An evaluation of MPM, GIMP and CMPM in geotechnical problems considering large deformations

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1 Delft Uiversity of Tehology A evaluatio of MPM, GIMP ad CMPM i geotehial roblems osiderig large deformatios Gozalez Aosta, Leo; Vardo, Phil; Hiks, Mihael Publiatio date 07 Doumet Versio Peer reviewed versio Published i Proeedigs of the 5th Iteratioal Coferee of the Iteratioal Assoiatio for Comuter Methods ad Advaes i Geomehais Citatio (APA) Gozalez Aosta, L., Vardo, P., & Hiks, M. (07). A evaluatio of MPM, GIMP ad CMPM i geotehial roblems osiderig large deformatios. I Proeedigs of the 5th Iteratioal Coferee of the Iteratioal Assoiatio for Comuter Methods ad Advaes i Geomehais : 9-3 Otober 07, Wuha, Chia Imortat ote To ite this ubliatio, lease use the fial ublished versio (if aliable). Please hek the doumet versio above. Coyright Other tha for stritly ersoal use, it is ot ermitted to dowload, forward or distribute the text or art of it, without the oset of the author(s) ad/or oyright holder(s), uless the work is uder a oe otet liese suh as Creative Commos. Takedow oliy Please otat us ad rovide details if you believe this doumet breahes oyrights. We will remove aess to the work immediately ad ivestigate your laim. This work is dowloaded from Delft Uiversity of Tehology. For tehial reasos the umber of authors show o this over age is limited to a maximum of 0.

2 9-3 Otober 07, Wuha, Chia A evaluatio of MPM, GIMP ad CMPM i geotehial roblems osiderig large deformatios L. Gozalez Aosta a *, P.J. Vardo a ad M.A. Hiks a a Geo-Egieerig Setio, Delft Uiversity of Tehology, the Netherlads * J.L.GozalezAosta-@tudelft.l Abstrat Stress osillatios are a ommo heomeo i the material oit method (MPM), sie this umerial method tyially uses regular fiite elemet (FE) shae futios to ma variables from surroudig odes to material oits ad vie versa, ideedetly of the loatios of the material oits withi a elemet. I itegratio this leads to a quadrature rule error ad, i strai ad stress alulatios, derivatives of tyial FE shae futios are disotiuous aross elemet boudaries ad do ot give aurate results away from Gauss oit loatios withi elemets. I geotehial aalysis, the ostitutive behaviour is geerally stress-deedet, ad therefore stress osillatios a ause severe iauraies i the alulated mehaial behaviour, iludig the develomet of wrog elasto-lasti quatities. Several attemts to imrove stress reovery ad redue quadrature errors have bee develoed, but seldom has a full omariso betwee methods bee made. I this aer, behmark small sale ad sloe stability roblems have bee examied i order to omare the relative erformae of the lassi material oit method (MPM), the geeralized iterolatio material oit (GIMP) method ad the ew omoud material oit method (CMPM). Keywords: CMPM, GIMP, Material Poit Method, Sloe Stability, Softeig, Stress Osillatio.. Itrodutio The material oit method (MPM) [-] has rove to be a valid umerial tehique to simulate geotehial behaviour, eseially whe simle ostitutive models are used, sie these models are related with variables less sesitive to stress osillatios, as i the ase of the Vo Mises riterio. I reality, the use of omlex ostitutive models is ofte eeded to esure the orret evolutio of the mehaial behavior of the soil, ad therefore the issue of stress osillatios should be addressed. The geeralized iterolatio material oit (GIMP) method [3] was reated to overome errors aused by the use of regular fiite elemet (FE) shae futios i MPM. I this method, FE shae futios are relaed by C shae futios, esurig otiuity i stresses whe material oits ross iter-elemet boudaries. As well as GIMP, the omoud material oit method (CMPM) [4] was develoed to redue the flutuatio i stresses aused by the ill-osed material oit loatios. CMPM uses C shae futios, ad the evaluatio of strais at eah material oit omes from the iterolatio of dislaemets from all odes of the elemet i questio ad the eighbourig elemets, istead of usig oly odes iside a suort domai as i GIMP. I the first setio of this aer, the theory of MPM, GIMP ad CMPM are outlied, followed by the aalysis of a behmark roblem to demostrate how stresses osillate usig these tehiques. A vertial uttig roblem is the aalyzed, to visualise how eah method redits the stresses develoed durig sloe failure.. Theoretial formulatio Sie MPM shares similarities with the FEM otiuum mehais framework [5-6], the weak form of mometum oservatio is, as tyial i FEM, give as s a udv σudv b udv τ uda () where is the material desity, a is the aeleratio, u is the virtual dislaemet, V is the body volume, is the Cauhy stress, b are the body fores, s are the resribed tratios, is the tratio boudary, ad is the solutio domai. Sie the domai is rereseted as oits of mass, the total mass i the domai a be omuted as [7]

