The Stable Explicit Time Stepping Analysis with a New Enrichment Scheme by XFEM

Transcription

1 Copyrigh 017 Tech Science Press CMC, vol.53, no.3, pp , 017 The Sable Explici Time Sepping Analysis wih a New Enrichmen Scheme by XFEM Xue-cong Liu 1, Qing Zhang 1,*, Xiao-zhou Xia 1 Absrac: This paper focuses on he sudy of he sabiliy of explici ime inegraion algorihm for dynamic problem by he Exended Finie Elemen Mehod (XFEM). A new enrichmen scheme of crack ip is proposed wihin he framework of XFEM. Then he governing equaions are derived and evolved ino he discreized form. For dynamic problem, he lumped mass and he explici ime algorihm are applied. Wih differen grid densiies and differen forms of Newmark scheme, he Dynamic Sress Inensiy Facor (DSIF) is compued by using ineracion inegral approach o reflec he dynamic response. The effeciveness of he proposed scheme is demonsraed hrough he numerical examples, and he criical ime sepping in differen siuaions are lised and analyzed o illusrae he facors ha affec he numerical sabiliy. Keywords: XFEM, DSIF, Newmark scheme, ime sepping, sabiliy. 1 Inroducion The fracure analysis of srucures and componens has been widely applied and highly valued in recen years, and modeling disconinuiies like crack is one of he imporan pars. In order o model he behavior of disconinuiies, he way of re-meshing is used which is o align he elemen edges wih he disconinuiies by classic finie elemen mehod (FEM). However, for he case of crack arbirary growh, he mesh changes a every sep and incurs high compuaion cos. Besides, oher soluions such as meshfree mehod, boundary elemen mehod and exended finie elemen mehod (XFEM) are available, and XFEM is considered o be he mos applicable. As proposed by Belyschko and Black (1999); Moes, Dolbow and Belyschko (1999); Belyschko and Moes (001), XFEM becomes a dominan numerical scheme nowadays. XFEM is based on he concep of pariion uniy, and he crack can be modeled independen of finie elemen mesh. All he elemens are divided ino he normal pars 1 Deparmen of Engineering Mechanics, Hohai Universiy, Nanjing 11100, P R China. * Corresponding Auhor. lxzhangqing@hhu.edu.cn (Qing Zhang).

2 188 Copyrigh 017 Tech Science Press CMC, vol.53, no.3, pp.03-, 017 and he enriched pars. For he enriched pars, he elemens can be influenced direcly by crack, so he enrich funcions are inroduced. As he mos used enrich funcions, Heaviside funcion and he Wesergaard sress funcion are used frequenly for he cracks and cracks ips, respecively. Dealing wih he dynamic fracure, Belyschko, Chen, Xu e al (003) proposed a kind of ip elemen in which he crack opens linearly and developed a propagaion crierion wih loss of hyperbolic. Laer on, a singular enrichmen funcion for ips is proposed for he elasodynamic cracks wih explici ime inegraion scheme [Belyschko and Chen (004)]. In order o deeply sudy he sabiliy and energy conservaion and o ge a more accurae resul, Réhoré, Gravouil and Combescure (005a, 005b) combined Space and Time XFEM (STX-FEM) o obain a unified space-ime discreizaion, and concluded ha he STX-FEM is a suiable echnique for dynamic fracure problems. On he oher hand, a new lumping echnique for mass marix was proposed in order o be more suiable for dynamic problems by Menouillard, Réhoré, Combescure e al (006); Menouillard, Réhoré, Noes e al (008) and he robusness and he sabiliy of his approach has been proved. As we noiced, he previous sudies are all based on he classical enrichmen scheme, and a large number of addiional degrees of freedom (DOFs) are required. In he meanime, various improved enrichmen mehods have been sudied. Song, Areias and Belyschko (006) has reinerpreed he convenional displacemen field, described disconinuiies by using phanom nodes and superimposed exra elemens ono he inrinsic grid for dynamic fracure problems. The mehod doesn require subdomain inegraion for he disconinuous inegrand and has a highly efficien bu neverheless quie accurae formulaion. Furher, Duan, Song, Menouillard e al (009) has shown is pracicabiliy on he shell problem as well as hree-dimension problem [Song and Belyschko (009)]. Besides, changing he basic enrichmen funcion is anoher soluion. Menouillard, Song, Duan e al (010) proposed a new enrichmen mehod wih only a singular enrichmen, which shows grea accuracy for saionary cracks. The similar research has been done by Rabczuk, Zi, Gereenberger e al (008). Wihou crack ip enrichmen, he Heaviside funcion has also been improved. Nisor, Panale and Caperaa (008) used only Heaviside funcion o model he dynamic crack growh. Kumar, Singh, Mishra e al (015) presened a new approach based Heaviside funcion along wih a ramp funcion which conains informaion like crack lengh and angle. A similar mehod was proposed by Wen and Tian (016); Tian and Wen (016), which is based on an exra-dof-free pariion of uniy enrichmen echnique, and no more exra DOFs are added in he dynamic crack growh simulaion. For all he sudy discussed above, he sabiliy of he mehod is always concerned. Generally, using explici scheme for dynamic problem, one goes hrough a very small ime sepping ha leads o high compuaion cos, while wih a larger one he numerical resul may be divergen. So in he presen paper, we will focus on he sabiliy of he numerical scheme. Based on he analyical soluion of he displacemen fields near crack ip, a new enrichmen scheme is used for he elemens influenced by crack ip. The Newmark scheme is adoped for ime inegraion, and differen parameers are esed o invesigae heir influence on he sabiliy. In addiion, DSIF is calculaed as an imporan parameer which represens he variaion of he sress field around he crack ip, and also

