Advanced Eulerian Techniques for the Numerical Simulation of Impact and Penetration using AUTODYN-3D

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1 9 th Intenational Symposium on Inteaction of the Effects of Munitions with Stuctues Belin, 3-7 May 1999 Advanced Euleian Techniques fo the Numeical Simulation of Impact and Penetation using AUTODYN-3D Gabi Luttwak 1 Rafael, Box 2250, Haifa 31021, Isael Malcolm S. Cowle Centuy Dynamics Inc., 2333 San Ramon alley Blvd, Suite 185, San Ramon, CA Tel: , Fax: , msc@autodyn.com Abstact This pape descibes two new capabilities that have been implemented in the non-linea dynamics pogam AUTODYN-3D. A second ode Eule pocesso using a Godunov solve, thee dimensional inteface econstuction, and dynamic memoy management. A Lagange to Eule emap. The numeical simulation of the fomation and penetation of an explosively fomed pojectile is used to demonstate these new capabilities. 1.0 Intoduction Impacts and penetations typically poduce lage mateial flow and defomation. In a Lagangian calculation the gid moves with the mateial velocity. This can cause sevee gid distotions of the Lagangian gid. In many cases Euleian coodinates ae a natual way to handle such poblems. In the past, howeve, seveal dawbacks have pevented the efficient utilization of the multi-mateial Euleian technique: The advection of mateial though an Eule gid can geneate numeical diffusion, which atificially smoothes gadients in the calculation. It is difficult to adequately follow mateial intefaces, which cut though the Eule cells, especially if the mateial flow is in a diection diagonal to the gid. Most impotantly, an Eule gid must cove egions not only whee mateial initially exists, but also egions into which mateial is expected to flow duing a calculation. Fo thee-dimensional calculations the esulting memoy equiements ae usually so pohibitive, that only coase-gid calculations can be computed.. A new second ode Eule pocesso has been implemented in the non-linea dynamic analysis pogam AUTODYN- 3D. This pocesso employs novel techniques to emedy the above deficiencies: A second ode Godunov scheme [1] is used to decease numeical diffusion. Multi-mateial intefaces ae followed using a fast and obust algoithm simila to Young s technique [2]. A unique dynamic memoy-management scheme allows almost no memoy to be used fo empty (void) cells. Thus, an Euleian calculation utilizes only slightly moe memoy esouces than an equivalent Lagangian calculation. Despite poblems with gid distotions, the Lagangian appoach still has seveal advantages ove Eule, paticulaly in modeling complex o thin mateial layes and mateial stength and failue. Theefoe, it is often pefeable to pefom the ealy stages of the calculation using a Lagangian gid, while gid distotions ae easonable. An option 1 Wok pefomed at Centuy Dynamics, while on sabbatical leave fom Rafael

2 has been developed fo AUTODYN-3D, which allows such calculations to be emapped into a multi-mateial Eule gid (conseving mass, momentum and enegy) when distotions become too sevee. An example illustates these capabilities. The fomation of an Explosively Fomed Pojectile (EFP) is computed using Lagangian coodinates. The esults of this calculation ae then emapped into an Euleian gid, and the impact of the EFP into a taget is computed. 2.0 Theoy To obtain the diffeential equations of motion, the consevation laws of mass momentum and enegy ae applied to an abitay small closed contol volume, : d dt d dt d dt ρdv = ρudv = ρe dv = T ρ( u u ) ds ρu g ( u u ) g ds + σ ds ρuet ( u ug ) ds + ( σ u ) ds The contol volume may be moving with an abitay speed, u g. In Euleian coodinates, the gid is fixed, u g =0. In Lagangian coodinates, the gid moves with the mateial velocity, u g = u. The total enegy e T is expessed as the sum of intenal enegy e and kinetic enegy. The stess tenso is decomposed into a hydostatic pessue p, and stess deviatos s. The pessue is obtained fom the equation of state, while the stess deviatos ae deived fom mateial constitutive elations: 2 e u T = e + 2 ; p = p( ρ, e) ; σ = pi + s Fo compessible flow, the esulting equations of motions ae hypebolic in chaacte. Duing the solution of these equations, shocks and contact discontinuities can be geneated, acoss which the diffeential equations do not hold. In such situations, applying the consevation equations acoss the discontinuity yields the Rankine-Hugoniot elations. The diffeence equations ae obtained by applying the integal consevation equations diectly ove the computational cell. In this way, the esulting equations ae locally consevative. Howeve, like the diffeential equations, these diffeence equations ae not valid at discontinuities. One solution to this poblem is to explicitly follow each shock pesent in a calculation and solve the Rankine-Hugoniot elations acoss the shock font. This technique is called shock fitting. Unfotunately, the complex fomations and inteactions of shocks that typically develop in multidimensional solutions make this technique fa too complicated and impactical to implement fo geneal-pupose calculations. An altenative, moe pactical appoach, is to modify the diffeence equations to captue the shocks. The most popula of these shock captue techniques is on Neumann s Pseudoviscosity [3]. With this technique, a tem is added to the diffeential equations, which causes gadients at shocks to be spead ove 2-3 computational cells. This method is usually used in fist ode solutions as esults tend to be somewhat noisy nea discontinuities, educing the accuacy of calculations. In the second ode Godunov scheme, an algoithm that coectly captues shocks without using atificial damping tems is used. 2.1 The second ode Godunov scheme In a fist ode diffeence scheme, vaiables ae assumed to be constant within a computational cell. With this assumption, thee will commonly be a discontinuity at the bounday between two neighboing cells. In gas-dynamics, the solution at a discontinuity between two constant states is known as the Riemann poblem. Godunov [4] poposed that the Riemann poblem be solved to obtain face values of vaiables. The esulting face values ae then used to integate the consevation equations ove the contol volume. In fist ode solutions, this technique diffuses gadients moe than the on-neumann pseudo-viscosity method. To ovecome this poblem, an Lee [1] extended Godunov s method to second ode. This technique has been implemented in AUTODYN-3D. To obtain second ode accuacy, each vaiable is assumed to vay linealy ove a cell. This equies caying not only a cell-centeed value fo each vaiable, but also its gadient. The gadient is used to extapolate face values of vaiables fom the cell-centeed values. In geneal, the extapolated face values ae diffeent on each side of a face, so an appoximate Riemann poblem is solved to advance the solution at each face to the middle of the time step.

3 To maintain stability, the advection tems must be diffusive up to the tuncation eo. Consequently a second ode tanspot scheme is much less diffusive than the equivalent fist ode solution. Howeve, in the neighbohood of shocks, whee the gadients ae infinite, a highe ode scheme may pefom wose. Such schemes can also atificially ceate new extema, o enhance existing extema in the distibution of the vaiables. To pevent such non-monotonic behavio, an Lee poposed a slope limiting technique, wheeby the density gadient (slope) in a cell is limited, so that the extapolated values of the density at the neighboing cell centes emain bounded by the values of the densities in the neighboing cells. The second ode Godunov scheme implemented in AUTODYN-3D closely follows the wok of Hancock [5]. Thus, gadients ae set to zeo at shocks and discontinuities, and nea fee sufaces and mateial intefaces (i.e. in these aeas, the solution evets to fist ode). The consevation equations ae integated ove the computational cells in thee-dimensional space (athe than caying out one-dimensional diection splitting which, while technically simple and fomally second ode accuate, is touble pone). Hancock s pesciptions fo slope-limiting ae used. Fo elasticplastic flow, the slopes of the stess deviatos ae not computed, so thei advection is only fist ode accuate (fo typical poblems involving high-speed impact and detonation, whee the stess deviatos account fo less than 10% of the pessue, this appoach can be justified). The Riemann solve uses a two-shock appoximation. Since the shock Hugoniot fo weak waves is tangent to the isentope, this appoximation holds easonably well eveywhee, except nea stong aefactions. In compession, and locally, ove the time step, a linea elationship is assumed between the shock speed, U s, and the mateial velocity, u: U s = c 0 + Fo most solids, such a elation is expeimentally obseved ove a lage ange of pessues. Fo expansions, U s is taken to be equal to the sound speed c 0. This appoach does not limit the equation of state used. The ole of the Riemann solve is somewhat simila to the atificial viscosity used in fist ode codes. 