A weighted least squares particle-in-cell method for solid mechanics

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2009; 0:1 6 [Verson: 2002/09/18 v2.02] A weghted least squares partcle-n-cell method for sold mechancs P. C. Wallstedt, J. E. Gulkey Unversty of Utah, USA SUMMARY A novel meshfree method s ntroduced that ncorporates features of the Materal Pont (MPM) and Generalzed Interpolaton Materal Pont (GIMP) methods and can be used wthn an exstng MPM/GIMP mplementaton. Weghted least squares kernel functons are centered at statonary grd nodes and used to approxmate feld values and gradents. Integraton s performed over cells of the background grd and materal boundares are approxmated wth an mplct surface. The new method avod nearest-neghbor searches whle sgnfcantly mprovng accuracy over MPM and GIMP. Implementaton s dscussed n detal and several example problems are solved, ncludng one manufactured soluton whch allows measurement of dynamc, non-lnear, large deformaton performance. Copyrght c 2009 John Wley & Sons, Ltd. key words: generalzed nterpolaton materal pont method, meshfree, marchng cubes, weghted least squares, mplct surface, MPM, GIMP, PIC 1. Introducton For many decades the Fnte Element Method (FEM) has been trusted wth predctve computatons for a wde range of structures and systems; t s known to be robust and accurate. However, there are classes of problems that reman dffcult to solve wth FEM. For example, the smulaton of extruson and moldng operatons produces extremely large deformatons of the mesh, whle smulaton of falure processes nvolves trackng of arbtrary and complex cracks and materal nterfaces. The most relable way of dealng wth movng dscontnutes n FEM s to re-mesh at every step of the problem. Ths leads to greater computatonal effort and reles crtcally on automated meshng algorthms that may only be relable for smple domans or for lnear trangles and tetrahedra. Durng the last two decades a plethora of methods has been developed to crcumvent the lmtatons of FEM. The Smooth Partcle Hydrodynamcs (SPH) method was developed by Lucy (1), Monaghan (2), and coworkers to model astrophyscs problems and later extended to Correspondence to: phlp.wallstedt@utah.edu james.gulkey@utah.edu Department of Mechancal Engneerng, Unversty of Utah, Salt Lake Cty, UT Copyrght c 2009 John Wley & Sons, Ltd. Receved Revsed

2 2 WALLSTEDT AND GUILKEY sold mechancs by Lbersky, Petschek and Randles (3; 4). Although t orgnally suffered from nstablty and lack of convergence, SPH has snce been refned and corrected by Swegle (5), Johnson and Bessel (6), Dlts (7) and others. Whle these modfcatons mproved the accuracy and stablty of SPH, they also ntroduced some nconvenence, such as keepng track of addtonal stress ponts. By reformng the Dffuse Element Method (DEM) of Nayroles and coworkers (8) n terms of Movng Least Squares (MLS), Belytschko, Lu and Gu (9; 10; 11) ntroduced the Element Free Galerkn (EFG) method whch acheved substantal mprovements n accuracy. Several other famles of MLS-based methods appeared soon after ncludng the Reproducng Kernel Partcle Method (RKPM) of Lu and Jun and coworkers (12; 13; 14; 15) and the Meshless Local Petrov Galerkn (MLPG) method of Atlur et al. (16). Excellent overvews of meshfree methods nclude the papers of Belytschko and coworkers (17), and Fres and Matthes (18), and the text of Lu (19). MLS-based methods were seen to be nstances of a more general partton of unty framework (20; 21) and convergence propertes were proven for broad classes of MLS-based methods (22). The MLS-based methods were shown to be useful for new classes of problems for whch FEM was ll-suted; development and applcaton of these methods s on-gong. Some lmtatons exst, however, such as the applcaton of essental boundary condtons (23) and the substantally larger computatonal cost relatve to FEM. The cell-based ntegraton schemes used by some MLS-based methods are partcularly troublesome for dynamc problems (24). The materal pont method (MPM) was descrbed by Sulsky et al. (25; 26) as an extenson to the FLIP (Flud-Implct Partcle) method of Brackbll (27), whch tself s an extenson of the partcle-n-cell (PIC) method of Harlow (28). Interestngly, the name materal pont method frst appeared n the lterature two years later n a descrpton of an axsymmetrc form of the method (29). MPM s extraordnarly easy to mplement and use for arbtrarly complcated domans such as foams, geologc formatons, or bologcal structures and can be ntalzed n seconds from magng data such as a CT scan. For those famlar wth SPH, MPM may be loosely thought of as SPH wth stress ponts, where the stress ponts and kernel functons are located at nodes of a statonary cartesan grd. A major correcton and mprovement to MPM was offered by Bardenhagen and Kober (30) and called the Generalzed Interpolaton Materal Pont (GIMP) method. GIMP retans the same generalty as MPM, though at addtonal computatonal cost. For realstc problems GIMP approaches frst order accuracy (31) and does not enjoy the reproducblty of MLSbased methods. MPM and GIMP have been studed and used by numerous nvestgators; a subset of these contrbutons ncludes: analyss and mprovement of tme ntegraton propertes by Bardenhagen (32), Sulsky and Kaul (33), and Wallstedt and Gulkey (31); membranes and flud-structure nteracton by York, Sulsky and Schreyer (34; 35); mplct tme ntegraton by Gulkey and Wess (36), and Sulsky and Kaul (37); conservaton propertes and plastcty by Love and Sulsky (38; 39); contact by Bardenhagen et al. (40); cracks and fracture by Narn (41); trackng of partcle extents by Ma et al. (42); enhanced velocty projecton and verfcaton va the method of manufactured solutons by Wallstedt and Gulkey (43; 31), In ths work a new method s proposed that fts wthn the GIMP framework: t s based on a cartesan grd and t does not search for nearest neghbors or requre a fnte element mesh. The new method s based on a Weghted Least Squares approxmaton of data surroundng each grd node and s referred to n the remander of ths paper as WLS. The method gves