3 The Sable Explici Time Sepping Analysis 189 can deermine he sabiliy of he simulaion. This paper is organized as following: Secion illusraes he governing equaions and he XFEM formulaion; The explici ime algorihm and he lumping echnique are inroduced in Secion 3; The DSIF is shown in Secion 4; In Secion 5, several numerical examples are provided, he feasibiliy and sabiliy of he simulaion are discussed. Governing equaions and XFEM formulaion.1 Governing equaions Le be a homogeneous wo-dimensional domain wih cracks. is bounded by, which is composed of, and as described in Fig.1. denoes he prescribed displacemens and denoes he racion in he domain. is referred o as he crack surface and assumed o be racion-free. The srong form of he linear momenum equaion and he boundary condiions give σ b u in (1a) u u on u σn on σn 0 on ε s u σ= C: ε σ c u where is prescribed he Cauchy sress ensor, displacemen field and acceleraion field, respecively. is he mass densiy, is he prescribed displacemen, u c u b n c u u (1b) (1c) (1d) (1e) (1f) and are he vecor of is he body force vecor, is he uni ouward normal, s is he symmeric operaor, C is he elasic module ensor and is he srain ensor. The small srains and displacemens are considered here as shown in Eq.(1e). Combine he equilibrium momenum equaion and he consiuive relaion ino he weak form, we have T u : C: ud + uud = ubd u d () s s. The XFEM formulaion Consider a ypical finie elemen mesh wih four-node elemens as shown in Fig., he geomery of crack is independen of he mesh. As in he classical XFEM, each node enriched by Heavisde funcion Hxhave wo addiion DOFs. H x is defined as a uni magniude for he elemens cu compleely by crack, and akes ±1 on he wo sides of he crack. For he case of he nodes wih crack ip enrichmen, eigh addiion DOFs are needed for each. The basic enrichmen funcions are inspired from he analyic soluion of near ip displacemen fields of mode I and mode II cracks, and can be wrien as he ε

4 190 Copyrigh 017 Tech Science Press CMC, vol.53, no.3, pp , 017 funcions ψ k r, [Belyschko and Black (1999)] ψk r, = r cos, sin, s in cos( ), cos sin( ) k 1,,3, 4 (3) where r, are he local polar coordinaes a he crack ip. We noe ha he second funcion in Eq. (3) is commonly referred o branch funcion, while he ohers are coninuous and added o improve he accuracy. So can be presened wih oher funcion se or jus wih he branch funcion, and i is well-documened and verified by Dolbow, Noes and Belyschko (000). ψ k r, c n Figure 1: u u Domain wih cracks and prescribed boundary condiions Node se by crack ip enrichnmen Node se by Heavisde enrichnmen Figure : Typical discreizaion of a domain wih crack and enriched nodes by XFEM As menioned above, he enrichmen funcions are developed based he displacemen fields near he crack ip, and can have differen forms. In he presen paper, a new form of enrichmen funcions is used, which derives from he displacemen fields direcly. By shifing he enrichmen funcions, we are able o correspond he enriched nodes displacemen o he rue displacemen wih XFEM. The displacemen can be wrien approximaely as

