A PRIMAL-DUAL ACTIVE SET ALGORITHM FOR THREE-DIMENSIONAL CONTACT PROBLEMS WITH COULOMB FRICTION

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1 Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number A PRIMAL-DUAL ACTIVE SET ALGORITHM FOR THREE-DIMENSIONAL CONTACT PROBLEMS WITH COULOMB FRICTION S. HÜEBER, G. STADLER AND B.I. WOHLMUTH Abstract: In tis paper, efficient algoritms for contact problems wit Tresca and Coulomb friction in tree dimensions are presented and analyzed. Te numerical approximation is based on mortar metods for nonconforming meses wit dual Lagrange multipliers. Using a nonsmoot complementarity function for te 3D friction conditions, a primal-dual active set algoritm is derived. Te metod determines active contact and friction nodes and, at te same time, resolves te additional nonlinearity originating from sliding nodes. No regularization and no penalization is applied, and local superlinear convergence can be observed. In combination wit a multigrid metod, it defines a robust and fast strategy for contact problems wit Tresca or Coulomb friction. Te efficiency and flexibility of te metod is illustrated by several numerical examples. Keywords: 3D Coulomb friction, contact problems, dual Lagrange multipliers, inexact primal-dual active set strategy, semismoot Newton metods, nonlinear multigrid metod. AMS Subject Classification (2000): 65K10, 65C20, 65N30, 65N55, 74M Introduction Solving contact problems wit friction in 3D is a callenging task in mecanics and of crucial importance in various applications. Te main difficulty lies in te conditions for contact and friction, wic are inerently nonlinear and complicate te teoretical analysis and te design of an efficient numerical solution algoritm. For a general introduction on contact problems wit and witout friction, we refer to [KO88,HHNL88,Wri02,Lau02,Wil03, EJK05]. Very often, laws named after Tresca and Coulomb are used to model friction. It is well known tat contact wit Tresca friction leads to a classical variational inequality. For Coulomb friction, te friction bound depends on te solution, and tis more realistic law results in a quasi-variational inequality. Tis work was supported in part by te Deutsce Forscungsgemeinscaft, SFB 404, B8, and SPP Received September 29,

2 2 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH A widely used approac for contact problems wit Coulomb friction is to apply a sequence of Tresca friction problems togeter wit a fixed point iteration (see, e.g., [HKD04,HDK02,LPR91,KS05,FK06,NJH80]). Tus, a crucial component for te solver is a fast and robust algoritm for Tresca frictional contact problems. Wile in two dimensions, Tresca friction corresponds to linear pointwise inequality constraints for te boundary stresses, te situation in 3D is more involved due to te quadratic inequality constraint, one as to deal wit. Contributions to teoretically sound numerical algoritms for friction in 3D are quite rare. We refer to te recent papers [HKD04,DHK + 05], were discrete 3D-Coulomb friction problems are approximated using FETI domain decomposition and algoritms for quadratic programming wit simple inequality constraints. Since quadratic rater tan linear constraints arise in 3D, tey are approximated as intersections of rotated squares in order to make te application of optimization algoritms for simple bounds possible. Improvements of tis approximation tecnique are proposed in [Kuč06a,Kuč06b]. A different idea is followed in [KK01,FK06], were monotone multigrid metods are used to construct a globally convergent solver for discrete Tresca and Coulomb frictional contact problems. Te implementation of tese metods relies on a multilevel ierarcy of spaces and requires te use of modified coarse grid basis functions and suitable coarse grid constraints. Already in an early paper [AC91], Newton-type metods for contact problems wit friction are used. Similar metods are also studied in te more recent contributions [CKPS98,CP99], were te performance of generalized Newton-type metods for frictional contact problems is sown to be superior to interior point metods. Te metods presented in [AC91, CKPS98, CP99] rely on reformulations of contact and friction conditions using nonsmoot equations and on generalized differentiability concepts and, tus, are related to te present paper. However, we use different reformulations of te complementarity conditions, study and exploit te structures arising in Newtontype steps and relate tem to (inexact) primal-dual active set strategies. In addition, we apply our algoritm to two-body contact problems wic are discretized in terms of mortar tecniques and introduce stabilized inexact Newton versions. Recently, te application of mortar metods as a discretization for te continuous problem as gained a considerable amount of interest (see, e.g.,

3 PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT 3 [BB00,HL02,HW05,FW05,FW06,Lau02,PL04]). By using a dual basis function for te Lagrange multiplier, te weak form of te constraints can be written as independent constraints for eac node on te slave side resulting in a quasi-variational inequality. Using a suitable local basis transformation, only te coefficients on te slave side are affected by te non-penetration condition and te friction constraints. Tus, te algebraic structure of a twobody contact problem is exactly te same as for a one-body contact problem. For details we refer to [WK03, HW05]. In te mixed formulation wic is motivated by te mortar approac, te Lagrange multipliers are treated as independent variables. Tis motivates te application of algoritms, suc as te primal-dual active set strategy wic use bot types of variables. We refer to [HIK03] were te relation between tis algoritm and a semismoot Newton metod applied to a complementarity function is studied. Tese metods ave been extended and applied successfully to contact problems witout friction [HW05,HKK04] and in 2D to friction [HMW06,Sta04,KS05]. However, te application to friction problems in 3D is not at all straigtforward. In tis paper, we derive a primal-dual active set metod by applying a semismoot Newton metod to a nonlinear complementarity function tat expresses te 3D Tresca friction conditions. In eac step of te algoritm, we solve a linear problem wit Robin boundary conditions on a suitable subset of te friction boundary and wit Diriclet conditions on te complement. Tis linear system can be solved using any efficient iterative or fast direct solver. Here, we apply a multigrid metod as iterative solver. By solving tese systems only approximately, we obtain an inexact primal-dual active set strategy. Te resulting algoritm can be regarded as a nonlinear multigrid metod; it also applies to problems wit Coulomb friction and yields a reliable and fast convergence. Finally, we present and study a full Newton approac for te Coulomb friction problem tat avoids fixed point ideas. Our implementation is based on te finite element toolbox UG, see [BBJ + 97], and te direct solver PARDISO, see [SGF00], is used for te full Newton approac. Te outline of tis paper is as follows: Te two-body Coulomb frictional contact problem is stated in Section 2, and te discretization based on mortar metods is briefly discussed. In Section 3, we develop and analyze our algoritm for 3D friction problems. For simplicity of presentation, we restrict ourselves to te case of a one-body contact problem wit Tresca friction. In Section 4, we study te performance of te algoritm for a test problem wit