3 A WEIGHTED LEAST SQUARES PIC METHOD FOR SOLID MECHANICS 3 up some of the generalty and ease-of-use of GIMP whle makng substantal gans n accuracy. WLS takes advantage of the prevous decade s worth of development of meshfree methods whle avodng the dsadvantage of nearest-neghbor searches. In contrast to many meshfree partcle methods, WLS performs ntegraton over a background, as does EFG. Some of the volume decomposton features of WLS are analogous to a method that was presented by Belytschko and coworkers (44) for creatng structured fnte element models from sold models by means of mplct surface defntons. Whle ther technques focused on fnte element dscretzatons, the cell decomposton and mplct surface estmaton also apply to the Cartesan background grd used n the present method. Ths paper s organzed as follows: Several components of the algorthm are frst descrbed such as the PIC framework, weght functons, least squares scheme, mplct surface defnton, marchng cubes polygonzaton, and off-object node correcton. Then these ndvdual components are combned n a detaled descrpton of the dscrete governng equatons of the man algorthm. Lastly, several example problems are defned and solved, followed by a dscusson of the utlty of the method. 2. Governng Equatons An Updated Lagrangan formulaton allows straghtforward modelng of hstory-dependant materals wth complcated consttutve models. The equaton of moton s σ + ρb = ρa (1) where σ s Cauchy Stress, ρ s densty, b s acceleraton due to body forces, and a s acceleraton. Essental boundary condtons may be set on velocty and acceleraton, resultng n prescrbed values of dsplacement u = x X, where X s poston n the reference confguraton and x s poston n the current confguraton. The deformaton gradent s defned as F = x X. Surface tractons are σn = t where n s normal to the object surface and t s the tracton vector on the surface. The conservaton of mass s ρ 0 = ρj (2) where ρ 0 s densty n the reference confguraton and J = det(f) s the Jacoban. A FEM mplementaton mght update the poston and velocty of nodes and compute F from u: F = (I u x ) 1. However, the current method (as well as GIMP) updates poston, velocty, and deformaton gradent. The result s that spatal dervatves are only frst order and the WLS bass defned later n the manuscrpt can be planar. The stress can be a functon of vrtually any varable used n the smulaton, but n ths work t s only a functon of the deformaton gradent F. The rates of poston, velocty, and deformaton gradent are ẋ = v, v = a and Ḟ = ( v)f, respectvely. The weak form of the governng equatons s obtaned by multplyng Equaton 1 by a monotoncally decreasng weght functon wth a narrow bass φ, and ntegratng over the current confguraton Ω wth boundary Γ; see, for example, the text of Belytschko, Lu and Moran (45). Integratng by parts over the hghest order term results n: σ( φ) dω + tφ dγ + ρbφ dω = ρaφ dω (3) Ω Γ Ω Ω

4 4 WALLSTEDT AND GUILKEY 3. Descrpton of the Method The FEM reles on a connected mesh of nodes whle meshfree methods typcally rely on a cloud of dsconnected nodes (along wth a background ntegraton grd and octree search structure). However, the WLS method of ths work s embedded wthn a Partcles-In-Cell (PIC) framework that s also used for the MPM and GIMP methods. In ths way WLS represents a hybrd of PIC and meshfree schemes. Values of poston, velocty, deformaton gradent, and other varables are assgned to dsconnected partcles that are sutably spaced throughout an object such that suffcent nformaton s present. A statonary cartesan grd facltates a bucket-sortng scheme so that partcle values are collected, va weghted least squares, to the regularly-spaced nodes of the grd to create felds of acceleraton and velocty throughout the doman. Gradents of stress and velocty on the grd are used to update partcle poston, velocty and deformaton gradent. The statonarty of the local least squares equatons makes the weghted desgnaton n WLS approprate, rather than movng least squares n whch values are collected to moble, dsconnected nodes based on nearest neghbors. A contguous, non-overlappng ntegraton scheme s defned by usng the cartesan background grd for ntegraton. As an object moves through the grd ts boundares pass through several cells of the grd. Such boundary cells are subdvded nto nteror and exteror portons and ntegraton s performed over the nteror porton of each cell only. The subdomans over whch numercal ntegraton s performed dffer n ths method from FEM and from GIMP, as llustrated n Fgure 1. (a) Fnte Elements (b) GIMP Partcles (c) WLS Cell Volumes Fgure 1. Comparson of Volume Partton Strateges Fnte elements always form a contguous non-overlappng partton of an object nto subdomans, whch provdes accurate and relable ntegraton, albet at the cost of complcated mesh generaton. The GIMP method only forms a contguous non-overlappng partton of the materal for the ntal poston; durng problem evoluton, gaps or overlaps may develop n the partton. Despte ths shortcomng, GIMP s great advantage s the easy ntalzaton of complcated domans. The WLS method of ths paper forms ntegraton domans that reman contguous and nonoverlappng by usng the cell structure of a PIC method. When a materal boundary cuts through the mddle of a cell, the cell s splt n two and only a porton of the cell s allowed to contrbute to the ntegraton. The new cell ntegraton method represents an mprovement

5 A WEIGHTED LEAST SQUARES PIC METHOD FOR SOLID MECHANICS 5 of accuracy compared to GIMP, whle retanng the exstng PIC framework, data structures and ntalzaton procedures. However, t nvolves the extra complcaton of marchng cubes polygonzaton and t tends to gnore sharp corners, n a manner smlar to GIMP. The WLS method makes use of several mathematcal components from the computer graphcs and meshfree communtes; detaled explanatons of these are gven n the followng sectons Weght Functons Two dfferent shape or weght functons are used wthn WLS. The shape functons are centered on nodes of the background grd and are defned n terms of the cell sze h. A wde second degree splne wth 27 node support (n 3D) allows nodes to see partcles that are farther away. A narrower pecewse lnear functon wth 8 node support s used to nterpolate grd data back to partcles. Ths see wde, apply narrow concept ensures that only nodes wth suffcent data are allowed to nfluence partcle updates. All weghts and shapes are expressed n 1D as functons of r = x x and are mplemented such that the frst avalable choce n the lst s always used; note that sgn(r) = ( 1 f r < 0, 1 otherwse)..75 r2 h r < h 2 2 (1.5h r ) W (r) = 2 2h r < 3h (4) otherwse S (r) = G (r) = { 1 r h r < h 0 otherwse { sgn(r) h r < h 0 otherwse Tensor products of these are used as shapes and weghts n hgher dmensons and the shapes and weghts become functons of vectors. A shorthand s frequently used for these functons nvolvng ndces of partcles or quadrature ponts. For example, f partcles are ndexed wth p, then W p = W (x p x ), S p = S (x p x ), and G p = G (x p x ) Weghted Least Squares Framework Although many bass functons can be chosen wthn a weghted least squares framework, the hyperplane s used throughout ths method. The partcles that fall wthn the non-zero regon of a weght functon centered at each node contrbute to the equaton at the node: f LS (r) = c 0 + c 1 x + c 2 y + c 3 z (7) where x, y, and z are components of r = x p x. The error norm that defnes the WLS formulaton s: L 2 = ( ) W p f LS 2 (r p ) f p (8) p (5) (6)