5 The Sable Explici Time Sepping Analysis 191 u( x) Ni( x) ui N j ( x) H( x) H( x j ) a j Nk ( x) βt F( x) F( xk ) b k (4) ii jj kk where I is he se of all nodes in he mesh, J is he se of nodes enriched by Heaviside funcion and K is enriched by F(x). is he sandard FEM shape funcions. and b k N i are he addiion DOFs associaed wih x Hx and F(x), respecively. he ransformaion marix beween he local coordinae and global coordinae. he crack ip enrichmen funcion in marix form β T F(x) 1 r cos ( cos ) sin ( cos ) F ( x) G sin ( cos ) cos ( cos ) (5) E where G is he shear modulus of maerial, G. (1 ) is he modulus of elasiciy, is he Poisson s raio. is he Kolosov consan, E a j is is 3-4, planesrain 3-., planesress 1 I is imporan o poin ou ha alhough sill has 4 base funcions, he number of addiional DOFs is reduced from 8 o for he nodes enriched by crack ip. The wo base funcions in firs row of marix F (x) are associaed wih he horizonal axis in he local coordinae a crack ip, and he second row corresponds o he verical axis. Wihou concerning he damping effec, we subsiue he displacemen field Eq.(4) ino he weak form Eq.(). I hen yields a sysem of linear algebraic marix equaions, which can be expressed as MU KU f (6) where M is he mass marix, K is he siffness marix and f is he force marix: M M M M= M M M M M M uu ua u au aa a u a K K K K= K K K K K K uu ua u au aa a u a F(x) (7.a) (7.b) T U u a, U u a, T u a T f f f f (7.c) The sub-marices and vecors ha come in he foregoing equaions are defined as below for four-node elemen ( i, j 1,,3, 4 ):

6 19 Copyrigh 017 Tech Science Press CMC, vol.53, no.3, pp , 017 e T d,,,, rs r s M Ν N (8) ij i j r s u a T d,,,, rs r s K B B (9) ij i j r s u a e u f Nbd N d (10.a) i i i a f N H( x) H( x ) bd N H( x) H( x ) d (10.b) i i i i i f N β F( x) F( x ) bd N β F( x) F( x ) d (10.c) i i T i i T i where a N i and N i are he marices of addiional shape funcions in XFEM, a Β i and B i are he marices of shape funcion derivaives and can be expressed as: N a i H( x) H( xi ) 0 Ni 0 H( x) H( xi ) (11.a) N N β F( x) F ( x ) (11.b) i i T i B Ni H ( x) H ( xi), x 0 0 N H ( x) H ( x ) N H ( x) H ( x ) N H ( x) H ( x ) a i i i y i i, y i i, x, (1.a) 0 x Bi 0 Ni T ( x) ( xi ) y β F F (1.b) y x

7 The Sable Explici Time Sepping Analysis The explici ime inegraion 3.1 Time inegraion As he mos commonly used for dynamic problems, he Newmark scheme is chosen as he ime inegraion algorihm. As we know, he ime inegraion algorihm can be divided ino wo ypes: he explici and he implici. Wih he implici mehod, here is no inrinsic limi o he ime sep, bu i needs o solve he global equaions by ieraing in each sep. For dynamic problems, los of ieraive calculaions are needed, using he implici scheme can bring many disadvanages such as vas compuaion and low efficiency. Compared wih he implici scheme, he explici scheme which is chosen in he presen paper can solve he equaions independenly wih no ieraive. For explici scheme, wo parameers and scheme and Eq.(6), he derived equaion can be given as are considered. Combine he Newmark U U U 1 U U (13.a) MU F KU (13.b) U U U U 1 (13.c) where is he ime sep, U is he vecor of displacemen, U and U are he vecor of velociy and acceleraion a ime, respecively. For a numeric scheme, he sabiliy, consisency and convergence are he main reference sandards. From now on, we will focus on he sabiliy of Newmark scheme because of he insabiliy is a sufficien condiion for non-convergence. The sabiliy condiions of Newmark scheme are deduced in deail by Réhoré, Gravouil and Combescure (004) wih heir cusom noaions: 1 1. if, i is an uncondiionally sable scheme. 1. If and, he sable condiion is 1. max where max is he maximum frequency of he srucure.