4 4 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH Tresca friction. Using fixed point ideas, tis approac is extended to contact problems wit Coulomb friction in Section 5. In Section 6, we apply our algoritms to two-body contact problems wit Coulomb friction in 3D using a fixed point metod. Finally in Section 7, we present a full Newton approac for Coulomb friction in 3D and compare numerically its performance wit te discussed fixed point-based metods. 2. Contact problem wit Coulomb friction in 3D In tis section, we formulate te contact problem wit Coulomb friction in linear elasticity. In addition, we briefly describe our nonconforming discretization based on mortar metods and te basis transformation wic transforms te system for two bodies into a system aving exactly te same structure as a one-body contact problem Problem statement. Let Ω l IR 3, l {m, s}, denote two elastic bodies. Te superscript l will refer to s, for te slave, or to m, for te master body, as is common in te mortar setting. We assume tat te boundaries Ω l are divided into tree disjoint measurable parts Γ l D, Γl N and Γ l C wit meas(γl D ) 0. We impose omogeneous Diriclet conditions on Γl D and Neumann data, i.e., a surface traction p l (L 2 (Γ l N ))3 on Γ l N. Moreover, we denote by f l (L 2 (Ω l )) 3 te volume forces acting on Ω l. We use te constitutive law for linear elasticity, namely σ l := λ l tr(ε l )Id + 2µ l ε l in Ω l, were σ l denotes te stress field and ε l := 1/2( u l + u l ) te linearized strain tensor tat bot depend on te displacement u l. Te Lamé constants λ l, µ l > 0 are given by µ l := E l /(2(1 + ν l )) and λ l := (E l ν l )/(1 (ν l ) 2 ) wit Young s modulus E l > 0 and Poisson ratio ν l (0, 0.5). Moreover, tr denotes te matrix trace operator and Id te identity matrix. Ten, te displacement u l satisfies te following equations Divσ l + f l = 0 in Ω l, (2.1a) u l = 0 on Γ l D, (2.1b) σ l n l p l = 0 on Γ l N, (2.1c) were n l denotes te unit normal outward vector on te boundary Ω l. To state te contact and friction conditions, we introduce for eac point of Γ s C te vectors τ 1, τ 2 tat span te tangential plane and use n := n s. We

5 PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT 5 assume tat {n, τ 1, τ 2 } is an ortonormal basis in IR 3 for eac point of Γ s C. In order to formulate te non-penetration condition of te two bodies, we use a predefined relation between te points of te possible contact zones Γ l C. Tis relation is realized by a smoot mapping R : Γs C Γm C satisfying R(Γ s C ) Γm C. We assume tat te mapping R is well defined and maps any x Γ s C to te intersection of te normal on Γs C at x wit Γm C. Ten te contact conditions on Γ s C are given by [u] n d 0, σ n 0, ([u] n d)σ n = 0 on Γ s C, (2.2) were d 0 denotes te gap between te two elastic bodies, σ n := n σ s n te normal component of te boundary stress, [u] := (u s (x) u m (R(x))) te jump and [u] n := [u]n te jump in normal direction. For te boundary stress in normal direction on te possible contact part, we ave to satisfy te condition σ n = n σ s (x)n = n σ m (R(x))n on Γ s C. (2.3) Additionally, we ave to ensure τ k σs (x)τ k = τ k σm (R(x))τ k, k = 1, 2. Finally, te Coulomb friction law states tat σ τ Fσ n, σ τ < Fσ n [u] τ = 0, on Γ s C. (2.4) σ τ = Fσ n β 0 : σ τ = β[u] τ, Above, denotes te Euclidean norm in IR 3, σ τ := σ s n σ n n te tangential component of te boundary stress and [u] τ te relative tangential displacement given by [u] τ := [u] [u] n n. Furtermore, F : Γ s C IR, F 0, is te friction coefficient Mortar discretization. By using dual Lagrange multipliers, we can apply locally a suitable basis transformation for te finite element basis. Te sape functions on te master side are modified by adding a linear combination of nodal sape functions on te slave side. In te new basis te sape functions on te master side satisfy a weak continuity condition, i.e., te jump of tese basis functions tested wit a Lagrange multiplier being defined on te slave side is zero. Tus, te coefficients in tis new basis on te slave side of te contact zone describe te relative displacement between te contact interfaces. In tis basis, te two-body contact problem as te same structure as in te one body case, since all constraints at te contact zone are restricted to te degrees of freedom on te slave side. For details,

6 6 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH see [WK03, HW05]. Motivated by tese considerations, in te sequel we will write u for te coefficient vector wit respect to te new constrained basis. Terefore, te following discussion olds for bot, te one-body and te twobody case. We denote te multiplier corresponding to te discretization of σ s n on te slave side by λ. In te new basis, (2.1) is satisfied in te discrete form A u + B λ = f, (2.5) wit B of te form (0, D) were due to te use of dual Lagrange multipliers te matrix D is diagonal. Let us denote by S te set of all nodes of te finite element mes belonging to Γ s C and by N te set containing all oter nodes. Ten for eac p S, te entry of te diagonal matrix D is given by D[p, p] = Id 3 φ p ψ p ds = Id 3 φ p ds =: D p Id 3, Γ C Γ C were Id 3 denotes te identity matrix in IR 3 3, and φ p and ψ p te primal and dual basis function associated wit te node p, respectively. If te displacements u are known, te Lagrange multiplier can be computed directly by using (2.5) λ = D 1 (f A u ) S, (2.6) were te subscript S on te rigt and side indicates tat we use only te entries of te vector corresponding to te nodes p S. We next introduce te scaled normal and tangential components of te multiplier λ and te normal and tangential components of te relative deformation u. For a node p S, let ( λ ) λ τ,p,s := p D p τ 1p λ IR 2 and λ n,p,s := λ p p D p τ D pn p IR, 2p ( ) u u τ,p := p τ 1p u p τ IR 2 and u n,p := u p n p IR. 2p Te discretized (and scaled) gap at te node p S is defined as d p := 1 d ψ p ds. D p Ten, te discrete conditions for normal contact are given by Γ s C u n,p d p, λ n,p,s 0, λ n,p,s (u n,p d p ) = 0 (2.7)

7 PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT 7 for eac p S, see also (2.2). We note tat we use different scaling factors for te primal and dual variables and tat D p is proportional to te local messize. Tis is motivated by te fact tat te H 1/2 -norm for te Lagrange multiplier and te H 1/2 -norm for te displacements ave te same error reduction. We remark tat te proposed scaling factors yield better numerical convergence rates for te inexact version of te algoritms. Te discrete Coulomb friction conditions are given by λ τ,p,s Fλ n,p,s, λ τ,p,s < Fλ n,p,s u τ,p = 0, for all p S. (2.8) λ τ,p,s = Fλ n,p,s β 0 : λ τ,p,s = βu τ,p, Here for simplicity, F is assumed to be constant and independent of te solution. Differently from te Coulomb friction conditions (2.4), were te friction bound is Fσ n, for Tresca friction tis bound is a function g H 1/2 (Γ s C ), g 0, given a priori. Te corresponding discrete variable is g p := g φ p ds. Γ s C Now, for Tresca friction conditions we ave to replace te friction bound Fλ n,p,s in (2.8) by g p. 3. Treatment of Tresca friction in 3D In tis section, we derive our algoritm for te treatment of friction in te tree-dimensional case. To simplify te presentation, we restrict ourselves to te case u n,p = 0 for all nodes p S, wit Tresca friction. In te following sections, we will return to te more general Signorini contact condition (2.3) combined wit Coulomb friction. We remark tat due to te discussion in Subsection 2.2, te following part olds as well for a one-body contact problem as for a two-body contact problem. In our iterative algoritm, we ave to solve in eac iteration step a linear problem wit boundary conditions of Diriclet-, Neumann- or Robin-type on te contact zone Γ C. Te detection of te zones for te different types of boundary conditions is based on a primal-dual approac. Te Lagrange multiplier playing te role of te dual variable can be locally computed in a postprocess from te primal variable u, see (2.6). Tis algoritm for 3D friction is closely related to te primal-dual active set metod for 2D friction. Consequently, it perfectly fits into te abstract framework used in [HW05,

