Modeling of Finite Gradient Elasticity Using Optimal Transportation Method

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1 9 th Iteratioal Cogress of Croatia Society of Mechaics September 2018 Split, Croatia Modelig of Fiite Gradiet Elasticity Usig Optimal Trasportatio Method Boris JALUŠIĆ *, Christia WEIßENFELS +, Jurica SORIĆ *, eter WRIGGERS + * Faculty of Mechaical Egieerig ad Naval Architecture, Uiversity of Zagreb, Ivaa Lučića 5, Zagreb, Croatia. s: {boris.jalusic, jurica.soric}@fsb.hr + Istitut für Kotiuumsmechaik, Leibiz Uiversität Haover, Appelstraße 11, Haover, Germay. {weissefels, wriggers}@ikm.ui-haover.de Abstract. A meshfree Optimal Trasportatio Method (OTM) is applied for modelig of gradiet hyperelasticity usig higher-order theory based o oly oe microstructural parameter. The OTM method is utilized for the purpose of structural behaviour modelig udergoig large deformatio. The idepedet displacemet variables are approximated usig iterpolatig maximum-etropy fuctios. This eables the satisfactio of the boudary coditios i a simple ad straightforward maer without the eed for the calculatio of additioal parameters. Novel stabilizatio algorithm employig oe material poit is preseted. The aalysis of umerical performace of the proposed approach is demostrated by a represetative umerical bechmark example. The obtaied results are commeted ad arise umerical problems are disscussed. 1 Itroductio At the momet, a wide variety of differet meshfree methods are beig employed for modelig of various egieerig problems. The reaso behid this applicatio are beefits ad advatages i compariso with stadard mesh-based methods, like Fiite Elemet Method (FEM) [1]. The meshless umerical approaches are able to overcome problems such as elemet distortio ad time-demadig mesh geeratio process. Nevertheless, the umerical itegratio ad calculatio of meshfree shape fuctios ad its derivatives still represet a major task due to the associated high computatioal costs [2]. I the preset cotributio the OTM formulatio [3] is adapted for modelig of deformatio of homogeeous structures employig strai gradiet elasticity theory [2]. I additio, the strai gradiet elasticity based o the theory with oly oe microstructural parameter is cosidered. The gradiet theory is utilized i order to more accurately capture the material behaviour, to remove discotiuities from strai ad stress fields ad to simulate the size effects. Size effects deped o the microstructure of the material ad ca be observed whe the size of the specime is sufficietly small, i.e. approximately the size of the microstructure. At this legth scale, specimes with similar shape but differet dimesios show differet mechaical behaviors. The solutio of fourth-order differetial equatios arisig i cosidered theories requires a high-order cotiuity of approximatio fuctios. Hece, usig the FEM for solvig this

2 type of problems is ot a wise choice sice stadard formulatios eed to possess C 1 cotiuity, which leads to complicated shape fuctios with large umber of odal degrees of freedom [4], eve if mixed elemets are utilized [5]. I compariso, the required cotiuity i the meshfree methods is obtaiable i a simple by addig more odes withi the approximatio domai [6]. The applied OTM method is based o the combiatio of the optimal trasportatio theory [7] with material-poit samplig usig the max-etropy meshless approximatio [8]. It eforces the mass trasport ad essetial boudary coditios exactly ad should be free from tesio istabilities. Furthermore, the method coserves liear ad agular mometum [3]. The ivestigated global domai is approximated with two differet sets of poits, the discretizatio odes ad material poits. At the odes the displacemets as the primary variables are computed by solvig the discretized equatio of motio. O the other had, at the material poits data associated with material is determied. This icludes desity, strai, stress, secod-order stress, etc. Furthermore, the material poits are used as itegratio poits. The algorithms are based o the explicit time itegratio. I every time step the positio of discretizatio poits ad the material poits is calculated usig updated Lagragia scheme. The OTM algorithm is combied with the stadard meshfree search algorithm where a arbitrary umber of odes ca be assiged to a material poit. Thus, the shape fuctios ca become highly oliear if a larger umber of odes are located withi the approximatio domai. For the itegratio of the goverig equatios oly oe material poit is used as i the origially proposed paper [3] to lower the computatioal time effort. It has bee observed that oe poit is eough whe solvig the problem usig classical cotiuum theory [9], but ot might be whe o-local (higher-order) cotiuum equatios is employed. Hece, curretly the ovel stabilizatio algorithm usig pealty method similar to oe i [9] is beig developed that could alleviate the methods stability issues. The OTM framework for the modelig of deformatio resposes of homogeeous material usig gradiet elasticity is preseted ad explaied at large i Sectio 2. Herei, the cosidered stabilizatio algorithm is also preseted. I Sectio 3. the proposed stabilizatio algorithm is tested o a problem of the truss subjected to tesile test usig uiform displacemets at both eds. I Sectio 4 cocludig remarks ad further research guidelies are give. 2 Optimal Trasportatio Method for Fiite Gradiet Elasticity 2.1 Goverig equatios for a higher-order oliear cotiuum The three dimesioal higher-order oliear cotiuum which occupies the global computatioal domai surrouded by the global outer boudary is cosidered. Accordig to [10], the goverig equatio is the weak form of the dyamic equilibrium writte as ηdiv(div ) dv ηdiv dv ηbdv ηφ d V =0, (1) V V V V 2