8 194 Copyrigh 017 Tech Science Press CMC, vol.53, no.3, pp , 017 As menioned by Réhoré, Gravouil and Combescure (004), if he parameers are chosen o be 1 4 and 1, he updaing equaions are uncondiionally sable. However, according o Eq.(13), if he acceleraion iem U sill need o be solved ieraively. Thus, for an explici ime inegraor, he presen paper. We noiced ha if ino he cenral difference mehod. 0 and exiss in Eq.(13.a), he equaions 1 0 is used in, he Newmark scheme evolves Furhermore, due o such a resricion of sabiliy condiion, here mus be a criical ime sep c. Wih he ime sep beyond he criical value, he numerical insabiliy and convergence problem will happen a some poin. In conras, he numerical resuls is very sable wihin he criical ime. We will focus on figuring ou he criical ime sep, and finding ou he facors ha can affec i in his paper. Tess wih differen grid densiies and differen parameers in he Newmark scheme will be conduced. 3. The lumped mass The marix above in Eq.(8) is known as he consisen mass marix, which includes sandard erms, block-diagonal enriched erms, and coupling erms [Menouillard, Réhoré, Combescure e al (006)]. However, for he problem of dynamic, he lumped mass marix is used more frequenly in order o simplify he numerical calculaions. Due o he exisence of addiional DOFs which have no clear physical significance, he disribuion of mass is no jus as a simple average as in radiional FEM. Menouillard, Réhoré, Noes e al (008) had in-deph sudy of he lumping echnique for he mass marix based on he conservaion of mass and momenum, and proved is effeciveness wih explici scheme for dynamics by XFEM. Besides, he lumping echnique was also researched by Zi, Chen, Xu e al (005); Elguedj, Gravouil and Maigre (009); Song and Belyschko (009); Jim, Zhang, Fang e al (016). In his paper, he lumped mass proposed by Menouillard, Réhoré, Combescure e al (006) is used m diag m 1 H n mes( ) node d (14) e e where is he elemen being considered, m is he elemen s mass, mes()is he area of elemen in D, nnode is he number of nodes in elemen, and H is he Heaviside funcion.

9 The Sable Explici Time Sepping Analysis DSIF In his secion, he DSIF is illusraed. Based on energeic consideraion, he SIF is used as a parameer of he srengh of singulariy, and some quesionable relevan quaniies of crack ip such as sress fields are avoided. There are a few schemes o calculae he SIF, such as he displacemen exrapolaion mehod, he virual crack exension mehod, he virual crack closure mehod and he ineracion inegral mehod. Due o he research of Nagashima, Omoo and Tani (003), he ineracion inegral mehod has he highes accuracy and is used here. In he ineracion inegral mehod, he auxiliary fields are inroduced and superimposed ono he acual fields. For dynamic loading case, an iem relaed o ineria is added, and he ineracion inegral wih force-free on crack surface can be given as da I u u q u u qda A where aux aux aux aux ij i,1 ij i,1 ik ik 1 j, j A i i,1 aux aux aux,, u and, u ij ij i ij, ij i (15) are he acual sae and he auxiliary sae, respecively. I is he ineracion inegral beween he acual sae and he auxiliary sae, A is he inegral domain. q is he weigh funcion which is going o be 0 ouside he conour boundary and is one inside in he presen paper. The DSIF can be wrien as K K I II I I mode I mode II E * E * (16) where * E is equal o E for he plane sress and for he plane srain E * E 1. The basic algorihm used here for he DSIF is concluded as following: (1) Give an inegral rang R, hen search for all he inegral elemens; () Loop hrough all he inegral elemens; (3) Loop hrough all he Gauss poins in each inegral elemen; (4) Calculae he acual sae and he auxiliary sae of each Gauss poin; (5) Ge he value of DSIF hrough Eq.(15) and Eq.(16). where R is he raio beween he acual inegral radius r and he minimum size L min of all elemens as shown in Fig.3.