8 8 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH HMW06] and inerits te advantages of tese metods, i.e., teir simple implementation and teir numerical efficiency. Te metod can be regarded as an active set strategy or alternatively as a semismoot Newton metod. Our derivation below is based on te latter aspect tat also guarantees fast local convergence of te iterates, see Teorem Derivation of te iteration sceme. To start wit, we review te conditions for 3D Tresca friction as given by (2.8) using g p instead of Fλ n,p,s. Since for g p = 0 tese conditions simplify to omogeneous Neumann boundary conditions in tangential direction, we first assume tat g p > 0 and, on page 13 comment on te case g p = 0. By a straigtforward calculation, it can be verified tat (2.8) wit g p instead of Fλ n,p,s is equivalent to C(u τ,p, λ τ,p,s ) = 0 for all p S, c τ > 0, were te nonlinear complementarity function C(, ) is defined by C(u τ,p, λ τ,p,s ) := max(g p, λ τ,p,s + c τ u τ,p )λ τ,p,s g p (λ τ,p,s + c τ u τ,p ). (3.1) We remark tat te equivalence between (2.8) wit g p instead of Fλ n,p,s and (3.1) and te results to be presented below old for eac c τ > 0. In te sequel, we take (3.1) as te starting point for our algoritm. Our main idea is, to apply a Newton-type algoritm for te solution of C(u τ,p, λ τ,p,s ) = 0. Unfortunately, bot te Euclidean norm and te max-function are not smoot. However, tey are semismoot in te sense of [QS93, HIK03] wic justifies te application of a semismoot Newton metod. As generalized derivative for f(b) := max(a, b), we use G f (b) = 0 if a b and G f (b) = 1 if a < b. We remark tat te arbitrary coice for te case a = b does not influence te convergence rate of te proposed algoritm. Note tat, in te first term of (3.1) te Euclidean norm appears for nonzero arguments only. Tis is due to te fact tat if λ τ,p,s +c τ u τ,p = 0, we obtain max(g p, λ τ,p,s + c τ u τ,p ) = g p and te Euclidean norm vanises. Terefore, te only non-differentiability tat matters in (3.1) is te max-function. In te semismoot Newton step te derivative of te Euclidean norm only occurs for points tat are differentiable in te classical sense.

9 PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT 9 We now compute, for p S, te generalized derivative G C of C(, ). For te variation (δu τ,p, δλ τ,p,s ) IR 2 IR 2, we obtain G C (u τ,p, λ τ,p,s )(δu τ,p, δλ τ,p,s ) = max(g p, λ τ,p,s + c τ u τ,p )δλ τ,p,s + χ A λ τ,p,s (λ τ,p,s + c τ u τ,p ) λ τ,p,s + c τ u τ,p (δλ τ,p,s + c τ δu τ,p ) g p (δλ τ,p,s + c τ δu τ,p ). (3.2) Here, χ A denotes te caracteristic function of te set A τ = {p S : λ τ,p,s + c τ u τ,p > g p }, i.e., χ A = 1 if λ τ,p,s + c τ u τ,p > g p, and χ A = 0 if λ τ,p,s + c τ u τ,p g p. We note tat λ τ,p,s (λ τ,p,s + c τ u τ,p ) is a 2 2-matrix, eiter zero or of rank one. Performing a semismoot Newton step at a current iterate (u τ,p, λ τ,p,s ), one derives te new iterates (uk τ,p, λk τ,p,s ) from Solve G C (u τ,p, λ τ,p,s )(δuk τ,p, δλk τ,p,s ) = C(u τ,p, λ τ,p,s ) and update (u k τ,p, λk τ,p,s ) = (u τ,p, λ τ,p,s ) + (δuk τ,p, δλk τ,p,s ). (3.3) Te caracteristic function χ A in (3.2) separates te nodes of S into I k τ, te inactive set, and A k τ, te active set, according to I k τ := {p S : λ τ,p,s + c τu τ,p g p 0}, A k τ := {p S : λ τ,p,s + c τ u τ,p g p > 0}. (3.4a) (3.4b) Using tis notation and (3.2) in (3.3), a straigtforward computation sows tat te new iterate (u k τ,p, λ k τ,p,s) satisfies ( Id2 M p were Mp matrix Fp e p := u k τ,p =0 for p Ik τ, (3.5a) ) λ k τ,p,s c τ Mp u k τ,p= p for p A k τ, (3.5b) := e p (Id 2 Fp given by λ g p and te vector p IR 2 τ,p,s + c F τu p τ,p, p := e p ), wit te scalar value e p and te 2 2- Fp (λ ( := λ τ,p,s λ τ,p,s + c ) τu τ,p g p λ τ,p,s + c τu τ,p, (3.6) τ,p,s + c τu ). (3.7) τ,p

10 10 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH Note tat, wile on te inactive set I k τ Diriclet conditions are imposed, te condition on te active set A k τ is of Robin-type since it involves te displacement u k τ,p and te surface traction λ k τ,p,s. Te sets A k τ and I k τ approximate te sets of slippy and of sticky nodes at te solution, respectively. In te more general setting of Signorini conditions, we ave also to set te boundary condition in normal direction. Te Robin condition (3.5b) can be easily andled if we rewrite (3.5b) as λ k τ,p,s + L p u k τ,p = r p. (3.8) We note tat r p enters in te rigt and side of te linear system and Lp gives a contribution to te system matrix. Comparing (3.8) wit (3.5b) and (3.7), we get under te assumption tat Id 2 Mp is regular L p r p ( ) :=c τ (Id 2 Mp ) 1 Mp = c τ (Id2 Mp ) 1 Id 2, (3.9a) := (Id 2 Mp ) 1 p. (3.9b) We mention tat tis Robin condition only guarantees positive definiteness of te system matrix if L p is positive definite. Note tat for L p = 0, we find a pure Neumann condition. Te degeneration of Robin type to Diriclet type boundary conditions is not included in te form (3.9). However, tis is not necessary for our problem, since nodes p wit a Diriclet condition belong to te set A k τ and terefore are not andled by (3.8). Obviously one can see tat te matrix Id 2 Mp is not necessary regular and terefore not invertible. In te next subsection, we are going to present tree possible modifications of te Robin system (3.5b) suc tat a regular matrix Id 2 Mp is obtained. Two of tese modifications give a positive definite and symmetric matrix L p. We remark tat all modifications converge in te limit case to te original system (3.5b) and tus preserves te local convergence properties of te numerical algoritm Modifications of te Robin system. To obtain a robust and convergent sceme, we ave to replace te matrix Fp by a scaled matrix F, l = 1, 2, 3. Te index l stands for one of te tree possibilities considered in tis paper. Te scaling can be seen as a projection of te Lagrange multiplier onto its feasible set given in (2.8). We remark tat our numerical experience sows tat tis scaling is essential for te robustness of te iteration sceme. According to te definition of Mp and (3.7), we replace in (3.5b) Mp by

