Interaction Methods for the SPH Parts (Multiphase Flows, Solid Bodies) in LS-DYNA

Transcription

1 13 th Internatonal LS-DYNA Users Conference Sesson: Flud Structure Interacton Interacton Methods for the SPH Parts (Multphase Flows, Sold Bodes) n LS-DYNA Jngxao Xu, Jason Wang Lvermore Software Technology Corporaton Abstract The nterestng and complex behavour of fluds emerges manly from nteracton processes. Smooth partcles hydrodynamcs s a meshfree, Lagrangan partcle method and a smple, yet flexble method for modelng flud flows and sold bodes n a robust way. It has been appled extensvely to the multphase flows, heat conducton, hgh explosve problems and so on. In ths paper, dfferent nteracton methods avalable n the LS-DYNA for SPH parts whch have wde range of densty and materal propertes are studed and compared. Node to node contacts ft well for the nteracton between two SPH parts wth hgh densty rato, the standard SPH nterpolaton method has better accuracy around the nterfaces when two SPH parts have smlar densty and materal propertes. Dfferent nteracton approaches can be combned together n one model to reach the best results. Also the nteractons between Lagrangan elements wth SPH partcles are dscussed. Some examples are presented to show how to use dfferent approaches wth dfferent combnaton of LS-DYNA keywords. Introducton SPH s a Lagrangan method for solvng partal dfferental equatons. Essentally, the doman s dscretzed by approxmatng t by a seres of roughly equ-spaced partcles. They move and change ther propertes (such as temperature) n accordance wth a set of ordnary dfferental equatons derved from the orgnal governng PDEs. SPH was frst appled by Lucy (1977) to astrophyscal problems, and was extended by Gngold (1982). Cloutman (1991) used SPH to model hypervelocty mpacts. Lbersky and Petschk have shown that SPH can be used to model materals wth strength. In recent years t has been developed as a method for ncompressble sothermal enclosed flows by Monaghan (1994). As a Lagrangan method, the nteracton between SPH partcles and FEM elements can be easly handled by a normal node to surface contact n LS-DYNA. Because of ts ablty to handle large dstortons by avodng the need for ntensve FEM remeshng, ts reasonable precson and stablty compared wth classcal methods FEM, SPH s a compettve approach compared to fnte elements (FE) and s ncreasngly beng used n some fast-transent dynamcs problems. Several authors have proposed to couple FE and SPH whch seems a reasonable approach n order to beneft from the advantages of both formulatons. In LS-DYNA, hybrd elements that enable couplng effects between SPH partcles and FEM sold are mplemented. In ths method, hybrd elements are confgured to facltate couplng effect of sold element and smoothed partcle hydrodynamcs (SPH). Ths method can be used to adaptvely transform a Lagrangan sold Part or Part Set to SPH partcles. Also hybrd elements are defned n a computer aded engneerng (CAE) grd model as a buffer or nterface between the SPH partcles and FEM solds 1-1