10 196 Copyrigh 017 Tech Science Press CMC, vol.53, no.3, pp , 017 Crack Crack Tip r= RL min The inegral elemens Disribuion of Gauss poin 5 Numerical examples 5.1 Saionary mode I crack Figure 3: The inegral elemens for DSIF Firs, in order o illusrae he effeciveness of he new enrichmen scheme presened above, we sudy he problem of an infinie plae conains a semi-infinie crack as shown in Fig.4. A prescribed verical loading of 0 =500MPa is applied o he upper surface. The evoluion of he loading is a ype of Heaviside sep funcion. For he geomery of configuraion, a recangular plae of size L H=10m 4m wih an iniial edge crack of E =10Gpa, lengh a=5m is used. Some maerial parameers are: Young s modulus Poisson s raio =0.3, he densiy =8000 kg/m 3. A mesh of uniform square is used for ess. A heoreical soluion of his problem wih a saionary crack was obained by [Freund (1990)] and i is given by K I 0 c(1 ) d 1 (17) where c d is he dilaaional wave speed, =0 is he ime ha he sress wave reach he crack ip from he edge. The heoreical soluion is used o compare our presen resuls, however, i has some limiaion, such as being effecive only in 3c when he

11 The Sable Explici Time Sepping Analysis 197 refleced sress wave reaches he crack ip, where c = H c d. Fig.5 presens he values of DSIF wih differen inegral domains. The ime sep is chosen as =0.1 s. The DSIF is normalized by K 0 a and he numerical ime is normalized by c. A he beginning, ime c, due o he sress wave has no reach he inegral domain, he value of DSIF is 0. Then, afer sress wave reaches he crack ip, he resuls wih differen inegral domains are in good agreemen wih each oher as shown in Fig.5. The heoreical soluion for comparison has also been ploed, and shows good accuracy o he presen resul. Fig.6 presens he resuls of DSIF wih differen ime sepping while good consisency and he resuls are no sensiive o he ime sep. The resuls inspire us o improve he compuaional efficien wih a larger sep ime which is less han he R =5. I shows criical ime. However, wih a much larger ime sepping for es, =0 s, he numerical resul is rapidly divergen. As a consequence, i mus have a criical ime sepping, which we will discuss i laer. In addiion, he resul of DSIF wih four crack ip enrichmen funcions of sandard XFEM is also ploed for comparison in Fig.6. I can be seen ha he new enrichmen scheme offers he almos same accuracy as XFEM wih sandard enrichmen funcions. 0 a L H Figure 4: The geomery and loading of a homogeneous maerial plae wih crack

12 198 Copyrigh 017 Tech Science Press CMC, vol.53, no.3, pp , R=5,=0.1s R=4,=0.1s R=3,=0.1s R=,=0.1s Anlyiacl resuls R=5,= 10s* R=5,= 10s R=5,= 5 s R=5,= 1 s R=5,= 0.1s ( * Sandard XFEM) K I /K K I /K / c / c Figure 5: The DSIF wih differen inegral pah Figure 6: The DSIF wih differen ime sep 5. Finie size edge plae wih an arbirarily oriened cenral crack In his example, A plae wih a cenral crack under uniaxial ensions is considered. As shown in Fig.7, he dimensions of he plae is h=0.04 m and b=0.0 m, and he lengh of he cenral crack is a=0.0048m. The maerial s properies are: E=199.99Gpa, =0.3, =5000kg/m 3. A prescribed verical loading of 0 =100MPa is applied o he upper surface and he lower surface. A mesh of uniform elemens is used. Firs of all, a horizon cenral crack =0 is considered. For he cases of differen inegral domains, Fig.8 shows ha he range R has lile effec on he crack ip s DSIF. The resuls agree very well wih he conclusion ha he SIF under differen inegral pah are he same. The resuls are also compared wih he sandard XFEM wih four crack ip enrichmen funcions, and shows good accuracy o he presen resul. Fig.9 presens he numerical resuls wih differen ime sep, he same conclusion can be drawn as in Fig.6. In addiional, he numerical resul given by Song, Areias and Belyschko (006) is compared, and shows he similar accuracy. Secondly, he cenral cracks of differen inclined angles are considered. The lengh of crack is he same a= m, and he angles, =15, 30, 45, 60, 75 are esed. The problem has been discussed by Phan, Gray and Salvadori (010) wih Symmeric- Galerkin Boundary Elemen Mehod and by Liu, Bui, Zhang, e al (01) wih Smoohed Finie Elemen Mehod. For he case of mode I, as depiced in Fig.10a, he values in he

13 The Sable Explici Time Sepping Analysis 199 peak of DSIF curves decrease by he increase of for a small period of ime afer he sress wave arrive in he ip. Fig.10b reveals ha DSIF in mode II are pracically he same for he pair of =15 and =75, and he pair of =30 and =60. A he meanwhile, he curve of =45 has he highes peak value. The resuls are compared wih Phan, Gray and Salvadori (010) and shown he good accuracy. y h a x b Figure 7: The recangular plae wih crack of differen angle Figure 8: The DSIF wih differen inegral pah of he lef crack ip