11 PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT 11 M and p by given by M := e p (Id 2 F ), := e p F (λ τ,p,s + c τu τ,p ), l = 1, 2, 3. In a second step, we replace te matrix Id 2 Mp a scaling factor β form (3.8) of te Robin boundary conditions is written as λ k τ,p,s wit r L M by Id 2 β wit > 0, suc tat te resulting matrix is regular. Ten te + L u k τ,p = M ) 1 Id 2 ), ( := c τ (Id2 β r := (Id 2 β M ) 1 (3.10) for l = 1, 2, 3. We mention tat similar modifications as done for te matrix Fp are used in primal-dual algoritms for te minimization of functionals involving Euclidean norms, see, e.g., [ACCO00, CGM99, HS06]). Tese type of problems ave a similar structure as (3.5). Clearly, tis relation is due to te fact tat te Tresca friction problem also corresponds to te minimization of a nonsmoot energy functional. F Since for all modifications Fp and β 1 as (u k τ,p, λ k τ,p,s) converges to te solution te modifications do not degrade te local superlinear convergence. Next we present te tree possible modifications used in tis paper. First modification. We use a parameter βp,1 1 in Id 2 βp,1 M p in (3.10) only if (λ τ,p,s ) (λ τ,p,s + c τu τ,p ) < 0. Tis condition is equivalent to te fact tat te angle between te two vectors is greater tan 90 degree. Since in te limit case bot vectors are parallel, te modification only applies wen te iterates are far away from te solution. In tis modification, te L p,1 unsymmetric matrix L p is replaced by We project te Lagrange multiplier onto its feasible set and define, wic is unsymmetric as well. Using α p F p,1 := λ τ,p,s ( λ := (λ τ,p,s ) ( λ τ,p,s + c ) τu τ,p λ τ,p,s + c τ u τ,p, τ,p,s λ τ,p,s + c τ u τ,p max ( g p, λ τ,p,s ) λ τ,p,s + c τ u τ,p. (3.11) δ p ) := min { λ τ,p,s }, 1, g p

12 12 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH it is easy to see tat γ 1 Fp,1 = αp δp is an eigenvalue of te matrix β p,1 := 1 α p δ p F p,1 wit te eigenvector λ τ,p,s. Te second eigenvalue is given by = 0 were te γ2 Fp,1 corresponding eigenvector is te vector wic is ortogonal to λ τ,p,s +c τuτ,p. Terefore, te eigenvalues of M p,1 are γ 1 Mp,1 = e p (1 αp δp ) and γ 2 Mp,1 = ep. Since 1 αp 1 and 0 δp 1, we get due to 0 < e p < 1 te relation 0 γ 1 Mp,1 < 2 and 0 < γ 2 Mp,1 < 1. Using (3.10) wit { 1 if αp < 0, yields a unsymmetric matrix 1 else. L p,1 wit positive eigenvalues. Second modification. In contrast to te first modification, we use a symmetric matrix F p,2. Here we need a parameter β p,2 1 in (3.10) for all cases wit (λτ,p,s ) (λ τ,p,s + c τu τ,p ) 0. We replace Fp by te symmetrization of (3.11), namely ( F p,2 := λ τ,p,s λ τ,p,s + c ) τu ( τ,p + λ τ,p,s + c )( ) τu τ,p λ τ,p,s 2 max ( g p, λ τ,p,s ) λ. (3.12) τ,p,s + c τ u τ,p One can proof tat te eigenvalues of F p,2 are γ 1,2 = F 1 p,2 2 (α p ± 1)δ p [ 1, 1] and terefore te eigenvalues of te matrix M p,2 are γ 1,2 = e M p,2 p (2 (αp ±1)δp )/2. Using te same arguments as before we get 0 γ 1,2 < 2. M p,2 Setting β p,2 := 2 2 (αp 1)δp results in a symmetric and positive definite matrix L p,2. Tird modification. In te tird modification, we use te matrix ( ) F p,3 := λ τ,p,s λ τ,p,s max ( g p, λ τ,p,s ) 2 (3.13) instead of F p wit eigenvalues γ 1 Fp,3 = 0 and γ 2 Fp,3 = (δ p eigenvalues of te matrix. Obviously tis matrix is symmetric and positive semidefinite ) 2. Terefore we get for te M p,3 due to 0 < e p < 1 te relation 0 γ Mp,3 < 1

13 PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT 13 and te matrix L p,3 defined by (3.10) wit βp,3 = 1 is symmetric and positive definite. We remark tat te matrix F p,3 converges in te limit case to te matrix Fp since in te solution for a node p A τ, we ave λ τ,p,s + c τ u τ,p λ τ,p,s + c τ u τ,p = λ τ,p,s. g p 3.3. Extension to te case g p = 0. In te above discussion, we assumed g p > 0 for all p S. Te reason for tis assumption is tat for g p = 0 one cannot stringently deduce λ τ,p,s = 0 (wic follows directly from (2.8)) from C(u τ,p, λ τ,p,s ) = 0. However, if Tresca friction combined wit fixed point ideas is used to solve Coulomb friction problems, g p = 0 naturally occurs for all non-contact points, wic makes te case g p = 0 rater important. Fortunately, after some minor canges, nodes p wit g p = 0 can also be andled using (3.5) wit (3.11)-(3.13). In te following, we consider p S wit g p = 0. First, we assume λ τ,p,s + c τ u τ,p > 0. Ten, p Ak τ since g p = 0. In te case λ τ,p,s + c τu τ,p = 0 M we ave to set p A k τ. In bot cases, we set = 0 and (3.5b) leads to te desired omogeneous Neumann condition λ k τ,p,s = 0. We mention tat in te case λ τ,p,s + c τu τ,p > 0 and λ τ,p,s > 0, we get due to (3.4b) and (3.6) automatically p A k τ and e p = 0 and terefore M = 0. In particular, tese matrices are well defined for tis case. So only for te cases λτ,p,s + c τ u τ,p = 0 or λ τ,p,s = 0, we ave to enforce te node p to be in A τ General Remarks. Computing generalized derivatives of nonsmoot functionals is a delicate issue. Wile in [AC91] mainly intuitive arguments are used, te related papers [CP99, CKPS98] use te concept of Bouliganddifferentiability. Tis concept allows for te use of globalization (e.g., linesearc) strategies, but calculating te searc direction requires to solve a nonlinear system in eac Newton step. In tese papers, tis problem is circumvented by substituting tis nonlinear system by a linear one. Despite tis euristic step, te autors report on good numerical results. Te concept of semismootness [QS93,HIK03] used in te present paper as te advantage tat te searc direction can be found by solving a linear system. Neverteless, one can also proof local superlinear convergence of te iterates, see Teorem 4.1.

