressures in the explosive depending on the number of elementsThe rather unexpected behaviour of the models with 499 and 859 elements has to be clarified. Thereason could lie in the location of the detonation point in conjunction with the location of theintegration points.Figure 15 shows the pressure depending on the distance from the initiation point. The curve doesnot differ for models with fine discretisation. It is observed that the pressures behind the pressurepeak in the model explosive6 are definitely smaller than the results of Shin. Therefore, the areaunder the pressure-distance curve is also smaller. This area reaches only 67 % of the area of Shin.The reason could be the missing burn fraction (see chapter 4.1).26

Pressure [Pa]2.4E+102.0E+101.6E+101.2E+108.0E+09explosive1, t=7e-6explosive1, t=1.4e-5explosive4, t=7e-6explosive4, t=1.4e-5explosive6, t=1.4e-5Shin t=1.4e-5Shin t=7e-6Area Shin 5.54e8Area explosive6 3.71e8(between 0.051 and 0.97)4.0E+090.0E+000 0.02 0.04 0.06 0.08 0.1 0.12Distance [m]Figure 15: Pressure distance curve in the explosive – comparison with Shin [21]The model explosive7 uses an ALE mesh instead of an Eulerian mesh. The results are almostidentical for the maximum pressure as well as for the bending of the curve. A calculation with aLagrangian mesh fails.The next question is how the pressures are developed at the border of the explosive. At this timeexperimental results for this region are not available. Here, the numerical results of EUROPLEXUScould be compared with numerical results of LS-DYNA. For this calculation a conical model isused (see chapter 4.3, CON20, Table 4). In LS-DYNA the boundary condition like the FSRcondition is relatively complex (BOUNDARY_SPC). To avoid an influence of the usage of thisboundary condition the model in LS-DYNA is built as a full 3D model. The LS-DYNA model has alength of 15 cm, a height of 5 cm and a width of 5 cm. The boundaries are built as fixed. Therefore,after the reflection of the wave at the boundaries the pressure patterns are no longer spherical. Theelement size in LS-DYNA is chosen as 1 mm with an ALE multi-material formulation (see Figure16).27

Figure 16: Model in LS-DYNAThe calculations show that the numerical results of both programs are nearly the same inside theexplosive as well as in the air near the explosive (see Figure 17). The calculation withEUROPLEXUS is done without the burn mass fraction.2.0E+101.5E+10EUROPLEXUS x=0.105EUROPLEXUS x=0.125EUROPLEXUS x=0.136LS-DYNA x=0.105LS-DYNA x=0.125LS-DYNA x=0.136Pressure [Pa]1.0E+105.0E+09ReflectionReflection0.0E+001.4E-5 1.6E-5 1.8E-5 2.0E-5 2.2E-5 2.4E-5 2.6E-5 2.8E-5 3.0E-5Time [sec]Figure 17: Pressure time curve in the explosive and in the air near the explosive; length of theexplosive = 0.124 m28

5.0E+094.0E+09EUROPLEXUS x=0.105EUROPLEXUS x=0.125EUROPLEXUS x=0.136LS-DYNA x=0.105LS-DYNA x=0.125LS-DYNA x=0.136Pressure [Pa]3.0E+092.0E+09ReflectionReflection1.0E+09Reflection0.0E+001.4E-5 1.6E-5 1.8E-5 2.0E-5 2.2E-5 2.4E-5 2.6E-5 2.8E-5 3.0E-5Time [sec]Figure 18: Detail of the pressure time curve4.3 Cone with two Symmetry AxesA two dimensional model does not consider the behaviour of the fluid in the third direction.Therefore, a three dimensional model for the simulation of the detonation should be used. Thecalculation costs are minimized by using a conical model (pyramid, see Figure 19). This model canbe built with one hexahedral element in the radial direction and symmetry axes at the edges of theelements. The element on the top is then a pyramid. To get elements with similar geometry thelength of the elements should be increased with an increasing distance from the centre.The size of the opening angle depends also on the size of the elements. This angle should be chosenso that the aspect ratio of the elements is not too large. Normally, the time step size for this model isrelatively small because of the very small element on the top of the cone.29

lengthExplosiveopeningxd pyrad ex,in d ex,endd air,ind air,endFigure 19: 3D simplified model with hexahedrons or tetrahedronsAlternatively four tetrahedral elements can be used instead of one hexahedral element. This resultsin a higher number of elements. The meshing can be done with scripts that convert the hexahedronsin tetrahedrons. These scripts are presented in [7]. The script pxhex2te converts the hexahedrons intetrahedrons; the script pxqua2tr converts the quadrangles to triangles (necessary for the surfaces).For the simulation a model is used which has in most cases a TNT charge of a volume of 8000 cm 3 ,accordingly a cube of 20 x 20 x 20 cm or a sphere with radius of 12.4 cm. The mass of this chargeis 12.8 kg. An ALE-calculation is used for all further investigations. The explosive is build as anEulerian mesh, the air is build as an ALE mesh.The symmetry at the surfaces can be considered by defining symmetry planes with the CONTSPLA AUTO command. This command defines symmetry conditions orthogonal to the surface ofthe defined nodes. Another possibility is the definition with the FSR-command as a sliding surface.Both methods give the same results.The calculations performed with the conical model are summarized in Table 4 and shortly describedhereafter.30

Cased pyr /d ex,in /d ex,end /d air,in /d air,end opening/ Charge Element type t endlengthCON1 0.05/0.05/0.08/0.2/0.2 0.2/4.0 12.8 FL34 (NF34) 4.0e-3CON2 0.01/0.02/0.02/0.02/0.05 0.2/4.0 12.8 FL34 (NF34) 4.0e-3CON3 0.008/0.01/0.01/0.01/0.05 0.1/4.0 12.8 FL34 (NF34) 4.0e-3CON4 0.008/0.005/0.005/0.005/0.02 0.03/1.3 12.8 FL34 (NF34) 4.7e-4CON5 0.008/0.005/0.005/0.005/0.02 0.03/1.3 12.8 FL35/FL38 4.7e-4CON6 0.002/0.002/0.002/0.002/0.01 0.01/1.3 12.8 FL35/FL38 4.7e-4CON7 0.001/0.001/0.001/0.001/0.005 0.01/1.3 12.8 FL35/FL38 4.7e-4CON8 0.001/0.001/0.001/0.001/0.01 0.05/3.0 12.8 FL35/FL38 1.5e-3CON9 0.001/0.001/0.001/0.001/0.01 0.05/3.0 1.0 FL35/FL38 1.5e-3CON10 0.001/0.001/0.001/0.001/0.005 0.05/3.0 1.0 FL35/FL38 1.5e-3CON11 0.0005/0.0005/0.0005/0.0005/0.01 0.025/1.5 1.0 FL35/FL38 1.5e-3CON12 0.005/0.001/0.001/0.001/0.01 0.1/3.0 1.0 FL35/FL38 1.5e-3CON13 0.005/0.001/0.001/0.001/0.04 0.2/3.0 1.0 FL35/FL38 1.5e-3CON14 0.005/0.001/0.001/0.001/0.02 0.2/1.5 1.0 FL35/FL38 1.5e-3CON20 0.0002/0.0001/0.001/0.001/0.005; 6.5e-3/0.4 1.0 FL35/FL38 7.0e-5Table 4: Calculations with a conical meshCON1This calculation uses the modified tetrahedral elements (see [7]) also for the tip of the model. Themesh is relatively coarse. The development of the air pressure wave is presented in Figure 20. Thepressure-time curve for a distance of 1 m shows the increase of the pressure until a value of1.83 10 6 Pa (time step size for evaluation 2 10 -6 sec) which is equivalent to 18.3 times theatmospheric pressure. The resulting pressure does not depend on the time step size for theevaluation. By choosing every calculated time step for the evaluation (1.75 10 -7 sec) the maximumpressure is also 1.830958 10 6 Pa. After the pressure peak the pressure is decreasing up to 232.2 Pa.This value of the “negative” pressure is relatively high. The pressure-time curve for other distancesfollows also these trends.31

12.0E+061.5E+061.0E+0615.0E+050.012323221 1 13Time [s]-5.0E+050.0 5.0E-04 1.0E-03 1.5E-03 2.0E-03 2.5E-03 3.0E-03 3.5E-03 4.0E-03FLUID_3D_SF7EUROPLEXUS-1- Distance = 1m -2- Distance = 2m -3- Distance = 3m14 JUNE 2007DRAWING 1Figure 20: Development of the pressure depending on the distance to the chargeCON2This model uses smaller tetrahedral elements than the mesh CON1.The finer mesh results in a steeper air blast wave at a distance of 1 m as well as in a distance of 2 m.The steeper wave causes also a higher pressure (See Figure 21). Figure 22 shows the maximumpressure depending on the distance to the charge. These curves can be obtained by using aCAST3M macro that selects all nodes lying near a line. The macro pxpdroi1 (See [7]) requires atolerance in which the nodes are lying. Then, with the macro pxordpoi the selected nodes areordered. The line which is used to select the nodes is one of the edges of the CON models.With EUROPLEXUS it is possible to get out the data depending on the space variable. This can bedone at a certain time step. From the results of different time steps the maximal values, the impulseand the positive phase duration can be extracted with the FORTRAN-executableAIRBLASTRESULT (see Appendix).A comparison of the different discretisations (see Figure 22) with the results of Kingery [14] showsthat a finer tetrahedral mesh produces smaller maximum pressures (disregarding the model CON1).Except for distances less than 25 cm, these pressures are smaller than the experimental pressures.The experience shows that Eulerian meshes normally react smoother than the reality.32

Figure 21: Comparison of a coarse with a fine mesh – pressure time curveCON3 and CON4These calculations use finer meshes. To reduce the computation time the length of model CON4(see Figure 19) is limited to 1.3 m.For the results of the models CON1 to CON4 it is important to note that the definition of the pressureat a certain point is arguable. There are several tetrahedrons at the same distance with differentorientations, and these elements have different pressures. Thus, the results with the tetrahedrons is afield that requires further work, as probably, tetrahedrons will be used for the calculation of the airinside the structures. For the employment of the tetrahedrons in a productive framework, theseelements should show safer and more reliable results.CON5This calculation uses hexahedral elements instead of the tetrahedrons. The mesh is relatively coarse.In comparison to the tetrahedrons the maximum pressures versus the distance show a relativelygood correlation with the analytical results.CON6 and CON7Both models use finer meshes than the model CON5.The behaviour of models with hexahedral elements regarding the dependency on the element sizeshows the reverse trends to those of the tetrahedral elements (See also Figure 22). A finer meshresults in a higher maximum pressure. This should be the normal behaviour of a refinement of theelements. The results of the model CON7 show the best correlation, even though the differences33

etween the experimental and the numerical results are still quite big. Further work has to be doneto check this discrepancy.Max. pressure [Pa]1E+88E+76E+74E+7CON1 ▲CON2 ▲CON3 ▲CON4 ▲CON5 ■CON6 ■CON7 ■Kingery2E+70E+00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Distance [m]Figure 22: Comparison of max. pressure – distance relationships from different conical models.CON8This calculation is done with a length of 3.0 m. The element sizes are nearly the same as in modelCON7. In comparison to the results of model CON7 the difference is small (see Figure 23).Max. pressure [Pa]1E+78E+66E+64E+6KingeryCON7CON8 ■CON9 ■ 1kgCON10CON11CON122E+60E+00.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Scaled distance [m/kg 1/3 ]Figure 23: Comparison of peak pressures – scaled distance relationships from different conicalmodels.34

CON9This model has approximately the element sizes of the mesh CON8 but has a charge of only 1 kg.The resulting pressures in a scaled size are smaller than those of model CON8.CON10This is a finer mesh for the air of the model CON9. However, all values are nearly the same.CON11CON11 uses a finer mesh in the explosive and near the explosive. The pressures near the explosiveare higher than in the previous models, but the difference in a larger distance is small.CON12Figure 24 shows the pressure versus the distance at a time of 1.5 10 -3 sec for the model CON9. Thepressures in the first elements are approximately 10 10 4 times higher than expected. Controlling theaspect ratio of the elements at the tip (pyramid and hexahedrons), it is realised that the elements arenot conformant. Therefore, model CON12 is tried, which uses a larger angle of the cone than themodel CON9 and has also a bigger pyramid at the tip. The largest aspect ratio of the hexahedrons isnow 3.0 and of the pyramid 15. The aspect ratio of the pyramid depends on the aspect ratio of thewhole model ( = length2 ⋅ opening). The pressures in the pyramid and in the three first hexahedrons arelarger than expected (blue rhombus points in Figure 24).To calculate the aspect ratio of an element the procedure aspectra can be used (see appendix).Figure 24: Pressure at the top of the conical model35

CON13The angle of this model is increased. The aspect ratio of the hexahedrons is 1.5, of the pyramid 7.5.The results for the elements near the tip show that the pressure value in the hexahedrons isrepresented better if the aspect ratio is smaller.CON14The pressures in the pyramid are decreasing with an aspect ratio of 3.75. The calculations do notshow better results.CON22, CON23, CON24These models use only cubic elements. Therefore, the element size near the explosive is similar tothe other models. The size of the elements increases with the distance to the explosive. Then, theonly parameter for this model is the opening of the cone (or the angle of the cone). The number ofelements is relatively small but the time step size depends on the smallest element.Case opening/ length Number of ChargeelementsCON22 0.1/4.0 386 12.8CON23 0.05/4.0 711 12.8CON24 0.02/4.0 1595 12.8Table 5: Calculations with a conical meshThe peak pressures show a big dependency on the element size. (see Figure 25). The influence ofthe element size on the impulse is much smaller (see Figure 26). Therefore, for further calculationsit has to be checked if the element size is small enough.36

4E+7Max. pressure [Pa]3E+72E+7Kingery 12.8 kgSPHE9CON8 ■CON22CON23CON241E+70E+00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Scaled distance [m/kg 1/3 ]Figure 25: Peak Pressure – Models CON22, CON23, CON242.5E+3Impulse [Pa sec]2.0E+31.5E+31.0E+3SPHE9CON8 ■CON22CON23CON24Kingery 12.8kg5.0E+20.0E+00.1 0.2 0.2 0.3 0.3 0.4 0.4Scaled distance [m/kg 1/3 ]Figure 26: Impulse – Models CON22, CON23, CON2437