2 Sesson: Flud Structure Interacton 13 th Internatonal LS-DYNA Users Conference When smulatng fluds, t s mportant to capture nteracton effects accurately n order to reproduce real world behavor. Smoothed Partcle Hydrodynamcs has shown to be a smple, yet flexble method to cope wth many flud smulaton problems n a robust way. Unfortunately, the results obtaned when usng SPH to smulate mscble fluds are severely affected, especally f densty ratos become large. In SPH, partcles have a spatal dstance covered by smooth length over whch ther propertes are smoothed by a kernel functon. Problems arse when rest denstes and masses of neghborng partcles vary wthn the smoothng length, as n such cases the smoothed quanttes of a partcle show falsfed values. The undesrable effects reach from unphyscal densty and pressure varatons to spurous and unnatural nterface tensons, as well as severe numercal nstabltes. A node to node penalty based contact was ntroduced to avod those nterface effects n LS-DYNA. We have couples of keyword optons avalable n LS-DYNA for the nteractons between SPH parts also the nteractons between SPH part and Sold part: Standard SPH nterpolaton method (normal way); Node to node penalty based contact through keyword *DEFINE_SPH_TO_SPH_COUPLING; Combnaton of both method through keyword *SECTION_SPH_INTERACTION; Node to surface contact between SPH part and Sold part; Couplng between SPH parts and sold parts through keyword *DEFINE_ADAPTIVE_SOLID_TO_SPH; Couplng between SPH partcles wth ALE elements through keyword *ALE_COUPLING_NODAL_PENALTY. Dfferent nteracton approaches can be combned together n one model to reach the best results. Some examples are demonstrated to show how to use dfferent approaches wth dfferent combnaton of LS-DYNA keywords. Standard SPH formulaton Fundamentals of the SPH method Partcles methods are based on quadrature formulas on movng partcles ( x ( t), w ( t)) P, P s the set of the partcles. x (t) s the locaton of partcle and w (t) s the weght of the partcle. The quadrature formulaton for a functon can be wrtten as: The quadrature formulaton (1) together wth the defnton of smoothng kernel leads to the defnton of the partcle approxmaton of a functon. The nterpolated value of a functon: u(x ) at poston X usng the SPH method s: Where the sum s over all partcles nsde and wthn a radus 2h, W s a splne based nterpolaton kernel of radus 2h. It mmcs the shape of a delta functon but wthout the nfnte 2 tals. It s a C functon. The kernel functon s defned as followng: 1-2 h f ( x) dx w ( t) f ( x ( t)) P ( u ( x )) w ( t) u( x ) W ( x x, h) W ( x x 1 x x, h) h h( x, y) (1) (2) (3)

3 13 th Internatonal LS-DYNA Users Conference Sesson: Flud Structure Interacton W ( x x, h) when h 0, s Drac functon, h s a functon of x and the so-called smoothng length of the kernel. And the cubc B-splne functon s defned: x and s d d ( d) C (2 d) when 0 d 1 when 1 d 2 elsewhere (4) The gradent of the functon smoothng length: u(x ) s gven by applyng the operator of dervaton on the h ( u ( x ) w u( x ) W ( x x, h) Evaluatng an nterpolated product of two functons s gven by the product of ther nterpolated values. (5) Fg 1. Support sze of 2d kennel functon Contnuty equaton and Momentum equaton The partcle approxmaton of contnuty equaton s defned as: d dt m v v W ' (6) 1-3

4 Sesson: Flud Structure Interacton 13 th Internatonal LS-DYNA Users Conference It s Gallean nvarant due to that the postons and veloctes appear only as dfferences, and has good numercal conservaton propertes. v s the velocty component at partcle. The dscretzed form of the SPH momentum equaton s developed as: dv dt m (7) ( ) W, The above formulaton ensures that stress s automatcally contnuous across materal nterfaces. Dfferent types of SPH momentum equatons can be acheved through applyng the dentty equatons nto the normal SPH momentum equaton. Symmetrc formulaton of SPH momentum equaton can reduce the errors arsng from partcle nconsstency problem. From equaton (7), the followng partcle body forces were derved: F F pressure vs cos ty m m p p W ( r, h) 2 v v 2 2 W ( r Where r x x, s the vscosty coeffcent of the flud. The pressure p are computed va the consttutve equaton: p ( ) k 0 (9) where k s the stffness of the flud and 0 s ts ntal densty. Fnally, for the acceleraton of a partcle, we have: Where, h) pressure vs cos ty external a 1/ ( F F F ) (10) external F are external forces such as body forces or forces due to contacts. (8) Multple Fluds and Sold bodes The above equatons (1)--(10) were desgned to handle sngle phase flud and can be easly extended n order to handle multple fluds and Sold bodes wth dfferent rest densty. Cares must be taken to avod the nterface nstablty due to the large densty rato across the nterfaces. Interacton through standard SPH nterpolaton As shown n Fg 2, the standard way to handle the nteractons between dfferent SPH parts s through the SPH nterpolaton functons (.e treated as one part for multple SPH fluds) and no 1-4