3 where is the desity ad φ defies the secod-order time derivative of deformatio map (positio vector φ( X, t) x i the curret cofiguratio) defied as φ 12φ φ 1 φ. (2) 2 t Furthermore, the displacemet vector is defied ad calculated as ux (, t) φ( X, t) X. I relatio (1) η deotes the admissible test fuctio, the iola stress tesor ad the double stress tesor. The tesors ad are defied as the partial derivatives of free eergy desity W W ( F) W G. The free eergy desity is depedet o the first-order deformatio gradiet F φ ad the secod-order deformatio gradiet x G φ. Thus, the secod- ad third-order stress tesors are computed as x x W, W. (3) F G The utilized free eergy desity fuctio correspods to the Neo-Hooke material used i [11] ad is take as 1 1 F F F, (4) 2 2 l : 3 2l W 2 J J W G l G G, (5) 2 where ad deote the Lame`s costats, J the determiat of F ad l the parameter associated with the size of the microstructure of the higher-order cotiuum. I order to derive the appropriate equatio to discretize the relatio (1) is firstly itegrated by parts (the first part two times), after that the divergece theorem is applied ad the decompositio of the gradiet operator is doe to properly idetify the higherorder boudary coditios [10] associated with the used cotiuum leadig to ( η)dv η t d ηt d ηbdv ηφ d V =0, (6) x x x V V V where t ad t represet the stadard tractio vector ad the higher-order tractio vector, respectively. The weak form i (6) has to fulfill the essetial boudary coditios u u, (7) u u, (8) x x 3

4 ad atural boudary coditios t =( Div ) N xn( I N N) : I : ( N N) ( x( N) ( I N N)) : I t (9) t N N t, (10) = : outer boudary. 2.2 Optimal Trasportatio Meshfree algorithm I compariso to classical meshfree methods like Elemet Free Galerki (EFG) [12] method ot oly the positio of odes but also the positios of material poits are updated durig calculatio withi the updated Lagragia framework. I this cotributio the OTM algorithm is derived usig the cetral differece time itegratio scheme (2) i the preseted weak form (6). Furthermore, the domai aroud each material poit is defied as a port (itegratio) domai usig the appropriate search algorithm. At every material poit the test fuctio η ad the deformatio map φ are approximated usig the max-etropy shape fuctios [8] ad odal values N p ηp N I x p η I, N I 1 N p φ x φ. (11) p I p I I 1 Herei N p is the umber of odes i the curret port domai for each material poit. Usig the approximatios (11) equatio (6) is trasformed to a algebraic equatio M 2 I 2 I ( t) φ F G R ( t) φ. (12) I relatio (12) the icremets used are defied as φ φ φ ad -1 φ φ φ. The stadard ad higher-order forces at the outer boudary are give as ad, equal to mp N p N ( x ) t v, (13) p1 I 1 I p p mp N p N ( x ) N t v, (14) p1 I 1 while the global mass matrix I p p M ad residual vectors R, F ad G are 4