SummaryThe pressures resulting from a calculation with tetrahedrons are too small. The results with hexahedronsare also small, but the difference to the analytical-experimental results is less. The differencedepends on the size of the elements. Smaller elements result in higher pressure and a bettercorrelation with the experimental results. The smallest element sizes are of the order of 1 mm. It isnot possible to calculate a fluid-structure interaction problem of realistic dimensions with a mesh ofthis size. So, the question which kind of simulation of the loading is most appropriate to be used isimportant (see chapter 3).4.4 Cubic Charge with two Symmetry AxesTo check if the boundary condition of the cone model represents a full 3D model two differentmodels are used. Both models are one eighth of the geometry with FSR conditions at the symmetryfaces. Another advantage with such a model is the possibility to compare the calculations with otherfinite element programs like LS-DYNA.In the first model the charge and the air are built as cubes. A regular mesh with hexahedrons isapplicable with the same size for all elements. The second model uses a spherical charge (seechapter 4.5).An interaction with a structure is not of interest for these calculations. Therefore, the explosive andthe air are built with fluid meshes. An ALE mesh is used for the air, an Eulerian mesh is used forthe explosive (see Figure 27).Figure 27: Model CUB1 – explosive as Eulerian (red), air as ALE (grey)The calculations performed are summarized in Table 6 and shortly described hereafter.38

Case Size of the model Description t end Steps CPU [s] ElementsCUB1 0.5 X 0.5 X 0.5 m Charge 12.8 kg TNT,element size 0.02 mCUB2 1.0 X 1.0 X 1.0 m Charge 12.8 kg TNT,element size 0.02 mCUB3 0.5 X 0.5 X 0.5 m Charge 12.8 kg TNT,element size 0.01 mCUB4 1.0 X 1.0 X 1.0 m Charge 1.0 kg TNT,element size 0.0167 m4.7e-4 415 181.5 156254.7e-4 270 790.6 1250004.7e-4 911 2275.9 1250006.0e-4 291 1367.7 216000Table 6: Calculations with cubic chargesCUB1In this calculation a small model with a coarse mesh is used. The charge has a mass of 12.8 kg. Thedevelopment of the pressure can be shown in Figure 28. At t=2 10 -5 the explosive has burned downexcept for the explosive in the corners of the cube. The pressure in the air is the atmosphericpressure of 10 5 Pa. After the explosion (e.g. t=6 10 -5 ) the pressure of the gas produced from theexplosive is decreasing and the pressure wave is running into the air. At t=10 -4 there can beobserved a discontinuous pressure along the three axes at the same distance from the detonationinitiation point. The reason could be the cubical shape of the charge. At t=1.4 10 -4 the pressurewave is reaching the surface (FSR condition) and is reflected.39

a) t=2e-5 b) t=6e-5c) t=1e-4 d) t=1.4e-4Figure 28: CUB1 – Pressure at different time stepsCUB2This calculation uses a bigger model but also with the coarse mesh. Figure 29 shows the maximumpressure versus the distance from the charge. The maximum pressures resulting from model CUB1and from model CUB2 are the same in the range of the size of model CUB1. The pressures areoverestimated up to a distance of 0.46 m; in relation to the size of the explosive (0.13 m) this rangeof overestimation is negligible. The resulting maximum pressures after the range of 0.4 m aredefinitely smaller than the experimental results.CUB3The size of the model is the same as that used in model CUB1 but with a finer mesh. The pressuresresulting from this calculation are bigger than in model CUB1. For CUB1 and CUB2 only the pressuresalong the orthogonal to the surface of the explosive are considered. The values are smaller ifthe pressures along the diagonal of the cube are used. The difference between both locations is verybig. Therefore to compare the results in a region near the explosive with experimental results acubic charge is not suitable. The differences between experimental and numerical results can becompared better with a spherical model.40

CUB4This model uses a finer mesh than model CUB2. The ending time of the calculation is longer so thatthe wave can be also observed in regions with a bigger distance from the charge.In comparison to the model CUB2 the pressures at CUB4 are smaller because of the smaller charge.A better method to compare is the use of the scaled distance, and in fact the values of the pressureare of the same order of magnitude in this case. There is also a big difference between the pressuresalong the diagonal and the pressures along the orthogonal to the charge surface.In comparison to the results with LS-DYNA (cub2) the results of EUROPLEXUS for the pressureare higher and represent the experimental values better but still not in a very satisfactory manner.Figure 29: Maximum pressures versus distance – cubical models41

Figure 30: Maximum pressures versus scaled distance – cubical models4.5 Spherical Charge with two Symmetry AxesThe weaknesses of using a cubical charge can be avoided by using a spherical charge. There are noproblems with a spherical charge from the different ending times of the detonation at the cornersand with effects resulting from the different surfaces.As mentioned initially, these first investigations disregard the interaction with the structure. So it ispossible to use a spherical volume for the surrounding air, too. This allows an easier modelling ofthe mesh, which can be done in several ways. At this point, the first approach of a sphericallysymmetric regular mesh with the centre of the charge is in CAST3M neither possible in an easyway nor is it necessary in considering the behaviour of the explosive and the air.The method used instead converts a cubical model via the INCL operator to a spherical model. Theresulting model seems to be relatively regular. The method can be done with the following steps:1. Modelling of a cubic surface for the outer charge (half volume) (Figure 31a, green).2. Modelling of a cubic surface for the inner charge. This part will be meshed with regular hexahedrons(half volume) (Figure 31a, blue).3. Modelling of a cubic surface for the air volume around of the charge (half volume) (Figure 31a,red).4. Projection of the outer charge surface and of the air surface with the PROJ operator to spheres(Figure 31b).5. Filling the volumes between the surfaces with hexahedrons (Figure 31c).42

6. Complete the volumes by adding the turned model.7. Defining of bounding box for an one-eighth model.8. Choosing the elements inside of the bounding box with the INCL operator (Figure 31d).9. Defining the bounding surfaces with the POIN operator.a) Cubic surface for outer charge (green)for inner charge (blue) and for air (red)b) Projection to a spherec) Filling the volume between the surfaces d) Elements inside the bounding boxFigure 31: Modelling of the spherical modelAlternatively an eighth of the whole spherical model can be used form the beginning. This saves therequired memory.Another possibility is the modelling with two macros (pxhex2te and pxqua2tr, see [7]). With them itis possible to rotate first a line around a point to get a part of a circle. A second macro is available toturn this part of a circle around a line to get a part of a sphere. However, the resulting model seemsto be more irregular then the model built with the method described before.The calculation cases performed are summarized in Table 7 and shortly described hereafter,together with the modifications introduced in the code to achieve this type of calculations. Theregions with different element densities are shown in Figure 32. The charge of all models is 12.8 kgTNT, the end of the calculation depends on the size of the model.43

Figure 32: Modelling of the spherical model – number of elements in the different modelsCase Size of the modelnel0/nel1/nel2Length[m]Distance withaspect ratio = 10Number ofelementsSPHE1 10/10/10 1 (5.55) 1625SPHE2 10/20/70 1 0.79 6875SPHE3 10/20/90 1 0.62 8375SPHE4 10/20/150 1 0.37 12875SPHE5 10/20/200 1 0.28 16625SPHE6 10/40/200 1 0.28 18125SPHE7 20/40/200 1 0.56 73000SPHE8 10/10/70 2 1.70 6125SPHE9 20/20/140 2 1.70 12125Table 7: Calculations with spherical chargesSPHE1In this calculation a small model with a coarse mesh is used. The results show a relatively lowmaximum pressure (See Figure 33).44

Figure 33: Maximum pressures versus distance – spherical modelsSPHE2, SPHE3, SHPE4 and SPHE5These calculations use finer meshes. The maximum pressures are increasing. The models SPE2 andSPE4 give realistic results. The models SPHE4 and SPHE5 give larger maximum pressure than theexperimental values. An Eulerian calculation should normally result in smaller values than theexperimental values (in contrast to a Lagrangian calculation which reacts stiffer with smallerelement sizes).SPHE6The larger values for the maximum pressure in model SPHE5 start near the charge. Therefore, themodel SPHE6 tests if a smaller element size in the outer charge (See d) red range) can result inmore realistic values.The overestimating of the pressure begins in this model earlier than in the model SPHE5. The sizeof the elements in the explosive should not be the reason for the overestimation.SPHE7This calculation uses a finer mesh in the inner charge (See Figure 31 d) green range). With thesmaller elements in the inner charge also the number of the elements in tangential direction isincreased.The results are quite better. For every model with small elements there is a limit from which the cal-45

culation fails. This limit depends on the radial and the tangential size of the elements. It seems thatthe elements are not usable when they are too thin. The pressures are overestimated if the aspectratio is about 10. When exceeding this ratio, the pressure is also not symmetric. It appears to behigher near the edges and smaller in the diagonal of the model. This difference between the edgeand the diagonal begins approximately at a distance of 0.45. The aspect ratio is there 8.0. It seemsthat the boundary conditions can not act with this exceptional element sizes. Respecting this, alsothe values for the calculation SPHE3 would not be applicable over a distance of 0.60 m. This has tobe considered for further calculations with the conical model and the spherical model. The aspectratio reaches the critical value 10 at a certain distance. This distance is calculated in Table 7.SPHE8This model is a coarse spherical model with a size of 2.0 m. The results follow the results of themodels with a size of 1 m. The pressure is approximately half the experimental value.SPHE9This calculation uses a finer mesh than model SHPE8. The pressures are a little bit larger than thepressures in model SHPE8.4.6 Comparison between the Different Models4.6.1 Maximum PressureThe comparison will be done with the fine meshes of the different models summarized in Table 8.Case Element size Charge[kg TNT]Size of the model[m]Steps CPU [s] ElementsCON4 (Tet) 0.02-0.005 12.8 1.3 8956 9 914.3 1582CON8 (Hex) 0.001-0.01 12.8 3.0 2205 13 3625 860CON9 (Hex) 0.001-0.01 1.0 3.0 2205 13 3482 805SPHE3 0.01 12.8 1.0 698 98 8375SPHE9 0.013 12.8 2.0 917 691 49000Table 8: Comparison of different modelsThe summery of the different curves (Figure 34) shows that the conical model with the tetrahedralelements results not in a realistic maximum pressure-distance curve. The closest agreement with the46

experimental values is obtained with the finest conical mesh. This mesh has a smallest element sizeof 1 mm! The calculation time is very high because of the small time step sizes that are necessarywith the small elements. The difference between the spherical and the conical model shows that theelement size has a big influence for the maximum pressures.1E+7Max. pressure [Pa]8E+66E+64E+6KingerySPHE3SPHE9CON4 ▲CON8 ■CON9 ■ 1kg2E+60E+00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Scaled distance [m/kg 1/3 ]Figure 34: Maximum pressures versus scaled distance4.6.2 ImpulseThe impulse is the parameter which has a capital importance for the loading of a structure. Thenumerical results are shown in Figure 36. The impulse has been calculated with an integration ofthe pressure- time curve over the time. The impulse is larger for the models with an explosive of12.8 kg and is smaller for an explosive of 1.0 kg. This depends on the positive phase duration whichhas also a big difference in the scaling. After the detonation the compressed combustion gas needstime to expand. The dashed line for the model CON9 (Z

Time of duration t d [sec]2.0E-31.6E-31.2E-38.0E-4SPHE9CON8 ■CON9 ■ 1kgKingery 1kgKingery 12.8kg4.0E-40.0E+00 0.2 0.4 0.6 0.8 1 1.2Scaled Distance [m/kg 1/3 ]Figure 35: Positive phase duration versus scaled distance1E+3Impulse [Pa sec]8E+26E+24E+2.SPHE3SPHE9CON4 ▲CON8 ■CON9 ■ 1kgKingery 1kgKingery 12.8kg2E+20E+00.1 0.3 0.5 0.7 0.9 1.1Scaled Distance [m/kg 1/3 ]Figure 36: Impulse versus scaled distance48

4.6.3 Arrival TimeThe arrival time also depends on the distance. The results for the different models are shown inFigure 37. The arrival time for the numerical investigations is defined by the arrival of the increasedpressure. The arrival time of Kingery is defined as the time from the initiation of the detonation tothe arrival of the pressure wave at this point. The arrival times for all calculations are higher thanthe experimental arrival times. The numerical results show only a small dependence on the elementsize.1.5E-3Arrival time t a [sec]1.0E-35.0E-4SPHE3SPHE9CON4 ▲CON8 ■CON9 ■ 1kgKingery 1kgKingery 12.8 kg0.0E+00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Scaled Distance [m/kg 1/3 ]Figure 37: Arrival time versus scaled distance4.6.4 Positive Phase DurationThis parameter is shown in Figure 35, and is defined by the length of positive pressure phase. Theresults of model SPHE9 represent the experimental results rather well. Discrepancies occur becauseafter a distance of 1.4 m the reflected wave at the boundary condition arrives before the positivepressure is ended. Also the conical model CON8 represents sufficiently the positive phase duration.The conical model with 1 kg TNT has a very small time t d . The scaling rule for t d is an issue to beclarified like for the time of arrival.49