14 14 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH Furtermore, we remark tat tere is some freedom in coosing te nonlinear complementarity function to express te complementarity conditions for te Tresca friction law. For g p > 0, we can also work wit (3.1) by C(u τ,p, λ τ,p,s ) := λ τ,p,s g p (λ τ,p,s + c τ u τ,p ) max(g p, λ τ,p,s + c τ u τ,p ). (3.14) Complementarity functions closely related to (3.14) for dealing wit friction conditions ave been used in [AC91, CP99, CKPS98]. A semismoot Newton iteration for te solution of C(u τ,p, λ τ,p,s ) = 0 results in an iteration rule tat also uses te active and inactive sets defined in (3.4), but results in a modified iteration step on A k τ. Our numerical experience yields tat algoritms based on (3.1) perform more robust compared to tose based on (3.14) Algebraic representation. In tis subsection, we give te matrix representation of te algebraic system, we ave to solve in eac iteration step. As mentioned above, we restrict ourselves to te case u n,p = 0. Te tangential conditions are eiter Diriclet conditions for te inactive nodes p Iτ k or Robin conditions for te active nodes p A k τ. To rotate te stiffness matrix A arising form standard linear elasticity and introduced in (2.5), we define te matrix wit te normal vectors by N := 0 0 n p IR S 3S,.. and te matrix wit tangential vectors by τ 1 T := p τ 2p IR2S 3S,.. were as above, for eac p S, n p, τ 1p and τ 2p denote te unit outward normal and te tangential vectors, respectively. Moreover, for p A k τ te matrices L are assembled into L A k τ := diag { L } IR 2Ak τ 2Ak τ. p A k τ Similarly, we define te vector r A k τ by te entries r for p A k τ. We now use te decomposition of S into A k τ and I k τ. Te vector u can ten be

15 PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT 15 partitioned into (u N, u, u I k τ Aτ), te multiplier λ k into (λ, I k λ τ A k) and te τ rigt and side f into (f N, f, f I k τ A k). Correspondingly, we can decompose τ te matrices A, N and T, and te arising linear system as te form A NN A NI k τ A NA k τ 0 Id I k N A k τ T A k τ A A k τ N T A k τ A A k τ I T k τ A k τ A + L T A k τ A k τ A k τ A k τ u k N u k I k τ u k A k τ = T A k τ f A k τ + r A k τ (3.15) We remark tat te Diriclet boundary condition (3.5a) on I k τ is reflected in te second row and te Robin-type conditions for p A k τ, see (3.5b) or (3.8), are included in te fourt row. In te case of te more general Signorini condition, we ave to replace te tird row by a more complex one, see [HW05]. 4. Algoritm and numerical examples for Tresca friction In tis section, we give te inexact primal-dual algoritm based on an efficient multigrid metod to solve te resulting linear problems. Furtermore, we present a numerical example to study te performance of our algoritm and compare te tree modifications of Section 3.2. Wile in te previous section, we ave derived te iteration rule for Tresca friction witout taking into account Signorini contact, te algoritm presented can andle Tresca friction and Signorini contact (2.2). Te discrete Signorini conditions (2.7) are realized by applying te primal-dual active set strategy on te nonlinear complementarity function f N 0 0. λ n,p,s max(0, λ n,p,s + c n (u n,p d p )) = 0, (4.1) wit c n > 0. Te strategy uses, in eac iteration te active and inactive sets A k n I k n := {p S : λ n,p,s + c n(u n,p d p) > 0}, := {p S : λ n,p,s + c n (u n,p d p ) 0}. We next state te inexact primal-dual active set (Ipdas) strategy for contact wit Tresca friction. Algoritm 1: Ipdas for contact wit 3D Tresca friction

16 16 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH (0) Set k = 1, coose c n > 0, c τ > 0 and m IN. Initialize u 0,0 and λ 0 as an initial solution. (1) Define te active and inactive sets by A k n := { p S : λ n,p,s + c ( ) } n u,m n,p d p > 0, In k := { p S : λ n,p,s + c ( ) } n u,m n,p d p 0, A k τ := { p S : λ τ,p,s + c τu,m τ,p g p > 0 }, I k τ := { p S : λ τ,p,s + c τ u,m τ,p g p 0 }. (2) For i = 1,..., m, compute u k,i = MG(uk,i 1 (3) If u k,m u k,0 / uk,m < ε u stop. (4) Compute te Lagrange multiplier as (5) Set u k+1,0, A k n, Ik n, Ak τ, Ik τ, u,m, λ ). λ k = D 1( f S A S u k,m = u k,m, k = k + 1 and go to step (1). ). Above, we denote by u k,i = MG(u k,i 1, A k n, Ik n, Ak τ, Ik τ, u,m, λ ) te iterate after one multigrid step for te linear system (3.15). By A S, we denote te rows of te stiffness matrix A corresponding to te nodes in te set S. In te case m IN, we solve te linear system inexactly, i.e., we update te active and inactive sets after m multigrid steps. For m =, we get te exact version of our algoritm. Recalling te derivation of our iteration rule as a semismoot Newton metod, we obtain te following local convergence result, see [HIK03,QS93]. Teorem 4.1. For m =, Algoritm 1 converges locally superlinear. We mention tat in te multigrid approac, we call te coarsest grid, on wic we solve exactly, te grid on level 0. We get te next finer grid on level l+1 by decomposing eac element on level l into 8 subelements. For all examples presented in tis paper, we use a W-cycle wit tree pre- and postsmooting steps for te multigrid. As smooter, a symmetric Gauß Seidel iteration is applied.

17 PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT Example. As a first example for a tree-dimensional contact problem wit Tresca friction, we consider a one-body contact problem. Te problem setting. We consider a linearly elastic cube Ω = [0, 1] 3 wit material parameters E = 200 and ν = 0.3 and take te xy-plane as te rigid obstacle wic implies tat te initial gap is d = 0. Te cube is subject to te Diriclet distortion u = (0, 0.2, x) on its upper surface [0, 1] 2 {1}, see te left part of Figure 1. For Tresca friction, te friction bound g is given a priori and does not depend on te distortion as in te case of Coulomb friction. For tis problem, we coose g = 800xy(1 x)(1 y). In te rigt of Figure 1, we sow te distorted body wit te effective von Mises stress σ eff given by σeff 2 := 3 i,j=1 σ ij δ ij p 2, were te pressure p is given by p := 1 3 tr(σ). Te Tresca friction law. To get a better understanding of te different types of nodes tat occur for contact problems wit Tresca friction, we sow in Figure 2 te nodes of te contact surface on Level 5. Different types of nodes are marked differently, see te legend in Figure 2. Note tat, for eac node te displacement is parallel to te multiplier λ τ as required. We remark tat, using te Tresca friction law, nodes can stick in tangential direction witout being in contact wit te obstacle. Figure 1. Problem definition wit prescribed Diriclet distortion (left) and distorted body wit effective von Mises stress σ eff ; te lower surface in te plot is subject to unilateral contact and Tresca friction. Performance of Algoritm 1. To investigate te performance of te algoritm, we first solve te linear system arising in eac iteration step exactly.