5 mp Np Np M N ( x ) I N ( x ) m, (15) p1 I1 J1 I p J p p mp N p R N ( x ) b m, (16) p1 I 1 I p p mp N p F B ( x ) v, (17) p1 I 1 I p p mp N p G H ( x ) : v. (18) p1 I 1 I p p I relatios (13) - (18) the m p represets the mass ad v p the volume at each material poit. The matrices B I ad H I cotai the first- ad secod-order derivatives of the +1 shape fuctios of ode I. By solvig (12) the icremetal positio vector φ I is determied ad the odal positio vector is updated φ φ φ. (19) I I I Furthermore, i order to reduce the computatioal time the lumped mass cocept [13] withi the OTM framework is utilized. 2.3 Stabilizatio procedure For the purpose of preservig the same computatioal time ad icreasig the accuracy of the method the ovel stabilizatio procedure is proposed. The procedure is based o method of pealizig the egative effects resultig from uderitegratio [14]. The developed stabilizatio algorithm is based o the assumptio of oly quadratic distributio of displacemets withi the ifluece domai. This is doe by computig the ormalized error at every material poit iside the ifluece domai. The calculated error is the multiplied with the appropriate pealty parameter ad subtracted from the odal residual vector r I assembled aalogously usig the vectors R, F ad G but utilizig the material poits withi the domai of ifluece. The ormalized error criterio for every poit p withi the ifluece domai of the ode I is e Ip x x x x dx dx I p I p = d X X X I 1 p 1, (20) where the distace vectors calculated from the OTM method are defied as 5

6 1 1 : x I p p X I p p I p X I p (21) I order to eforce the error to be zero the pealty method is applied ad the stabilized odal residual vector is computed as if N I stab I I N I p Ip p1 r r x e. (22) 3 Numerical example 3.1 Truss subjected to large displacemets A three-dimesioal truss subjected to mixed boudary coditios as show i Figure 2 is cosidered. The truss i subjected to large displacemets at the frot ad back side, while the other compoets of boudary coditios are the stadard tractios ad higher-order trasctio which are take as zero. The geometry of the truss is defied by the legth L 1. For the aalysis of deformatio the bechmark material properties 6 6 of the truss are take as E 1 10, 0.3 ad The large desity is chose here i order to compute the example faster i a quasi-static case to test the stabilizatio algorithm for the higher-order cotiuum. For the purpose of comparig the solutios, firstly the deformatio of the geometrically idetical truss uder the assumpsio of the classical cotium without ay microstructural effects icluded is computed. I Figures 3 ad 5, the deformed shapes of the truss with the cotours of the displacemets i the z directio ad the Vo Mises stress are show. As see from the deformed shape i Figure 3, the truss at both eds reached the fial values of the displacemet boudary coditios. Figure1: Truss with boudary coditios Secodly, a higher-order cotiuum with a very small value of the microstructural parameter of l/ L 0.05 is applied i computatios. Here it should be metioed that by itroducig the higher-order cotiuum ad withi that the calculatio of secod- 6

7 order derivatives, o solutio ca be achieved without the applicatio of stabilizatio procedure. Hece, the uder itegratio of the weak form has a very large effect o the deformatio of the structure, umerical calculatios ca ot be performed because of very large errors arisig durig computatio process. Figure2: Truss classical cotiuum distributio of displacemet u z Figure4: Truss classical cotiuum distributio of Vo Mises stress Figure3: Truss higher-order cotiuum distributio of displacemet u z Figure5: Truss higher-order cotiuum distributio of Vo Mises stress I Figures 3 ad 5, the obtaied deformed shapes of the higher-order cotiuum with very high value of the stabilizatio parameter are portrayed. By the fial positios of the odes, it is clear that the applicatio of the preseted stabilizatio procedure causes the destructio of the displacemet boudary coditios ad deformed cotiuum forms which are ot accurate. Hece, here additioal umerical studies are eccesary i order to determie why the metioed problems persist. Furthermore, as a alterative to max-etropy approximatio fuctios differet more robust approximatio fuctios should be take i cosideratio ad the obtaied results should be compared. 4 Coclusio The meshfree OTM formulatio has bee applied for the modelig of deformatio of the homogeeous structure. The ifluece of the material microstructure is take ito accout usig oe parameter withi the free eergy desity of the utilized higher-order cotiuum. Furthermore, as a remedy to the methods stability issues coected to uderitegratio of the weak form ovel stabilizatio procedure is employed. The OTM algorithm is tested o oe bechmark truss example. Usig the preseted stabilizatio algorithm, solutios ca be obtaied, however some problems have bee idetified. The first problem is the determiatio of the value of the pealty parameter if higher-order 7