4.6.5 Comparison with results of other authorsExperiments and calculations with spherical explosions without reflections are relatively rare in theliterature.Clutter [9] uses only one element with a different EOS for the explosive. He compares the results ofthe impulse and the peak pressure in a logarithmic scale. The values for the peak pressure are higherfor small scaled distances and are smaller for large scaled distances. The impulses for scaleddistances smaller than 0.8 m/kg 1/3 are definitely too large (up to 2.5 times of the Kingery data). Alsothe values for larger distances are overestimated. A fluid calculation should results in smaller peakpressures and smaller impulses.Fairlie [12] presents calculations with AUTODYN for a small amount of TNT (8 g). Thecomparison of the peak pressures shows a good correlation with the values of Kingery (3%difference). The difference for the impulse is 22%, in the same range as the herein presentedcalculations.The investigation from Alia [1] uses LS-DYNA with a multi material formulation. The amount of454 g C-4 (equivalent to 609 g TNT) is modelled with the JWL-equation. The overpressure at acertain point is smaller than the pressures resulting from Kingery (25%). The impulse is alsosmaller (40%).In the calculations presented the difference between the numerical results and the experimentalresults of Kingery are in a range of 30-40% for the peak pressures as well as for the impulses. Thesmallest elements in the CON-models have a length of 1 mm due to the small angle of the cone.This raises the question whether the modelling of large structures is possible with a JWL-equation.Therefore, the next chapter presents several possibilities for the simulation of air blast waves.4.7 Influence of several parametersAs seen above, there are differences between the experimental results and the numericalcalculations. The computed maximum pressure is too small, the arrival time longer, and the impulsehigher. The potential influence of the several parameters involved should be determined.4.7.1 Specific heat ratio (CON15)Kingery defines the parameter γ in the range of 1.176 to 1.402. The cited literature is still not available,but as stated, γ varies with the overpressure. Without the knowledge of the dependence on theoverpressure it is not possible to define a more accurate material law for the air. Nevertheless, theinfluence of a varying γ can be tested with a calculation which chooses γ to be 1.176 (smallest50

value in [13]). These results show that the maximum pressures are smaller, and the arrival time ishigher. To conclude, it seems that the influence of the parameter γ towards a better material law,would not lead to better results.4.7.2 Values for γ, E 0, ρThere are different values for the parameters γ , E 0 and ρ of the ideal gas equation (16) in theliterature. These differences should be considered and assessed.Case Used values (CON8) (CON15 CON16γ 1.35 1.176 1.4E 0 2.1978e5 4.37e5 2.5e5ρ 1.300 1.300 1.293Table 9: Parameters γ, E 0 and ρThe maximum pressure resulting from the model CON16 is higher, the arrival time is smaller butthe difference with model CON8 is very small. The difference from model CON15 to model CON8is very small, too. Therefore the choice of the parameter values for the air doesn’t have a biginfluence.4.7.3 Parameters for the explosiveThe parameters for the JWL equation (15) could also have an influence on the results. Thecalculations are made with the parameters used by AUTODYN and Shin [21] (see Table 2)(CON17, CON18). It is found that, overall, the influence of the chosen parameters is small. Theresults for the standard parameters of AUTODYN are nearly the same. The maximum pressure withthe values of Shin is at a distance of 0.40 m approximately 10 % higher. The arrival time is 5 %smaller; the impulse is 1 % higher.4.7.4 Burn mass fractionThe burn mass fraction has an influence on the behaviour in the explosive (see chapter 4.1). Themodel CON8 is calculated with the burn mass fraction (B S = 2.5, model CON19) to test theinfluence on the behaviour of the air. The difference between the calculations with and without burnmass fraction is very small and represents for the overpressure, the impulse and the duration of thepositive phase only 1%. The arrival time is approximately 10 % higher.51

5 Bubble modelSeveral models with a compressed bubble are used to test whether a bubble model can represent theair blast waves resulting from an explosion (see Figure 38). The 3D models are similarly built as themodels in chapter 4.5. The region with the explosive is increased and replaced with the compressedbubble. The different parameters of the model are described in Table 10. The model bubble1 uses20 elements in the circumference direction (for the eighth of the sphere); all other models use 40elements. Model bubble1 has 13000 elements; all other models have 104000 elements.Figure 38: Model compressed bubbleThe resulting values (particularly the maximum pressure and the impulse) can be compared with theexperimental values resulting from an explosion. This is done here with the values of Kingery [14].Those values show for the impulse a maximum at a scaled distance of 0.8 m. The curve fromKinney does not consider this maximum and therefore, the impulse is smaller and with them thecorresponding TNT equivalent (see also Figure 4).The impulse is the most important parameter for the calculation of the interaction between the airblast wave and the structure. Therefore, an impulse-distance curve should be found by varying thecharge that represents the impulse-distance curve of the calculation with the compressed bubble.52

Modell airl bBubbleBubbleBubbleTNT equivalent CorrespondingElement length[m][m]pressure[Pa]density[kg/m 3 ]e int[J/kg]of the energy of blast wave with at the border ofthe bubble TNT equivalent the bubble [m]bubble1 3 0.5 1e7 13.0 2.16e6 3.22 2.0 0.083bubble2 3 0.5 1e7 13.0 2.16e6 3.22 2.0 0.042bubble3 5 0.5 1e7 13.0 2.16e6 3.22 2.0 0.042bubble4 5 0.5 1e9 130.0 2.16e7 325 n/a 0.075bubble5 5 0.5 2.5e6 6.5 1.10e6 0.78 0.6 0.075bubble6 5 0.5 1.97e7 18.4 3.06e6 6.48 3.7 0.075bubble7 5 0.5 3.93e7 26 4.32e6 13.0 6.5 0.075bubble8 5 0.5 4.9e6 9.19 1.52e6 1.59 1.1 0.075bubble9 10 1.0 1.0e7 13 2.2e6 26.2 16 0.15bubble10 10 1.0 2.5e6 6.5 1.1e6 6.36 5 0.15bubble11 10 1.0 1.25e6 4.6 7.77e5 3.05 2.5 0.15bubble12 10 1.0 4.9e6 9.19 1.52e6 12.7 9 0.15bubble13 10 1.0 3.6e6 7.8 1.32e6 9.28 6.8 0.15Table 10: Compressed bubble modelsThe comparison of the maximum pressures shows that the maximum pressure is smaller in thecompressed bubble model than in the equations of Kingery. The peak pressures of the modelsbubble1 and bubble3 show nearly the same behaviour because the element size is nearly the same.The model bubble3 is bigger. The model bubble2 uses a finer mesh. Therefore the pressure peak ishigher and represents the experimental values of Kingery better. The difference in the impulsesbetween the models with different element sizes is relatively small.53

1E+3Impulse [Pa sec]8E+26E+24E+2.bubble1bubble2bubble3Kinney 2kgKingery 2kg2E+20E+00 0.5 1 1.5 2 2.5 3Scaled Distance [m/kg 1/3 ]Figure 39: Comparison of several compressed bubble models (impulse)2E+6Max. pressure [Pa]2E+61E+68E+5Kingerybubble1bubble2bubble34E+50E+00 0.5 1 1.5 2 2.5 3 3.5 4Scaled distance [m/kg 1/3 ]Figure 40: Comparison of several compressed bubble models (maximum pressure)54

Several calculations are done with models with a radius of 0.5 m of the compressed bubble and withdifferent overpressures. The best fitted curve for the model bubble1 is the impulse-distance curvewith approximately 2 kg TNT (see Figure 39). All the resulting impulse-distance curves are fitted tocurves from Kingery to get a corresponding TNT equivalent. The values of the TNT equivalents fordifferent overpressures in the bubble are shown in Figure 41.The bubble with the overpressure corresponds to an amount of energy that can be calculated withVE =5bub( pbub−10 )γ −1(22)This energy can also be converted in a TNT equivalent. 1kg TNT has the energy of 4520 kJ. This isthe energy that is introduced in the model.The corresponding energy and the introduced energy (both in TNT equivalent) are shown in Figure41. The curves show in the logarithmic scale of the pressure an increasing dependency on thepressure. The reference pressure in Figure 41 is not the overpressure in the bubble. The TNTequivalent corresponding to the impulse-distance curve can also be transformed in a bubbleoverpressure with the same energy. This pressure is used in the Figure 41 due to a better calculationof the necessary bubble overpressure by a given charge.14Energy [kg TNT equivalent]1210864Resulting pressure wave in TNTEnergy of overpressure in TNT201E+6 1E+7 1E+8Pressure = energy of the corresponding TNT equivalent (resulting pressure wave) [Pa]Figure 41: Comparison of the energy of the fitted curve with the energy of the bubble (modelsbubble3-bubble8)55

The corresponding TNT-equivalent is divided by the TNT equivalent of the overpressure. Thisdivision gives a value α bub shown in Figure 42. This value α bub shows a linear dependency also fordifferent sizes of the bubble, tested here with the radius of 0.5 m and 1.0 m. With this straight line itshould be possible to calculate an overpressure for a bubble with the volume V bub for a given chargeof TNT. The idea is to describe an equation with the following formp = f( V , W)(23)The equation of the line in Figure 42 can be described by this trend linebubbubα = −0.263⋅ log( p ) + 2.41(24)bubTNT1Energy of overpressure/Energy resulting pressure wave0.80.60.40.2lbub=0.5 mlbub=1.0 m06 6.2 6.4 6.6 6.8 7 7.2 7.4Pressure = energy of the corresponding TNT equivalent [log Pa]Figure 42: Factor α bubThe following procedure can be used to get the overpressure of the bubble for a given TNTequivalent.1. Calculation of the energy of the detonation withe = W ⋅ 4520 kJ / kg(25)2. Calculation of the overpressure energy of this amount of TNT withTNTpe( γ 1)TNTTNT= ⋅ − (26)Vbub56

3. Calculation of the factor α bub with equation (24).4. The pressure of the bubble can be calculated withppTNTbub= + p0(27)αbub5. The values for the internal energy e int,bub and the density ρ bub can be calculated bymultiplying the values of the uncompressed air with the factor f bubfpbubbub= (28)p0With this method it is possible to calculate by a given bubble size and a given charge theoverpressure in the bubble.57

6 Control VolumeThis model foresees the loading of a cut surface with a pressure depending on the AIRB-function(see chapter 7). The difference to the model in chapter 7 is the loaded element which is here a fluidelement. Additional to the external forces (continua element) the flux between the CL3D elementand the fluid element is to be considered.6.1 Flux between the CL3D and the fluid elementThe flux calculation for a fluid element is done by using the differences in the pressure, density andthe energy between the fluid element and the adjoined elements. Therefore, it is obligatory in thecase of AIRB to calculate the density and the energy for the CL3D elements, if they are adjoinedwith a fluid element. As an assumption, an adiabatic equation of state will be used to calculate thevalues.γ⎛ 1 ⎞ ⎛ 1 ⎞p0⎜ ⎟ = p⎜ ⎟⎝ρ0⎠ ⎝ρ⎠γ(29)The initial density ρ0is used from the input. The pressure p in equation (29) can be calculated withthe ideal gas equation resulting inp1ργρ= ( γ − 1) i(30)ρ0γ 00Then, the current density can be computed byγ −1⎡ pρ⎤0ρ = ⎢ ⎥⎣( γ −1)i0⎦1/ γ(31)The pressure p that exists at the surface is known and is calculated in the subroutine CL_AIRB byadding the initial pressure (atmospheric pressure). Then, the current energy can be calculated withthe ideal gas equation.pi =( γ −1)ρ(32)6.2 Several modelsThe model for the investigations with the compressed bubble is modified for calculations with acontrol volume. Instead of the compressed bubble the air is loaded by a pressure-time function (5 kgTNT) with the AIRB command. The models cv8 and cv9 are built so that all elements are almost58

quadratic. Therefore, the size of the elements increases with the distance to the charge. Model cv6uses the same thickness for all elements.Figure 43: Model control volume (cv8)Modell airl bTNTElements in radial(m)(m)equivalentdirectioncv6 2.5 1 5 30cv8 4.0 1 5 30cv9 4.0 1 5 40Table 11: Control volume modelsThe model cv6 has a coarse mesh and shows in the investigated region a sufficient representation ofthe experiments (see Figure 44 and Figure 45). The models cv8 and cv9 are longer and the elementsare more conform. The coarse mesh cv8 gets very small pressures and impulses. The finer modelcv9 represent the experimental peak pressures from Kingery better. The peak pressure isoverestimated for larger scaled distances. No model of the models tested herein can represent theimpulse.59

These models have also another problem if they are used for realistic calculations. The volume thatis cut away is missing in the model. The cut surfaces are additional boundaries with specialproperties. Therefore, these models are here not more investigated.4E+6Max. pressure [Pa]3E+62E+6Kingerycv6cv8cv91E+60E+00.6 0.8 1 1.2 1.4 1.6 1.8 2Scaled distance [m/kg 1/3 ]Figure 44: Maximum pressure for several control volume models60

400Impulse [Pa sec]300200.Kingery 5 kgKinney 5 kgcv6cv8cv910000.6 0.8 1 1.2 1.4 1.6 1.8 2Scaled Distance [m/kg 1/3 ]Figure 45: Impulse for several control volume models61

7 Implementation of an Air Blast Loading Function7.1 MotivationCalculations using the explosive with the JWL-equation may produce accurate results but need alsosmall elements. This results in small time step sizes and large computation times. There are severalpossibilities to reduce this cost, as shown in chapter 3. One of these is the use of a load-timefunction instead the fluid-structure interaction.This load-time function can be used if reflections of the air blast wave do not have an influence andif there is no need to consider the shadowing of the structure. The load-time function depends onthe size of the charge, the distance from the charge and some other conditions. The advantage ofthis method is the reduction of the computation time.Calculations with this method can be used to get an idea of the behaviour of the structure and to validateresults of a fluid-structure-interaction for special cases.7.2 Used FunctionThe modified Friedlander equation from Baker [2] (see equation(4)) can be used for the implementation.The parameters of this equation can be chosen from Kingery [14] or from Baker [2]. Theparameter b for the (decay) can be calculated with the knowledge of the maximum negativepressure or with the knowledge of the impulse. The value of b, as calculated in chapter 2.2.5, willbe used here.7.3 ImplementationThe load-time function is implemented as a new impedance. Impedances enable the input of boundaryconditions for special elements (CLxx) lying over the common elements. The command for theimpedance air blast wave (IMPE AIRB) allows the definition of a size of the charge, the origin ofthe detonation and the starting time of the detonation. Different conditions with different parametersare possible. Spherical and hemispherical explosion can be considered, also with reflectionconditions. The following conditions are possible by choosing the CONF parameter:1 = spherical detonation (full space), reflection conditions, equation form Kingery2 = spherical detonation (full space), no reflection conditions, equation form Kingery3 = spherical detonation (full space), no reflection conditions, equation form Baker4 = hemispherical detonation (half space), reflection conditions, equation form Kingery62

5 = hemispherical detonation (half space), no reflection conditions, equation form KingeryChanges are made in the following files• material_i_airb. This file includes two subroutines: MI_AIRB to read and check the inputparameter and CL_AIRB to calculate the pressure at a certain time step for the chosen element.In addition the maximum time step size is specified equal to the twentieth part of the duration ofthe positive phase.• fl38. This file calculates for a FL38 element the flux between the fluid elements. The flux forthe IMPE AIRB elements is calculated also in this file by using the formula in chapter 6.1.• cl3d and cl3i. The nodal forces resulting from the detonation pressure are calculated in thesefiles.7.4 Verification with ExamplesThe implemented function has to be verified with different examples. The function for an air blastwave in free air (without reflection) can be used for the control volume model (see chapter 3) andcan be compared with a model with the explosive implemented with the JWL-equation.The reflected pressures can be validated with experimental and numerical data from the literature.63

8 Mesh generation for LS-DYNATo compare the results of EUROPLEXUS with LS-DYNA an output is written that converts objectsin EUROPLEXUS towards an LS-DYNA input file. This file contains the nodes and the elementsof the objects. Different objects can be written in the same file with different part numbers for theelements. There is not an output of the materials and the loads.The mesh is written in a file called PXTOLS-DYNA.k.64