18 18 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH no contact and slip no contact and stick contact and slip contact and stick distortion λ τ Figure 2. Visualization of te solution at te nodes p S for Tresca friction on Level 5 (left), legend (upper rigt) and cutout (lower rigt). We note tat in eac step, our algoritm updates te contact/non-contact sets A k n and Ik n as well as te slip/stick sets Ak τ and Ik τ and at te same time perform a Newton step to adopt te distortion s direction u τ to te direction of λ τ for sliding nodes. Table 1. Performance of Algoritm 1 for te exact strategy (m = ). Initialization: u 0,0 = 0, λ 0 = 0 for all levels. Tolerance ε u = 10 9 wit te modification (3.11). l A k n /Ak τ for k = 2,3,4,5,6,7,8,9 0 2/0 2/0 1 6/4 5/8 5/8 5/8 5/8 5/8 2 15/16 12/16 12/16 12/16 12/ /32 38/37 33/39 33/39 33/39 33/ /66 132/ / / / / / / / / / / / / /446 For our tests, we initialize te algoritm on eac level wit u 0,0 = 0 and λ 0 = 0. Tis leads to A1 n = A1 τ =. We terminate te Ipdas-iteration if te relative cange in te solution is less tan In te complementarity function, we use c n = c τ = 100; Algoritm 1 yields a fast and stable converge

19 PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT 19 error Convergence rate modification 2. modification 3. modification iteration steps Factor α p 1. modification 2. modification 3. modification iteration steps Scaling Factor β p 1. modification 2. modification iteration steps Figure 3. Convergence of λ k on Level 5, c n = c τ = 100 and ε u = (left); Beavior of α (middle) and β (rigt) at te node (0.0625, 1, 0) S; u 0,0 = (1, 1, 0), λ 0 = ( 1, 1, 0). on all levels. Table 1 sows te number of iterations needed on different refinement levels, and te number of nodes belonging to te active sets A k n and A k τ for te first modification (3.11). We remark tat for te second (3.12) and te tird modification (3.13) only minor differences occur. Note tat in eac iteration step one linear system as to be solved. Te number of iterations increases only weakly on finer levels; it seems to depend linearly on te level. Usually, after te exact active sets for bot friction and contact condition are found, te metod requires about 3 4 more iterations to converge. In tese steps, te algoritm adjusts, for p S, te direction of te tangential traction λ τ,p to te tangential displacement u τ,p. Comparison between te modifications. Now we compare te convergence and te beavior of te factors αp and β for te tree modifications. In te left of Figure 3, we sow te errors λ k λ on Level 5 in a logaritmic plot for te initialization u 0,0 = (1, 1, 0) and λ 0 = ( 1, 1, 0). Here, λ denotes te Lagrange multiplier of te solution. We remark tat additional iteration steps, compared to Table 1, are needed due to te different initialization and te smaller tolerance ε u = In te middle of Figure 3, we present te cosinus αp of te angle between te vectors λ τ,p,s and λ τ,p,s + c τu τ,p and on te rigt te beavior of te scaling factor β, l = 1, 2, for te node p = (0.0625, 1, 0) A k τ for all k. We observe for all modifications a superlinear convergence. Comparing te beavior of te factor α te first and te second modification sow almost te same beavior. Altoug te factor α tends faster towards one for te tird modification,

20 20 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH Table 2. Comparison between te exact (m = ) and te inexact (m = 1) strategy wit c n = c τ = 100 using (3.11). Tolerance for Ipdas strategy: ε u = strategy exact inexact nested level l DOF K l MG-steps M l MG-steps M l MG-steps we observe a slower convergence. For te factor β we observe a better beavior for te first modification tan for te second one. From now on, we use modification one for all computations. Exact versus inexact metod. We next compare te exact version, i.e., m =, of Algoritm 1 wit te inexact version. In te inexact version, we use m = 1, i.e., we update te active and inactive sets after eac multigrid step. We denote by K l te iteration step in wic te correct active and inactive sets are found for te first time and never cange later in te iteration. For te inexact approac, we denote tis step by M l. Table 2 sows te numbers K l and M l on eac level and te necessary numbers of multigrid iterations to solve te full nonlinear problem on level l. We observe tat te numbers K l and M l are almost te same. Tey seem to depend linearly on te level l. Terefore, tere is no need to solve te linear system exactly. Furtermore, we compare te inexact approac, were we start wit u 0,0 = 0 and λ 0 = 0 on eac level, wit te nested approac in wic we inerit u 0,0 and λ 0 on level l+1 from level l. Te values K l and te necessary numbers of multigrid iteration steps are sown in te last column of Table 2. We note tat te inexact primal-dual active set strategy can be interpreted as a non-linear multigrid metod. Influence of te parameter c τ. In a last test, we investigate te influence of te parameter c τ on our algoritm. Recall tat c τ can be seen as weigt for te tangential distortion u τ,p in te sum wit te tangential component λ τ,p,s of te Lagrange multiplier. Tus, it plays a similar role for te tangential component as c n for te normal component. In Table 3, we compare te numbers K l and te numbers MG of necessary multigrid steps for different values of c τ, were we fix c n = 100 and use te inexact approac m = 1. As

21 PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT 21 Table 3. Comparison between different values for c τ for te inexact strategy (m = 1) using c n = 100 and ε u = Initialization: u 0,0 = 0, λ 0 = 0 for all levels l. c τ level l K l MG K l MG K l MG M l MG M l MG can be seen, te algoritm beaves quite stable and independent of c τ, if c τ is large enoug. For tis example, we find c τ 10. For c τ = 1 te algoritm does not converge on Level 5. A very similar beavior is observed in [HW05] wit respect to te parameter c n. Tus, in general it appears advantageous to coose bot c τ and c n in order to balance te different scales of te distortion u and te Lagrange multiplier λ. 5. Fixpoint algoritm and numerical examples for Coulomb friction In tis section, we extend Algoritm 1 to contact problems wit Coulomb friction and give a numerical example sowing te performance of te metod. For Coulomb friction, te friction bound g p = Fλ n,p,s needs to be iteratively adjusted using te normal component of te Lagrange multiplier. Terefore, we get a furter outer loop for te update of te friction bound. Algoritm 2: Ipdas fixed point strategy for contact wit 3D Coulomb friction (0) Set k = 1 and coose c n > 0, c τ > 0, m IN and k f IN. Initialize u 0,0 and λ 0. (1) If mod kf (k 1) = 0, set k c = k 1 and update te friction bound by g k c p = F max{ 0, λ k c n,p,s}, p S.

22 22 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH (2) Define te active and inactive sets by A k n := { ( ) } p S : λ n,p,s + c n u,m n,p d p > 0, In k := { p S : λ n,p,s + c ( ) } n u,m n,p d p 0, A k τ := { p S : λ τ,p,s + c τ u,m τ,p g k c p > 0 }, Iτ k := { p S : λ k τ,p,s + c τu,m τ,p g k c p 0}. (3) For i = 1,..., m, compute u k,i = MG(uk,i 1 (4) Compute te Lagrange multiplier as (5) If k > k f and u k,m (6) Set u k+1,0, A k n, Ik n, Ak τ, Ik τ, u,m, λ ). λ k = D 1( f S A S u k,m u k c,m ). / u k,m < ε u stop. = u k,m and k = k + 1 and go to step (1). In tis algoritm, we denote by mod te modula-operator. Comparing tis algoritm wit Algoritm 1 for Tresca friction, we remark tat ere we update te friction bound after k f steps of te (inexact) active set strategy. Tis update is done in step (1). Since we do not solve te resulting linear problems exactly, it is not guaranteed tat λ k n,p,s 0 for all p S. Terefore, we set g p = F max { 0, λn,p,s} k. For te coice m = kf = 1, te friction bound and te active and inactive sets are updated after eac multigrid step. As stopping criterion, we use te relative error between te actual solution u k,m and te solution for te last friction bound u k c,m. For te coice m = and k f =, we get te exact version of te algoritm. In tis case, we solve te resulting Tresca friction problem exactly for eac friction bound. Obviously, tis approac is rater costly. However, for a small friction coefficient F it can be sown tat tis discrete fixed point mapping is contractive and tus converges; see [NJH80] Example. In tis section, we study te performance of Algoritm 2 for Coulomb friction. Tis friction model is pysically more realistic tan te Tresca model, since only points tat are in contact wit te obstacle are points for wic friction occurs. Points on te contact boundary wit a