5 13 th Internatonal LS-DYNA Users Conference Sesson: Flud Structure Interacton contact treatments are needed on the nterfaces of the dfferent SPH parts. In SPH, partcles have a spatal dstance (smoothng length) over whch ther propertes are smoothed by a kernel functon (such as densty, pressure). Smoothed quanttes of a partcles show falsfed values when denstes and masses of neghborng partcles vary largely wthn the smoothng length. As shown n Muller et al (2005), mscble fluds wth a densty rato larger than 10 cannot be realstcally smulated f the standard SPH densty summaton s used. The reason s that n SPH, the macroscopc flow s manly governed by the densty computaton. Over or underestmatng the densty leads to erroneous pressure values, whch mght result n unnatural acceleraton caused by erroneously ntroduce pressure rato (Ihmsen et al 2011). Also lead to a spurous nterface tenson and a large gap between the fluds. The erroneous quanttes lead to undesrable effects, reachng from unphyscal densty and pressure varatons to spurous and unnatural nterface tensons, and even to severe numercal nstabltes Another ssue wth the nteracton through standard SPH nterpolaton s that dfferent SPH flud parts may stck together after the nteracton due to the SPH functon nterpolatons. To actvate ths opton, CONT parameter n *CONTROL_SPH has to be set as 0, and no contacts are allowed between those SPH parts. Part I 2h Part II Fg 2. Interacton through SPH nterpolaton (treated as one part and no contact s needed) Interacton through node to node contacts A penalty based node to node contact model s ntroduced on the nterfaces of the dfferent SPH parts. As shown n Fg 3, all the SPH nterpolatons (densty, pressure and so on) are carred out nsde the local domans of each SPH part. No spurous nterface tenson or nterfaces nstablty happened n ths model. The contact forces on the nterfaces wll be appled to the external forces as n equaton (10). 1-5

6 Sesson: Flud Structure Interacton 13 th Internatonal LS-DYNA Users Conference Fg 3. Interacton through node to node contacts In ths system, the repulsve contact force actng on partcle due to contact, s drectly proportonal to the dsplacement or overlap between partcles δ: (11) where δd2h and s the lnear-sprng constant or stffness. If the contact s modeled usng only ths lnear-sprng, no energy wll be consumed and the contact wll be ferfectly elastc. In realty, some knetc energy s dsspated n plastc deformaton, and/or converted to heat or sound energy. To account for those energy losses, a contact dampng force based on a dashpot model s also ncluded: The contact dampng force s proportonal to the relatve velocty of the contactng partcles, where the constant of proportonalty s known as the dampng coeffcent, v v 1 v2. A *DEFINE_SPH_TO_SPH_COUPLING keyword s needed between any two SPH parts for the contact nteracton, also parameter CONT n *CONTROL_SPH need to be set as 1 to deactvate the nteracton through standard nterpolaton. (12) Interacton through both normal nterpolaton method and node to node contact method n one model Combne the CONT=1 opton n *CONTROL_SPH keyword wth keyword *SECTION_SPH_INTERACTION to support the partal nteractons between SPH parts through normal nterpolaton opton and partal nteracton between SPH parts through node to node contacts n one model. All the SPH parts that are defned wth *SECTION_SPH_INTERACTION keyword wll ntegrate wth each other through normal nterpolaton method automatcally, and node to node contacts are needed for the nteractons between SPH parts defned wth *SECTION_SPH_INTERACTION keyword and other SPH 1-6

7 13 th Internatonal LS-DYNA Users Conference Sesson: Flud Structure Interacton parts, also for the nteractons between any other SPH parts that not defned through *SECTION_SPH_INTERACTION keyword. Normally, nteracton through the standard nterpolaton method produces more consstent results across the nterface when SPH parts nteracted have smlar denstes and materal propertes, however the smoothed quanttes of a partcles show falsfed values when denstes and masses of neghbourng partcles vary largely wthn the smoothng length. The erroneous quanttes lead to undesrable effects, reachng from unphyscal densty and pressure varatons to spurous and unnatural nterface tensons, and even to severe numercal nstabltes. Interacton through node to node contact allow the users to select the desred amount of contact force between two SPH parts by choosng the desred penalty scale factors accordng to the smulaton problem at hand and t help to avod the nstabltes due to large densty ratos at the nterfaces. Also for any two SPH parts wth total Lagrangan formulaton defnton, only node to node contact can be used as nteracton method, snce the neghbourng lsts are only updated at begnnng for SPH partcles wth total Lagrangan formulaton defnton. Wth *SECTION_SPH_INTERACTON keyword, users can take advantage of both nteracton methods n one model based on the SPH parts propertes as shown n Fg 4. Fg 4. Interacton through both standard nterpolaton methods and contact methods SPH couplng wth Sold elements For the normal nteracton between SPH partcles and FEA elements (Solds and Shells), node to surface contacts n LS-DYNA can be used, snce both methods are based on the Lagrangan descrpton. When modellng SPH partcles as fluds flow and FEA elements as structure n the models, the Flud Structure Interacton problems can be easly handled by the node to surface contact. Keyword *DEFINE_ADAPTIVE_SOLID_TO_SPH s used to adaptvely transform a Lagrangan sold Part or Part Set to SPH partcles, when the Lagrangan sold elements comprsng those parts fal (Shown n Fg 5). One or more SPH partcles (elements) wll be 1-7

8 Sesson: Flud Structure Interacton 13 th Internatonal LS-DYNA Users Conference generated for each faled element. The SPH partcles replacng the faled element nhert all of the propertes of the faled sold element, e.g. mass, knematc varables, and consttutve propertes. Fg 5. Transform Sold elements nto SPH partcles Wth ICPL=0, ths keyword s used for debrs smulaton, no couplng happens between newly generated SPH partcles and sold elements, user need to defne node to surface contact for the nteracton between those two parts. When ICPL=1 and IOPT=1, the newly generated SPH partcles are bonded wth sold elements as one part through the couplng (Hybrd elements). Wth ICPL=1 and IOPT=0, ths keyword s used as Hybrd Element couplng SPH wth Sold (as shown n Fg 6). Hybrd elements SPH Partcles Sold elements 1-8 Fg 6. Example of Hybrd elements a stranst layers between SPH partcles and Sold elements In ths hybrd elements method, we have the SPH formulaton whch can endure qute large deformaton and at the same tme we have the Sold meshes whch clearly descrbe the materal nterfaces. Sold elements constran SPH nodal locatons. SPH elements provde penalty force aganst sold nodal motons. Hybrd elements are used as transt layers between SPH elements and Sold elements, for a porton of grd model comprses SPH partcles because the lkelhood of endurng large deformaton, whle the rest of the model comprses FEM sold elements, hybrd elements are placed between the solds and the partcles, each hybrd element comprses two layers: sold layer and partcle layer.

9 13 th Internatonal LS-DYNA Users Conference Sesson: Flud Structure Interacton SPH couplng wth ALE, DEM methods Keyword *ALE_COUPLING_NODAL_PENALTY provdes a penalty base contact nteracton between SPH partcles and ALE materals (master segments). Also keyword *DEFINE_SPH_DE_COUPLING defnes a penalty based contact between SPH partcles and DEM partcles. Ths opton uses the node to node contacts to couple SPH solver wth dscrete element sphere (DES) solver. Examples Water, ar mpactng wth rgd rng 3D tank wth fluds whch has the dmenson of 1.0X0.8X0.01 (Fg. 7) was calculated to valdate the node to node contact n LS-DYNA for multple SPH parts wth hgh densty rato across the nterfaces. The fluds n the tank were water and ar wth ar on the top, the densty rato between those two fluds s more than Both water and ar were model wth SPH partcles. A rgd rng modeled wth cylnder shell mpacted the fluds n the tank wth the speed of 50 n Y drecton. The results from the SPH partcles were compared wth the results from the ALE method wth the same dmenson and parameters (see Fg.9 and Fg.10). In the model, automatc_ node_ to_surface contacts were used for the nteracton between ar, water partcles and rgd shells, a node to node contact was used for the nteracton between ar partcles and water partcles. The contact between two SPH partcles from dfferent parts was detected when the dstance of two partcles s less than SRAD*(sum of smooth lengths from two partcles)/2.0. SRAD s parameter ranged from 0 to 1.0 and s used to adust the detectng crtera due to ntal penetraton. The standard nteracton through SPH nterpolaton wll not work for ths case. A proper penalty scale factor has to be used for better performance. As show n Fg. 8, a double value of penalty scale factor wll cause more noses around the nterface of the two SPH fluds. The fnal deformed shape of water was comparable wth the results from ALE elements (Fg. 9). The velocty hstorys for the rgd rng from both SPH model and ALE model were plotted and compared n Fg. 10, two results were close. 1-9