9 References[1] Alia, A.; Souli, M.: High explosive simulation using multi-material formulations, AppliedThermal Engineering 26, pp. 1032-1042, 2006.[2] Baker, Wilfrid E.: Explosions in the Air, University of Texas Pr., Austin, 1973.[3] Baker, W.E.; Cox, P.A.; Westine, P.S.; Kulesz, J.J.; Strehlow, R.A.: Explosion Hazards andEvaluation, Elsevier, Amsterdam, 1983.[4] Birnbaum, Naury K.; Clegg, Richard A.; Fairlie, Gerg E.; Hayhurst, Colin J.; Francis, NigelJ.: Analysis of blast loads on buildings, Preprint from “Structures under Extreme LoadingConditions – 1996”, Montreal, Quebec, Canada, 1996.[5] Brode, Harald L.: Numerical solutions of spherical blast waves, Journal of Applied Physics26, No 6, pp. 766-775, 1955.[6] Casadei, F.: “Use of EUROPLEXUS for Building Vulnerability Studies. Progress Report 1”,Technical Note I.05.50, July 2005.[7] Casadei, F.; Anthoine, A.: Use of EUROPLEXUS for Building Vulnerability Studies,Progress Report 2, Technical Note I.006972, March 2007.[8] CASTEM 2000, Guide d’ Utilisation, CEA, France, 1990.[9] Clutter, J.; Keith, Stahl, Michael: Hydrocode simulations of air and water shocks for facilityvulnerability assessments, Journal of Hazardous Materials, 106A, pp. 9-24, 2004.[10] Dobratz, B.M.; Crawford, P.C.: LLNL Explosives Handbook: Properties of Chemical Explosivesand Explosive Simulants, University of California, Lawrence Livermore NationalLaboratory, Report UCRL-5299, Rev. 2; 1985.[11] EUROPLEXUS, User’s Manual, online version.[12] Fairlie, Greg E.: Efficient analysis of high explosive air blast in complex urban geometriesusing the AUTODYN-2D & 3D hydrocodes, analytical and experimental methods, 15 th Int.Symposium on the Military Aspects of Blast and Shock, 14-19 September 1997, Banff,Canada.[13] Kingery, C.N.; Pannill, B.F.: Parametric Analysis of the Regular Reflection of Air Blast, BRLReport 1249, June 1964 (AD 444997).[14] Kingery, Charles N.; Bulmash, Gerald: Airblast Parameters from TNT Spherical Air Burstand Hemispherical Surface Burst, Defence Technical Information Center, Ballistic ResearchLaboratory, Aberdeen Proving Ground, Maryland, 1984.[15] Kinney, Gilbert F.; Graham, Kenneth J.: Explosive Shocks in Air, Springer, Berlin, 1985.[16] Krauthammer, T.; Altenberg, A.: Negative phase blast effects on glass panels, InternationalJournal of Impact Engineering, 24 (1), pp. 1-18; 2000.[17] Larcher, Martin; Herrmann, Nico; Stempniewski, Lothar: Explosionssimulation leichterHallenhüllkonstruktionen, Bauingenieur 6, pp. 271-277, 2006.[18] Lu, J.P.; Christo, F.C.; Kennedy, D.L.: Detonation modelling of corner-turning shocks inPBXN-111, 15 th Australian Fluid Mechanics Conference, Sydney, 2004.[19] Protective Design Center, United States Army Corps of Engineers: CONWEP, ConventionalWeapons Effects, https://pdc.usace.army.mil/software/conwep/, 22.May 2007.[20] Rose, T.A.; Smith, P.D.: Influence of the principal geometrical parameters of straight citystreets on positive and negative phase blast wave impulses, International Journal of Impact65

Engineering 27, pp. 359-376, 2002.[21] Shin, Young S.; Chisum, James E.: Modelling and simulation of Underwater shock problemsusing a coupled Lagrangian-Eulerian analysis approach, Shock and Vibration, Vol. 4, No. 1,pp. 1-10, 1997.[22] Smith, P.D.; Hetherington, J.G.: Blast and Ballistic Loading of Structures. Laxton's, 1994.[23] Tarver, C.M.; Hallquist, J.O.; Erickson, L.M.: Modeling short pulse duration shock initiationof solid explosives, Proceedings of the 8 th International Detonation Symposium, NavalSurface Weapons Center, Albeuquerque, NSWC MP 86-194, pp. 951-961, 1985.[24] Zukas, Jonas A.; Walters, William P.: “Explosive Effects and Applications”, Springer Verlag,New York, 1998.66

10 Apendix10.1 EUROPLEXUS Codedyms.ffSUBROUTINE DYMS(NUMN,X,NBELEM,INDOX)** writes on file 'pxtodyna.k' an input file for LS-DYNA* containing the mesh.* note that this file only centents the mesh and no* materials and loads.* For all objects defined by the {\tt ELEM}-lecture the* nodes and elements are written in this input file.* The objects defined by the {\tt SETS}-lecture are written* in additional element and node sets.*USE M_DYMS*INCLUDE 'NONE.INC'*INCLUDE 'CAREL.INC'INCLUDE 'GIBTYP.INC'INCLUDE 'CONTRO.INC'*INTEGER NUMN(*),INDOX(*),NBELEM(*)REAL*8 X(IDIM,*)* local variables*CHARACTER ENAM*4*INTEGER, PARAMETER :: MXNDEL=20CHARACTER*7 IJ(MXNDEL)*INTEGER I, IEL, ITYP, NBNOD, NAD, II, NLM, IZO,NBLM, IDEBU,& NBN, ITYFIC, PIDREAL*8 XX, YY, ZZ*INTEGER, ALLOCATABLE :: ENS(:)INTEGER :: NIN,LONENS*ITYFIC=3CALL OPNFIC(49,ITYFIC,'PXTOLS-DYNA.k')*** nodes*WRITE (49,1001)1001 FORMAT('*NODE')DO 100 I=1,NPTLXX=X(1,I)YY=X(2,I)IF(IDIM.EQ.2) THENWRITE(49,1002) I,XX,YYELSEZZ=X(3,I)WRITE(49,1003) I,XX,YY,ZZENDIF1002 FORMAT(I8,1P2E16.08)1003 FORMAT(I8,1P3E16.08)100 CONTINUE** elements*WRITE (49,1101)1101 FORMAT('*ELEMENT_SOLID')* CALLGIBLEC(A(N71),A(N72),A(N73),LOOP,ENS,LONENS,NIN)* WRITE(49,999) name_dyms(1)* 999 FORMAT(A20)* WRITE (*,*) NELEMDO 210 IEL=1,NELEM* WRITE (*,*) IELITYP=INDOX(IEL)NBNOD=NCEL(1,ITYP)IF(NBNOD.GT.MXNDEL) THENCALL ERRMSS('DYMS','TOO MANY NODES IN ANELEMENT')STOPENDIFNAD=INDOX(IEL+2*NELEM)-1PID=1151 GOTO (161,162,163,164,165,166,167,168,169), NBNOD160 CALL ERRMSS('DYMS','TOO MANY NODES INOUTPUT')STOP* -- 1 node161 WRITE(49,1201)IEL,PID,(NUMN(NAD+I),I=1,NBNOD)1201 FORMAT(I8,I8,I8)GO TO 200* -- 2 nodes162 WRITE(49,1202)IEL,PID,(NUMN(NAD+I),I=1,NBNOD)1202 FORMAT(I8,I8,2I8)GO TO 200* -- 3 nodes163 WRITE(49,1203)IEL,PID,(NUMN(NAD+I),I=1,NBNOD)1203 FORMAT(I8,I8,3I8)GO TO 200* -- 4 nodes164 WRITE(49,1204)IEL,PID,(NUMN(NAD+I),I=1,NBNOD)1204 FORMAT(I8,I8,4I8)GO TO 200* -- 5 nodes165 WRITE(49,1205)IEL,PID,(NUMN(NAD+I),I=1,NBNOD)1205 FORMAT(I8,I8,5I8)GO TO 200* -- 6 nodes166 WRITE(49,1206)IEL,PID,(NUMN(NAD+I),I=1,NBNOD)1206 FORMAT(I8,I8,6I8)GO TO 200* -- 7 nodes167 WRITE(49,1207)IEL,PID,(NUMN(NAD+I),I=1,NBNOD)1207 FORMAT(I8,I8,7I8)GO TO 200* -- 8 nodes168 WRITE(49,1208)IEL,PID,(NUMN(NAD+I),I=1,NBNOD)1208 FORMAT(I8,I8,/,8I8)GO TO 200* -- 9 nodes169 WRITE(49,1209)IEL,PID,(NUMN(NAD+I),I=1,NBNOD)1209 FORMAT(I8,I8,/,9I8)67

GO TO 200*200 CONTINUE*210 CONTINUE*CALL CLOFIC(49)*ENDm_material_i_airb.ffMODULE M_MATERIAL_I_AIRB** material of type "air blast" (16_26)*USE M_MATERIALSUSE M_MATERIALS_ARRAY*IMPLICIT NONESAVE*PRIVATEPUBLIC :: MI_AIRB, CL_AIRB*CONTAINS*============================================SUBROUTINE MI_AIRB(INUMLDC)* ------------------------------------------------------------------* impedance : air blast m.larcher 04-07* ------------------------------------------------------------------* iop=26 (airb)* xmut(1) = x-coordinate of explosive charge* xmut(2) = y-coordinate of explosive charge* xmut(3) = z-coordinate of explosive charge* xmut(4) = mass of explosive charge (in kg)* xmut(5) = initial time of the explosion* xmut(6) = choose of different explosion models* 1 = unconfined (full space), reflected (kingery)* 2 = unconfined (full space), not reflected (kingery)* 3 = unconfined (full space), not reflected (baker)* 4 = half-confined (half space), reflected (kingery)* 5 = half-confined (half space), not reflected(kingery)*INCLUDE 'NONE.INC'*INCLUDE 'CARMA.INC'INCLUDE 'POUBTX.INC'**----- variables globales :INTEGER, INTENT(IN) :: INUMLDC**----- variables locales :INTEGER, PARAMETER :: NMOT=6 , LENX=6 , LENI =1CHARACTER*4 :: MOT(NMOT)INTEGER :: KOPT(NMOT)LOGICAL :: IMPRIM*DATA MOT/'X ','Y ','Z ','MASS','TINT','CONF'/*INTEGER, PARAMETER :: N_MSG = 6 , LG_FMT = 350CHARACTER(LG_FMT) :: GET_FMT(N_MSG)*IF FRANCAISCHARACTER(32), PARAMETER :: NOM='IMPEDANCE(EXPLOSION EN AIR)'DATA GET_FMT(:) />'X-COORD DE LA CHARGE',>'Y-COORD DE LA CHARGE',>'Z-COORD DE LA CHARGE',>'MASSE DE LA CHARGE',>'TEMPS INITIAL DE L EXPLOSION',>'CONFINEMENT (1=LIBRE, REFLECHIE, KINGERY,>2=LIBRE, NO REFLECHIE,KINGERY,>3=LIBRE, REFLECHIE,BAKER,>4=DEMI-ESPACE, REFLECHIE,KINGERY,>5=DEMI-ESPACE, NO REFLECHIE,KINGERY)'/CELSECHARACTER(32), PARAMETER :: NOM='IMPEDANCE(AIR BLAST)'DATA GET_FMT(:) />'X-COOR OF THE CHARGE',>'Y-COOR OF THE CHARGE',>'Z-COOR OF THE CHARGE',>'MASS OF THE CHARGE',>'INITIAL TIME OF THE EXPLOSION',>'CONFINEMENT (1=UNCONFINED,REFLECTED,KINGERY,>2=UNCONFINED, NOT REFLECTED,KINGERY,>3=UNCONFINED, REFLECTED,BAKER,>4=HALF-SPACE, REFLECTED,KINGERY,>5=HALF-SPACE, NOT REFLECTED,KINGERY)'/CENDIF*CALL CREATE_MATERIAL (LENX, LENI, 0)NEWMAT%NAME = NOMNEWMAT%TYPE = 16 !! IMPEDANCENEWMAT%MATENT(1) = 26 !! NUMERO DEL'IMPEDANCENEWMAT%NUMLDC = INUMLDCNEWMAT%LGECR = LGECR(NEWMAT%TYPE)**----- lecture des parametresCALLLIRVAL(NMOT,MOT,NEWMAT%MATVAL,KOPT)**----- donnees completes ? (1, 2, 4 indispensables)IF((KOPT(1)+KOPT(2)+KOPT(4)) /= 3 ) THENWRITE(BLABLA,1001)CALL ERRMSS('MAT_AIRB',BLABLA)STOP 'MAT_AIRB 3'ENDIF** default valuesIF(KOPT(3) == 0) NEWMAT%MATVAL(3)=0.D0 ! ZIF(KOPT(5) == 0) NEWMAT%MATVAL(5)=-HUGE(0.D0) ! TINTIF(KOPT(6) == 0) NEWMAT%MATVAL(6)=1.D0 !CONF**----- impressionsWRITE(BLABLA,1000)NEWMAT%NUMLDC,NEWMAT%NAMECALL MECTSG(BLABLA)IF(IMPRIM(8)) THENCALLMECVAL(GET_FMT(1),NEWMAT%MATVAL(1))CALLMECVAL(GET_FMT(2),NEWMAT%MATVAL(2))CALLMECVAL(GET_FMT(3),NEWMAT%MATVAL(3))CALLMECVAL(GET_FMT(4),NEWMAT%MATVAL(4))CALLMECVAL(GET_FMT(5),NEWMAT%MATVAL(5))68