3 9-3 Otober 07, Wuha, Chia m x dv () where m is the mass of the oit, is the umber of oits i the domai, ad x is the ositio of ay oit. Substitutig Eq. () ito Eq. () gives s m u( x ) a( x ) m σ( x ) u( x ) m b( x ) u( x ) m τ ( x ) u( x ) (3) Usig the Galerki method, the virtual dislaemet is aroximated as x v v v u u N (x ) (4) where N v is the shae futio, subsrit v deotes grid ode v, is the umber of odes i the elemet, ad x is the material oit ositio i the elemet. Substitutig Eq. (4) ito Eq. (3) ad followig stadard FE roedures [8], leads to m Nv x ) a mσ( x ) Nv ( x ) mb( x ) Nv ( x ) s ( m τ ( x ) N ( x v ) (5) It is evidet that oservatio of mometum is ahieved by usig shae futios (terms, 3 ad 4 i Eq. (5)) ad shae futios derivatives (term i Eq. (5)). This leads to a variatioal iosistey, whih beomes more imortat durig material oit elemet rossig, due to term beig disotiuous betwee elemets whe usig C 0 shae futios. I geeral, MPM is limited to liear (C 0 ) shae futios, sie the use of quadrati elemets a lead to errors whe egative arts of the shae futios are used, i artiular i the mass matrix. I additio, for imliit formulatios, durig the alulatio of the stiffess matrix the o-gaussia ositio of the material oit i the umerial itegratio a ause some terms of the strai-dislaemet matrix to beome egative, leadig to a urealisti elemet deformatio. It has bee roosed to redue the imbalae aused by the use of C 0 shae futios by usig modified liear shae futios, so that they beome C shae futios, imrovig the satial itegratio [9-0]. This aroah is used i GIMP, where the origial FE shae futios ad their gradiets are relaed by a set of otiuous shae futios ad shae futio gradiets with a oloal suort, dimiishig the imbalae ause by loal FE shae futios ad elimiatig elemet rossig oise. If a regular grid is used, osiderig oits er elemet for the iitial oditio ad avoidig distortio of the material oit domai, GIMP shae futios a be omuted, for examle for D elemets, as [] (7 6 )/ / 6 0 N i where is the loal oordiate. Fig. shows the GIMP shae futio give by Eq. (6). (6) Fig.. GIMP shae futio suort domai.

4 9-3 Otober 07, Wuha, Chia CMPM uses C shae futios to alulate the strais ad stresses, by ireasig the solutio domai usig odes from the urret ad eighbourig elemets, leadig to a smooth distributio withi a elemet. CMPM does ot elimiate elemet rossig oise, but it redues osillatios aused by disotiuous stresses ad lower order shae futio derivatives, leadig to a smaller stress disotiuity betwee elemets. Fig. shows two ossible ositios for a material oit (labeled m) i the domai; i a boudary elemet ad i a etre elemet. To iterolate strais i a D elemet, the boudary ad etre shae futios (N i b ad N i resetively) are omuted as [] b ξ ξ 3 N 8 (7) b N ξ ξ 3 4 b N 3 ξ ξ 8 N N N N ξ ξ ξ ξ 3ξ ξ ξ 9 3 3ξ 3 (8) (a) (b) Fig.. Shae futios for (a) a boudary elemet, ad (b) a etre elemet. It is imortat to emhasise that CMPM shae futios are oly used to evaluate strais, ad that regular FE shae futios are still used to omute the goverig equatio matries. I the followig setios, imliit ad a exliit odes are used to solve behmark ad sloe stability roblems (resetively). Note that stress osillatios aused by quadrature errors ad shae futio disotiuities our i both these methods (imliit ad exliit), but i the imliit method extra osillatios are observed due to the stiffess matrix itegratio. 3. Stress osillatio i MPM, GIMP ad CMPM A illustratio of stress osillatios usig MPM, GIMP ad CMPM is reseted i this setio. A axisymmetri aalysis of a thik-walled hollow ylider has bee arried out usig 4 oded quadrilateral elemets. Radial dislaemet is alied iremetally at the ier boudary (r = r i ) ad the outer boudary is omletely fixed (r = r e ), as show i Fig. 3. The ylider is assumed to be liear elasti, with E = 5000 kpa ad v = 0.3. The solutio is obtaied usig a quasi-stati imliit ode. Fig. 3. Thik-walled ylider uder iteral ressure.