23 PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT 23 positive distance to te obstacle are traction-free, i.e., we apply omogeneous Neumann conditions Figure 4. Distorted body wit effective von Mises stress σ eff (left); te lower surface in te plot is subject to contact and Coulomb friction. Visualization of te friction bound Fλ n,p (small balls) and of λ τ,p (rigt) no contact contact and slip contact and stick distortion λ τ Figure 5. Visualization of te solution at te nodes p S for Coulomb friction on level 5 (left), explaining legend (upper rigt) and cutout (lower rigt). Recall tat in te Coulomb law, te friction bound becomes g p = Fλ n,p,s and tus depends on te actual distortion. In te following tests, we consider

24 24 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH te same geometry and data as for te example of Section 4.1 and coose te friction coefficient F 1. Te distorted cube can be seen in te left plot of Figure 4. Note tat its distortion is significantly different from te one obtained wit te Tresca law; (Figure 1). In te rigt of Figure 4, we visualize te constraint λ τ,p,s Fλ n,p,s tat olds for all p S. Te few nodes were tis inequality olds in a strict sense are nodes in contact wit te obstacle tat, at te same time, stick to tis obstacle, i.e., u n,p = d p = 0 and u τ,p = 0. We remark tat te solution as a singularity at te node (1, 1, 0). Te Coulomb friction law. As for Tresca friction, (Figure 2), we also visualize te different types of contact nodes for te Coulomb law. Note tat, in contrast to te Tresca model, te Coulomb law only allows tree types of nodes, since nodes tat are not in contact wit te obstacle are not subject to any (friction) constraints. Te size of te nodes in Figure 5 is proportional to te normal contact force λ n,p Level 0 (k max = 32, K l = 10) A n A τ Level 1 (k max = 38, K l = 9) A n A τ Level 2 (k max = 45, K l = 10) A n A τ Level 3 (k max = 47, K l = 21) Level 4 (k max = 46, K l = 12) Level 5 (k max = 45, K l = 14) A n A τ A n A τ A n A τ Figure 6. Beavior of Algoritm 2: Numbers A k n and Ak τ of active nodes in eac iteration step k on eac level l. We used te parameters c n = c τ = 100, m = k f = 1 and ε u = Initialization: u 0,0 = 0 and λ0 = 0. Performance of te algoritm. We apply te inexact version of Algoritm 2 wit m = k f = 1 to solve te contact problem wit Coulomb friction. We use c n = c τ = 100, ε u = and initialize te iteration wit u 0,0 = 0 and λ 0 = 0 on eac level. Figure 6 sows te beavior of our algoritm on various

25 PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT 25 Table 4. Beavior of Algoritm 2: Necessary numbers k max and K l on eac level l for c n = c τ = 100, m = k f = 1 and ε u = level l DOF u0,0 = 0, λ0 = 0 nested approac K l k max K l k max levels. Te number of nodes contained in te active sets are plotted over te iteration steps k. We denote by k max te necessary number of iteration steps and by K l te iteration step in wic te correct active sets are found for te first time and do not cange afterwards. Te number K l is marked by a dased vertical line in Figure 6. We observe tat bot k max and K l appear to be quite independent of te level l. Looking at Figure 6 more closely, we note tat on eac level tere are only minor canges of te active nodes after k = 10. Table 4 sows a comparison between te initialization u 0,0 = 0 and λ 0 = 0 on eac level and te nested approac. As expected, te correct active sets are found earlier, and as a result fewer iterations are required for te nested approac. 6. Numerical examples for two-body contact wit Coulomb friction Now, we consider a curved contact interface subjected to Coulomb friction. A two-dimensional cross section of our geometry is sown in Figure 7. Te lower domain Ω m, assumed to be te master side, models a sperical sell tat is fixed at te outer boundary. Against tis sell, we press te body modeled by te domain Ω s, wic is assumed to be te slave side. At te top surface of Ω s, we apply te surface traction (0, 0, 150 exp( 100r 2 )), were r denotes te distance to te midpoint of te top surface of Ω s. Te geometry is given by r i = 0.7, r a = 1.0, r = 0.6, = 0.5 and d = 0.3. In Ω s, we use a Young modulus E s = 300 and a Poisson ratio ν s = 0.3, wile in Ω m we ave E m = 400 and ν m = 0.3. Discussion of te results & comparison for various friction coefficients. Te results for te friction coefficient F = 0.7 are sown in Figures 7-9.

26 26 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH d Ω s r i r r a Ω m Figure 7. Problem definition (left), deformed mes wit effective von Mises stress σ eff on Level 3 (middle), two-dimensional cross section of te deformed mes on Level 2 (rigt) for F = Figure 8. Visualization of te nodes being in contact togeter wit te deformation and stress vectors on Level 3 for F = 0.7 (left), and a cutout (rigt); for te legend we refer to Figure λ n F*λ n λ τ Figure 9. Visualization of λ n (left) and λ τ (middle) wit te friction bound 0.7λ n (dotted), and two-dimensional visualization of te Lagrange multipliers (rigt) on Level 3. Figure 7 sows te deformed body wit te effective von Mises stress σ eff on Level 3 and a two-dimensional cross section on Level 2. In Figure 8, we

27 PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT Friction coefficient F= λ n F*λ n λ τ Figure 10. Visualization of te nodes being in contact, togeter wit te deformation and te stress vectors (left), visualization of λ τ (middle) wit te friction bound 0.5λ n (dotted) and a two-dimensional visualization of te Lagrange multipliers (rigt). sow te nodes being in contact, teir relative tangential slip [u τ,p ] (lines) and te tangential contact pressure λ τ,p (arrows). We remark tat te nodes in te middle wit only an arrow are te sticky nodes, te oters are slippy. Figure 9 sows on te left te normal part of te contact pressure λ n and in te middle te norm of te tangential part λ τ. Te dots represent te friction bound 0.7λ n. Te two-dimensional plot on te rigt sows te normal and tangential part of te Lagrange multiplier and te friction bound for all nodes p S over teir distance to te midpoint of te contact zone on te surface of Ω s. Te small oscillations in te plot occur due to te fact tat all nodes p S over te wole contact zone are presented and tat we work wit an unstructured mes. Te results for te same problem but wit te smaller friction coefficient F = 0.5 are sown in Figure 10. Te number of sticky nodes decreases for smaller F. Level 1 (k max = 31, K l = 9) Level 2 (k max = 24, K l = 8) Level 3 (k max = 26, K l = 9) A n A τ A n A τ A n A τ Figure 11. Beavior of Algoritm 2: Numbers A k n and Ak τ in eac iteration step k on Levels 1, 2, 3 for F = 0.7. We use te parameters c n = c τ = 100, m = k f = 1 and ε u = 10 9.