CALLMECVAL(GET_FMT(6),NEWMAT%MATVAL(6))ENDIF**IF FRANCAIS1000 FORMAT('LOI NUMERO',I5,' : ',A)1001 FORMAT('LA DIRECTIVE "AIRB" ESTINCOMPLETE')CELSE1000 FORMAT('LAW NUMBER',I5,' : ',A)1001 FORMAT('THE DIRECTIVE "AIRB" ISINCOMPLETE')CENDIF*END SUBROUTINE MI_AIRB*============================================SUBROUTINE CL_AIRB (MAT_CUR, ECR, D, P,DTAIRB)* ------------------------------------------------------------------* cond. aux limites air blast m.larcher 04-07* ------------------------------------------------------------------* mat_cur : current material* matval(1) = x-coordinate of explosive charge* matval(2) = y-coordinate of explosive charge* matval(3) = z-coordinate of explosive charge* matval(4) = mass of explosive charge (in kg)* matval(5) = initial time of the explosion* matval(6) = choose of different explosion models* 1 = unconfined (full space), reflected (kingery)* 2 = unconfined (full space), not reflected(kingery)* 3 = unconfined (full space), not reflected (baker)* 4 = half-confined (half space), reflected (kingery)* 5 = half-confined (half space), not reflected(kingery)* d : distance between charge and clxx element centroid* (already computed in the clxx element)* p : blast pressure (output)* ecr(1) : pressure (output)* dtairb : maximum time step (output)*IMPLICIT NONE*INCLUDE 'TEMPX.INC' ! T = CURRENT TIMEINCLUDE 'TEMPS1.INC' ! TINIZI = INITIAL TIME**--- variables globalesTYPE(MATERIAL), INTENT(INOUT) :: MAT_CURREAL(8), INTENT(IN) :: DREAL(8), INTENT(OUT) :: PREAL(8), INTENT(OUT) :: DTAIRBREAL(8), INTENT(INOUT) :: ECR(*)*--- variables localesREAL(8) :: T_START ,T_D , T_CURR, T_NEGREAL(8) :: PARAM1, PARAM2, PARAM3,PARAM4REAL(8) :: PARAM5, PARAM6, PARAM7,PARAM8REAL(8) :: P_MAX, B_BLAST , Z_BLAST, U_T_D,Y_CONWEP,> P_MAX1, T_MAX1, T_D1, U_P_MAX, U_T_START,P_NEGREAL(8), DIMENSION(9) :: POLY_T_DREAL(8), DIMENSION(10) :: POLY_T_STARTREAL(8), DIMENSION(12) :: POLY_P_MAXINTEGER :: I*P = 0.D0 ! initializationDTAIRB = 1e99 ! initialisation - only in the positive phasethe time step size will be changed*----- air blast pressure*----- parameters of detonation for all modelsZ_BLAST = D/MAT_CUR%MATVAL(4)**0.333333D0B_BLAST = 5.2777D0*(Z_BLAST**(-1.1975D0))P_NEG = (0.35/Z_BLAST)*1.E5IF(P_NEG MAT_CUR%MATVAL(4)**(1./3.)*1E-3END IF*----- time of explosion-------------------------------------------IF (MAT_CUR%MATVAL(5) -0.06017700, 0.0696360, 0.0215297, -0.01616589, -0.0023253,> 0.00147752/)U_T_START = -0.2024257+1.37784*LOG10(Z_BLAST)END IFY_CONWEP = 0.0DO I = 1, 10Y_CONWEP = Y_CONWEP +POLY_T_START(I)*U_T_START**(I-1)END DOT_START = 1D-3 * 10 ** Y_CONWEP*----- time in the pressure curveT_CURR = T - MAT_CUR%MATVAL(5) - T_STARTIF (T_CURR>0) THEN*----------------------------------------------------------------------*begin different modelsP_MAX=0T_D=0P_MAX1=0T_D1=0*--------------------------------------------------------------------*baker incidentIF (MAT_CUR%MATVAL(6)==3) THENPARAM1 = 808.D0*(1.D0+(Z_BLAST/4.5D0)**2.0)PARAM2 = SQRT(1.D0+(Z_BLAST/0.048D0)**2.0)PARAM3 = SQRT(1.D0+(Z_BLAST/0.32D0)**2.0)PARAM4 = SQRT(1.D0+(Z_BLAST/1.35D0)**2.0)*------- max pressure for this distanceP_MAX = 1D5 *PARAM1/(PARAM2*PARAM3*PARAM4)PARAM5 = 980.D0*(1.D0+(Z_BLAST/0.54D0)**10.D0)PARAM6 = 1.D0+(Z_BLAST/0.02D0)**3.D0PARAM7 = 1.D0+(Z_BLAST/0.74D0)**6.D0PARAM8 = SQRT(1.D0+(Z_BLAST/6.9)**2.0)T_D = 1D-3 * MAT_CUR%MATVAL(4)**(1.D0/3.D0) *>PARAM5/(PARAM6*PARAM7*PARAM8)GOTO 100069

END IF*--------------------------------------------------------------------* kingery sphericalIF((MAT_CUR%MATVAL(6)==1) .OR.(MAT_CUR%MATVAL(6)==2)) THENIF(Z_BLAST 0.00291430135946, 0.00187957449227,0.0173413962543,> 0.00269739758043, -0.00361976502798, -0.00100926577934/)ELSE IF(Z_BLAST>2.28) THENU_T_D = -3.130058+3.152472*LOG10(Z_BLAST)POLY_T_D = (/0.62103, 0.096703, -0.00801302,0.00482705,>0.00187587, -0.002467385, -0.000841116668,0.00061932910, 0.0 /)ELSEU_T_D = -1.33361206714+9.2996288611*LOG10(Z_BLAST)POLY_T_D = (/ 0.23031841078, -0.0297944268969,0.0306329542941,> 0.0183405574074, -0.0173964666286, -0.00106321963576,> 0.0056206003128, 0.0001618217499, -0.0006860188944 /)END IF*--------------------------------------------------------------------* kingery hemisphericalELSEIF(Z_BLAST-0.00475933, -0.00428144, 0.0, 0.0, 0.0/)ELSE IF(Z_BLAST>2.78) THENU_T_D = -3.53626+3.463497*LOG10(Z_BLAST)POLY_T_D = (/0.6869066, 0.09330353, -0.00058494, -0.0022688499,> -0.000295908, 0.0014802986, 0.0, 0.0, 0.0 /)ELSEU_T_D = -2.124925+9.2996288*LOG10(Z_BLAST)POLY_T_D = (/0.31540924, -0.0297944, 0.0306329,0.018340557,>-0.0173964, -0.00106321, 0.0056206, 0.000161821, -0.00068601889/)END IFEND IF*--------------------------------------------------------------------* kingery spherical, reflectedIF(MAT_CUR%MATVAL(6)==1) THENU_P_MAX = -0.214362789151+1.35034249993*LOG10(Z_BLAST)POLY_P_MAX = (/3.22958031387, -2.21400538997,0.35119031446,> 0.657599992109, 0.0141818951887, -0.243076636231,> -0.00158699803158, 0.0492741184234,0.00227639644004,> -0.00397126276058, 0.0 , 0.0/)*--------------------------------------------------------------------* kingery spherical, incidentELSE IF (MAT_CUR%MATVAL(6)==2) THENU_P_MAX = -0.214362789151+1.35034249993*LOG10(Z_BLAST)POLY_P_MAX = (/2.611368669, -1.69012801396,0.00804973591951,> 0.336743114941, -0.00516226351334, -0.0809228619888,> -0.00478507266747, 0.00793030472242,0.0007684469735,> 0.0, 0.0, 0.0/)*--------------------------------------------------------------------* kingery hemispherical, reflectedELSE IF(MAT_CUR%MATVAL(6)==4) THENU_P_MAX = -0.240657322658+1.36637719229*LOG10(Z_BLAST)POLY_P_MAX = (/3.4028321, -2.2103087, -0.218536586,0.89531958,> 0.24989, -0.569249, -0.1179168, 0.2241311, 0.0245620, -0.0455116,> -0.001909307, 0.003614711/)*--------------------------------------------------------------------* kingery hemispherical, incidentELSE IF (MAT_CUR%MATVAL(6)==5) THENU_P_MAX = -0.214362789151+1.35034249993*LOG10(Z_BLAST)POLY_P_MAX = (/2.780769, -1.6958988, -0.1541937,0.514050,> 0.0988534, -0.2939126, -0.02681123,0.109097,0.001628467,> -0.0214631, 0.0001456723, 0.001678477/)END IF*--------------------------------------------------------------------* kingery calculationDO I = 1, 12P_MAX1 = P_MAX1 +POLY_P_MAX(I)*U_P_MAX**(I-1)END DOP_MAX= 1D3 * 10 ** P_MAX1DO I = 1, 9T_D1 = T_D1 + POLY_T_D(I)*U_T_D**(I-1)END DOT_D= 1D-3 * 10 ** T_D1*--------------------------------------------------------------------*end different models1000 P = P_MAX*(1.D0-T_CURR/T_D)*EXP(-B_BLAST*T_CURR/T_D)IF (T_CURRT_D . AND . T_CURRT_D+T_NEG/2. . AND .T_CURRT_D+T_NEG) THENP=0 ! AFTER LOADINGEND IFEND IFIF(P

10.2 Miscellaneous codeairblastresult.f* Reads a file which is build with europlexus LIST-operator.* Writes as an output a file with the peak pressure, theimpulse,* the arrival time and the duration of the positive phase.* airblastresult dyna7d.txt* arrival time - arrival of the first pressurePROGRAM airblastresult* x(location_number,time)=location,y(location_number,time)=pressurereal*8 :: x(100000,1000), y(100000,1000), ymax (1000)real*8 :: t(1000), tdist(1000), flagmax(1000), ta(1000)real*8 :: t0, tend, impulse,tdinteger ijk(4,100000)integer :: n, ival, i, xi, yi, icom, flag, endimpcharacter(len=30) :: text1n=0don=n+1xmax=0read(5,100,END=999) text1, ival, text2, icom100 FORMAT(A8, I8, A11 , I8)if(ival1) tdist(n)=t(n-1)tdist(n)=t(n)-tdist(n)*-----------------------------------------*Reading of the values*-----------------------------------------DO i = 1, ivalread(5,*,END=999) x(i,n), y(i,n)2001 FORMAT(2E14.5)if((y(i,n)>1e5).and.(ta(i)==0))ta(i)=t(n)if(y(i,n)>ymax(i)) thenif(flagmax(i)1).and.(y(i,n)

x6 y6 z6 = coor p6;x7 y7 z7 = coor p7;x8 y8 z8 = coor p8;*di1 = (((x1-x2)*(x1-x2))+((y1-y2)*(y1-y2))+((z1-z2)*(z1-z2)))**0.5;di2 = (((x4-x2)*(x4-x2))+((y4-y2)*(y4-y2))+((z4-z2)*(z4-z2)))**0.5;di3 = (((x4-x3)*(x4-x3))+((y4-y3)*(y4-y3))+((z4-z3)*(z4-z3)))**0.5;di4 = (((x1-x3)*(x1-x3))+((y1-y3)*(y1-y3))+((z1-z3)*(z1-z3)))**0.5;di5 = (((x1-x5)*(x1-x5))+((y1-y5)*(y1-y5))+((z1-z5)*(z1-z5)))**0.5;di6 = (((x6-x2)*(x6-x2))+((y6-y2)*(y6-y2))+((z6-z2)*(z6-z2)))**0.5;di7 = (((x4-x8)*(x4-x8))+((y4-y8)*(y4-y8))+((z4-z8)*(z4-z8)))**0.5;di8 = (((x7-x3)*(x7-x3))+((y7-y3)*(y7-y3))+((z7-z3)*(z7-z3)))**0.5;di9 = (((x6-x5)*(x6-x5))+((y6-y5)*(y6-y5))+((z6-z5)*(z6-z5)))**0.5;di10= (((x6-x8)*(x6-x8))+((y6-y8)*(y6-y8))+((z6-z8)*(z6-z8)))**0.5;di11= (((x7-x8)*(x7-x8))+((y7-y8)*(y7-y8))+((z7-z8)*(z7-z8)))**0.5;di12= (((x7-x5)*(x7-x5))+((y7-y5)*(y7-y5))+((z7-z5)*(z7-z5)))**0.5;lis1 = PROG di1 di2 di3 di4 di5 di6 di7 di8 di9 di10 di11 di12;min2 = mini(lis1);max2 = maxi(lis1);asra=0;SI (min2 > 0.0);asra = max2/min2;FINS;finproc asra;aspect_ratio.dgibi*Example for the calculation of the aspect ratio*Construction d'une sphere a partir d'un cubeopti donn 'aspectratio.procedur';opti dime 3 elem cub8;*Nombre de bissectionsnel0a =10;nel0 = 10;r0 = .25;sizeex = 0.5;*Referenceo0 = 0. 0. 0.;x0 = (sizeex) 0. 0.;xa0 = 0 (sizeex) (sizeex);xb0 = (sizeex) (sizeex) (sizeex);xc0 = (sizeex) 0 (sizeex);xd0 = 0 0 (sizeex);y0 = 0. (sizeex) 0.;z0 = 0. 0. (sizeex);symp1 = (sizeex/2.) (sizeex/2.) (sizeex/2.);*Cube intermediaire (centre=o0 et arete=r0)vol0 = o0 droi nel0a x0 tran nel0a z0 volu tran nel0a y0coul bleu homo o0 r0;cub0 = (o0 droi nel0 y0 tran nel0 x0)et (o0 droi nel0 z0 tran nel0 y0) et(o0 droi nel0 x0 tran nel0 z0)syme 'POINT' symp1 homo o0 r0;cub1 = (o0 droi nel0 y0 tran nel0 x0)et (o0 droi nel0 z0 tran nel0 y0) et(o0 droi nel0 x0 tran nel0 z0)syme 'POINT' symp1;*Pojection sur la sphere de rayon unitairespe1 = cub0 proj 'CONI' o0 'SPHE' o0 x0;*Remplissagevol1 = cub0 volu nel0 spe1 coul roug;vges = vol1 et vol0;elim 1.d-8 vges;maxar=0;REPE I0 (NBEL vges);ar = aspectra(vges ELEM CUB8 &I0);SI (ar > maxar);maxar=ar;FINS;FIN I0;MESS maxar;calc_b.cpp// calc_b.cpp : calculation of the b-factor for the Friedlander// equation and the impulses resulting of a given b#include "stdafx.h"#include #include #include #include #include using namespace std;/*Calculation of the maximum pressure*/double p0(double z,int flag){double pp;if(flag==1){ //Kinneydouble zaehler=808.*(1.+pow(z/4.5,2.));double nenner1=sqrt(1.+pow(z/0.048,2.));double nenner2=sqrt(1.+pow(z/0.32,2.));double nenner3=sqrt(1.+pow(z/1.35,2.));pp=zaehler/(nenner1*nenner2*nenner3);}else{ //Kingerydouble t=log10(z);double u=-0.214362789151+1.35034249993*t;double y=2.611368669-1.69012801396*u+0.00804973591951*pow(u,2.)+0.336743114941*pow(u,3.)-0.00516226351334*pow(u,4.)-0.0809228619888*pow(u,5.)-0.00478507266747*pow(u,6.)+0.00793030472242*pow(u,7.)+0.0007684469735*pow(u,8.);pp=pow(10.,y);}return pp;}/*Calculation of the length of the positive phase*/double td(double w,double z,int flag){double tdd;if(flag==1){ //Kinneydouble zaehler=980.*(1.+pow(z/0.54,10.));double nenner1=1.+pow(z/0.02,3.);double nenner2=1.+pow(z/0.74,6.);double nenner3=sqrt(1.+pow(z/6.9,2.));tdd=pow(w,1./3.)*zaehler/(nenner1*nenner2*nenner3);}72

else{ //Kingerydouble t=log10(z);double y,u;if(z

for(b=0;bfabs(i00-iSoll)){iDistMin=fabs(i00-iSoll);bDistMin=b;}//cout

maxar=0;REPE I0 (NBEL vges);ar = aspectra(vges ELEM CUB8 &I0);SI (ar > maxar);maxar=ar;FINS;FIN I0;MESS maxar;geom_new = (air1 et exp1 et ages et nxp1);elim geom_new 1e-8;TASS geom_new;OPTI sauv form 'bubble1.msh';sauv form geom_new;bubble4.epx*bubble test, wheather a compressed bubble can represent anexplosion*models bubble1 to bubble3 and bubble5 to bubble13 similar$ECHOCAST 'bubble3.msh' geom_newTRID NONL ALEOPTI NF34OPTI TOLC 1e-1OPTI TION 1e-10$DIMEPT6L 100000FL38 110000 ZONE 2ECROU 1100000NBLO 100000NALE 5000 NBLE 5000TERM$GEOMFL38 air1FL38 exp1TERM*GRIL EULE LECT air1 exp1 TERMALE TOUSAUTO AUTR$MATE$ airflut RO 1.3 EINT 2.1978E5 GAMM 1.35 PB 0ITER 1 ALF0 1 BET0 1 KINT 0 AHGF 0 CL 0.5CQ 2.56 PMIN 0 PREF 1.e5 NUM 1LECT air1 TERMflut RO 130 EINT 2.1978E7 GAMM 1.35 PB 0ITER 1 ALF0 1 BET0 1 KINT 0 AHGF 0 CL 0.5CQ 2.56 PMIN 0 PREF 1.e5 NUM 1LECT exp1 TERM$LINK COUPFSR LECT ages TERMECRI DEPL VITE CONT ECRO TFRE 10.E-3FICH ALIC TEMP tfreq 5e-7ELEM LECT fe1 fe2 TERMFICH ALIC TFRE 1E-4$OPTI NOTELOG 1$CALC TINI 0 TEND 1e-2*============================================SUITPost-treatment (time curves from alice temps file)ECHO*RESU ALIC TEMP GARD PSCRSORT GRAPAXTE 1.0 'Time [s]'*COUR 1 'press_1' ECRO COMP 1 GAUSS 1 ELEM LECT fe1TERMCOUR 2 'press_2' ECRO COMP 1 GAUSS 1 ELEM LECT fe2TERMtrac 1 2 TEXT axes 1.0 'PRESS [MPa]'*==========================================SUITPost-treatment (time curves from alice temps file)ECHO*RESU ALIC GARD PSCR*SORT GRAP*AXTE 1.0 'Time [s]'*SCOURBE 100 ' 1.000E-04 ' NSTO 2 SAXE 1 'x' LECT nxp1TERM ECRO COMP 1SCOURBE 101 ' 2.000E-04 ' NSTO 3 SAXE 1 'x' LECT nxp1TERM ECRO COMP 1*And so on….SCOURBE 199 ' 1.000E-02 ' NSTO 101 SAXE 1 'x' LECT nxp1TERM ECRO COMP 1list 101 102 103 104 105 106 107 108 109 110 axes 1 'PRESS[MPa]'list 111 112 113 114 115 116 117 118 119 120 axes 1 'PRESS[MPa]'list 121 122 123 124 125 126 127 128 129 130 axes 1 'PRESS[MPa]'list 131 132 133 134 135 136 137 138 139 140 axes 1 'PRESS[MPa]'list 141 142 143 144 145 146 147 148 149 150 axes 1 'PRESS[MPa]'list 151 152 153 154 155 156 157 158 159 160 axes 1 'PRESS[MPa]'list 161 162 163 164 165 166 167 168 169 170 axes 1 'PRESS[MPa]'list 171 172 173 174 175 176 177 178 179 180 axes 1 'PRESS[MPa]'list 181 182 183 184 185 186 187 188 189 190 axes 1 'PRESS[MPa]'list 191 192 193 194 195 196 197 198 199 axes 1 'PRESS[MPa]'trac 101 TEXT axes 1.0 'PRESS [MPa]'*============================================SUITDYNA 4 - POSTTREATMENT FROM ALICE FILE$ECHOCONV WIN$75

RESU ALIC GARD PSCR*OPTI PRIN*SORT VISU NSTO 1*============================================PLAYCAME 1 EYE -1.13949E+01 2.30078E-01 5.73754E-01! Q 6.80412E-01 3.18025E-02 -7.31191E-01 3.72550E-02VIEW 9.92653E-01 9.77586E-02 7.13025E-02RIGH -7.20556E-02 4.19024E-03 9.97392E-01UP -9.72049E-02 9.95201E-01 -1.12035E-02FOV 2.48819E+01scengeom navi freerefe fram bbox centline heouiso fili fiel ecro 1 !1 scal user prog 1.e7 pas 1.e7 1.4e8 termtext iscalima onsler cam1 1 nfra 1trac offs fich avi nocl nfto 100 fps 5 kfre 10 comp -1rend* obje lect struc term rend!trac offs fich avi cont noclrendfreq 1gotr loop 98 offs fich avi cont noclrend* obje lect struc term rendgotrac offs fich avi contrend* obje lect struc term rendENDPLAY*===========================================FINcon12.dbigi*CON12 Conical model with 1 kg TNT*opti donn 'D:\Users\larchma\cast3m\pxpdroi1.procedur';opti donn 'D:\Users\larchma\cast3m\pxordpoi.procedur';OPTI echo 1;OPTI dime 3 elem cub8;***************************************************** Parameters *********************************************************************************************opening=0.1;length=3.0;dex=0.05;dexin= 0.001;dexfi= 0.001;daiin= 0.001;daifi= 0.04;*Length of the pyramiddpy=0.005;****************************************************DENS 100;opex=dex*opening/length;oppy=dpy*opening/length;p1 = 0 0 0;*Points of the pyramidppy2 = (dpy) (oppy) (oppy);ppy3 = (dpy) (0-oppy) (oppy);ppy4 = (dpy) (0-oppy) (0-oppy);ppy5 = (dpy) (oppy) (0-oppy);lpyr1 = p1 d ppy2;lpyr2 = p1 d ppy3;lpyr3 = p1 d ppy4;lpyr4 = p1 d ppy5;pyra = manu pyr5 p1 ppy5 ppy4 ppy3 ppy2;*Points of the explosivepe2 = (dex) (opex) (opex);pe3 = (dex) (0-opex) (opex);pe4 = (dex) (0-opex) (0-opex);pe5 = (dex) (opex) (0-opex);*Points of the airp2 = (length) (opening) (opening);p3 = (length) (0-opening) (opening);p4 = (length) (0-opening) (0-opening);p5 = (length) (opening) (0-opening);*defining the lines between the pyramid and the explosivelpe1 = ppy2 d ppy3;lpe2 = ppy3 d ppy4;lpe3 = ppy4 d ppy5;lpe4 = ppy5 d ppy2;*defining the lines around explosiveDENS dexin;le1 = ppy2 d 'DINI' dexin 'DFIN' dexfi pe2;le2 = ppy3 d 'DINI' dexin 'DFIN' dexfi pe3;le3 = ppy4 d 'DINI' dexin 'DFIN' dexfi pe4;le4 = ppy5 d 'DINI' dexin 'DFIN' dexfi pe5;*defining the lines between the explosive and the airDENS 100.;lea1 = pe2 d pe3;lea2 = pe3 d pe4;lea3 = pe4 d pe5;lea4 = pe5 d pe2;*defining the surfaces around the explosiveae1=dall lpe1 le1 lea1 le2;ae2=dall lpe2 le2 lea2 le3;ae3=dall lpe3 le3 lea3 le4;ae4=dall lpe4 le4 lea4 le1;ae5=dall lpe1 lpe2 lpe3 lpe4;ae6=dall lea1 lea2 lea3 lea4;*defining the surfaces around the pyramideapyr1 = surf (lpyr1 et lpyr2 et lpe1) plane;apyr2 = surf (lpyr2 et lpyr3 et lpe2) plane;apyr3 = surf (lpyr3 et lpyr4 et lpe3) plane;apyr4 = surf (lpyr4 et lpyr1 et lpe4) plane;apyrsum = (apyr1 et apyr2 et apyr3 et apyr4 et ae6);elim 1e-8 apyrsum;*defining the volume of the explosivegeomex= (ae1 et ae2 et ae3 et ae4 et ae5 et ae6) coul roug;elim 1e-8 (geomex);vex = (geomex) volu;*defining the lines around air*DENS 0.01;la1 = pe2 d 'DINI' daiin 'DFIN' daifi p2;la2 = pe3 d 'DINI' daiin 'DFIN' daifi p3;la3 = pe4 d 'DINI' daiin 'DFIN' daifi p4;la4 = pe5 d 'DINI' daiin 'DFIN' daifi p5;*defining the lines at the endDENS 100.;lend1 = p2 d 'DINI' 100 'DFIN' 100 p3;lend2 = p3 d 'DINI' 100 'DFIN' 100 p4;lend3 = p4 d 'DINI' 100 'DFIN' 100 p5;lend4 = p5 d 'DINI' 100 'DFIN' 100 p2;76

*defining the surfaces around the aira1=dall lea1 la1 lend1 la2;a2=dall lea2 la2 lend2 la3;a3=dall lea3 la3 lend3 la4;a4=dall lea4 la4 lend4 la1;a6=dall lend1 lend2 lend3 lend4;*defining the volume of the explosivegeomai= (a1 et a2 et a3 et a4 et ae6 et a6) coul bleu;elim 1e-8 geomai;vai = geomai volu;vai1=vai;vex1=vex;a6=(a6 ELEM 1);* Erstellen der Huelle fuer IMPE*geom2 = (a1 et a2 et a3 et a4 et a5 et ae1 et ae2 et ae3 et ae4);*Points for the controllfp1 = vai1 poin proche (0.2 0 0);pp0 = fp1 et fp1;REPE I0 (NBEL vai1);TEST0 = pp0 INCL (vai1 ELEM CUB8 &I0) 'VOLU';SI ((NBEL TEST0)> 0);quit I0;FINS;FIN I0;MESS &I0;fe1 = vai1 elem CUB8 &I0;fp2 = vai1 poin proche (0.4 0 0);pp0 = fp2 et fp2;REPE I0 (NBEL vai1);TEST0 = pp0 INCL (vai1 ELEM CUB8 &I0) 'VOLU';SI ((NBEL TEST0)> 0);quit I0;FINS;FIN I0;MESS &I0;fe2 = vai1 elem CUB8 &I0;fp3 = vai1 poin proche (0.6 0 0);pp0 = fp3 et fp3;REPE I0 (NBEL vai1);TEST0 = pp0 INCL (vai1 ELEM CUB8 &I0) 'VOLU';SI ((NBEL TEST0)> 0);quit I0;FINS;FIN I0;MESS &I0;fe3 = vai1 elem CUB8 &I0;fp4 = vai1 poin proche (0.8 0 0);pp0 = fp4 et fp4;REPE I0 (NBEL vai1);TEST0 = pp0 INCL (vai1 ELEM CUB8 &I0) 'VOLU';SI ((NBEL TEST0)> 0);quit I0;FINS;FIN I0;MESS &I0;fe4 = vai1 elem CUB8 &I0;fp5 = vai1 poin proche (1.0 0 0);pp0 = fp5 et fp5;REPE I0 (NBEL vai1);TEST0 = pp0 INCL (vai1 ELEM CUB8 &I0) 'VOLU';SI ((NBEL TEST0)> 0);quit I0;FINS;FIN I0;MESS &I0;fe5 = vai1 elem CUB8 &I0;fp6 = vai1 poin proche (1.2 0 0);pp0 = fp6 et fp6;REPE I0 (NBEL vai1);TEST0 = pp0 INCL (vai1 ELEM CUB8 &I0) 'VOLU';SI ((NBEL TEST0)> 0);quit I0;FINS;FIN I0;MESS &I0;fe6 = vai1 elem CUB8 &I0;*lines for the output of the pressurevges = (vai1 et vex1);nxpl = pxpdroi1 vges p1 (p1 plus p2) 0.0005;nxpl = pxordpoi nxpl p1;*areas on the sidesasum1 = (apyr1 et ae1 et a1);asum2 = (apyr2 et ae2 et a2);asum3 = (apyr3 et ae3 et a3);asum4 = (apyr4 et ae4 et a4);asum = (asum1 et asum2 et asum3 et asum4);geom_new = (vai1 et vex1 et pyra et nxpl);TASS geom_new;OPTI sauv form 'CON12.msh';sauv form geom_new;*list (nbel vai1);list (nbel vex1);list (nbno geom_new);list(mesu(vex));con12.epscon12 conical model with 12.8 kg TNT$ECHOCONV WINCAST geom_newTRID NONL ALEOPTI NF34OPTI TOLC 1e-1*OPTI PART$DIMEPT6L 10000FL38 1300FL35 1 ZONE 4TABL 100 100ECROU 1000000NBLO 100000NALE 5000 NBLE 5000TERM$GEOMFL38 vai1FL38 vex1FL35 pyraTERM*77