5 9-3 Otober 07, Wuha, Chia Differees betwee MPM, GIMP ad CMPM stress osillatios are show i Fig. 4, i whih a dislaemet of.5 m has bee alied i a sigle ste ad the stress omuted aross the domai. Fig. 4. Radial stress osillatio usig MPM, GIMP ad CMPM. Fig. 4 shows that radial stresses omuted usig GIMP ad CMPM are loser to the aalytial solutio, whereas MPM stresses osillate more. It is also evidet that the CMPM solutio osillates sigifiatly less tha that for GIMP, although the GIMP solutio is otiuous aross elemet boudaries. However, the GIMP solutio is iomlete at the domai boudaries, sie iterolatio usig ghost odes was ot imlemeted i this examle. Sie a imliit ode is used, stiffess matries are used to solve the equatio of motio. Followig a similar aroah as i FEM, the stiffess matrix is omuted as K m S B T DB W V (9) where S is the umber of material oits withi a elemet, B is the strai-dislaemet matrix, D is the elasti stress-strai matrix, W are the material oit itegratio weightigs, iitially seleted to have a value of, ad V are the material oit volumes. As a be see i Eq. (9), the elemet stiffess matrix is a futio of the umber of material oits iside a elemet, leadig to a variatio of the elemet stiffess durig elemet rossig ad thereby to the osillatio of stresses. A alterative weightig value is here roosed to be a futio of both the elemet volume ad the volume of the material oits withi the elemet: V W (0) where V is the elemet volume ad e is the umber of material oits i the elemet. I Fig. 5, the deviatori stress (q) of a sigle material oit (m-a from Fig. 3) is lotted for 50 stes, eah of.5 m. As a be see i Fig. 5a, stress osillatio is observed usig all methods, but GIMP ad CMPM follow the aalytial solutio more losely. It is also otieable that, durig elemet rossig, a jum i the stresses ours. The effet of usig the modified weightig value is show i Fig. 5b. The results irove osiderably, due to stiffess i the domai beig more evely distributed. e V (a) (b) Fig. 5. m-a deviatori stress usig (a) regular stiffess matrix, ad (b) modified stiffess matrix.

6 9-3 Otober 07, Wuha, Chia 4. Geotehial aliatio To demostrate the use of MPM, GIMP ad CMPM i geotehial roblems, a vertial uttig stability roblem has bee aalyzed, usig a Vo Mises ostitutive model iororatig ost-eak softeig as desribed i [3]. Fig. 6 shows the iitial geometry of the roblem, haraterized by a height of m ad a width of m. The material is a elasto-lasti strai-softeig soil, with elasti arameters E = 400 kpa ad v = The eak ad residual ohesios are 6 kpa ad kpa resetively, ad the softeig modulus is -5 kpa. I the domai, the left vertial boudary is fixed to revet material oit dislaemet i the horizotal diretio, ad the bottom boudary is fixed to revet material oit dislaemet i the vertial diretio. The sloe is loaded by gravity ad the failure ours due to its self weight. A exliit form of the ode is used to solve the roblem. m m 4. Results Fig. 6. Sketh of the iitial material oit loatios ad bakgroud mesh. The results after 0 m of horizotal dislaemet at the toe are show i Fig. 7. As a be see, the stresses osillate the most whe usig MPM, whereas GIMP ad CMPM show a smoother develomet of the stresses ad lasti strais aroud the develoig bad. Durig the earlier stes i the aalysis, CMPM shows the smoothest results, but, sie regular FE shae futios are used to erform the umerial itegratio, the CMPM auray dereases after several mesh rossigs our. Nevertheless, CMPM results are better tha MPM beause the imbalae aused by elemet rossig is atteuated by the use of o-loal suort (i.e. odes outside the elemet). After large deformatio, GIMP exhibits the smoothest results sie the stresses are otiuous. The lasti deformatios show are osistet betwee GIMP ad CMPM, but more diffuse with MPM. MPM GIMP CMPM Fig. 7. Horizotal stress, shear stress ad aumulated lasti strai otours with MPM, GIMP ad CMPM.