28 28 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH Table 5. Beavior of Algoritm 2: Necessary numbers k max and K l on eac level for c n = c τ = 100, m = k f = 1, ε u = 10 9 and F = 0.7. level DOF u0,0 p = 0.1 n p, λ 0 p = n p nested approac K l k max K l k max Performance of te algoritm. We consider te performance of Algoritm 2 for te friction coefficient F = 0.7. For te initialization we set u 0,0 p = 0.1n p for p S, up 0,0 = 0 for p / S and λ 0 p = n p on eac level. Using te parameters c n = c τ = 100, m = k f = 1 andε u = 10 9, we get te performance of Algoritm 2 sown in Figure 11. Here, te number of nodes in A k n and Ak τ are sown for Level 1-3. Again, te dased vertical line marks te step K l in wic te correct active sets are found for te first time and remain uncanged. A comparison between tis approac and te nested approac is sown in Table 5. Te results obtained sow qualitatively te same beavior as for te cube; see Section Friction coefficient F= Figure 12. Nonsymmetric boundary data: Deformed mes wit effective von Mises stress σ eff on Level 3 (left), visualization of te nodes being in contact togeter wit te deformation and te stress vectors (middle) and of λ τ (rigt) wit te friction bound 0.7λ n (dotted). Problem wit nonsymmetric boundary traction. For te next example, we use te same data and geometry as above (see Figure 7), but use a nonsymmetric boundary traction, namely, we apply te surface traction

29 PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT 29 (15, 0, 150 exp( 100r 2 )) at te top surface. Te results for F = 0.7 are presented in Figure Full Newton approac for multibody contact wit Coulomb friction Wile our algoritm in Section 5 for te solution of te Coulomb friction problem is based on fixed point ideas, we now present a full Newton approac for te Coulomb frictional contact problem. Te main advantage of tis approac is its fast convergence wic is due to te fact tat te friction bound is updated in te Newton iteration and not via a fixed point loop. In tis section, we apply a fast direct solver [SGF00] to solve te linear system in eac Newton step. To derive te full Newton iteration, one replaces g p in (3.1) eiter by g p (u n,p, λ n,p,s ) := Fmax(0, λ n,p,s ) (see also [CKPS98,CP99]) or by g p (u n,p, λ n,p,s ) := Fmax(0, λ n,p,s +c n (u n,p d p )) before deriving te Newton iteration step. Te equivalence of tese two coices follows from (4.1). In te sequel, we use te latter replacement since ten te Newton-type iteration automatically takes te form of an active set metod tat, in eac iteration estimates te tree relevant sets for Coulomb friction (no contact, contact and stick, contact and slip). Te resulting nonlinear complementarity function is D(u, λ ) :=max ( F(λ n,p,s + c n (u n,p d p )), λ τ,p,s + c τ u τ,p ) λ τ,p,s Fmax ( 0, λ n,p,s + c n (u n,p d p ) ) (λ τ,p,s + c τ u τ,p ). (7.1) Similarly as in Section 3, we derive a semismoot Newton step for te solution of D(u, λ ) = 0 and (4.1). Using te notation gp := F(λ n,p,s + c n (u n,p d p)), we obtain te following settings. On I k n := {p S : g p 0} (estimation for set of nodes not in contact): λ k n,p,s = 0 and λ k τ,p,s = 0. Note tat I k n A k τ, were A k τ := {p S : λ τ,p,s+c τ u τ,p gp 0}. We remark tat te setting for λ k τ,p,s is derived directly from (7.1) for tat case, namely from λ τ,p,s + c τ u τ,p λ τ,p,s = 0. On I k τ := {p S : λ τ,p,s + c τ u τ,p gp of sticky nodes): u k n,p = d p and u k τ,p + ( Fu τ,p /g p < 0}: (estimation for set ) λ k n,p,s = u τ,p. (7.2)

30 30 S. HÜEBER, G. STADLER AND B.I. WOHLMUTH Note tat I k τ Ak n, were Ak n := {p S : g p > 0}. > 0} (estima- On A k τn := A k τ A k n = {p S : λ τ,p,s + c τ u τ,p gp tion for set of slippy nodes): were v p and g p L p. u k n,p = d p λ k τ,p,s + L u k τ,p + Fv λ k n,p,s = r and + gp v, (7.3) := (Id 2 Mp ) 1 (λ τ,p,s + c τu τ,p )/ λ τ,p,s + c τu τ,p IR2 is used for te friction bound g p in te matrices Mp and Note tat I k n, I k τ, and A k τn represent a disjoint splitting of S. Comparing (7.2) and (7.3) wit (3.5a) and (3.5b), respectively, we observe tat te main difference is te term involving λ k n,p,s on te left and side of (7.2) and (7.3). Again we apply te modifications stated in Subsection 3.2 and replace te matrices M L Mp and L p by and, respectively. Terefore, we use instead of vp te vector ṽ := (Id 2 β M ) 1 (λ τ,p,s+c τ u τ,p )/ λ τ,p,s+c τ uτ,p. Now, we briefly state te algebraic representation of te above system, were we use in addition to te notation introduced in Section 3 and U := diag { u τ,p G := diag { g p Id 2 }p S IR 2S 2S. }p S, V := diag{ } ṽ IR 2S S. p S We use a subblock notation for te matrices, e.g., G I n τ.... Defining Ñ := N In, k we obtain after eliminating te Lagrange multiplier, te following linear system to be solved in eac full Newton step for te Coulomb problem as AÑÑ AÑI k τ AÑA k τn 0 N I k τ N A k τn K I k τ A I k τ Ñ K I k τ A I k τ I k τ G I k τ T I k τ K I k τ A I k τ A k τn u k Ñ u k I k τ u k A k τn = f Ñ d I k τ d A k τn K I k τ f I k τ j I k τ T A A k τn A k τn Ñ T A A k τn A k τn I k τ T A A k τn A k τn A +L k τn A T k τn A k τn T A k τn f A k τn + r A k τn wit te notation T A k τn := T A k τn FV A k τn N A k τn, K I k τ := FU I k τ N I k τ

31 and PDAS ALGORITHM FOR 3D COULOMB FRICTIONAL CONTACT 31 r A k τn := r A k τn + G A k τn V A k τn, j I k τ := G I k τ U I k τ. We observe, comparing tis system wit (3.15), tat ere te normal and tangential components are pointwise coupled by te lines four and five. u s D = 0 us D = 0 Ω s Ω s Ω m Figure 13. Problem definition (left), deformed mes wit effective von Mises stress σ eff (middle), two-dimensional cross section of te deformed mes (rigt) for F = 0.7. Numerical Example. As example, we consider te situation presented in te left of Figure 13, were a two-dimensional cross section of te problem definition is sown. Te ring is fixed on its upper outer edge and te tool on its bottom. Note tat te bodies penetrate in teir reference configuration. We use Young modulus E m = and a Poisson ratio ν m = 0.3 for te inner tool modeled by Ω m, and E s = , ν s = 0.3 for te outer ring being te slave domain Ω s. Te friction coefficient is F = 0.7. Te deformed mes wit effective von Mises stress σ eff is sown in te middle and te rigt of Figure 13. Figure 14 sows te possible contact nodes on te ring being te slave side. Te nodes witout a line are nodes not being in contact wit te inner tool. Comparison between fixpoint Newton and full Newton approac. To sow te beavior of te full Newton approac, we compare te convergence rates of te Lagrange multiplier wit tose obtained from te fixpoint Newton approac given by Algoritm 2 wit m = and k f = 1. Our algoritm is initialized wit u 0,0 = 0 and λ 0 = 0, and te parameters c n = c τ = 10 8 and te tolerance ε u = 10 9 are used. Te finite element mes consists of degrees of freedom. In te left of Figure 15, te relative error λ k λ / λ is sown for te fixed point and te full Newton approac