14 00 Copyrigh 017 Tech Science Press CMC, vol.53, no.3, pp , R=5, = 0.01s R=5, = 0.0s R=5, = 0.001s Song (006) K I /K Time (s) Figure 9: The DSIF of he lef crack ip wih differen ime incremens (a).5.0 Phan(010) presen resul (b) 1. Phan(010) presen resul K I /K K II /K Time(s) Time(s) Figure 10: The DSIF of crack ip wih differen roaion angle: (a) Mode I; (b) Mode II 5.3 The sable explici ime sepping analysis This par focuses on he main facors ha influence he criical ime sep. The grid densiy and ieraion form are he wo main subjecs considered here. The experimen configuraion model is presened in Fig.7 wih =0. The maerial properies and he oher parameers are he same as ha used in las example. The mehod of numerical approximaion is used o ge he criical ime. Firsly, he resuls wih differen grid densiies were obained. Three uniform meshes are considered, which are of CCT: 49 99, CCT1: 4 49, CCT: 13 4 elemens. Wih parameers =0, =1/ of Newmark scheme, he criical ime sep of differen meshes can be urned ou. As shown in Fig.11(a), he criical ime we go is abou = s wih elemens. When he ime sep is less han c, he numerical calculaion resuls are compleely consisen and do no produce divergence. Conversely, c c

15 The Sable Explici Time Sepping Analysis 01 divergence is presened in he calculaion when c. The divergence occur a abou μs, μs, μs, μs when is s, s, s, s, respecively. As a comparison, Fig.11(b) is presened wih he mesh of I is seen ha he criical ime is s which is improved han he one in Fig.11(a). The divergence occurs a abou μs, μs, μs, is s, s, s, s, respecively. KI/K μs when r54e8f r55e8f r54.8e8f r54.9e8f 4.85f 4.85f f KI/K e7ki 1.01e7ki e7ki 1.005e7f 1.005e7f (us) Figure 11: Numerical sabiliy wih differen ime sepping (R=5, 49 99, (b) CCT1: =0, Table 1: The criical ime sepping for differen densiies of grid (R=5, c ( μs CCT:49 99 CCT1:4 49 CCT:13 4 ) lump fem ( μs ) lump c fem % % % =1/): (a) CCT: =0, =1/) To clarify his case furher, we repeaed he above seps wih CCT: 13 4 elemens, and he comparison resuls are shown in he Table 1. The criical ime sep is abou s in he case of CCT, which is larger han he case of CCT1. I is hence concluded ha he criical ime sep decreases wih he increase of grid densiy. Besides, he criical ime sep of he sandard FEM for he lumped mass is also lised. Wih more lump elemens, he criical ime sep fem is decreased, and his is consisen wih he case of c. The values of lump c fem are similar, which range from % o %. As Menouillard, Réhoré, Combescure e al (006); Elguedj, Gravouil and Maigre (009) suggesed, lump for he saionary crack, he value /3 is wihin he numerical range 3 fem lised in his paper. So he numerical sabiliy can be guaraneed.

16 0 Copyrigh 017 Tech Science Press CMC, vol.53, no.3, pp , 017 In addiion, we ook ino accoun he effec of Newmark scheme for he criical ime sep. Four cases are concerned. Before sudying he impac of ieraive forma on criical ime sep, all he cases are lised under he same condiions: CCT1, a mesh of 4 49 elemens, R=5, in Fig.1. An approximaely idenical resul can be obained. The sabiliy condiions of he Newmark scheme are also verified direcly. = s. We lised he firs 30 microseconds wih differen parameer values In Fig.11(b), we presened he es resul wih =1/. As a comparison, he resul wih =/3 is shown in Fig.13. The divergence occur a abou μs, μs, μs when is s, s, s, respecively. The criical ime we obained is abou c = s, which is smaller han he case of =1/. For furher invesigaion, he cases of =3/4, =1 are esed. The resuls are lised in Table.. The criical ime sep of he sandard FEM for he lumped mass are also lised. In Table., i is seen ha he criical ime sep decreases wih he increase of. So does. lump Furhermore, we observed ha he values of c fem are nearly he same (abou 91%), lump and he parameer have nearly no influence on he values of. c fem lump fem gama1ki gama3ki gama34ki gama1ki 3 5 KI/K (us) Figure 1: Numerical resuls wih differen parameers (CCT1: 4 49, R=5, =0, = s)