2.2 Multi-mateial inteface tacking In an Euleian calculation the gid emains fixed in space, and mateials flow though it. This means that mateial intefaces may cut though cells. One of the most difficult tasks in a multi-mateial Euleian scheme is to adequately follow these mateial intefaces. A numbe of diffeent appoaches have been used to addess this poblem. Some solutions (e.g. Paticle In Cell (PIC) [6] ) use Lagangian paticles to tack mateial intefaces. These paticles move with the mateial and consequently suffe fom distotion poblems simila to those chaacteistic of Lagangian gids. A ecent altenative is the Level Set [7] method in which the inteface is located at some constant value of a function defined at the cell vetices. Typically the function may be chosen as the distance to the inteface. By advancing the values of this function in time, the inteface position can be detemined. Unlike Lagange paticle methods, the Level Set method can deal easily with changes in the topology of the intefaces. Howeve, the location of intefaces obtained using eithe of these two appoaches may not be consistent with the patial volume and mass pesent in a cell, as computed by the advection tems of the consevation equations. Because of these limitations, the following olume of Fluid method (OF) [15] has been implemented in AUTODYN. Mateial inteface locations ae needed to compute the faction of each mateial that is tanspoted acoss a cell face duing the advective phase. OF methods detemine the location of any mateial intefaces within a dono cell by analyzing the volume factions of mateials in neighboing cells, essentially pefoming an inteface econstuction. Simple pefeential tanspot [8] and dono-accepto schemes use only the two cells adjacent to the face (i.e. a onedimensional econstuction) to detemine the location of mateial intefaces. SLIC (Simple Line Inteface Calculation) [9] is a futhe impovement of this appoach in which the ode of fluxing mateials is chosen to be dependent on thei pesence in thee cells (the dono cell, the accepto cell and the cell behind the dono cell). Robust and efficient, SLIC gives good esults if the flow is nomal to the gid (mateial intefaces ae always constucted nomal to the gid), but can cause lage eos when the flow is diagonal though the gid. su

4 SLIC Multidimensional Reconstuction This poblem is lagely ovecome by using a moe complex multidimensional inteface econstuction [2,10-12], whee the shape of the inteface cutting the cell is assumed to be plana, but not necessaily nomal to the gid. If the nomal diection to this plane is known, the position of the mateial inteface can be exactly detemined fom the volume a, of one of the mateials in the cell [12]. The nomal is assumed to be in the diection of the gadient of the elative patial volume, ( a /). The gadient is computed using the values of ( a /) in the dono cell and all its neighbos [2]. This high-esolution algoithm is ideal fo a well-behaved inteface whee at least one neighbo to the dono cell is full of mateial and one is empty. If this is not the case (e.g. if thee is a single dop of mateial in a cell), the nomal to the inteface is not well defined, in which case SLIC is used. SLIC is also eveted to if thee ae moe than two mateials (including voids) in a dono cell. 2.3 Dynamic memoy management A Lagangian gid defoms with the mateials it is modeling, wheeas an Euleian gid is fixed in space and the mateials flow though it. Consequently an Eule gid must span all the space eached by any of its mateials duing the calculation. Fo poblems in which mateials move many times thei chaacteistic lengths, the numbe of Euleian cells equied will fa exceed the numbe of Lagangian cells necessay to descibe the same poblem. Fo thee-dimensional poblems this atio may well be moe than an ode of magnitude. To account fo the advection vaiables, an Euleian cell also equies about twice the stoage of a Lagangian cell. Consequently, memoy equiements fo unning many Euleian calculations with adequate gid esolution exceed the available esouces. To ovecome these difficulties AUTODYN-3D takes advantage of moden compile technology in using a novel dynamic memoy-management scheme. AUTODYN-3D, esion 4, has been completely ewitten in Fotan 90, allowing a simple and moe tanspaent use of pointes. Each cell in an Eule gid has a pointe, pointing to a location in memoy whee gid and mateial vaiables fo the cell ae stoed (gid coodinates ae not included in these data, they ae stoed in sepaate aays ). Undestanding that empty (void) cells equie no gid o mateial data, the pointes fo all empty cells point to a common location in memoy containing