GRIL EULE LECT vex1 pyra TERMALE TOUSAUTO AUTR$MATE$ airflut RO 1.3 EINT 2.1978E5 GAMM 1.35 PB 0ITER 1 ALF0 1 BET0 1 KINT 0 AHGF 0 CL 0.5CQ 2.56 PMIN 0 PREF 1.e5 NUM 11a 3.738e11 b 3.747e9 r1 4.15 r2 0.90ros 1630LECT vai1 TERM$ explosiveflut ro 1630 eint 3.68e6 gamm 1.35 PB 0ITER 1 ALF0 1 BET0 1 KINT 0 AHGF 0 CL 0.5CQ 2.56 PMIN 0 PREF 1.e5 NUM 11a 3.738e11 b 3.747e9 r1 4.15 r2 0.90d 6930 TDET 0.0 pini 1e5xdet 0.0 ydet 0.0 zdet 0.0LECT vex1 pyra TERM$LINK COUPFSR LECT asum1 asum2 asum3 asum4 p1 a6 TERMECRI DEPL VITE CONT ECRO TFRE 10.E-3FICH ALIC TFRE 2E-6$OPTI NOTELOG 1$CALC TINI 0 TEND 1.5e-3*============================================SUITPost-treatment (time curves from alice temps file)ECHO*RESU ALIC GARD PSCR*SORT GRAP*AXTE 1.0 'Time [s]'*COUR 1 'press_1' ECRO COMP 1 GAUSS 1 ELEM LECT fe1TERMCOUR 2 'press_2' ECRO COMP 1 GAUSS 1 ELEM LECT fe2TERMCOUR 3 'press_3' ECRO COMP 1 GAUSS 1 ELEM LECT fe3TERMCOUR 4 'press_4' ECRO COMP 1 GAUSS 1 ELEM LECT fe4TERMCOUR 5 'press_5' ECRO COMP 1 GAUSS 1 ELEM LECT fe5TERMCOUR 6 'press_6' ECRO COMP 1 GAUSS 1 ELEM LECT fe6TERMCOUR 7 'int1' subc 1 1e5COUR 8 'int2' subc 2 1e5COUR 11 'int3' INT 7COUR 12 'int4' INT 8COUR 21 'press_1' ECRO COMP 1 GAUSS 1 ELEM LECT 1TERMCOUR 22 'press_2' ECRO COMP 1 GAUSS 1 ELEM LECT 2TERMCOUR 23 'press_3' ECRO COMP 1 GAUSS 1 ELEM LECT 3TERMCOUR 24 'vi_1_160' VITE LECT 160 TERMCOUR 25 'vi_2_160' VITE LECT 160 TERMCOUR 26 'vi_1_156' VITE LECT 156 TERMCOUR 27 'vi_1_155' VITE LECT 155 TERMSCOURBE 33 '1E-4' NSTO 51 SAXE 1 'x' LECT nxpl TERMECRO COMP 1SCOURBE 34 '5E-4' NSTO 251 SAXE 1 'x' LECT nxpl TERMECRO COMP 1trac 1 2 3 4 5 6 TEXT axes 1.0 'PRESS [MPa]'trac 11 12 TEXT axes 1.0 'PRESS*time [MPa]'trac 21 22 23 TEXT axes 1.0 'PRESS [MPa]' xmin 0 xmax 2e-4nx 10trac 24 25 26 27 TEXT axes 1.0 'Velocity [m/sec]'trac 33 34 TEXT axes 1.0 'distance [m]'SCOURBE 101 ' 2.00E-06 ' NSTO 2 SAXE 1 'x' LECT nxplTERM ECRO COMP 1SCOURBE 102 ' 4.00E-06 ' NSTO 3 SAXE 1 'x' LECT nxplTERM ECRO COMP 1** and so on****************************************SCOURBE 849 ' 1.500E-03 ' NSTO 751 SAXE 1 'x' LECT nxplTERM ECRO COMP 1*COUR 1 'press_1' ECRO COMP 1 GAUSS 1 ELEM LECT fe1TERMlist 101 102 103 104 105 106 107 108 109 110 axes 1 'PRESS[MPa]'** and so on****************************************list 841 842 843 844 845 846 847 848 849 axes 1 'PRESS[MPa]'*=============================================FINcon22.dgibi*CON22 Conical model with 12.8 kg TNT*quadratic elements*opti donn 'D:\Users\larchma\cast3m\pxpdroi1.procedur';opti donn 'D:\Users\larchma\cast3m\pxordpoi.procedur';OPTI echo 1;OPTI dime 3 elem cub8;***************************************************** Parameters *******************************************************************************************opening=0.1;length=10.0;dex=0.124;dexin= 0.0001;*Length of the pyramiddpy=0.001;****************************************************DENS 100;opex=dex*opening/length;dexfi= opex*2.;daiin= opex*2.;daifi= opening*2.;oppy=dpy*opening/length;p1 = 0 0 0;*Points of the pyramidppy2 = (dpy) (oppy) (oppy);ppy3 = (dpy) (0-oppy) (oppy);ppy4 = (dpy) (0-oppy) (0-oppy);ppy5 = (dpy) (oppy) (0-oppy);78

lpyr1 = p1 d ppy2;lpyr2 = p1 d ppy3;lpyr3 = p1 d ppy4;lpyr4 = p1 d ppy5;pyra = manu pyr5 p1 ppy5 ppy4 ppy3 ppy2;*Points of the explosivepe2 = (dex) (opex) (opex);pe3 = (dex) (0-opex) (opex);pe4 = (dex) (0-opex) (0-opex);pe5 = (dex) (opex) (0-opex);*Points of the airp2 = (length) (opening) (opening);p3 = (length) (0-opening) (opening);p4 = (length) (0-opening) (0-opening);p5 = (length) (opening) (0-opening);*defining the lines between the pyramid and the explosivelpe1 = ppy2 d ppy3;lpe2 = ppy3 d ppy4;lpe3 = ppy4 d ppy5;lpe4 = ppy5 d ppy2;*defining the lines around explosiveDENS dexin;le1 = ppy2 d 'DINI' dexin 'DFIN' dexfi pe2;le2 = ppy3 d 'DINI' dexin 'DFIN' dexfi pe3;le3 = ppy4 d 'DINI' dexin 'DFIN' dexfi pe4;le4 = ppy5 d 'DINI' dexin 'DFIN' dexfi pe5;*defining the lines between the explosive and the airDENS 100.;lea1 = pe2 d pe3;lea2 = pe3 d pe4;lea3 = pe4 d pe5;lea4 = pe5 d pe2;*defining the surfaces around the explosiveae1=dall lpe1 le1 lea1 le2;ae2=dall lpe2 le2 lea2 le3;ae3=dall lpe3 le3 lea3 le4;ae4=dall lpe4 le4 lea4 le1;ae5=dall lpe1 lpe2 lpe3 lpe4;ae6=dall lea1 lea2 lea3 lea4;*defining the surfaces around the pyramideapyr1 = surf (lpyr1 et lpyr2 et lpe1) plane;apyr2 = surf (lpyr2 et lpyr3 et lpe2) plane;apyr3 = surf (lpyr3 et lpyr4 et lpe3) plane;apyr4 = surf (lpyr4 et lpyr1 et lpe4) plane;apyrsum = (apyr1 et apyr2 et apyr3 et apyr4 et ae6);elim 1e-8 apyrsum;*defining the volume of the explosivegeomex= (ae1 et ae2 et ae3 et ae4 et ae5 et ae6) coul roug;elim 1e-8 (geomex);vex = (geomex) volu;*defining the lines around air*DENS 0.01;la1 = pe2 d 'DINI' daiin 'DFIN' daifi p2;la2 = pe3 d 'DINI' daiin 'DFIN' daifi p3;la3 = pe4 d 'DINI' daiin 'DFIN' daifi p4;la4 = pe5 d 'DINI' daiin 'DFIN' daifi p5;*defining the lines at the endDENS 100.;lend1 = p2 d 'DINI' 100 'DFIN' 100 p3;lend2 = p3 d 'DINI' 100 'DFIN' 100 p4;lend3 = p4 d 'DINI' 100 'DFIN' 100 p5;lend4 = p5 d 'DINI' 100 'DFIN' 100 p2;*defining the surfaces around the aira1=dall lea1 la1 lend1 la2;a2=dall lea2 la2 lend2 la3;a3=dall lea3 la3 lend3 la4;a4=dall lea4 la4 lend4 la1;a6=dall lend1 lend2 lend3 lend4;*defining the volume of the explosivegeomai= (a1 et a2 et a3 et a4 et ae6 et a6) coul bleu;elim 1e-8 geomai;vai = geomai volu;vai1=vai;vex1=vex;*lines for the output of the pressurevges = (vai1 et vex1);nxp1 = pxpdroi1 vges p1 (p1 plus p2) 1e-8;nxp1 = pxordpoi nxp1 p1;nxp2 = pxpdroi1 vges p1 (p1 plus p3) 1e-8;nxp2 = pxordpoi nxp2 p1;REPE I0 (NBNO nxp1-2);p111 = nxp1 poin (&I0+2);p222 = nxp1 poin (&I0+3);p333 = nxp2 poin (&I0+2);p444 = nxp2 poin (&I0+3);x1 y1 z1 = coord p111;x2 y2 z2 = coord p222;x3 y3 z3 = coord p333;x4 y4 z4 = coord p444;yy=((y4-y2)+(y3-y1))/2.;ar=(x2-x1)/yy;mess x1 yy ar;FIN I0;*areas on the sidesasum1 = (apyr1 et ae1 et a1);asum2 = (apyr2 et ae2 et a2);asum3 = (apyr3 et ae3 et a3);asum4 = (apyr4 et ae4 et a4);asum = (asum1 et asum2 et asum3 et asum4);geom_new = (vai1 et vex1 et pyra et nxp1);TASS geom_new;OPTI sauv form 'con22.msh';sauv form geom_new;*list (nbel vai1);list (nbel vex1);list (nbno geom_new);list(mesu(vex));cub.dgibi*Cubical model CUB1 to CUB3opti donn 'D:\Users\larchma\cast3m\pxpdroi1.procedur';opti donn 'D:\Users\larchma\cast3m\pxordpoi.procedur';OPTI echo 1;OPTI dime 3 elem qua4;*****************************************Parameter ***********************************************************************sizex = 0.5; !size of the airdeel = 0.02; !density of the elementssize=0.1; !size of the explosive****************************************DENS deel;p1 = 0 0 0;p2 = (sizex) 0 0;p3 = (sizex) (sizex) 0;p4 = 0 (sizex) 0;79

l1 = p2 d p3 d p4 d p1;l3 = p1 d p2 d p3 d p4 d p1;a1 = surf l3 plane;p10 = 0 0 (sizex);*a2 = l1 tran p10;a3 = a1 plus p10;elim (a2 et a3);l2 = p1 d p2;a4 = l2 tran p10;elim (a4 et a2);p11 = 0 0 (0-sizex);a1 = orie a1 p11;*elim (a1 et a2 et a3 et a4);geom1= (a1 et a2 et a3 et a4) coul roug;opti elem cub8;v1 = (geom1) volu;* Erstellen der Huelle fuer IMPEgeom2 = (geom1 et v1);*Explosiveep1 = p1;ep2 = (size) 0 0;ep3 = (size) (size) 0;ep4 = 0 (size) 0;ep5 = 0 0 (size);el1 = ep1 d ep2 d ep3 d ep4 d;ea1 = surf el1 plane;ve1 = ea1 volu (size/deel) tran ep5;elim 0.001 (v1 et ve1);air1=diff v1 ve1;exp1=ve1;vges= (air1 et ve1);nxpl = pxpdroi1 vges p1 (p1 plus (sizex 0 0)) 0.0005;nxpl = pxordpoi nxpl p1;nxp2 = pxpdroi1 vges p1 (p1 plus (sizex sizex sizex)) 0.0005;nxp2 = pxordpoi nxp2 p1;geom_new = (air1 et geom1 et exp1 et nxpl et nxp2);TASS geom_new;OPTI sauv form 'cubX.msh';sauv form geom_new;list (nbno geom1);list(mesu(exp1));con.epsCUBX - cubical model$ECHOCONV WINCAST geom_newTRID NONL ALEOPTI NF34OPTI TOLC 1e-1$DIMEPT6L X ! CUB1:100000, CUB2, CUB3, CUB4:300000,CUB4:FL38 840000 FL35 1 ZONE 4ECROU 2200000NBLO 100000 NALE 10000NBLE X ! CUB1:20000, CUB2, CUB3:150000,CUB4:230000TERM$GEOMFL38 air1FL38 exp1TERM*GRIL EULE LECT exp1 TERMALE TOUSAUTO AUTR$MATE$ airflut RO 1.3 EINT 2.1978E5 GAMM 1.35 PB 0ITER 1 ALF0 1 BET0 1 KINT 0 AHGF 0 CL 0.5CQ 2.56 PMIN 0 PREF 1.e5 NUM 11a 3.738e11 b 3.747e9 r1 4.15 r2 0.90ros 1630LECT air1 TERM$ explosiveflut ro 1630 eint 3.68e6 gamm 1.35 PB 0ITER 1 ALF0 1 BET0 1 KINT 0 AHGF 0 CL 0.5CQ 2.56 PMIN 0 PREF 1.e5 NUM 11a 3.738e11 b 3.747e9 r1 4.15 r2 0.90d 6930 TDET 0.0 pini 1e5xdet 0.0 ydet 0.0 zdet 0.0LECT exp1 TERM$LINK COUPFSR LECT geom1 TERMECRI DEPL VITE CONT ECRO TFRE 10.E-3FICH ALIC TFRE 2E-6$OPTI NOTE LOG 1$CALC TINI 0 TEND 4.75e-4finexplosive.dgibi*Conical Model for calculations only with the explosiveopti donn 'D:\Users\larchma\cast3m\pxpdroi1.procedur';opti donn 'D:\Users\larchma\cast3m\pxordpoi.procedur';OPTI echo 1;OPTI dime 3 elem cub8;dex=0.124; ! length of the cone**********************************************Parameter *******************************************************************************opex=2e-2; ! openingdexin= 0.001; ! element size in the centerdexfi= 0.01; ! element size at the enddpy=0.001; ! Length of the pyramid element*********************************************DENS 100;oppy=dpy*opex/dex;p1 = 0 0 0;*Points of the pyramidppy2 = (dpy) (oppy) (oppy);ppy3 = (dpy) (0-oppy) (oppy);ppy4 = (dpy) (0-oppy) (0-oppy);ppy5 = (dpy) (oppy) (0-oppy);lpyr1 = p1 d ppy2;80

