Time-adaptive non-linear finite-element analysis of contact problems

Transcription

1 Proceedings of the 7th GACM Colloqim on Comptational Mechanics for Yong Scientists from Academia and Indstr October -, 7 in Stttgart, German Time-adaptive non-linear finite-element analsis of contact problems Matthias Grafenhorst * and Stefan Hartmann Micro Abstract In recent ears, mortar finite element methods have been sccessfll applied as space discretiation scheme to a wide range of contact problems. The finite deformation contact formlation taen p is based on a mortar approach sing dal Lagrange mltipliers. If the constittive models are of rate-tpe, the entire sstem of eqations represents a non-smooth DAE-sstem. This sstem will be investigated in connection with higher-order time integration methods sing DIRK-methods. Institte of Applied Mechanics, Clasthal Universit of Technolog, Clasthal-Zellerfeld, German * Corresponding athor: matthias.grafenhorst@t-clasthal.de Introdction In this contribtion, we show the applicabilit of diagonall implicit Rnge-Ktta schemes (DIRK) for the time discretiation of geometricall non-linear problems in comptational contact mechanics. The spatial discretiation framewor is based on a mortar finite element method sing a dal Lagrange mltiplier method as the constraint enforcement techniqe. Space and time discretiation The nmerical treatment of contact problems in the field of solid and strctral dnamics with constittive eqations of evoltionar tpe, is detailed in this wor as a soltion procedre of differential-algebraic eqations (DAEs) related to the consistent application of the method of vertical lines, see [] and the literatre cited therein. After performing the spatial discretiation sing standard finite elements and a mortar finite element discretiation techniqe for the contact interface, see [ 4, 8 ], we arrive at a DAE-sstem which is defined b Mü(t) = g (, λ, q, t) M ü(t), g (, λ, q, t) = f int(, q, t) f et (, t) + f c (, λ), () q(t) = r Q (, q, t), () C n (, λ) = {C ni (, λ i )} =, () C T (,, λ) = {C Ti (,, λ i )} =, (4) (t ) =, (t ) = v, q(t ) = q. (5) In this global DAE-sstem, the nowns are the nodal displacements (t) R n, the internal variables q(t) R n Q, which describe the histor of the material behavior and are given b ordinar differential eqations (), and the global Lagrange mltiplier λ(t) R n S, n S = n n S, carring for all n n S slave nodes with the individal nodal Lagrange mltiplier vector λ i R, i S. The non-linear force residal g R n comprises contribtions from internal and eternal forces f int R n and f et R n as well as contact forces f c R n which will be discssed in more detail below. B sorting the nnown nodal displacements T = { T N, T S, M} T, n = n N + n S + n M, into the three grops S (slave), M (master) and N (remaining) comprising nodes or degree of freedoms, contact forces are given as f c (, λ) = ˆB()λ = ˆD T () λ. (6) ˆM T ()

2 The so-called mortar matrices are given b [ ˆD() = ˆDi ()] [ ] R n S n S and ˆM() = ˆMi () R n S n M (7) The nodal blocs ˆD i R and ˆM i R are defined b ˆD i () = ˆM i () = h() γ h() c M i N () dγ I, i S, S, (8) M i ( N () χ h γ c ) dγ I, i S, M, (9) with the identit matri I R. In (8) and (9), N () and N () represent the slave side and master side shape fnctions, M i are the dal Lagrange mltiplier shape fnctions as particlar choice for the Lagrange mltiplier interpolation, see e.g. []. Moreover, χ h γ c : h() I h() defines a discrete approimation of the actal smooth contact mapping for a transformation between the slave srface h() and the master srface h() in the crrent configration, see e.g. []. Similarl, spatial discretiation of the wea non-penetration constraint ields the weighted nodal gap, g i () = n T i S ˆD i () M ˆM i () for each slave node i S and the spatial discretiation of the wea frictional sliding conditions according to Colomb friction, ields a discrete relative tangential velocit given b ṽ Ti = {ṽξi ṽ ηi } { = τ T i S ˆD i () M () } ˆM i () R, i S, () sch that the reqirement of frame indifference is satisfied. In (), n i represents at each slave node i S an averaged nodal nit normal, see []. Frther, at each slave node i S, two orthonormal nit tangent vectors τ ξi and τ ηi are constrcted from the averaged normal n i, which are compiled in the matri τ i = [ τ ξi, τ ηi ] R. According to [9, ] for resolving the contact non-linearit introdced b the discretied KKT-tpe constraints for normal and tangential contact, a primal-dal active set strateg is emploed which reformlates the KKT conditions as eqalit constraints, see () and (4), with the help of a nodal-wise defined non-linear complementarit fnction for normal contact C ni (, λ i ) = λ ni ma(, λ ni c n g i ), c n > () and the nonlinear complementarit fnction for the frictional sliding contact C Ti (,, λ i ) = { Cξi C ηi } = ma (µ c (λ ni c n g i ), λ Ti + c τ ṽ Ti ) λ Ti µ c ma(, λ ni c n g i ) {λ Ti + c τ ṽ Ti }, R, c n >, c τ >. () The epression λ ni = n T i λ i, denotes the proection of the nodal Lagrange mltiplier λ i in the normal direction and λ Ti = τ T i λ i R, defines the proection of λ i in the tangential plane. Frther details on the actal nmerical implementation for the applied mortar method can be fond in [ 4, 8 ]. For the temporal discretiation of the DAE-sstem, high-order, stiffl accrate, diagonall implicit Rnge-Ktta schemes (DIRK) are applied, see e.g. [5]. Appling this procedre to the DAE-sstem defined b ()-(5) ields in each stage the sstem of non-linear

3 eqations, G (U ni, λ ni, Q ni ) = M U ni U S ni t nā ii + g (U ni, λ ni, Q ni, T ni ) + M ü(t ni) =, (4) L(U ni, Q ni ) = Q ni Qni S r Q (U ni, Q ni, T ni ) =, t n a ii (5) C n (U ni, λ ni ) =, (6) C T (U ni, λ ni ) =. (7) These eqations depend on the stage vales U ni, λ ni and Q ni, with the stage time T ni = t n +c i t n, the time step-sie t n, the weighting factors c i, a i, ā i = s = āiā of the implicit s-stage DIRK-method and the nown starting vectors i Uni S = n + c i t n v n + t n ā i A n, = i Qni S = q n + t n a i Q n. (8) = After solving (4)-(7) b sing a semi-smooth Mltilevel-Newton algorithm, the stage derivatives A ni and Q ni will be compted b A ni = U ni U S ni t nā ii and Q ni = Q ni Q S ni t n a ii. (9) De to the stiff accrac, the soltion variables in the last stage concide with the the new soltion at time t n+. Moreover, this propert ensres constraint enforcement at the end-point t n+. Ironing eample For the nmerical eample, a half-clindrical elastic die (Neo Hooean model, K =.75 5 MPa, c = 4 4 MPa) intrding and sliding into a viscoelastic bloc (finite strain viscoelasticit model, see [7]) with material parameters specified in Tab. is investigated, see also [8, ]. The geometr, applied bondar and load conditions of the qasi-static problem are depicted in Fig.. For the space discretiation, 8-node heahedral QP-elements are considered ielding bilinear interpolation on contact srfaces. Frthermore, we assme the contact interaction to be frictionless. The initial conditions are ero. For the illstrated problem, the integration methods detailed in Table are analed in view of their epected and achieved temporal convergence order for the displacement field and the internal variables, see Fig.. De to the lac of smoothness in the time domain, we report comparable to problems in plasticit, see e.g. [], an order redction where the third order of the Ale/Cash method is not attained. The other methods reach their theoretical orders. For frther information regarding the for applied DIRK-methods, we refer to []. The reslting deformation, von Mises stress and Lagrange mltiplier distribtion are illstrated for t = s in Fig. for the reference soltion (Ale/Cash, t n =.5 4 s). Conclsions One maor advantage of high-order DIRK time-integration methods is a comptationall efficient time-step control provided b an embedded techniqe, see [6]. The error of the varios stage Table. Material parameters for the finite strain viscoelasticit model according to [7] K c c c µ η s N/mm N/mm N/mm N/mm N/mm N s/mm MPa

4 h() h() R in mm R T = (t) (t) in mm (t) (t) T = t in s (a) Geometr, bondar conditions (nits in mm) (b) Loading path Figre. Ironing eample ~λ in MPa in mm.7e- -.e- -.4e- -.7e e- -6.8e e e- -.e+ -.4e+ (a) Displacement σv in MPa 5.e+ 4.44e+.89e+.e+.78e+.e+.67e+.e+ 5.56e-.e (b) Von Mises stress (c) Lagrange mltiplier Figre. Displacement, von Mises stress and Lagrange mltiplier distribtion for reference soltion (Ale/Cash, tn =.5 4 s) at t = s qantities is approimated b an embedded Rnge-Ktta method which is one order lower than the original method. In this case, onl fnction evalations are necessar for determining the error and the time-step is controlled so that the error remains nder a prescribed tolerance. As a prereqisite for this efficient adaptive time-step control, a corresponding temporal convergence behavior has to be ensred. This seems to be the case for DIRK-methods for frictionless contact problems. References [] P. Ellsiepen and S. Hartmann. Remars on the interpretation of crrent non-linear finiteelement-analses as differential-algebraic eqations. International Jornal for Nmerical Methods in Engineering, 5:679 77,. [] P. Farah, A. Popp, and W. Wall. Segment-based vs. element-based integration for mortar methods in comptational contact mechanics. Comptational Mechanics, 55():9 8, 5. Table. Applied DIRK-methods abbreviation method order p stages s Ale/Ell CAA Ale/Cash Bacward-Eler Aleander/Ellsiepen trapeoidal rle or CAA Aleander/Cash

5 e-6 e-5 e-7 e-6 e-8 e-7 err e-9 errq e-8 e- e- Ale/Ell Trape/CAA Ale/Cash e- e- e- t n in s (a) Global error in displacement field e-9 e- Ale/Ell Trape/CAA Ale/Cash e- e- e- t n in s (b) Global error for internal variables Figre. Convergence behavior for temporal order analsis [] M. Gitterle. A dal mortar formlation for finite deformation frictional contact problems inclding wear and thermal copling. PhD thesis, Technische Universität München, Institte for Comptational Mechanics,. [4] M. Gitterle, A. Popp, M. W. Gee, and W. A. Wall. Finite deformation frictional mortar contact sing a semi-smooth newton method with consistent lineariation. International Jornal for Nmerical Methods in Engineering, 84(5):54 57,. [5] M. Grafenhorst, J. Rang, and S. Hartmann. Time-adaptive finite element simlations of dnamical problems for temperatre-dependent materials. Jornal of Mechanics of Materials and Strctres, ():57 9, an 7. [6] E. Hairer and G. Wanner. Solving Ordinar Differential Eqations II. Springer, Berlin, nd revised edition, 996. [7] S. Hartmann. Comptation in finite strain viscoelasticit: finite elements based on the interpretation as differential-algebraic eqations. Compter Methods in Applied Mechanics and Engineering, 9(-4):49 47,. [8] A. Popp. Mortar Methods for Comptational Contact Mechanics and General Interface Problems. PhD thesis, Technische Universität München, Institte for Comptational Mechanics,. [9] A. Popp, M. W. Gee, and W. A. Wall. A finite deformation mortar contact formlation sing a primal-dal active set strateg. International Jornal for Nmerical Methods in Engineering, 79():54 9, sep 9. [] A. Popp, M. Gitterle, M. W. Gee, and W. A. Wall. A dal mortar approach for d finite deformation contact with consistent lineariation. International Jornal for Nmerical Methods in Engineering, 8():48 465,. [] A. Popp, A. Seit, M. W. Gee, and W. A. Wall. Improved robstness and consistenc of D contact algorithms based on a dal mortar approach. Compter Methods in Applied Mechanics and Engineering, 64:67 8, sep. [] M. A. Pso and T. A. Larsen. A mortar segment-to-segment contact method for large deformation solid mechanics. Compter Methods in Applied Mechanics and Engineering, 9(6-8):6 69, 4. [] B. Yang, T. A. Larsen, and X. Meng. Two dimensional mortar contact methods for large deformation frictional sliding. International Jornal for Nmerical Methods in Engineering, 6(9):8 5, 5.