dummy data fo void cells. Each time a cell is emptied, the block of memoy pointed to fo that cell is eleased and the pointe is ediected to the common memoy location fo void cells. When a peviously empty cell has mateial flow into it, a new block of memoy is assigned fo its gid and mateial vaiables and the pointe fo the cell (peviously pointing to the common memoy location) is ediected to the newly assigned memoy. A simila technique is used fo multi-mateial cells. As mateials flow in and out of cells, the amount of memoy pointed to is adjusted accoding to the numbe of mateials in each cell. Fo efficiency, system memoy is allocated as equied in lage supeblocks. Released blocks ae not etuned to the system but eused, using stack logic. Also, a list is kept of the non-empty cells, so that most computations ae not caied out fo empty cells. 2.4 Lagange to Eule Remap The advanced multi-mateial Euleian techniques descibed above impove the efficiency and the quality of Euleian calculations. Howeve, if gid distotions ae small, a Lagangian calculation still has some definite advantages ove Eule. In Eule, the advection phase poses a numbe of poblems. The advection tems have to be computed, and this consumes both memoy and CPU esouces. Even using a second ode scheme, the advective phase of an Eule calculation adds moe diffusion to solutions compaed to Lagange. Cetain mateial popeties cannot be advected with a high degee of accuacy in Eule. It is moe efficient to follow a Lagangian, body-fitted bounday, than a geneal multi-mateial inteface, which cuts though Euleian cells. It is difficult to tack thin layes of mateial moving though an Eule gid with good accuacy Fo these easons, it is pefeable to use a Lagangian appoach wheneve pactical. The Lagange to Eule emap algoithm in AUTODYN-3D allows this to be done easily. Often one wants to use a Lagangian solution in the ealy stages of a calculation befoe gid distotions begin to cause seious poblems, then switch to an Euleian solution fo the late stages. The emap algoithm actually seves a dual pupose, as it can also be used to set up and initialize multi-mateial Eule egions at the stat of a calculation. Geneating a Catesian Eule gid is faily staightfowad, but filling the gid with vaious shaped stuctues can be vey difficult. Simple standad fills, such as blocks, cylindes o sphees, can eadily be geneated, but custom fitted shapes ae usually much moe difficult. If such shapes ae geneated and filled using body-fitting Lagange gids (a much simple task), these gids can then be immediately emapped into an Eule gid.

5 The emap algoithm is based on applying the consevation laws of mass, momentum and enegy. As this emap is applied only once in a calculation, a fist ode scheme is used that is an extension of the 2D algoithm used by Hancock [5,13] (the genealization to second ode is staightfowad [14] ). Fist AUTODYN computes the common volume δ ij, between each Lagangian cell i, and evey Euleian cell j. The Lagangian cells in AUTODYN ae hexahedons, but the scheme woks fo any polyhedon. The density ρ j in the Euleian cell j is computed as: ρ j 1 = ρiδ j i ij Simila elations apply fo momentum and enegy totals. Remapping of vaiables is diffusive (this is equied fo stability), so consevation of momentum will decease kinetic enegy, Consequently, peseving total enegy esults in inceasing intenal enegy. To pevent unphysical heating in egions whee the kinetic enegy is lage, the intenal enegy is peseved sepaately. A volume-weighted aveage is used fo vaiables that do not have a simple consevation law (e.g. the stess deviatos). 3.0 Example: An Explosively Fomed Pojectile (EFP) To demonstate the new capabilities descibed in this pape, the fomation of an explosively fomed penetato (EFP) and its oblique impact on a taget plate has been computed using combined Lagange and Eule simulations. 3.1 EFP Fomation The fomation of an EFP is chaacteized by a elatively smooth defomation pocess that can be easily handled in Lagange coodinates, so a Lagangian calculation is pefomed fo this phase. Figue 1 shows the initial chage configuation. Figue 1: The initial chage configuation The explosive loading of a thin coppe line foms the EFP. As the line is loaded, its initial shape causes it to convege inwads towads its axis, foming a compact, high-speed pojectile that can impact tagets at lage stand-off distances. To pefom such a calculation in Eule would be much less efficient, because it would be necessay to esolve the thin line as it moved though the Eule gid. AUTODYN s Lagangian pocesso can easily handle the