7 9-3 Otober 07, Wuha, Chia 5. Colusios This aer has ivestigated the imbalae aused by the use of regular FE shae futios to solve the goverig equatio i MPM, leadig to the well-kow stress osillatio roblem. The use of CMPM ad GIMP imroves the auray of the results, sie C, derivative otiuous, shae futios are used to iterolate material oit strais, reduig the osillatio aused by quadrature rule errors ad elemet-rossig issues. It has bee show that CMPM a be used i imliit ad exliit odes, ad is able to omute more aurate stresses tha MPM ad GIMP. However, the auray of CMPM is oly higher tha GIMP util a ertai oit. This is beause CMPM uses regular FE shae futios to omute iteral ad exteral loads, ad this leads to osillatios whe elemet rossig ours. At this oit, GIMP offers better results. The additioal osillatios aused by the stiffess matrix itegratio i imliit odes was also demostrated. Sie the stiffess of a elemet is govered by the umber of material oits withi a elemet ad the material oit weightig value, durig elemet rossig a irregular stiffess distributio over the domai is obtaied, ausig jums i stresses. It has bee show that this roblem a be redued by the use of a modified weightig value that is a futio of the iitial elemet volume ad the sum of the material oit volumes iside the urret elemet, leadig to a smoother stiffess distributio. Referees [] Sulsky D, Che A, Shreyer HL. A artile method for history deedet materials. Comut. Methods Al. Meh. Egg. 994; 8: [] Sulsky D, Zhou S, Shreyer HL. Aliatio of a artile-i-ell method to solid mehais. Comuter Physis Commuiatios. 995; 87: [3] Bardehage S, Kober E. The geeralized iterolatio material oit method. CMES: Comut. Model. Eg. Si. 004; 5: [4] Gozalez Aosta L, Vardo PJ, Hiks MA. Comoud material oit method (CMPM) to imrove stress reovery for quasi-stati roblems. Proedia Eg. 07; 75: [5] Che ZP, Qiu XM, Zhag X, Lia YP. Imroved oulig of fiite elemet method with material oit method based o a artile-to-surfae otat algorithm. Comut. Methods Al. Meh. Egg. 05; 93: 9. [6] Wag B, Vardo PJ, Hiks MA. Ivestigatio of retrogressive ad rogressive sloe failure mehaisms usig the material oit method. Comut. Geoteh. 06; 78: [7] Wag B. Sloe failure aalysis usig the material oit method. PhD thesis, Delft Uiversity of Tehology, 06. [8] Bathe KJ. Fiite elemet roedures. Pretie Hall, New Jersey, 98. [9] Steffe M, Wallstedt PC, Guilkey JE, Kirby RM, Berzis M. Examiatio ad aalysis of imlemetatio hoies withi the material oit method (MPM). Comut. Model. Eg. Si. 008; 3: [0] Steffe M, Kirby RM, Berzis M. Aalysis ad redutio of quadrature errors i the material oit method (MPM). It. J. Numer. Meth. Egg. 008; 76: [] Zhag X, Che Z, Liu Y. The material oit method. A otiuum-based artile method for extreme loadig ases. Aademi Press, 06. [] Sadeghirad A, Astaeh AV. A fiite elemet method with omosite shae futios. Egieerig Comutatios: It. J. Comuter-Aided Egg. Software. 0; 8: [3] Wag B, Vardo PJ, Hiks MA. Develomet of a imliit material oit method for geotehial aliatios. Comut. Geoteh. 06; 7: