A HYSTERETIC FORMULATION FOR ISOGEOMETRIC ANALYSIS AND SHAPE OPTIMIZATION OF PLANE STRESS STRUCTURES

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1 8 th GRACM Iteratioal Cogress o Comutatioal Mechaics Volos, 12 Jul 15 Jul 215 A HYSEREIC FORMULAION FOR ISOGEOMERIC ANALYSIS AND SHAPE OPIMIZAION OF PLANE SRESS SRUCURES A. N. Mosidis, V. K. Koumousis Istitute of Structural Aalsis & Aseismic Research Natioal echical Uiversit of Athes Zografou Camus, 1578, Athes, Greece argi m@hotmail.com ; vkoum@cetral.tua.gr Keords: Hsteretic Fiite Elemets, Bouc-We Model, Isogeometric Aalsis, Shae Otimizatio. Abstract. I this ork, a e hsteretic formulatio for the ielastic static ad damic aalsis of lae stress roblems ithi the frameork of Isogeometric Aalsis (IGA) is reseted. he Bouc-We model is utilized as a smooth hsteretic, rate ideedet model, caable of exressig the hsteretic behaviour that ca be easil exteded to accout for stiffess degradatio, stregth deterioratio ad ichig heomea. O the basis of the classical theor of lasticit, the geeralized 3D Bouc-We model is exressed i tesorial form icororatig the ield criterio ad liear or oliear, isotroic or kiematic hardeig la. Subsequetl, basis fuctios geerated from No-Uiform Ratioal B-Slies (NURBS) ad the aroriate cotrol oits defie the structure s geometr ad are emloed to build the elastic stiffess matrix. he, the elastic formulatio is exteded b cosiderig as additioal hsteretic degrees of freedom the lastic strais at the quadrature oits defied b the aroriate quadrature rule for the umerical itegratio. he evolutio of the lastic strais is determied b a Bouc We evolutio equatio ad the solutio rovides the dislacemets at cotrol oits of the structure ad the lastic strais at the quadrature oits. O the basis of the roosed formulatio the shae otimizatio roblem is formulated usig mathematical rogrammig. he obective fuctio is the miimum mass of the structure ad the cotrol oits of the boudar ad/or the cotrol eights are selected as the otimizatio desig variables. Furthermore, stress ad dislacemet costraits are imosed at secific oits. Fiall, umerical results are reseted that validate the roosed hsteretic formulatio. A good agreemet is achieved betee the stadard FEM ad the roosed scheme. 1 INRODUCION Isogeometric aalsis (IGA) is a eerg based comutatioal method for the solutio of boudar value roblems. he mai cocet of the IGA is that the same fuctios (B-slies ad NURBS) used i Comuter Aided Desig (CAD) to hadle the grahics are also used to iterolate the geometr ad the dislacemet field. Oe of the advatages of IGA relies o the fact that IGA offers greater recisio ad iteractio betee the overall modelig ad aalsis rocess, hereas the fiite elemet mesh is ol a aroximatio of the exact CAD geometr. Moreover, the beefits of shae otimizatio usig IGA are ver imortat because geometr ad meshig are tightl coected ad the geometr-to-mesh maig is automatic. I this ork, the Bouc-We model is utilized as a smooth hsteretic, rate ideedet model, caable of exressig the hsteretic behavior ad it is icororated i a IGA code. I this resect, the elastic formulatio is exteded b cosiderig as additioal hsteretic degrees of freedom the lastic strais at the quadrature oits defied b the aroriate quadrature rule for the umerical itegratio. he evolutio of the lastic strais is determied b a Bouc We evolutio equatio. I this a, the elastic ad hsteretic stiffess matrices of the structure are derived ad are assembled to form the equatios of motio. he evolutio equatios for the hsteretic degrees of freedom costitute a additioal set of first order oliear equatios hich together ith the equatios of motio determie the sstem of equatios that describes the ielastic roblem. he etire sstem of equatios is coverted ito state sace form ad the solutio is established o the basis of a Livermore itegratio scheme. Numerical results are reseted that ustif the validit ad accurac of the roosed formulatio. Moreover, a shae otimizatio roblem is formulated usig mathematical rogrammig based o NURBS geometries. 2 B-SPLINES AND NURBS GEOMERIES Let 1,, 1 be a o-decreasig sequece of real umbers, i.e. i i 1, i 1,,. he i are called kots ad is the kot vector [1]. he i th B-slie basis fuctio of -degree, deoted b N, is defied recursivel as:

2 N 1 A. N. Mosidis, V. K. Koumousis. if i otherise i1 u i i1 N N N 1 i 1, 1 i i i1 i1 (1) If 1 is the umber of kots, there exist basis fuctios. he half-oe iterval, th i kot sa. he kot vectors hich have the form: is called the a, a, 2,,, b, b (2) 1 1 that is the first ad last kot has multilicit 1, are defied as o-eriodic or oe. Moreover, a kot vector is uiform if all iterior kots are equall saced. Otherise it is o-uiform. he derivative of a basis fuctio is give b: N N 1 Ni 1, 1 (3) A th degree B-slie curve is defied b: i i i1 i1 C N P a b (4) i1 i th here P i are the cotrol oits ad N are the degree B-slie basis fuctios defied o the oeriodic kot vector of Eq. (2). he P i are the vertices of the cotrol olgo. he derivative of a B-slie curve is give b the exressio: C N P i (5) i1 A examle is reseted i Figure 1 for olomial order 3 ad the oe, o-uiform kot vector,,,,1,1,2,3,3,3,4,5,5,5,5. he cotrol olgo is i,,,,,,,,,, i i 1 P. Figure 1.(a) Basis fuctios, (b) B-slie curve (cotrol oits( ), kot locatios( )) A tesor roduct B-slie surface is defied b the relatio:

3 m A. N. Mosidis, V. K. Koumousis., N N, S P (6) i1 1, q mq1 th here P is the cotrol et, N are the degree B-slie basis fuctios defied o the o-eriodic kot vector 1,, 1 ad N q, are the q th degree B-slie basis fuctios defied o the oeriodic kot vector 1,, m q 1. Comig to the No Uiform Ratioal B-Slie (NURBS) curves, coic sectios ad circles are idel used i comuter grahics ad comuter aided desig. Oe of the greatest advatages of NURBS is their caabilit of th recisel reresetig coic sectios ad circles i cotrast to o-ratioal B-slies. A degree NURBS curve is give b: i1 N ipi i1 A C a b (7) N i here P i x zi are the cotrol oits (formig a cotrol olgo), i are the eights ad N th are the degree B-slie basis fuctios defied o the o-eriodic kot vector of Eq. (2). It is usuall assumed that a, b ad i for all i. From the geometric oit of vie, a NURBS curve is the roectio of a o-ratioal (olomial) B-slie curve defied i four-dimesioal (4D) homogeeous coordiate sace back ito three-dimesioal (3D) hsical sace. he o-ratioal B-slie i four-dimesioal sace is defied b the relatio: C N P i (8) here x,, z, b: i i i i i i i i i1 P are the eighted cotrol oits. he derivative of a NURBS curve is evaluated A A A C 2 C (9) A ad here the derivatives of are comuted b the Eq. (5). A NURBS surface of degree i the directio ad degree q i directio is defied b: m N N, q P i1 1,, m N N S (1) i1 1, q mq1 here P x,, z are the cotrol oits (formig a bidirectioal cotrol et), eights, are the N are the th degree B-slie basis fuctios defied o the o-eriodic kot vector 1,, 1 N vector 1,, m q 1 ad q, are the q th degree B-slie basis fuctios defied o the o-eriodic kot. Alterativel, a NURBS surface ca be rereseted usig homogeeous coordiates as: m S, N N P (11) here x,, z, P. i1 1, q

4 3 ISOGEOMERIC ANALYSIS AND SHAPE OPIMIZAION Isogeometric aalsis exteds the isoarametric cocet offerig better abilities to hadle the grahics [2]. hus, the solutio sace ad the geometr are iterolated usig the same scheme. he mai differece betee FEA ad IGA is that i classical FEA a iterolatio scheme is selected for the dislacemet field ad the same scheme is used to aroximate the geometr. O the cotrar, i IGA the aroriate B-slie/NURBS basis, hich exactl describes ad hadles modificatios of the geometr, is used to aroximate the uko solutio field. Eseciall, i the case of lae stress roblems, the solutio-dislacemet field is defied as:,, d m x N N, q i1 1 d m, m i i1 1 N N, q i1 1 u x u, R, d u (12) m here R, Ni, N, q N N, q ad d are the cotrol variables, i.e. i1 1 the value of dislacemet at the cotrol oit. It is ver coveiet to determie a scheme for the global umberig of basis fuctios such that: A 1 i (13) ad N, R, (14) A B meas of Eq. (14) the dislacemet field of Eq. (12) ca be stated i the folloig form: m u, NA, da (15) he ifiitesimal strai vector (lae stress coditios) is defied b the folloig relatio: A1 u x x d m x NA BA da B d 1 d A A1 x u A,2 A,1 u x x xx N A,1, m u,2, A N, N, (16) here x x 1 d d, d,, d, d is the vector of the uko cotrol variables. For a isotroic material, oe ca rite: m xx 1 xx xx E 2 1 x 1 x x 2 1 C he exressio for the ricile of virtual ork i this case is ritte as: (17)

5 it 1mq1 1 1 A. N. Mosidis, V. K. Koumousis. W W d d R 1 mq1 d B CBdtdetJdd d R 1 1 1mq1 1 1 ext C tdetjdd d R B CBdtdetJdd R x here d are the virtual dislacemets, are the corresodig virtual strais ad J x is the Jacobia matrix. he last of itegrals of Eq. (18) is evaluated umericall b alig a aroriate quadrature rule. 4 INELASIC BEHAVIOR AND BOUC WEN HYSEREIC MODEL he resose of most materials he stressed u to certai level, remais elastic. I this rage the exhibit o memor o their reached stress-strai state ad retur to zero stress strai he uloaded. I stress sace, the elastic domai is delimited b a exteral boudar, i.e. the ield surface, hich is defied b a ield fuctio of the form:, f (19) here,. Loadig further the material, lastic ieldig or lastic flo, maifested as ermaet strais at uloadig. his is described b the lastic flo rule: is the iitial ield stress, hereas a admissible stress state must satisf the coditio Qσ Q Q or ε or (2) l l l σ ( tesor comoets otatio) ( tesor otatio) ( matrixvector otatio) here Q is a lastic otetial fuctio ad is the lastic multilier. For most civil egieerig materials, ot icludig soils, a commo valid aroach is to associate the lastic otetial ith the ield fuctio, Q (associated flo rule) ad exress eq. (2) as: σ l l (18) or ε (21) σ ogether ith the evolutio of the lastic strai, a evolutio of the ield stress itself is also maifested (hardeig) ad the ield surface udergoes exasio ad/or traslatio. I the case of kiematic hardeig, hich is caable of redictig the Bauschiger effect, the ield fuctio is exressed i the form: f, (22) here α is a tesorial back stress hardeig arameter, that reresets the evolutio of the cetre of the ield surface i the stress sace. he back stress evolves as a fuctio of the lastic multilier, ad the hardeig fuctio G as: α G or G or a G (23) ( tesor otatio) ( tesor comoets otatio) ( matrixvector otatio)

6 I the case of Prager s liear kiematic hardeig, evolutio of the back stress is defied b the folloig liear relatio: C C G, here GC (24) l here C is the hardeig costat. I additio the total strai tesor is cosidered as the sum of a elastic comoet el ad a lastic l comoet (assumtio of additive decomositio) ad thus: el l el l or ε ε ε (25) Furthermore the stress icremet is liearl related to the elastic strai icremet i the lastic regio ad ca be exressed b the folloig costitutive relatio: σ C:ε C: ε ε el l C C C C or or el l el l kl kl kl kl kl (26) For lastic flo to occur, the stresses must remai o the ield surface (cosistec coditio) ad hece: d d or : dσ : dα (27) σ α I order to fid, eq. (26) is re-multilied b the flo vector ad usig Eq. (27) ad Eq. (21): 1 G C C (28) Relatio (28) holds ol he ieldig has occurred. hus, b itroducig the folloig Heaviside te fuctios: 1, 1, σ : dσ H1 H2 (29),, σ: dσ a sigle relatio is established for the lastic multilier i the hole stress sace, hich is the mai itervetio of the Bouc We model ([3], [4]): 1 HH 1 2 G C C (3) o derive the Bouc-We relatios, the to Heaviside fuctios are smootheed usig the folloig exressios: H 1 N, N 2 (31) ad (sice there is o lastic deformatio durig uloadig ad ):

7 H H sig C (32) Fiall, usig Eq. (21) ad Eq. (3), the folloig Bouc We model is derived: N l 1 1 sig C R 2 2 H1 H2 (33) here the iteractio matrix R is exressed as: 1 R G C C (34) ad determies the ecessar iterrelatios betee the lastic strai comoets to secure that the stresses remai o the ield surface accoutig also for the hardeig la (cosistec coditio). I this rate form the Bouc-We model ca icororate a ield criterio ad hardeig la, ecasulatig all differet asects of loadig ad uloadig hases. 5 SIFFNESS AND HYSEREIC MARICES he elastic deformatio field is exteded b itroducig a additioal matrix of hsteretic degrees of freedom hich herei are the lastic strais at quadrature oits: l l l l l l z l l l 1,1 1,2 1, mpl 2,1 2,2 2, mpl Pl,1 Pl,2 Pl, mpl (35) l here, is the lastic comoet of the strai vector (16) at the, i i oit of bidirectioal et of quadrature oits, here i 1, 2,,, 1, 2,, m ad m is the total umber of quadrature oits. At this oit, a hsteretic tesor roduct B-slie surface hsteretic field ca be cosidered so that: here xx m m m l l l l l l N N z,,,, q i NA NA za N z N ad l l i1 1 A1 A1 x A N are B-slie basis fuctios of reselected order l, q (36) ad q corresodig to kot l vectors ad hich are costructed i such a a that the hsteretic surface iterolatig the lastic comoet of the strai vector of quadrature oits does t result i erratic shaes (a more detailed resetatio o the surface iterolatio ca be foud i [1]). Moreover, 1, m hsteretic degrees of freedom ad A 1 i. he ricile of virtual ork (Eq. (18)), b meas of relatio (25), is exressed as: l C dd R z z z is the vector of (37) Substitutig relatios (16) ad (36) ito relatio (37) the folloig exressio is obtaied:

8 l d B CBd N z d d R l B CBdd B CN z dr (38) ad fiall the folloig costitutive equatio is deduced at the atch level: d ke d kh z ke kh R 2m2m 2m3PlmPl z (39) here k e is the isogeometric elastic ad k h the herei itroduced hsteretic stiffess matrix. he uko vector z, cotaiig all lastic strais at quadrature oits follos a evolutioar equatio of Bouc-We te give i relatio (33) ideedetl for ever three comoet lastic strai vector. It becomes evidet that the roosed formulatio ca be used also for other tes of elemets i a effort to icororate directl the hsteretic behavior. 6 SAE EQUAIONS - SOLUION PROCEDURE he equatio of motio is exressed as: md cd kedkh z R t (4) I additio to the liear equatios of motio (Eq. (4)), the ucouled o-liear evolutio equatios of the hsteretic degrees of freedom (Eq. (33)) for each quadrature oit are required. he sstem of differetial equatios of motio ad evolutio equatios ca be trasformed ito state sace form itroducig further the odal velocities as additioal ukos: X 1 X2 1 X 2m cx2kex1khx3pt X 3 f X1, X2, X3 (41) here 3 1 X is the vector of uko dislacemets, X the corresodig vector of uko velocities ad X the vector of the hsteretic degrees of freedom. he sstem of Eq. (41) suffices to determie the oliear damic behavior of the structure. he sstem of first order oliear differetial equatios ca be solved usig Ruge Kutta redictor-corrector te of itegrator schemes, such as the Livermore famil of solvers (Radhakrisha ad Hidmarsh 1993), alloig for robust ad ucoditioall stable solutios. A basic advatage of the roosed method is that the load is hadled through the sstem of first order differetial equatios. hus the method accouts for the iheret ielasticit i.e., the flo rule ad cosistec coditio after ieldig ad the correctio is strictl umerical to achieve the desired accurac, hereas i a radial retur scheme the redictor ste is liear ad all ielastic cosideratios are i the corrector ste. Usig the roosed elemet, shae otimizatio of elastic roblems ca be easil formulated b selectig the coordiates ad/or eights of the cotrol oits of the NURBS geometr as desig variables aimig at miimizig the mass of the structure uder secific loadig ad a set of stress ad/or dislacemet costraits imosed at secific oits [5]. 7 NUMERICAL EXAMPLES 1 st examle A Isogeometric Aalsis code as develoed to imlemet the roosed formulatio. I this examle, the otimizatio of a siml suorted beam ith a redefied legth of l 8m is iitiall examied. A elastolastic material is cosidered ith E 21GPa,.3, 235MPa, E 21GPa ad thickess t.8m. I additio, the material follos the vo Mises ield criterio ith liear kiematic hardeig la. he obective is to miimize the eight of the beam loaded b a cocetrated force P 1kN at the 2 t

9 midoit, subected to to o-liear costraits, the first of hich requires that the deflectio at the midoit of the beam umid.1m ad the secod demads that the vo Mises stress at the extreme bottom fiber of the midoit VM. he iitial shae of the beam is a rectagle of height h 2m. he biquadratic ( 2, q 2 ) NURBS surface is formed from kot vectors,,,.5,.5, 1, 1, 1 ad the cotrol eights are defied 1 (the NURBS degeerates to B-slie surface). he otimizatio variables are the vertical ositios of cotrol oits of the bottom ro, the vertical ositios of cotrol oits of the uer ro, hich are costraied to lie o a straight horizotal lie, hereas the itermediate oits are liked to the extreme oits. he otimizatio roblem is solved usig the fmico solver of MALAB hich is aroriate for the miimizatio of costraied oliear multivariable fuctios. he iitial ad the otimized shae of the beam are sho i Figure 2..4m l=8m P=1kN Figure 2. Iitial ad the otimized shae of the beam.998m Subsequetl, the otimized beam is refied b isertig e kots ad raisig the olomial order of the basis fuctios. he structure is subected to mootoicall icreasig cocetrated load P 5kN at the midoit, hich leads to gradual ieldig of the beam. I Figure 3, the alied load is lotted agaist the vertical deflectio of the midoit. he load-deflectio curve is comared ith the oe obtaied usig Abaqus. 5 alied load P (kn) Abaqus IGA, Bouc-We,5,1,15,2,25 Vertical deflectio at midoit (m) Figure 3. Load-deflectio curve at midoit of the beam From the revious curve, it is aaret that the solutio obtaied based o the Bouc-We hsteretic model imlemeted i a IGA code agrees ell ith the solutio obtaied usig Abaqus. Fiall, the distributio of the vo-mises stress of the beam is illustrated i Figure 4. 2 d examle I the 2 d Figure 4. Distributio of the vo-mises stress (kn/m 2 ) examle, the otimum outer shae of a oe saer-rech ith a redefied legth of

10 l 25cm is ivestigated. A elastolastic material is cosidered ith E 21GPa,.3, 235MPa, Et 21GPa ad thickess t 1cm. I additio, the material follos the vo Mises ield criterio ith liear kiematic hardeig. he obective is to miimize the eight of the saer loaded b a cocetrated force P 1kN at oit A (Figure 5), subected to a o-liear costrait requirig that the deflectio at oit A ua.1cm. he otimizatio variables are the vertical ositios of cotrol oits of the bottom ro, the uer ro is smmetric to the bottom oe, hereas the itermediate oits are liked to the extreme oits. Moreover, a additioal costrait aligs the cotrol oits of the hadle. he otimizatio roblem is solved b fmico solver of MALAB. he otimized shae of the saer is sho i Figure 5. 4cm A P=1kN l=25cm Figure 5. Iitial ad the otimized shae of the oe saer he erformace of the otimizatio algorithm is deicted i Figure 6. obective fuctio (cm 2 ) Otimizatio histor Figure 6. Evolutio of the otimizatio rocedure Subsequetl, the otimized saer is subected to mootoicall icreasig cocetrated load P 4kN at oit A, i excess of its omial value, hich leads to gradual ieldig. I Figure 7, the alied load is lotted agaist the vertical deflectio at oit A. he load-deflectio curve is comared ith the oe obtaied usig Abaqus. 4 alied load P (kn) 3,5 3 2,5 2 1,5 1,5 Abaqus IGA, Bouc-We,2,4,6,8 1 1,2 1,4 1,6 1,8 Vertical deflectio of oit A (cm) Figure 7. Load-deflectio curve of the oit A of the saer Fiall, the distributio of stress comoet σ xx of the saer is illustrated i Figure 8.

Figure 8. Stress comoet σ xx (kn/cm 2 ) 8 CONCLUSIONS A e hsteretic formulatio for the ielastic static ad damic aalsis of lae stress roblems ithi the frameork of Isogeometric Aalsis (IGA) is reseted.

11 Figure 8. Stress comoet σ xx (kn/cm 2 ) 8 CONCLUSIONS A e hsteretic formulatio for the ielastic static ad damic aalsis of lae stress roblems ithi the frameork of Isogeometric Aalsis (IGA) is reseted. Oe of the advatages of the roosed method relies o the fact that the resose is hadled through the sstem of first order differetial equatios. his rovides the dislacemets at cotrol oits, the elastic ad lastic strais ad the stresses at quadrature oits that satisf the ielastic costitutive relatios ad equilibrium ithout a additioal iterative rocess. herefore, b solvig the sstem of differetial equatios umericall, the scheme stas alas o the ield fuctio ad satisfies the flo rule b defiitio. Cosequetl, the local iteratios of Neto Rahso method are avoided, at the exese of the umerical solutio of first order evolutio equatios for the itroduced additioal hsteretic ukos. Moreover, the roosed formulatio utilizes the ielastic costitutive relatio i the ricile of virtual ork i a searable form distiguishig the elastic ad hsteretic art. he derived structural matrices are evaluated ol oce, at the begiig of the aalsis rocedure. hus, the roosed formulatio directl accouts for ielasticit i a atural a b solvig i couled form the liear equilibrium equatios together ith the o-liear evolutio equatios, avoidig the trial elastic redictios folloed b lastic correctios. Fiall, the roosed method avoids the tedious evaluatio of the odal iteral forces of stadard methods emloig umerical itegratio over the elemet volume i ever ier ste. For these reasos, the roosed formulatio turs out comutatioall more efficiet for the same accurac as comared to stadard methods. he IGA formulatio through the cotrol oits ad their eights is roved more efficiet for shae otimizatio roblems as comared to the stadard isoarametric formulatio. REFERENCES [1] Piegl, L., iller, W. (1996), he NURBS Book(Moograhs i Visual Commuicatio), Sriger Berli Heidelberg [2].J.R. Hughes, J.A. Cottrell ad Y. Bazilevs, Isogeometric aalsis: CAD, fiite elemets, NURBS, exact geometr ad mesh refiemet, Comuter Methods i Alied Mechaics ad Egieerig, 194 (39-41), , (25). [3] riatafllou, S. P., ad Koumousis, V. K. (212). A hsteretic quadrilateral lae stress elemet. Arch. Al. Mech., 82(1 11), [4] riatafllou, S. P., ad Koumousis, V. K. (214). Hsteretic fiite elemets for oliear static ad damic aalsis of structures. J. Eg. Mech., 1.161/(ASCE) EM , [5] Wolfgag A. Wall, Moritz A. Frezel, Christia Cro, Isogeometric structural shae otimizatio, Comuter Methods i Alied Mechaics ad Egieerig, 197 (33-4), (28).

Lecture #4: Integration Algorithms for Rate-independent Plasticity (1D)

Lecture #4: Integration Algorithms for Rate-independent Plasticity (1D) 5-0735: Damic behavior of materials a structures Lecture #4: Itegratio Algorithms for Rate-ieeet Plasticit (D) b Dirk Mohr TH Zurich, Deartmet of Mechaical a Process gieerig, Chair of Comutatioal Moelig