6 6 WALLSTEDT AND GUILKEY Dfferentatng L 2 wth respect to the coeffcents c and settng each equaton equal to zero results n the followng system: p W p W x p W y p W z p W x p W p x2 W xy c 0 p W xz p W y p W xy p W y2 p W yz c 1 c 2 = p W f p p W f px p W z p W xz p W yz p W z2 c p W f py (9) 3 p W f pz The system s rewrtten as Mc = l and M, c, and l are termed the moment, coeffcent, and load vectors/matrces, respectvely. Each local functon s found by the followng, whch s smply a dfferent form of Equaton 7. f LS (r) = M 1 l (1 x y z ) T A shorthand s defned for the value and gradent of f LS (r), at the locaton of the node: 3.3. Implct Surface Defnton (10) o f = f LS (r = 0) = c 0 (11) o f = f LS (r = 0) = ( c 1 c 2 c 3 ) T (12) In FEM, surfaces are modeled wth the edges or faces of elements, and are accurate nsofar as the element can match the curve of the object. In GIMP, surfaces are mpled but never defned. Ths means that GIMP does not resolve sharp convex or concave features; however t also avods many of the complcatons of FEM such as meshng and specal contact algorthms. The treatment of materal surfaces n WLS s one of ts most crtcal components. The method uses flagged ponts that are placed on materal surfaces and gven specal treatment n the algorthm. The surface partcles add an extra component of dffculty to the ntalzaton of the method, as compared to GIMP, but the dffculty s not severe. For geometrcally smple objects such as pressure vessels and beams the surface of the object s clearly defned. And for complcated domans arsng from three-dmensonal scan data, surface partcles can be located by placng them halfway between data samples that are n and out. A cell subdvson method s developed based on the marchng cubes polygonzaton algorthm of Lorensen and Clne (46). For a 3D doman, the marchng cubes algorthm requres a cartesan grd of ponts as nput. The value stored at each pont may be a smple bnary flag ndcatng whether the pont s nsde or outsde the object. However, a more sophstcated verson uses the value on each pont to represent a sgned dstance from the surface of the object, ndcatng how far the pont s located away from the object s surface. For WLS, a cartesan grd s already used n parts of the algorthm. A means of mposng a sgned dstance on each grd node s developed that makes use of exstng least squares machnery. A flag β p s defned on all partcles that s unty for surface partcles and s larger for nteror partcles; a value of 2 s used. The flag values are ntegrated to nearby grd nodes: β = p S p β p (13) Based on the values of β a sgned dstance d for every node n the grd s estmated as: { h β > 2 d = (14) h otherwse

7 A WEIGHTED LEAST SQUARES PIC METHOD FOR SOLID MECHANICS 7 Ths frst estmate s rough but t serves to ntalze all nodes as n or out. The next formula reles on the dea that the surface to be located s always one dmenson less than the doman of the problem. Therefore the postons of surface partcles can be ntegrated as though they are ordnary partcle varables. The formula shown below forms sums over surface partcles only to fnd an average dstance to the surface for each node that has one or more surface partcles wthn the non-zero regon of ts weght functon. x surf = p S px surf p p S p (15) Note that x surf s only defned on nodes near the surface, and for each such node the followng steps are performed: 1. Defne surface normal n at surface poston x surf as the gradent of the partcle flags: n = G (x surf )β (16) 2. Sgned dstance s projecton of dstance vector onto surface normal: d = sgn((x surf x ) n) x surf x (17) Thus the creaton of sgned dstances for all nodes takes place n two stages. In the frst stage all sgned dstances are estmated wth Equaton 14, and n the second stage those nodes that fall near the surface are gven more precse sgned dstances va Equatons Subdvson of Boundary Cells va Marchng Cubes The ntegraton doman for an object n WLS conssts of the sum of the volumes of each full or partally full cell that s occuped by the object. Cells wthn the object are completely flled, cells on the boundary are partally flled, and cells away from the object are empty. In order to subdvde each cell s volume n an effcent manner the Marchng Cubes algorthm of Lorensen and Clne (46) s used. The 2D varaton of ths algorthm s called Marchng Squares. For a 2D cell each of ts four nodes may flagged as n or out based on whether the node s located wthn the object or outsde of t. Thus there exst 2 4 = 16 combnatons of node flags, whch may be reduced, wth symmetrc reflectons and rotatons, to four unque cases nvolvng trangles, quadrlaterals, or pentagrams; see Fgure 2. In 3D there exst 2 8 = 256 combnatons of node flags whch reduce to ffteen unque cases. Ths exhaustve lstng of possble cases of cell subdvson makes the Marchng Cubes algorthm speedy and robust. (a) trangle (b) quad (c) pentagram (d) square (e) not allowed Fgure 2. Subdvded cell regons