lpyr2 = p1 d ppy3;lpyr3 = p1 d ppy4;lpyr4 = p1 d ppy5;pyra = manu pyr5 p1 ppy5 ppy4 ppy3 ppy2;*Points of the explosivepe2 = (dex) (opex) (opex);pe3 = (dex) (0-opex) (opex);pe4 = (dex) (0-opex) (0-opex);pe5 = (dex) (opex) (0-opex);*defining the lines between the pyramid and the explosivelpe1 = ppy2 d ppy3;lpe2 = ppy3 d ppy4;lpe3 = ppy4 d ppy5;lpe4 = ppy5 d ppy2;*defining the lines around explosiveDENS dexin;le1 = ppy2 d 'DINI' dexin 'DFIN' dexfi pe2;le2 = ppy3 d 'DINI' dexin 'DFIN' dexfi pe3;le3 = ppy4 d 'DINI' dexin 'DFIN' dexfi pe4;le4 = ppy5 d 'DINI' dexin 'DFIN' dexfi pe5;*defining the lines between the explosive and the airDENS 100.;lea1 = pe2 d pe3;lea2 = pe3 d pe4;lea3 = pe4 d pe5;lea4 = pe5 d pe2;*defining the surfaces around the explosiveae1=dall lpe1 le1 lea1 le2;ae2=dall lpe2 le2 lea2 le3;ae3=dall lpe3 le3 lea3 le4;ae4=dall lpe4 le4 lea4 le1;ae5=dall lpe1 lpe2 lpe3 lpe4;ae6=dall lea1 lea2 lea3 lea4;*defining the surfaces around the pyramideapyr1 = surf (lpyr1 et lpyr2 et lpe1) plane;apyr2 = surf (lpyr2 et lpyr3 et lpe2) plane;apyr3 = surf (lpyr3 et lpyr4 et lpe3) plane;apyr4 = surf (lpyr4 et lpyr1 et lpe4) plane;apyrsum = (apyr1 et apyr2 et apyr3 et apyr4 et ae6);elim 1e-10 apyrsum;*defining the volume of the explosivegeomex= (ae1 et ae2 et ae3 et ae4 et ae5 et ae6) coul roug;elim 1e-10 (geomex);vex = (geomex) volu;vex1=vex;*Points for the controllfp1 = vex1 poin proche (0.1 0 0);pp0 = fp1 et fp1;REPE I0 (NBEL vex1);TEST0 = pp0 INCL (vex1 ELEM CUB8 &I0) 'VOLU';SI ((NBEL TEST0)> 0);quit I0;FINS;FIN I0;MESS &I0;fe1 = vex1 elem CUB8 &I0;*lines for the output of the pressurevges = (vex1);nxpl = pxpdroi1 vges p1 (p1 plus pe2) 2e-5;nxpl = pxordpoi nxpl p1;*areas on the sidesasum1 = (apyr1 et ae1);asum2 = (apyr2 et ae2);asum3 = (apyr3 et ae3);asum4 = (apyr4 et ae4);asum = (asum1 et asum2 et asum3 et asum4);geom_new = (vex1 et pyra et nxpl et fe1 et asum);TASS geom_new;OPTI sauv form 'explosiveX.msh';sauv form geom_new;cv8.dgibi* Construction d'une sphere a partir d'un cube* control volume modelopti donn 'D:\Users\larchma\cast3m\pxpdroi1.procedur';opti donn 'D:\Users\larchma\cast3m\pxordpoi.procedur';opti dime 3 elem cub8;*Nombre de bissectionsnel0 = 30;sizeex = 1.0;sizeai = 4.0;*Cote du cube intermediairer0 = .25;dini = 3.141*sizeex/(4.*nel0);dfin = 3.141*sizeai/(4.*nel0);*Referenceo0 = 0. 0. 0.;x0 = (sizeex) 0. 0.;xa0 = 0 (sizeex) (sizeex);xb0 = (sizeex) (sizeex) (sizeex);xc0 = (sizeex) 0 (sizeex);xd0 = 0 0 (sizeex);y0 = 0. (sizeex) 0.;z0 = 0. 0. (sizeex);x1 = (sizeai) 0. 0.;y1 = 0. (sizeai) 0.;z1 = 0. 0. (sizeai);c0 = x0 plus y0 plus z0 / 2.;c1 = x1 plus y1 plus z1 / 2.;symp1 = (sizeex/2.) (sizeex/2.) (sizeex/2.);symp2 = (sizeai/2.) (sizeai/2.) (sizeai/2.);*Cube intermediaire (centre=o0 et arete=r0)cub0 = (o0 droi nel0 y0 tran nel0 x0)et (o0 droi nel0 z0 tran nel0 y0) et(o0 droi nel0 x0 tran nel0 z0)syme 'POINT' symp1 homo o0 r0;cub2 = (o0 droi nel0 y1 tran nel0 x1)et (o0 droi nel0 z1 tran nel0 y1) et(o0 droi nel0 x1 tran nel0 z1)syme 'POINT' symp2;*Pojection sur la sphere de rayon unitairespe1 = cub0 proj 'CONI' o0 'SPHE' o0 x0;spe2 = cub2 proj 'CONI' o0 'SPHE' o0 x1;a_press = spe1 coul 'BLEU';*Remplissageair1 = spe1 volu 'DINI' dini 'DFIN' dfin spe2 coul bleu;ages = enve air1;a_abso = ages diff a_press;*Points for the controllnxp1 = pxpdroi1 air1 x0 (x0 plus x1) 0.0005;nxp1 = pxordpoi nxp1 x0;81

geom_new = air1 et a_press et nxp1;elim geom_new 1e-8;geom_new = geom_new et a_abso;TASS geom_new;OPTI sauv form 'cv8.msh';sauv form geom_new;cv8.epxcontrol volume test$ECHOCAST 'cv8.msh' geom_newTRID NONL ALEOPTI NF34OPTI TOLC 1e-1OPTI TION 1e-10$DIMEPT3L 300000FL38 400000 CL3Q 6000 ZONE 3ECROU 3000000NBLO 100000NALE 50000 NBLE 5000TERM$GEOMFL38 air1CL3Q a_pressTERM*GRIL EULE LECT air1 TERMALE TOUSAUTO AUTR$MATE$ airflut RO 1.3 EINT 2.1978E5 GAMM 1.35 PB 0ITER 1 ALF0 1 BET0 1 KINT 0 AHGF 0 CL 0.5CQ 2.56 PMIN 0 PREF 1.e5 NUM 1LECT air1 TERMIMPE AIRB X 0 Y 0 Z 0 MASS 5.0 TINT -1.5e-4CONF 3 LECT a_press TERMLINK COUPFSR LECT a_abso TERMECRI DEPL VITE CONT ECRO tfreq 1e-2FICH ALIC TEMP tfreq 1e-8ELEM LECT 1 401 801 9265 35281 TERMFICH ALIC TFRE 1E-4$OPTI NOTELOG 1$CALC TINI 0 TEND 3e-2FINexplosive.epxexplosive1-6$ECHOCAST geom_newTRID NONL ALEOPTI NF34OPTI TOLC 1e-1OPTI TION 1e-10$DIMEPT6L 10000FL38 1300 FL35 1 ZONE 4TABL 100 100ECROU 1000000NBLO 100000NALE 5000 NBLE 5000TERM$GEOMFL38 vex1FL35 pyraTERM*GRIL EULE LECT vex1 pyra TERMALE TOUSAUTO AUTR$MATEflut ro 1630 eint 3.68e6 gamm 1.35 PB 0ITER 1 ALF0 1 BET0 1 KINT 0 AHGF 0 CL 0.5CQ 2.56 PMIN 0 PREF 1.e5 NUM 11a 3.738e11 b 3.747e9 r1 4.15 r2 0.90d 6930 TDET 0.0 pini 1e5xdet 0.0 ydet 0.0 zdet 0.0LECT vex1 pyra TERM$LINK COUPFSR LECT asum1 asum2 asum3 asum4 p1 ae6 TERMECRI DEPL VITE CONT ECRO TFRE 10.E-3FICH ALIC TFRE 1E-6$OPTI NOTE LOG 1$CALC TINI 0 TEND 1.4e-5FINsphe.dgibi*Construction d'une sphere a partir d'un cubeopti donn 'D:\Users\larchma\cast3m\pxpdroi1.procedur';opti donn 'D:\Users\larchma\cast3m\pxordpoi.procedur';opti dime 3 elem cub8;**********************************************Parameter ********************************************************************************Nombre de bissectionsnel0 = X;nerad1 = X;nerad2 = X;sizeai = 1.0; !Length of the model**********************************************Cote du cube intermediairer0 = .5;sizeex = 0.128;*Referenceo0 = 0. 0. 0.;x0 = (sizeex) 0. 0.;y0 = 0. (sizeex) 0.;z0 = 0. 0. (sizeex);x1 = (sizeai) 0. 0.;y1 = 0. (sizeai) 0.;z1 = 0. 0. (sizeai);c0 = x0 plus y0 plus z0 / 2.;c1 = x1 plus y1 plus z1 / 2.;82

*Cube intermediaire (centre=o0 et arete=r0)cub0 = o0 droi nel0 x0 tran nel0 z0 volu tran nel0 y0moin c0 coul bleu homo o0 r0;cub1 = (o0 droi nel0 y0 tran nel0 x0) et(o0 droi nel0 z0 tran nel0 y0) et(o0 droi nel0 x0 tran nel0 z0) moin c0 homo o0 r0;cub2 = (o0 droi nel0 y1 tran nel0 x1) et(o0 droi nel0 z1 tran nel0 y1) et(o0 droi nel0 x1 tran nel0 z1) moin c1 homo o0 r0;*Pojection sur la sphere de rayon unitairespe1 = cub1 proj 'CONI' o0 'SPHE' o0 x0;spe2 = cub2 proj 'CONI' o0 'SPHE' o0 x1;*Remplissagevol1 = cub1 volu nerad1 spe1 coul roug;vol1 = vol1 et (vol1 tour 180 o0 (x0 moin y0)) et cub0;vol2 = spe1 volu nerad2 spe2 coul bleu;vol2 = vol2 et (vol2 tour 180 o0 (x1 moin y1));*Bounding boxxbb = (sizeai*2.) 0. 0.;ybb = 0. (sizeai*2.) 0.;zbb = 0. 0. (sizeai*2);cubbb = o0 droi 1 xbb tran 1 zbb volu tran 1 ybbcoul vert homo o0 r0;par1 = vol1 incl cubbb 'VOLU';par2 = vol2 incl cubbb 'VOLU';elim 1.d-10 (par1 et par2);vges = (par1 et par2);elim vges 1e-10;p0 = 0. 0. 0.;px1 = 0. 0. sizeai;px2 = 0. sizeai 0.;py1 = 0. 0. sizeai;py2 = sizeai 0. 0.;pz1 = sizeai 0. 0.;pz2 = 0. sizeai 0.;symx = vges poin plan p0 px1 px2 1e-6;symy = vges poin plan p0 py1 py2 1e-6;symz = vges poin plan p0 pz1 pz2 1e-6;aimpe = vges poin sphe o0 x1 0.001;air1 = par2;exp1 = par1;nxpl = pxpdroi1 vges p0 (p0 plus (sizeai 0 0)) 0.0005;nxpl = pxordpoi nxpl p0;nxp2 = pxpdroi1 vges p0 (p0 plus (sizeai sizeai sizeai)) 0.0005;nxp2 = pxordpoi nxp2 p0;geom1 = (symx et symy et symz et aimpe);fp1 = air1 poin proche (0.5 0 0);pp0 = fp1 et fp1;REPE I0 (NBEL air1);TEST0 = pp0 INCL (air1 ELEM CUB8 &I0) 'VOLU';SI ((NBEL TEST0)> 0);quit I0;FINS;FIN I0;MESS &I0;fe1 = air1 elem cub8 &I0;elim geom_new 1e-10;TASS geom_new;OPTI sauv form 'dynaX.msh';sauv form geom_new;sphe.epxSPHE2 to SPHE7 - modell armelle$ECHOCONV WINCAST geom_newTRID NONL ALEOPTI NF34OPTI TOLC 1e-1*OPTI PART$DIMEZONE 4PT6L X !SPHE2 5000, SPHE3 10000, SPHE4,5,6,7 100000FL38 X !SPHE2,3 8400, SPHE4,5,6 43000, SPHE7 100000ECRO X !SPHE2 70000, SPHE3 100000, SPHE4,5,6 350000,SPHE7 750000NBLO X !SPHE2,3 1000, SPHE4,5,6 10000NALE X !SPHE2,3 500, SPHE4,5,6 10000, SPHE7 50000NBLE X !SPHE2,3 15000, SPHE4,5,6 60000, SPHE7 600000TERM$GEOMFL38 air1FL38 exp1* CL32 a6TERM*GRIL ALE TOUSAUTO AUTR$MATE$ airflut RO 1.3 EINT 2.1978E5 GAMM 1.35 PB 0ITER 1 ALF0 1 BET0 1 KINT 0 AHGF 0 CL 0.5CQ 2.56 PMIN 0 PREF 1.e5 NUM 11a 3.738e11 b 3.747e9 r1 4.15 r2 0.90ros 1630LECT air1 TERMflut ro 1630 eint 3.68e6 gamm 1.35 PB 0ITER 1 ALF0 1 BET0 1 KINT 0 AHGF 0 CL 0.5CQ 2.56 PMIN 0 PREF 1.e5 NUM 11a 3.738e11 b 3.747e9 r1 4.15 r2 0.90d 6930 TDET 0.0 pini 1e5xdet 0.0 ydet 0.0 zdet 0.0LECT exp1 TERM$LINK COUPFSR LECT geom1 TERMECRI DEPL VITE CONT ECRO TFRE 10.E-3FICH ALIC TFRE 2E-6$OPTI NOTELOG 1$CALC TINI 0 TEND 4.75e-4FINgeom_new = (air1 et exp1 et geom1 et nxpl et nxp2 et fe1);83

European CommissionJoint Research Centre – Institute for the Protection and Security of the CitizenTitle: Simulation of the effects of an air blast waveAuthor: Martin LarcherLuxembourg: Office for Official Publications of the European Communities2007 – 86 pp. – 21.0 x 29.7 cmScientific and Technical Research series – ISSN 1018-5593AbstractThis work is being conducted in the framework of the project RAILPROTECT, which deals with the security andsafety of rail transport against terrorist attacks. The bombing threat is only considered, and focus is placed onpredicting the effects of explosions in railway and metro stations and rolling stock and on assessing thevulnerability of such structures. As the aim of this project is to calculate the behaviour of structures under aloading produced by air blast waves, an indispensable starting point in this study is the ability to simulate thegeneration of such waves from a given quantity of explosive, and to follow their propagation through 3D spacesas they finally impinge onto the structures under consideration. The results of such numerical tests of free airblasts are presented in this report and are compared to experimental data available in the literature. Thisanalysis is preceded by an exposition of some basic concepts on blast wave characteristics, explosives, and adescription of the equation of state adopted herein for the modelling of the detonation of a solid explosive.85

The mission of the JRC is to provide customer-driven scientific and technical supportfor the conception, development, implementation and monitoring of EU policies. As aservice of the European Commission, the JRC functions as a reference centre ofscience and technology for the Union. Close to the policy-making process, it servesthe common interest of the Member States, while being independent of specialinterests, whether private or national.86