17 The Sable Explici Time Sepping Analysis e8ki 8.8e8ki 8.7e8ki 8.685e8ki 3 KI/K (us) Figure 13: Numerical sabiliy wih differen ime sepping(cct1: 4 49, R=5, =/3) Table : The criical ime sepping for differen parameers R=5, =0) c ( μs 1/ /3 3/4 1 =0, (CCT1: 4 49, ) lump fem ( μs ) lump c fem % % % % 6 Conclusions In he presen paper, we carried ou some numerical experimens of he sable explici ime sepping wihin he XFEM framework. A new enrichmen scheme for crack ip is proposed and is applicabiliy and availabiliy has been sufficienly verified. The DSIF is used as an imporan parameer of he dynamic response and is also a parameer of judging he sabiliy of numerical mehod. Objecive o sudying he facors ha can affec he sabiliy, differen densiies of grid and differen parameers of Newmark scheme have been esed. The conclusions are shown as: The grid densiy and he form of ieraive mehod have obvious effecs on sabiliy; The criical ime sepping c decreases wih he increase of grid densiy; The criical ime sepping c decreases wih he increase of he parameer beween 0.5 and 1 of Newmark scheme;

18 04 Copyrigh 017 Tech Science Press CMC, vol.53, no.3, pp , 017 A similar conclusion can be obained by he sandard FEM wih he lumped mass, and he values of are relaively sable. lump c fem Furhermore, he simulaion resuls are found in good agreemen wih each oher when hey are sable. Therefore, increasing ime sepping appropriaely in he range of criical value can improve he compuaional efficiency. Acknowledgmen: The auhors are graeful o he Naional Naural Science Foundaion of China (No , No ), he 1h Five-Year Supporing Plan Issue (No. 015 BAB07B10), Jiangsu Province Naural Science Fund Projec (No. BK ) and he Posgraduae Research and Innovaion Projecs in Jiangsu Province (No.014B 31614) for he financial suppor. References Belyschko, T.; Black, T. (1999): Elasic crack growh in finie elemens wih minimal remeshing. Inernaional Journal for Numerical Mehods in Engineering, vol. 45, no. 5, pp Belyschko, T.; Chen, H. (004): Singular Enrichmen Finie Elemen Mehod for Elasodynamic Crack Propagaion. Inernaional Journal of Compuaional Mehods. vol. 1, no.1, pp Belyschko, T.; Chen, H.; Xu, J.; Zi, G. (003): Dynamic crack propagaion based on loss of hyperboliciy and a new disconinuous enrichmen. Inernaional Journal for Numerical Mehods in Engineering, vol. 58, no. 1, pp Belyschko, T.; Moes, N.; Usui, S.; Parimi, C. (001): Arbirary disconinuiies infinie elemens. Inernaional Journal for Numerical Mehods in Engineering, vol. 50, no. 4, pp Dolbow, J.; Moes, N.; Belyschko, T. (000): Disconinuous enrichmen in finie elemens wih a pariion of uniy mehod. Finie Elemens in Analysis and Design, vol. 365, no.3-4, pp Duan, Q.; Song, J.H.; Menouillard, T. (009): Belyschko. Elemen-local level se mehod for hree-dimensional dynamic crack growh. Inernaional Journal for Numerical Mehods in Engineering, vol. 80, no. 1, pp Elguedj, T.; Gravouil, A.; Maigre, H. (009): An explici dynamics exended finie elemen mehod. Par 1: Mass lumping for arbirary enrichmen funcions. Compuer Mehods in Applied Mechanics and Engineering, vol.198, no. 30-3, pp Freund, L. B. (1990): Dymamic fracure mechanics. Cambridge Monographs on Mechanics and Applied Mahemaics. Jim, L.; Zhang, T.; Fang, E.; Song, J. H. (016): Explici phanom paired shell elemen approach for crack branching and impac damage predicion of aluminum srucures. Inernaional Journal of Impac Engineering, vol. 87, pp Kumar, S.; Singh, I.V.; Mishra, B.K.; Singh, A. (015): New enrichmens in XFEM o model dynamic crack response of -D elasic solids. Inernaional Journal of Impac Engineering, vol. 87, pp