8 8 WALLSTEDT AND GUILKEY Whle t s easy to perform ntegraton over cells that are empty or full, the complex polyhedra generated by Marchng Cubes may have varatons of topology that, n general, cannot be expressed n terms of a sngle hexahedral or tetrahedral fnte element. Although t may be theoretcally possble to compute the correct weghts and locatons of Gauss ponts for a 2D pentagram, for example, such s not generally done. And the task becomes unmanageable f Gauss ponts are to be computed for the ffteen dfferent polyhedra n 3D. In leu of exact ntegraton, two methods of approxmate ntegraton are dscussed, and one of these s ultmately chosen for mplementaton n WLS Mult-pont ntegraton over partally-flled cells The cells of the background grd may be ntegrated to arbtrarly hgh order by fllng them wth the requste number of Gauss ponts. For an nteror cell ths s guaranteed to exactly ntegrate a polynomal over the cell. But f such a cell s subdvded nto two regons, only one of whch s occuped by the object, then the ntegraton s no longer exact. However, such a scheme may be consdered an approxmaton. For example, consder a 2D cell whch s populated wth 16 Gauss ponts, and assume that eleven ponts are wthn the object and fve are outsde of t. The approxmaton of the ntegral over the cell s then found as the sum of all the weght-value products of the Gauss ponts, where the value of the outsde Gauss ponts s taken to be zero. Another closely related approxmaton s to space sxteen sample ponts evenly throughout the cell, rather than at the Gauss locatons, and set the outsde ponts to zero as above. The weght of each nner pont s smply one sxteenth of the cell volume. Such approxmaton schemes obvously lose ther hgh order. The mult-pont approxmate ntegraton scheme s used by Belytschko and coworkers for the Element Free Galerkn method. In ther overvew of meshfree methods Belytschko et al. state that the background grd strkes many researchers as unacceptably crude, for wthn a cell, quadrature s performed over any dscontntes and boundares whch do not concde wth the boundares of the cells. Our experence so far suggests that the effects are qute mnmal (17). However, n a broad revew of meshfree methods Fres and Matthes (18) are more crtcal, clamng that the ntegraton error whch arses from the msalgnment of the supports and the ntegraton domans s often hgher than the one whch arses from the ratonal character of the shape functons Low order ntegraton over cell sub-domans In ths approxmate ntegraton scheme the volume of ntegraton s consdered to be more mportant than the order of ntegraton. The volume of a cell s subdvded nto nner and outer portons and a sngle sample pont s located at the centrod of the nner regon. In a FEM mplementaton the deformaton gradent and stress are computed at Gauss ponts based on nformaton nterpolated from the nodes of the element. However, n WLS the stress at Gauss ponts s found n three stages. Frst, stress s computed on each partcle; second, functons of average stress n the neghborhood of each node are created at the node (see Sectons 4.1 and 4.3); and thrd, the stress at Gauss ponts s nterpolated from the stress functons at the cell nodes (see Equatons 21 and 24). Ths smoothng and averagng process causes nformaton from several nearby cells to be ncluded at each Gauss pont, rather than nformaton from only the enclosng cell. One advantage of ths approxmaton s that the centrod s unque for the regon so that ambguous confguratons are dsallowed. The unqueness of the ntegraton pont, the

9 A WEIGHTED LEAST SQUARES PIC METHOD FOR SOLID MECHANICS 9 statonarty of cell boundares, and the broad and smooth nformaton used at Gauss ponts, together suggest that zero energy modes, such as those occurrng wth sngle Gauss pont ntegraton of quadrlateral fnte elements, are unlkely to occur. Generally speakng, the mathematcal analoges between FEM, GIMP, and WLS reman tentatve due to the major dfferences between the methods. Each famly of methods must be ndependently analyzed for lmtatons and troublesome modes. Another advantage of the centrodal approxmaton s that the value of the ntegral vares smoothly as the object boundary passes through the cell. In the mult-pont approxmaton, Gauss ponts may abruptly turn on or turn off as the object surface passes through them. The sngle pont ntegraton method of ths secton s used throughout WLS Detals of Cell Subdvson The sgned dstance nformaton developed n secton 3.3 s used to subdvde cell edge segments and enclose a shape wthn them. If any two nodes are on the same edge of a cell they are termed adjacent. And f two nodes are adjacent, yet one of them s n, and the other s out, then t s clear that the surface of the object passes somewhere between them. Furthermore, the sgned dstance nformaton s used to approxmate the locaton on the segment at whch the surface crosses. In the followng equaton two specal node ndces are denoted wth captal letters: I s the ndex of the n node and O s the ndex of the out node. For any two adjacent nodes wth one n and one out, the followng expresson gves global coordnates for the locaton at whch the materal boundary ntersects the cell segment: x cut = x I d I x I x O d I d O (18) The cell boundary cut postons x cut are used, together wth the n node or nodes of the cell to form the shape ndcated by the Marchng Cubes scheme. For example, f one node of a cell s n and the other three nodes are out, then two cut ponts are created between the n node and ts two adjacent nodes. The two cut ponts plus the n node form a trangle, whose volume and centrod are used for ntegraton. Ths ntegraton scheme elmnates the gaps and overlaps of GIMP ntegraton but ncurs the extra nconvenence of ntalzng surface partcles and buldng a surface. Each WLS cell that s near a surface s dvded nto flled and empty portons. Ths results n a contguous, non-overlappng descrpton of the object, although corners tend to be rounded; see Fgure Surface Condtonng Whle the weghted least squares scheme provdes excellent results for values and gradents at nodes wth suffcent numbers of partcles, there are nodes near object boundares that may end up wth spurous values due to ll condtonng of the moment matrx. A relable way of detectng spurous nodes s descrbed n ths secton, followed by two remedal procedures. Some cases of ll-condtonng are easly detected, such as an nsuffcent number of partcles to form the planar bass (3 partcles are requred n 2D; 4 n 3D) or a matrx determnant that s wthn machne precson of zero. But other spurous nodes may arse that cannot be detected by these means. A more robust and general method of predctng ll-condtoned moment matrces s as follows. For weght functon W (r) of Equaton 4 a partcle has greater weght as t gets closer to the node; therefore the weght functon for each contrbutng partcle s a good approxmaton