8 cotiuum is computed. It is really hard to choose the value of the pealty due to the icorrect itegratio of the secod-order derivatives of the max-etropy shape fuctios that are complex ad their values is ot easy to cotrol. The secod problem observed is that the stabilizatio procedure destroys the essetial boudary coditios. I further research the metioed issues should be addressed. The chage to the more robust approximatio fuctios should be cosidered. This could improve the smoothess of secod-order derivatives ad more accurate calculatios should be carried out. Ackowledgemet: This work has bee ported i part by Croatia Sciece Foudatio uder the project The ivestigatios are also a part of the project ported by Alexader vo Humboldt Foudatio. Refereces [1]. Wriggers, Noliear Fiite Elemet Methods, Spriger Sciece & Busiess Media2008. [2] H. Askes, E.C. Aifatis, Gradiet elasticity i statics ad dyamics: A overview of formulatios, legth scale idetificatio procedures, fiite elemet implemetatios ad ew results, Iteratioal Joural of Solids ad Structures, 48 (2011) [3] B. Li, F. Habbal, M. Ortiz, Optimal trasportatio meshfree approximatio schemes for fluid ad plastic flows, Iteratioal Joural for Numerical Methods i Egieerig, 83 (2010) [4] S.-A. apaicolopulos, A. Zervos, I. Vardoulakis, A three-dimesioal C1 fiite elemet for gradiet elasticity, Iteratioal Joural for Numerical Methods i Egieerig, 77 (2009) [5] E. Amaatidou, N. Aravas, Mixed fiite elemet formulatios of strai-gradiet elasticity problems, Computer Methods i Applied Mechaics ad Egieerig, 191 (2002) [6] Z. Tag, S. She, S.N. Atluri, Aalysis of materials with strai-gradiet effects: A Meshless Local etrov-galerki(mlg) approach, with odal displacemets oly, Cmes-Computer Modelig i Egieerig & Scieces, 4 (2003) [7] C. Villai, Topics i optimal trasportatio theory, Graduate Studies i Mathematics, America Mathematical Society, rovidece, Rhode Islad, USA, [8] M. Arroyo, M. Ortiz, Local maximum-etropy approximatio schemes: a seamless bridge betwee fiite elemets ad meshfree methods, Iteratioal Joural for Numerical Methods i Egieerig, 65 (2006) [9] C. Weißefels,. Wriggers, Stabilizatio algorithm for the optimal trasportatio meshfree approximatio scheme, Computer Methods i Applied Mechaics ad Egieerig, 329 (2018) [10] R. Suyk,. Steima, O higher gradiets i cotiuum-atomistic modellig, Iteratioal Joural of Solids ad Structures, 40 (2003) [11].E. Fischer, C1 Cotiuous Methods i Computatioal Gradiet Elasticity, Uiversitat Erlage-Nürberg, Germay, [12] T. Belytschko, Y.Y. Lu, L. Gu, Elemet-free Galerki methods, Iteratioal Joural for Numerical Methods i Egieerig, 37 (1994) [13] L. Fedeli, A. adolfi, M. Ortiz, Geometrically-exact time-itegratio mesh-free schemes for advectio-diffusio problems derived from optimal trasportatio theory ad their coectio with particle methods, [14] G.C. Gazemüller, A hourglass cotrol algorithm for Lagragia Smooth article Hydrodyamics, Computer Methods i Applied Mechaics ad Egieerig, 286 (2015)