6 explosive-line inteaction and line defomation that occus. The steel casing suounding the explosive is not axially symmetic, so a thee-dimensional analysis is equied. Howeve, thee is a symmety plane at y=0, so only half the geomety is modeled. Figue 2 shows the shape of the line afte fomation, when it has eached the stand-off of the taget plate. Figue 2: The line (EFP) afte fomation By this time, the loading of the line by the detonation poducts is complete, so the explosive and casing ae emoved fom the calculation, leaving only the EFP gid. 3.2 EFP Penetation The next step is to compute the impact and the penetation of the EFP into a taget whee both pojectile and taget ae expected to undego lage defomations. This type of poblem is most commonly solved in Euleian coodinates. To achieve this, a Catesian Eule gid is geneated to cove the egion occupied by the pojectile and the taget, and egions into which they may be expected to move duing the calculation. The EFP is then emapped fom the Lagange gid into the Eule gid. Figue 3 shows the Lagange EFP gid supeimposed in the Eule gid befoe the emap. Figue 4 shows the EFP emapped into the Eule gid. Figue 3: Lagange EFP and Eule gid befoe emap Figue 4: EFP emapped into the Eule gid

7 A taget plate is geneated in a simila way. The plate, which is at 45 degees to the pojectile, is fist geneated as a Lagangian gid and then its contents emapped into the Catesian Eule gid. Figue 5 shows the Eule gid with both the pojectile and taget plate emapped into it. Figue 6 shows the mateial suface contous in the Eule gid. Figue 5: Pojectile and taget emapped into Eule Figue 6: Mateial contou plot afte the emap Notice that in figue 5, the majoity of the cells in the Eule gid ae empty. The dynamic memoy-management scheme implemented in AUTODYN-3D geatly educes memoy equiements in this situation, allowing the calculation to be caied out on a PC with only 64 Mb of memoy Figue 8 shows the Eule gid at two diffeent times duing the penetation calculation. Figue 9 shows mateial contou plots at the same two times. Figue 8: The Eule gid at two diffeent times duing the penetation calculation Figue 9: Mateial contou plots at two diffeent times duing the penetation calculation

8 4. Conclusions This pape has descibed and illustated some novel Euleian techniques implemented in the thee-dimensional code AUTODYN-3D. They include a second ode Godunov pocesso with a high-esolution multi-mateial inteface tacking algoithm and efficient dynamic memoy-management scheme, and a Lagange to Eule emap capability. An example has demonstated how, togethe, these new featues can alleviate taditional dawbacks associated with Euleian techniques. In paticula, dynamic memoy management pemits vitual Euleian gids to be set up that equie much less memoy than conventional Eule gids, and the emap capability allows the combined advantages of Lagange and Eule techniques to be bought to bea in solving complex inteaction poblems. REFERENCES [1] B. van Lee, J.Comp.Phys, 32, p101, (1979) [2] D.L.Youngs, An Inteface Tacking Method fo a 3D Hydodynamics, AWRE , (1987) [3] J. on Neumann and Richtmye, J. Appl. Phys., 21, p232, (1950) [4] S. K. Godunov, Mat.Sb., 47, p271, (1959) [5] S. L. Hancock, PISCES 2DELK Theoetical Manual, Physics Int. Co., (1985) [6] F.H. Halow, Poc. Symp. Appl. Math., 15, p269, (1963) [7] S. Oshe, J. A. Sethian, J.Comp.Phys., 79, p12, (1988) [8] W. E. Johnson Development and Application of Pogams Related to Hypevelocity Impacts, Systems Science, Softwae ep. 3SR-353, (1970) [9] W. F. Noh and P.R.Woodwad, SLIC,Simple line Inteface Calculation, in the Poc. of the 5 th Int. Symp. on the Num. Meth. in Fluid Dynamics, held at Enschede, The Nethelands,(1976), A. I. an De ooen et al. ed., Spinge elag, (1976) [10] R. De Ba The KRAKEN code, in L.L.N.L ep. UCID_17366,(1974) o A Method in Two-D Euleian Hydodynamics, UCID-19683, LLNL(1974) [11] D. L. Youngs, Time dependent multi-mateial flow with lage fluid distotion, in Num. Meth. in Fluid Dynamics, ed.k. W. Moton, B. W. Baines, Academic Pess, (1982) [12] G. Luttwak, Numeical Simulation of wate jet penetation, p191, Shock Waves in Condensed Matte-1983, ed.j.r.asay et al., Elsevie Science Publishes B.., (1984) [13] B. J. Thone, D. B. Holdidge, The TOOREZ Lagangian ezoning code, Sandia N.L. ep. SLA , (1974), see appendix by D.S.Mason. [14] G.Luttwak, Second Ode Discete Rezoning, p1777 in High Pessue Science and Technology-1993, Schmidt et al. ed., AIP-Pess (1994) [15] C.W.Hit,B.D.Nichols, J.Comp.Phys., 39, p201, (1981)