10 Sesson: Flud Structure Interacton 13th Internatonal LS-DYNA Users Conference Fg 7. Problem set up of water mpact t=2.7 ms t=2.7 ms 1-10 t=5.0 ms t=5.0 ms

11 13 th Internatonal LS-DYNA Users Conference Sesson: Flud Structure Interacton Fg 8. Upper: deformaton shape for ar and water model Lower: deformaton shape wth double value of penalty scale factor Fg 9. Fnal deformaton shape from SPH model (left) compared to ALE model (rght) Fg 10. Impact velocty from SPH model (B) compared to velocty from ALE model (A) Summary We present couples of keyword optons avalable n LS-DYNA for the nteractons between SPH parts also the nteractons between SPH part and Sold part: Standard SPH nterpolaton method (normal way); Node to node penalty based contact through keyword *DEFINE_SPH_TO_SPH_COUPLING; Combnaton of both method through keyword *SECTION_SPH_INTERACTION; Node to suface contact between SPH part and Sold part; Couplng between SPH parts and sold parts through keyword *DEFINE_ADAPTIVE_SOLID_TO_SPH; Couplng between SPH partcles wth ALE elements through keyword *ALE_COUPLING_NODAL_PENALTY. Dfferent nteracton approaches can be combned together n one model to reach the best results. Some examples are 1-11

12 Sesson: Flud Structure Interacton 13 th Internatonal LS-DYNA Users Conference demonstrated to show how to use dfferent approaches wth dfferent combnaton of LS-DYNA keywords. References Jngxao Xu, Jason Wang, Node to node contacts for SPH appled to multple fluds wth large densty rato, 9th European LS-DYNA Conference L.B. Lucy, A numercal approach to the testng of the fsson hypothess, Astron. J. 82 (12) (1977) L.D. Cloutman, SPH smulatons of hypervelocty mpacts, Lawrence Lvermore Natonal Laboratory, Rep. UCRL-ID , R.A. Gngold and J.J. Monaghan, Kernel estmates as a bass for general partcle methods n hydrodynamcs, J. Comput. Phys. 46 (1982) L.D. Lbersky and A.G. Petschek, Smooth partcle hydrodynamcs wth strength of materals, New Mexco Insttute of Mnng and Technology, Socorro, NM. J.J. Monaghan, Smulatng free surface flows wth SPH, J. Comp. Phys. 110 (1994) Paul W. Cleary, Modellng confned mult-materal heat and mass flows usng SPH, Appled Mathematcal Modellng, Volume 22, Issue 12, December 1998, Pages F. OTT and E. SCHNETTER, A modfed SPH approach for fluds wth large densty dfferences, X. Y. Hu and N. A. ADAMS, A mult-phase sph method for macroscopc and mesoscopc flows, Comput. Phys. 213, 2 (2006), B. Solenthaler and R. Paarola, Densty contrast SPH Interfaces, Eurographcs/ACM SIGGRAPH symposum on Computer Anmaton, Matthas Muller, Barbara Solenthaler, Rchard Keser and Markus Gross, Partcle-Based Fludflud Interacton, Eurographcs/ACM SIGGRAPH symposum on Computer Anmaton (2005), pp , 2, 4, 6. Markus Ihmsen, Julan Bader, Gzem Aknc and Mathas Teschner, Anmaton of ar bubbles wth SPH, Internatonal Conference on Computer Graphcs Theory and Applcaton, P. A. Cundall and O. D. L. Strack, A dscrete numercal model for granular assembles (1979), Geotechnque, 29(1),