19 The Sable Explici Time Sepping Analysis 05 Liu, P.; Bui, T.; Zhang, C.; Yu, T.T.; Liu, G.R.; Golub, M.V. (01): The singular edge-based smoohed finie elemen mehod for saionary dynamic crack problems in D elasic solids. Compuer Mehods in Applied Mechanics & Engineering, vol , no. 4, pp Menouillard, T.; Réhoré, J.; Combescure, A.; Bung, H. (006): Efficien explici ime sepping for he exended Finie Elemen Mehod(X-FEM). Inernaional Journal for Numerical Mehods in Engineering, vol. 68, no. 9, pp Menouillard, T.; Réhoré, J.; Moes, N.; Bung, H. (008): Mass lumping sraegies for X-FEM explici dynamics: Applicaion o crack propagaion. Inernaional Journal for Numerical Mehods in Engineering, vol. 74, no. 3, pp Menouillard, T.; Song, J.H.; Duan, Q.; Belyschko, T. (010): Time dependen crack ip enrichmen for dynamic crack propagaion. Inernaional Journal of Fracure, vol. 6, no. 1-, pp Moes, N.; Dolbow, J.; Belyschko, T. (1999): A finie elemen mehod for crack growh wihou remeshing. Inernaional Journal for Numerical Mehods in Engineering, vol. 46, no. 1, pp Nagashima, T.; Omoo, Y.; Tani, S. (003): Sress inensiy facor analysis of inerface cracks using X-FEM, Inernaional Journal for Numerical Mehods in Engineering, vol. 56, no. 8, pp Nisor, I.; Panale, O.; Caperaa, S. (008): Numerical implemenaion of he exended finie elemen mehod for dynamic crack analysis. Advances in Engineer Sofware, vol. 39, no. 7, pp Phan, A. V.; Gray, L. J.; Salvadori, A. (010): Transien analysis of he dynamic sress inensiy facors using SGBEM for frequency-domain elasodynamics. Compuer Mehods in Applied Mechanics & Engineering, vol. 199, no. 45-8, pp Rabczuk, T.; Zi, G.; Gersenberger, A.; Wall, W.A. (008): A new crack ip elemen for he phanom-node mehod wih arbirary cohesive cracks, Inernaional Journal for Numerical Mehods in Engineering, vol. 75, no. 5, pp Réhoré, J.; Gravouil, A.; Combescure, A. (004): A sable numerical scheme for he finie elemen simulaion of dynamic crack propagaion wih remeshing. Compuer Mehods in Applied Mechanics & Engineering, vol. 193, no. 4-44, pp Réhoré, J.; Gravouil, A.; Combescure, A. (005a): An energy-conserving scheme for dynamic crack growh using he exended finie elemen mehod, Inernaional Journal for Numerical Mehods in Engineering, vol. 63, no. 5, pp Réhoré, J.; Gravouil, A.; Combescure, A. (005b): A combined space-ime exended Finie Elemen Mehod. Inernaional Journal for Numerical Mehods in Engineering, vol. 64, no., pp Song, J. H.; Belyschko, T. (009): Dynamic fracure of shells subjeced o impulsive loads. Journal of Applied Mechanics, vol. 76, no. 5, pp Song, J. H.; Belyschko, T. (009): Cracking ong node mehod for dynamic fracure wih finie elemens, Inernaional Journal for Numerical Mehods in Engineering, vol. 77, no. 3, pp

20 06 Copyrigh 017 Tech Science Press CMC, vol.53, no.3, pp , 017 Song, J. H.; Areias, P. M. A.; Belyschko, T. (006): A mehod for dynamic crack and shear band propagaion wih nodes phanom nodes. Inernaional Journal for Numerical Mehods in Engineering, vol. 67, no. 6, pp Tian, R.; Wen, L. (016): Improved XFEM: An exra-dof free, well-condiioning, and inerpolaing XFEM. Compuer Mehods in Applied Mechanics & Engineering, vol. 85, no. 3, pp Wen, L.; Tian, R. (016): Improved XFEM: Accurae and robus dynamic crack growh simulaion. Compuer Mehods in Applied Mechanics & Engineering, vol. 308, pp Zi, G.; Chen, H.; Xu, J.; Belyschko, T. (005): The exended finie elemen mehod for dynamic fracures. Shock and Vibraion, vol. 1, no. 1, pp. 9-3.