10 10 WALLSTEDT AND GUILKEY for a dmensonless radus from the partcle to the node; see Fgure 3(a) and note how the weght contours are nearly crcular. If all sample partcles fall outsde a chosen weght, then remedal procedures are performed. Experence suggests that a trgger weght TW n the range 0.1 < TW < 0.25 wll ndcate an approprate amount of extra surface condtonng; the TW used throughout ths paper s 0.12 and an example of predcted spurous nodes s shown n Fgure 3(b). TW = 0.4 TW = 0.2 TW = 0.1 TW = 0.05 TW = Wndow Node Partcles Good Nodes Ill Nodes Wndow Node (a) Trgger Weght (b) Spurous Nodes Fgure 3. Trgger Weght for Spurous Nodes; (a) s a magnfed vew of the regon near the wndow node of (b). For every spurous node the rank of the moment matrx s reduced. Ths s accomplshed by replacng Equaton 7 wth q (r) = c 0. Then the nverse of the moment matrx s formed by settng all values to zero, except for Mxx 1 = 1/M xx. For the computaton of nternal forces, experence shows that moment matrx rank reducton s nsuffcent to remedy the effects of spurous nodes, and a more drastc step must be taken. In these cases the weghted least squares coeffcents for nternal force at a node are dscarded, and values at the node are found from an average of the 26 nodes that surround t n 3D, or the 8 nodes that surround t n 2D. Ths procedure elmnates the nfluence of spurous nodes on the gradent of stress. 4. Algorthm Sequence and Dscrete Equatons In ths secton each stage of the WLS tme step s descrbed and the least squares, marchng cubes, and matrx condtonng technques of prevous sectons are combned to form the algorthm Current partcle densty, body force, and stress From the current partcle values of poston x p, acceleraton due to body forces b p, and deformaton gradent F p, the Jacoban J p = det(f p ), densty ρ p = ρ 0 /J p, and body force

11 A WEIGHTED LEAST SQUARES PIC METHOD FOR SOLID MECHANICS 11 = ρ p b p are computed. Stress s computed on each partcle from a consttutve equaton σ p = σ(f p ), whch may be very general due to the Lagrangan formulaton. f ext p 4.2. Least squares moment matrx and object surface A least squares moment matrx s formed at each node to be used for subsequent calculatons. Partcles that fall wthn the non-zero regon of the weght functon of each node contrbute to ts matrx; see secton 3.2. Specal flagged surface partcles are used va the procedure of secton 3.3 to determne how far n or out of the object each node s located Node values va WLS o f ext The least squares values v o, ρ o,, and σ o are computed on the nodes va the weghted least squares framework of Equaton 10, keepng n mnd the shorthand of Equatons 11 and Integrate over full and partal cell volumes Let q be an ndex over the quadrature ponts (one for each non-zero volume cell) and an ndex over nodes. Values of densty, body force, and stress are found at each quadrature pont by: ρ q = α o S qρ α (19) S q f ext q = σ q = α S q o f ext α S q (20) α S q o σ α S q (21) where α s unty f a node s maxvw > TW and zero otherwse. By usng α n ths manner the values on quadrature ponts are only based on well-condtoned nodes. In practce the quadrature pont values are not stored; they are computed on-the-fly durng ntegraton. The followng ntegratons are performed: m = q S q ρ q V q (22) f ext = q S q f ext q V q (23) f nt = q σ G q V q (24) 4.5. Extrapolaton If a node s maxvw < TW, then the averagng process of Secton 3.5 s performed for f nt only.

12 12 WALLSTEDT AND GUILKEY 4.6. Equatons of moton All necessary data are now collected on the grd and the equaton of moton can be solved and partcle varables updated. The grd velocty s found drectly va weghted least squares: v = o v. The equaton of momentum s solved on the grd by: 4.7. Tme update a = f nt + f ext ρ (25) The tme update s a centered-dfference scheme commonly used n FEM. Grd acceleraton s used to update grd velocty: v n+ 1 2 = v n a t (26) Acceleraton s nterpolated back to partcles and used to update partcle velocty: v n+ 1 2 p = v n 1 2 p + S p a t (27) Poston s updated by nterpolated grd velocty, not by current partcle velocty. Ths at frst seems unnecessary but s done to ensure that partcles at the same pont n space have the same velocty: = x n p + The gradent of velocty s calculated on each partcle and used to update the deformaton gradent: x n+1 p v n+ 1 2 p = S p v n+ 1 2 t (28) G p v n+ 1 2 (29) F n+1 p = F n p + v n+ 1 2 p F n p t (30) The centered-dfference update scheme requres that velocty be ntalzed to a negatve half tme step. Veloctes at the 1/2 tme step are sometmes avalable, but for typcal smulatons they may be mpossble to fnd. Instead, the approach used n ths paper s to multply the grd acceleraton values of Equaton 25 by 1/2 for the frst tme step only. Ths propagates the 1/2 through the algorthm and corrects the frst order error that would otherwse be ncurred. 5. Example Problems A neo-hookean consttutve model (45) s used for the example problems of ths secton. The stran energy functon s: Ψ(F) = 1 2 λ(ln J)2 µ ln J µ ( trace ( F T F ) 2 ) (31) where µ and λ are standard Lamé constants. Then the stress n the current confguraton s defned to be: σ = λ ln J J I + µ ( FF T I ) (32) J

13 A WEIGHTED LEAST SQUARES PIC METHOD FOR SOLID MECHANICS 13 For the sake of stablty a tme step sze s chosen to ensure that nformaton cannot travel more than a characterstc dstance n a sngle tme step. However, the ntegraton regons produced from cell subdvson cannot be used as a characterstc dstance because several of them are guaranteed to be very small at any tme wthn the problem. Experence has consstently shown that the cell sze of the background grd represents a relable characterstc dstance, whch s confrmed by the temporal convergence measurements of the frst example to follow. The maxmum wave speed for a 3D sotropc elastc sold s defned as: v max = λ + 3µ, (33) ρ Note that v max approaches nfnty as Posson s rato approaches 1/2. An adaptve tme step sze may be set throughout the smulaton by: mn(h) t(x, t) = CF L v max + max( v p ) (34) where 0 < CF L < 1; see Fgure 6(b). A nomnal tme step sze may be chosen at the begnnng of the smulaton by assumng that materal propertes are constant n space and tme, that object veloctes are well below wave speeds, and that Posson s rato s close to zero: t 0 = CF L mn(h) E/ρ (35) where E s Young s modulus. For comparson purposes an explct non-lnear FEM code based on lnear trangles wth sngle Gauss pont ntegraton s constructed accordng to typcal desgns; see, for example, the text of Belytschko, Lu, and Moran (45). The dscrete equatons are lsted n Appendx A Oscllatng Rng In prevous work (31) a manufactured soluton s developed for a dynamc rng made of a neo-hookean materal. The full development of the soluton s not repeated here but ts key element s the radal dsplacement of a rng as a functon of tme and poston: u(r) = Acos(cπt) ( c 1 R + c 2 R 2 + c 3 R 3) (36) where c 1 = 6R I R O (R O 3R I ), c 2 = 3(R O + R I ) RO 2 (R O 3R I ), c 2 3 = RO 2 (R O 3R I ), (37) A s a user-specfed amplfcaton factor, c = E/ρ, t s tme, R s radus n the reference confguraton, R I s the nner rng radus and R O s the outer rng radus. The manufactured soluton s used to measure the accuracy propertes of the WLS method descrbed n ths work n comparson to an exstng algorthm, GIMP, and to a FEM code. For PIC methods there s some freedom n the way n whch partcles are arranged wthn the cell structure. The GIMP method typcally works best when partcles are arranged n a

14 14 WALLSTEDT AND GUILKEY cartesan manner, because gaps and overlaps between partcles are mnmzed. The cartesan arrangement of partcles s used to dscretze the rng for GIMP; see Fgure 4(a). In contrast to GIMP, the partcles of WLS do not represent volume n the algorthm; they can be treated as sample ponts only. Therefore t has been found that performance s mproved by usng a body-ftted arrangement of partcles such as shown n Fgure 4(b). For (a) Cartesan partcle placement used wth GIMP (b) Radal partcle placement used wth WLS Fgure 4. Alternate Partcle Arrangements the results that follow, the arrangement of partcles s chosen that produces the best accuracy for each method. Error s measured on partcles as the magntude of the dfference between computed and exact dsplacement: δ p = (x p X p ) u exact (X p, t). (38) The manufactured soluton s always smooth n space and tme, therefore a strct defnton of error may be used: the maxmum error from all partcles and all tme steps L = max(δ p ). Ths defnton demands that error over all partcles be reduced for a problem to be consdered more accurate, nstead of merely reducng error for a majorty of the partcles. The Posson s rato used for the rng must be zero, else the manufactured soluton cannot be expressed n a closed form. In 2D the number of partcles per cell s four wth the cartesan arrangement, and approxmately four n the radal partcle arrangement. Young s modulus s 10000, ntal densty s one, and the amplcaton factor s 0.1. The rng s ntal outer radus s one and ntal nner radus s 0.5. The CFL s 0.4, except for the measurements of temporal convergence. Representatve plots of the tme hstory are shown n Fgure 5. Results for temporal and spatal convergence of WLS are presented n Fgure 6 n comparson to FEM and GIMP. Temporal convergence measurements are made usng the nomnal tme step sze of Equaton 35. For ths problem WLS s an mprovement over GIMP. The method behaves n a second order manner for coarser meshes, n contrast to the frst-order (at best) behavor of GIMP. Ths suggests that the method s sutable for use wth geometrcally smple objects such as pressure vessels and beams for whch the accuracy of FEM s customarly expected Sngle Dsk Impact Although a contact model s not developed n ths paper, the problems for whch GIMP and WLS may be used frequently nvolve contact, mpact, and nter-penetraton scenaros

15 A WEIGHTED LEAST SQUARES PIC METHOD FOR SOLID MECHANICS (a) tme = 0ms (b) tme = 5ms (c) tme = 10ms Fgure 5. Radal (top) and Hoop (rght) stress (Pa) for oscllatng rng GIMP WLS FEM L Error e-05 1e-06 GIMP WLS FEM Cell Sze h L Error CFL (a) Spatal (b) Temporal Fgure 6. Convergence for the Oscllatng Rng where energy conservaton s a hgh prorty. Ths aspect of the algorthm s assessed va the frctonless mpact of a dsk aganst a wall as shown n Fgure 7. The Posson s rato s 0.3, the number of partcles per cell s about four, and Young s modulus and ntal densty are both The dsk s 0.4 unts n dameter and s ntally located wthn a unt square of 48x48 cells at (0.3,0.3). The dsk has ntal partcle veloctes of (0.2,-0.2) wth CF L = 0.4. Ths system s modeled wth WLS and GIMP and the stran and knetc energes are plotted n Fgure 8 as a functon of the tme. It can be seen that energy s handled correctly n WLS. The transfer of energy from knetc to stran, then back agan, occurs smoothly and wthout sgnfcant fluctuaton or nstablty. The correspondng energy plot s also shown for GIMP whch dsplays the same desrable

16 16 WALLSTEDT AND GUILKEY (a) tme = 0s (b) tme = 2s (c) tme = 3s Fgure 7. Stran Energy per volume for sngle dsk Energy Knetc Stran Total Energy Knetc Stran Total Tme (s) (a) WLS Tme (s) (b) GIMP Fgure 8. Energy for sngle-dsk mpact trends. For mult-body mpact t should be straghtforward to mplement contact n the manner commonly used for GIMP, where a separate computatonal grd s created for each object, and the contact s handled wth a method such as that descrbed by Bardenhagen (40) Dynamc Hole n Plate A square plate wth a crcular hole s suddenly subjected to a body force n the horzontal drecton. The body force s modeled wth a force vector ( X,0) n the reference confguraton that ncreases wth X-poston but remans constant n tme. The soluton nvolves a large deformaton, dynamc smulaton wth a neo-hookean consttutve model;

17 A WEIGHTED LEAST SQUARES PIC METHOD FOR SOLID MECHANICS 17 therefore no exact answer s avalable. The Posson s rato s 0.3, the number of partcles per cell s about four, Young s modulus s , and ntal densty s The plate s one unt hgh and wde and the hole radus s 0.5. The CFL s 0.4. The progresson of the smulaton s shown n Fgure (a) tme = 0µs (b) tme = 464µs (c) tme = 696µs Fgure 9. Von Mses stress (GPa) of plate wth hole In order to compare WLS to GIMP and FEM, a crtcal area of the problem - the surface of the hole - s examned. For the sake of stress analyss t must be found what values of stress develop on the hole s surface so a determnaton can be made about whether the plate s strong enough to wthstand the forces on t. In Fgure 10 the partcles or FEM Gauss ponts that are ntally located wthn a dstance of 0.35h from the hole surface, where h s the WLS cell sze, are compared for FEM, WLS and GIMP. The Von Mses stress s plotted wth respect to the angle along the hole surface. The angle of the bottom face of the bar s zero, and the angle of the left face s π/2. Von Mses Stress GIMP WLS FEM π/8 π/4 3π/8 Angle along hole surface Fgure 10. Comparson of Von Mses stress (GPa) at hole surface The results dsplayed n the fgure suggest that WLS s able to provde a surface stress whch s nearly as accurate as FEM. But GIMP shows some dffculty n predctng surface stress, and

18 18 WALLSTEDT AND GUILKEY the GIMP result has more scatter. Whle t would be dffcult to determne a relable falure pont for the surface of the hole usng GIMP, the WLS result reflects realstc stresses at the hole s surface. 6. Dscusson The WLS method ntroduced n ths paper acheves the objectve of mprovng the accuracy of sold mechancs smulatons that are performed wthn a PIC framework. The method takes advantage of the concepts of weghted least squares surface estmaton and mplct surface defnton to more precsely defne the regon of ntegraton. The method s motvated by two general requrements. It needs to provde the accuracy of fnte elements, whch are well-known and are trusted for predctve results. And t needs to be as easy to ntalze as other PIC methods, whle avodng nearest neghbor searches and handlng contact and nter-penetraton scenaros wth ease. The method of ths paper represents a compromse between the demands of these two famles of methods and affords several benefts of both. The method acheves accuracy that s sgnfcantly better than GIMP, but somewhat less than FEM. Its PIC form allows t to be used wthn an exstng MPM/GIMP mplementaton whle avodng nearest neghbor searches. However, the complexty of mplementaton and ntalzaton are both greater than requred for MPM/GIMP. In addton to the programmng effort requred for GIMP, WLS also requres mplementaton of least squares and marchng cubes routnes. Problems are ntalzed wth the nteror partcles of GIMP, but also requre surface partcles, whch may be calculated from three-dmensonal mage data wth addtonal effort. The PIC structure of the method allows t to be used sde-by-sde wth GIMP n space or tme to provde addtonal accuracy for certan objects. The nteror partcles of WLS may be located at the same postons as GIMP partcles and have the same ntal volumes; surface partcles have zero volume. Ths enables a one-way transfer of algorthm from WLS to GIMP f a problem begns to dsplay behavors for whch WLS s ll-suted, such as rupture of a surface. For example, an over-pressurzed tank can be modeled up untl ts pont of rupture wth WLS, then GIMP may assume control of the soluton usng the same partcles and contnue smulaton of the dsntegraton of the tank. 7. Appendx A: Dscrete equatons for the comparson fnte element code An explct non-lnear FEM code based on lnear trangles wth sngle Gauss pont ntegraton s constructed followng the text of Belytschko, Lu, and Moran (45). The dscrete momentum equaton at each node s Mass lumpng s defned from the reference confguraton as a = f nt /M + b(x, t) (39) M = 1 3 ρ0 e A 0 e (40)

19 A WEIGHTED LEAST SQUARES PIC METHOD FOR SOLID MECHANICS 19 where the ndex e loops over the elements surroundng node. The nternal forces are f nt = e σ(f e ) φ e (x ) (41) where the deformaton gradent on each element s ( 1 3 F e = I u φ e (x )) (42) The updates of nodal velocty and poston are v n+1/2 x n+1 = v n 1/2 + a n t (43) = x n + v n+1/2 t (44) References [1] Lucy LB. A numercal approach to the testng of the fsson hypothess. The Astron. J. 1977; : [2] Monaghan JJ. An ntroducton to sph. Comput. Phys. Comm. 1998; 48: [3] Lbersky LD, Petschek AG. Smooth partcle hydrodynamcs wth strength of materals. Advances n the Free Lagrange Method, Lecture Notes n Physcs 1990; :395. [4] Randles PW, Lbersky LD. Smoothed partcle hydrodynamcs: Some recent mprovements and applcatons. Computer Methods n Appled Mechancs and Engneerng 1996; 139: [5] Swegle JW, Hcks DA. Smooth partcle hydrodynamcs stablty analyss. J. Comput. Phys. 1995; 116: [6] Johnson GR, Bessel SR. Normalzed smoothng functons for sph mpact computatons. Int. J. Numer. Meth. Engng. 1996;. [7] Dlts GA. Movng least squares partcle hydrodynamcs : Conservaton and boundares. Int. J. Numer. Meth. Engng. 2000; : [8] Nayroles B, Touzot G, Vllon P. Generalzng the fnte element method: dffuse approxmaton and dffuse elements. Comput. Mech. 1992; 10: [9] Belytschko T, Lu YY, Gu L. Element free galerkn methods. Int. J. Numer. Meth. Engng. 1994; 37: [10] Lu YY, Belytschko T, Gu L. A new mplementaton of the element free galerkn methods. Comput. Methods Appl. Mech. Engng 1994; 113:

20 20 WALLSTEDT AND GUILKEY [11] Belytschko T, Gu L, Lu YY. Fracture and crack growth by element free galerkn methods. Modelng Smul. Mater. Sc. Engng. 1994; 115: [12] Lu WK, Adee J, Jun S. Reproducng kernel partcle methods for elastc and plastc problems. Advanced Computatonal Methods for Materal Modelng, vol. AMD 180 and PVP 268. ASME: New York, 1993; [13] Lu WK, Jun S, L S, Adee J, Belytschko T. Reproducng kernel partcle methods for structural dynamcs. Internatonal Journal for Numercal Methods n Engneerng 1995; 38: [14] Lu WK, Jun S, Zhang Y. Reproducng kernel partcle methods. Internatonal Journal for Numercal Methods n Fluds 1995; 20: [15] Jun S, Lu WK, Belytschko T. Explct reproducng kernel partcle methods for large deformaton problems. Internatonal Journal for Numercal Methods n Engneerng 1998; 41: [16] Atlur SN, Zhu T. A new meshless local petrov-galerkn (mlpg) method. Computer Modelng n Engneerng and Scences 1998; 22: [17] Belytschko T, Krongauz Y, Organ D, Flemng M, Krysl P. Meshless methods: An overvew and recent developments. Comput. Methods Appl. Mech. Engrg. 1996; 139:3 47. [18] Fres TP, Matthes HG. Classfcaton and overvew of meshfree methods. Techncal Report, Techncal Unversty Braunschweg [19] Lu GR. Mesh free methods: movng beyond the fnte element method. CRC Press LLC, [20] Duarte CA, Oden JT. Hp clouds - a meshless method to solve boundary-value problems. Techncal Report 95-05, Texas Insttute for Computatonal and Appled Mathematcs, Unversty of Texas at Austn [21] Melenk JM, Babuska I. The partton of unty fnte element method: Basc theory and applcatons. Comput. Methods Appl. Mech. Engrg. 1996; 139: [22] Lu WK, L S, Belytschko T. Movng least-square reproducng kernel methods () methodology and convergence. Computer Methods n Appled Mechancs and Engneerng 1997; 143: [23] Fernndez-Mndez S, Huerta A. Computer methods n appled mechancs and engneerng. Internatonal Journal for Numercal Methods n Engneerng 2004; 193: [24] Rabczuk T, Belytschko T, Xao SP. Stable partcle methods based on lagrangan kernels. Comput. Methods Appl. Mech. Engrg. 2004; 193: [25] Sulsky D, Chen Z, Schreyer H. A partcle method for hstory dependent materals. Computer Methods n Appled Mechancs and Engneerng 1994; 118: [26] Sulsky D, Zhou S, Schreyer H. Applcaton of a partcle-n-cell method to sold mechancs. Computer Physcs Communcatons 1995; 87:

21 A WEIGHTED LEAST SQUARES PIC METHOD FOR SOLID MECHANICS 21 [27] Brackbll J, Ruppel H. Flp: A low-dsspaton, partcle-n-cell method for flud flows n two dmensons. J. Comp. Phys. 1986; 65: [28] Harlow F. The partcle-n-cell computng method for flud dynamcs. Methods Comput. Phys. 1963; 3: [29] Sulsky D, Schreyer H. Axsymmetrc form of the materal pont method wth applcatons to upsettng and taylor mpact problems. Computer Methods n Appled Mechancs and Engneerng 1996; 139: [30] Bardenhagen S, Kober E. The generalzed nterpolaton materal pont method. Computer Modelng n Engneerng and Scences 2004; 5: [31] Wallstedt P, Gulkey J. An evaluaton of explct tme ntegraton schemes for use wth the generalzed nterpolaton materal pont method. Journal of Computatonal Physcs 2008; 227: [32] Bardenhagen S. Energy conservaton error n the materal pont method for sold mechancs. Journal of Computatonal Physcs 2002; 180: [33] Sulsky D, Schreyer H, Peterson K, Kwok R, Coon M. Usng the materal pont method to model sea ce dynamcs. Journal of Geophyscal Research 2007; 112:do: /2005JC [34] York AR, Sulsky DL, Schreyer HL. The materal pont method for smulaton of thn membranes. Internatonal Journal for Numercal Methods n Engneerng 1999; 44: [35] York AR, Sulsky DL, Schreyer HL. Flud-membrane nteracton based on the materal pont method. Internatonal Journal for Numercal Methods n Engneerng 2000; 48: [36] Gulkey JE, Wess JA. Implct tme ntegraton for the materal pont method: Quanttatve and algorthmc comparsons wth the fnte element method. Internatonal Journal for Numercal Methods n Engneerng 2003; 57: [37] Sulsky D, Kaul A. Implct dynamcs n the materal-pont method. Computer Methods n Appled Mechancs and Engneerng 2004; 193: [38] Love E, Sulsky DL. An uncondtonally stable, energymomentum consstent mplementaton of the materal-pont method. Computer Methods n Appled Mechancs and Engneerng 2006; 195: [39] Love E, Sulsky DL. An energy-consstent materal-pont method for dynamc fnte deformaton plastcty. Internatonal Journal for Numercal Methods n Engneerng 2005; 65: [40] Bardenhagen S, Gulkey J, Roessg K, Brackbll J, Wtzel W, Foster J. An mproved contact algorthm for the materal pont method and applcaton to stress propagaton n granular materal. Computer Modelng n Engneerng and Scences 2001; 2:

22 22 WALLSTEDT AND GUILKEY [41] Narn JA. Materal pont method calculatons wth explct cracks. Computer Modelng n Engneerng and Scences 2003; 4: [42] Ma J, Lu H, Komandur R. Structured mesh refnement n generalzed nterpolaton materal pont method (gmp) for smulaton of dynamc problems. Computer Modelng n Engneerng and Scences 2006; 12: [43] Wallstedt P, Gulkey J. Improved velocty projecton for the materal pont method. Computer Modelng n Engneerng and Scences 2007; 19: [44] Belytschko T, Parm C, Moës N, Sukumar N, Usu S. Structured extended fnte element methods for solds defned by mplct surfaces. Int. J. Numer. Meth. Engng. 2003; 56: [45] Belytschko T, Lu WK, Moran B. Nonlnear Fnte Elements for Contnua and Structures. John Wley and Sons, LTD, [46] Lorensen WE, Clne HE. Marchng cubes: A hgh resoluton 3d surface constructon algorthm. Computer Graphcs 1987; 21(4).