Smulaton of Under Water Exploson usng MSC.Dytran Peran Dng rjaan Bujk MSC.Software Corporaton 2300 Traverwood Drve nn rbor, MI 48105 (734) 994-3800 Ths paper descrbes the numercal smulaton of a cylnder submerged under water subjected to exploson usng MSC.Dytran. The cylnder s modeled usng a Lagrangan mesh. Multple Euler domans are used to the ar nsde the cylnder, the surroundng ar, water and the explosve. Snce the model ncludes ar, water and explosve, a multmateral Euler solver s requred. fast general couplng s used to smulate the nteracton between the Lagrangan mesh and Euler mesh. When by the mpact of the shock wave and subsequent gas bubble, the cylndrcal structure deforms, fals and water flow nto the cylnder. INTRODUCTION When a submerged structure subjected to UNDer water EXploson (UNDEX) loadng, t s mportant to predct the structural response to the shock wave. Further more, n the case of the exploson occurrng close to the structure, a hgh velocty water jet penetratng the gas bubble occurs. Ths water jet s extremely effcent n producng damage. Shockwave Pressure 1 st Bubble Pulse 2 nd Bubble Pulse 3 rd Bubble Pulse Tme Fg. 1 Pressure waves and bubble phenomena of UNDEX
Fgure 1 [1] llustrates the pressure-tme hstory, whch s observed n the water at a fxed dstance from the pont of exploson. Upon arrval of the shock wave, the pressure rses nstantaneously to the peak value and decreases at nearly exponental rate. Subsequent to the shock wave, other pressure pulses occur. These pulses arse from a much slower phenomenon, namely the pulsatng of the gas bubble, whch contans the gaseous products of the exploson. The hgh pressure of the gas causes an ntally rapd expanson of the bubble and the nerta of the outward movng water carres t far beyond the pont of pressure equlbrum. The outward moton stops only after the gas pressure has fallen substantally below the ambent pressure. Now the hgher surroundng pressure reverse the moton. gan the flow overshoots the equlbrum and when the bubble reaches ts mnmum sze, the gas s recompressed to a pressure of several hundred atmospheres. t ths pont we have effectvely a second exploson and the whole process s repeated. The bubble oscllates n ths way several tmes. The poston and the sze of the bubble are shown n Fg. 1 for a few specfc moments, whch correspond to the pressure-tme curve as ndcated above. The pressure-tme hstory reflects the low gas pressure durng the phase where the bubble s large and t shows the pressure pulses, whch are emtted from the bubble near ts mnmum. The perod of the bubble pulsatons s very long when compared wth the shock wave porton of the pressure-tme hstory of an exploson. In partcular, ths duraton s long enough for gravty to become effectve. Such a bubble has great buoyancy and, therefore, mgrates upward. However, t does not float up lke a balloon, but shoots up n jumps. In order to be able to predct structural behavor under UNDEX loadng wthout gong nto destructve testng, a lot of fnte element computer analyss have been made. In ths knd of analyss the structural part of the problem has been modeled by a FEM package and the loadng functon of the UNDEX had to be devsed from one- or twodmenson explosve shockwave theory or from experments that were performed wthout a structure. In these approaches the three-dmensonal flud-structure nteracton s not take nto account, whch can lead to erroneous results. Ths s certanly true for near feld UNDEX problems where the nteracton s hghly three-dmensonal and very large deformatons of the structure can occur. MSC.Dytran [2] s a commercal software package for short-term transent analyses that nvolve structural parts and/or computatonal flud dynamcs (CFD) parts. The CFD solver of MSC.Dytran uses a Euleran approach and employs a fnte volume method to dscretze the governng equatons. These equatons are the conservaton laws and are ntegrated n tme by a frst-order explct dynamc procedure. The Euler mesh s statonary n the space whle the flud can flow through the mesh. CFD smulatons provde flud velocty, pressure felds and other varables. In smulatons wth flud-structure nteracton the flud nsde a fnte volume doman s bounded by a surface that represents the nteractng structure. Ths surface s called a couplng surface and enables the flud to exert a force on a deformable structure. MSC.Dytran has been used n UNDEX smulatons snce 1992. By usng MSC.Dytran, Chsum and Shn [3] studed the behavor of the bubbles for the charge type, charge wegh, and hydrostatc pressure. However, only Euler Solver (CFD) was used n ther analyss. Unemoto et al. [4] took nto account the flud-structure nteracton by usng LE technque of MSC.Dytran, whch requres detaled meshng by the user. unque daptve Multple Euler Domans [5, 6] technology had been developed n MSC.Dytran for sngle and mult-materal Euler solvers. Multple Euler domans are automatcally generated around a couplng surface, and each Euler doman automatcally adapts tself when the couplng surface moves and deforms. The Euler materals can be ether hydrodynamc or have shear strength. Materal n- and out-flow can be defned, as well as flow between the Euler domans across porous or open areas n the couplng surfaces. When the structure fals, the Euler materals flow through the holes, faled surfaces and ruptured areas. Ths daptve Multple Euler Domans technology makes effcent modelng of UNDEX usng MSC.Dytran possble. The purpose of ths paper s to demonstrate the applcaton of Multple-materal Euler solver wth daptve Multple Domans to the UNDEX process. The problem smulates a cylnder submerged under water subjected to exploson usng MSC.Dytran. The cylnder s modeled usng a Lagrangan mesh. Multple Euler domans are used to the ar nsde the cylnder, the surroundng ar, water and the explosve. fast general couplng s used to smulate the nteracton between the Lagrangan mesh and Euler mesh. When by the mpact of the shock wave and subsequent gas bubble, the cylndrcal structure deforms, ruptures due to plastc stran falure and water flow nto the cylnder. FORMULTION OF THE PHYSICL PROBLEM In a typcal UNDEX smulaton there are ar, water, explosve and the metal of the cylnder.
The ar s assumed to be deal and to satsfy the equaton of state: p = ( γ 1) ρe (1) Here p, ρ and e are respectvely the pressure, densty and specfc nternal energy and γ s the rato of the heat capactes of the gas. The explosve s modeled as a compressed hot gas n ths smulaton. The water s assumed to be compressble but nvscd and rrotatonal. It s modeled wth a polynomal equaton of state as follows: ρ p = k( 1) (2) ρ Here k s bulk modulus, and 0 are respectvely the overall densty and reference densty. The materal flow s descrbed by the conservaton laws for mass, momentum and energy, that read: d dt d dt V d dt 0 dv + ρ( u n) d = V ρ 0 (3) ρ u dv + ρu ( u n) d = pn d (4) V ρ edv + ρe( u n) d = u pn d (5) Here V s a volume, s the boundary of ths volume, n s the normal vector along the surface, u denotes the velocty vector n the volume. In applyng the conservaton law of mass, mass s transported from one element to the other. Both the donatng element as well as the recevng element can have multple materals. Euler elements can contan up to fve materals n one element. Frst the materals common to both elements are transported out of the donatng element and f no common materal s left n the donatng element other materals are transported as well. Ths approach mnmzes unphyscal mxng and preserves materal nterfaces between ar and water as much as possble. To apply the conservaton laws for Euler elements that are only partally nsde the couplng surface the boundary of that part of an Euler element that s nsde the couplng surface has to be determned. Ths boundary conssts of the nterfaces between Euler elements and the ntersecton of the couplng surface wth the Euler element. These ntersectons are called polpacks. The conservaton laws are appled to that part of the Euler element that s nsde the couplng surface and surface ntegrals are computed by summng across the Euleran nterfaces and the polpacks. Flow and other communcaton from one Euler mesh to the other take place through porous shell elements, that are common to both couplng surfaces [6]. The pressure computaton n elements that contan a mxture of ar and water s based on one of the thermodynamc equlbrum prncples that amounts to pressure equlbrum. In an Euler element wth both ar and water there s a dstnct pressure nsde the ar and a dstnct pressure nsde the water. lthough masses are fxed durng the pressure computaton the volume of the ar and the volume of the water are not fxed and they are adjusted teratvely untl the pressure n the water equals the pressure n the ar. The cylnder conssts of shell elements that deform under stresses and support falure models. n explct fnte element solver solves the shell dynamcs, and an explct Euler solver models the materal nsde and outsde the cylnder. The nteracton between these two solvers takes place n two steps: (1) The mass n the Euler elements exerts a pressure load on the cylnder surface. These loads consttute the boundary condtons for the fnte element solver, resultng n new grdpont acceleratons and veloctes for the cylnder. From the updated plastc stran or updated stresses of the shell elements t s determned whch elements are falng. Fnally the cylnder grd ponts are moved wth the new veloctes. (2) The cylnder grd ponts move and so the Euler mesh has a new boundary. Consequently, the volume of mass n each element may change. Snce densty s mass dvded by the volume of the mass, denstes also change, and so pressure. NUMERICL MODELS The relatve poston of the cylnder, water surface and explosve s shown n Fg. 2.
r Water 1.5 m Cylnder 1 m Explosves 1.5 m Fg. 2 Postons of cylnder, water surface and explosve To model the flud nsde and outsde of the cylnder, two Euler domans are used. The outer doman has the cylnder surface as part of the boundary, materal s outsde the cylnder surface and there s no materal nsde the cylnder surface. The contents nsde the cylnder are modeled n the nner doman and ths doman s also enclosed by the cylnder surface. Materal of the nner doman s nsde the cylnder surface and there s no materal outsde the cylnder surface. Therefore both Euler domans use the cylnder surface as part of ther enclosure. The outer Euler doman and ts enclosng surface are shown n Fg. 3. The outer boundary of the outer doman s gven by a suffcently large fxed box. Pressure at the outer boundary s set to the hydrostatc pressure. Ths behaves lke open boundary. The Euler mesh contans the water and the ar on the top of the water. The densty of water s 1000 Kg/ m 3. The bulk modulus s taken as 2.2E9 Pa. Water hydrostatc pressure s defned startng from 1.0E5 Pa at the surface and ncreasng gong down. mnmum pressure of zero s defned for the water, so that f a porton of water got a negatve pressure, all of the water would flow out of that regon and a vod would be created. The densty of ar s 1.1848 Kg/ m 3. The rato of the heat capactes of the gas s constant as 1.4. Specfc nternal energy s taken as 2.14E5 Kg-m 2 /s 2. Intal ar pressure s set to 1.0E5 Pa. Outer Euler doman: Large fxed box wth open boundary Inner Euler doman r Cylnder surface r Water Cylnder surface Fg. 3 Outer Euler doman and ts enclosng surface Fg. 4 Inner Euler doman and ts enclosng surface
The explosve TNT s created n ths Euler mesh, too. The densty of the explosve s 1700 Kg/ m 3 and the mass s 0.445 Kg. The specfc nternal energy s 4.765E6 Kg-m 2 /s 2. The explosve can be modeled by a JWL or IG (Ignton and Growth) equaton of state n MSC.Dytran. However f we assume that the explosve s a ball, the radus of the ball s only 0.04 m. fner mesh has to be created to smulate ths small ball. In ths smulaton the explosve s defned as a compressed hot gas (γ=1.4). The mass and specfc nternal energy are those of the explosve charge. The radus of the gas ball s taken as 0.1m and the densty s adjusted to 105 Kg/ m 3 to keep equvalent mass of the explosve. Intal ar pressure s calculated from Eq. (1) to be 2.0E8 Pa. The nner Euler doman s shown n Fg. 4. The surface presents the outer boundary of the doman. The nner doman s ntalzed by ar. The outer and the nner domans have meshes that do not concde. The element sze for each doman s the same as 0.1 m n ths smulaton. Snce the purpose of ths paper s just to demonstrate the new technology n MSC.Dytran, the model has been 0.1 s 0.2 s 0.285 s 0.38 s 0.45 s 0.5 s Fg. 5 Isosurfaces of materal fractons
smplfed and no detals of the cylnder are modeled. The cylnder s modeled wth Lagrangan shell elements ncorporatng both a plastcty model as well as a falure model. It s 0.6 m long wth dameter of 1.0 m. The end covers are modeled as rgd bodes havng the approprate mass and center of gravty. Once any of these elements exceeds some falure crteron t fals. Snce the boundary of the fnte volume doman s provded by the shell elements of the cylnder, once shell element fal flow takes place between the nner Euleran doman and the outer Euleran doman. Gravty load s appled to the whole model. 0.1 s 0.2 s 0.285 s 0.38 s 0.45 s 0.5 s Fg. 6a Effectve stress plotted on the deformed shapes of the cylnder
0.1 s 0.2 s 0.285 s 0.38 s 0.45 s 0.5 s Fg. 6b Effectve plastc stran plotted on the deformed shapes of the cylnder
RESULTS ND DISCUSSION The smulaton s carred out 0.5 seconds for the duraton of the formaton and collapse of the frst and second bubble untl begnnng of formaton of the thrd bubble. The CPU tme to run ths model s 34 hours on a Wndows 2000 3.06GHz. Plots of the water bubble shapes are shown n Fg. 5. The frst bubble came n touch wth the structure and collapsed on causng a bubble jet. Damages on the cylnder caused by ths jet are observed. Then the second bubble collapsed on the cylnder and the formed bubble jet enlarge the damages. The effectve stress and plastc stran are plotted on the deformed and damaged shapes of the cylnder as shown n Fg. 6. Fgure 7 shows velocty dstrbuton of the flud. The frst and second bubble jets emtted near the bubble mnmum radus are clearly seen n the plot at 0.2 s and 0.38 s, respectvely. 0.2 s 0.38 s Fg. 7 Velocty dstrbuton of flud Sne the explosve s defned as a compressed hot gas, t may affect the shock wave characterstcs. However, t models the bubble formaton and collapse correctly together wth bubble pulse loadng. Wth regard to the purpose of ths smulaton, no attempt s made to study effect of the shock wave on the structure, whch s also very mportant. However, the method presented does not put any restrcton on ths knd of smulaton. CONCLUSION It s now possble to accurately smulate UNDEX, usng the Mult-materal Euler Solver of MSC.Dytran. The unque daptve Multple Euler Domans technology makes t possble to model effcently. lthough expermental data s not avalable, the smulaton results confrm expectatons. CKNOWLEDGMENTS Fgure 5-7 s made wth CEI-Ensght. The authors apprecate Wolter van der Veen and Paul Bus for ther valuable dscusson and suggeston.
REFERENCES 1. Swsdak, M. M., 1978, Exploson effects and propertes: PRT II - Exploson effect n water, Naval Surface Weapons Center. 2. lnk to the webste of MSC.Dytran can be found at the followng URL: http://www.mscsoftware.com 3. Chsum, J.E. and Shn, Y.S., Exploson Gas Bubble Near Smple Boundares, Shock and Vbraton, 1997, Vol. 4, No. 1, pp. 11-25. 4. Umemoto, K., Sakaue, H. and Gefken, P., Comparson of Numercal Smulaton by a Multmateral LE Flud-structure Interacton nalyss Code wth Interface Pressure Measurements, the 70 th Shock and Vbraton Symposum, 1999. 4. van der Veen, W.., Smulaton of a compartmented arbag deployment usng an explct, coupled Euler/Lagrange method wth adaptve Euler domans, NFEMS, March 2003, Florda. 5. van der Veen, W.., Crushng smulaton of a partally flled Fuel Tank beyond falure, 2005 SE World Congress, prl 2005, Detrot. (to appear) 6. Hrsch, C., 1990, Numercal Computaton of Internal and External Flows, John Wley&Sons.