Partitioned formulation of frictional contact problems using localized Lagrange multipliers

Transcription

1 COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2000; 00:1 6 Preared using cnmauth.cls [Version: 2002/09/18 v1.02] Partitioned formulation of frictional contact roblems using localized Lagrange multiliers José A. González 1, K. C. Park 2,, and Carlos A. Felia 2 1 Escuela Suerior de Ingenieros de Sevilla, Avda. de los Descubrimientos s/n, Sevilla, Sain 2 Deartment of Aerosace Engineering Sciences, Center for Aerosace Structures, University of Colorado, Boulder, CO , USA SUMMARY Interface treatment methods for the contact roblem between non-matching meshes have traditionally been based on a direct couling of the contacting solids emloying a master-slave strategy or classical Lagrange multiliers. These methods tend to generate strongly couled systems that is deendent on the discretization characteristics on each side of the contact zone. In this work a dislacement frame is intercalated between the interface meshes. The frame is then discretized so that the discrete frame nodes are connected to the contacting substructures using the localized Lagrange multiliers collocated at the interface nodes. The resulting methodology alleviates the need for master-slave book-keeing and rovides a artitioned formulation which reserves software modularity, facilitates non-matching mesh treatment and asses the contact atch test. Frictional contact roblems are used to demonstrate the salient features of the roosed method. Coyright c 2000 John Wiley & Sons, Ltd. key words: contact; friction; finite element method; localized Lagrange multiliers 1. INTRODUCTION In comutational structural mechanics, roblems resenting frictional contact surfaces combined with non-matching meshes remain as a difficult task. The source of this difficulty comes not only from the strong non-linearity of the frictional contact law, involving multivalued relationshis between kinematic and static variables, but also from the severe discontinuity forced by the different meshes used to model the contact interface. Most existing interface treatment methods for the contact roblem between non-matching meshes have traditionally been based on a direct couling of the contacting solids emloying a masterslave strategy, classical Lagrange multiliers or mortar-like methods [1, 2, 3, 4]. Corresondence to: K. C. Park, Center for Aerosace Structures and Deartment of Aerosace Engineering Sciences, University of Colorado at Boulder, Camus Box 429, Boulder, CO , U.S.A. kcark@colorado.edu Contract/grant sonsor: Consejería de Educación y Ciencia, Junta de Andalucía, Sain; contract/grant number: Convocatoria 2004 de concesión de ayudas ara el erfeccionamiento de investigadores en centros de investigación de fuera de Andalucía Received 17 January 2000 Coyright c 2000 John Wiley & Sons, Ltd. Revised 18 Setember 2002

2 2 J. A. GONZÁLEZ, K. C. PARK AND C. A. FELIPPA The resent work is an extension of the formulation roosed by Rebel, Park and Felia [5] to solve the contact roblem with non-matching meshes introducing an intermediate contact surface, or contact frame, endowed with indeendent degrees of freedom and treated with a FEM discretization. This frame is connected to the contacting substructures using localized Lagrange multiliers [6] and can be constructed in order to reserve the constant-stress interface atch test [7]. In the work of Rebel, Park and Felia a general formulation of the roblem is derived using the variational framework roosed by Park and Felia [8, 9] and then is articularized to the two dimensional case. However, they considered the contact zone to be a riori known after alying a two stages redictor-corrector algorithm and decided the contact oint states using a trial and error based algorithm. In the resent work, we extend [5] so that the contact frame may searate from the substructures in all directions, that is, the frame now is a comletely free system that moves while maintaining the contact zone as an unknown. Another new feature is in the way of finding contact states; in the resent work contact conditions are imosed mathematically using the augmented Lagrangian formulation and rojection functions, making the contact search to be art of the overall contact algorithm. 2. THE CONTACT FRAME Let us suose two solids in contact and denote their domains Ω and Ω. To formulate the contact roblem, instead of considering the direct interaction between the two bodies during the contact rocess, we will insert a deforming non-hysical surface Λ between them and reformulate the contact roblem in terms of interaction of the two solids with this new element, called the contact frame, using localized Lagrange multiliers on each side of the frame. The contact tractions acting on the frame are reresented in the exloded view of Figure 1 where the localized Lagrange multiliers connecting solid Ω with the frame are named using the vector quantity λ = (λ n, λ t1, λ t2 ) t and the multiliers connecting solid Ω are named λ = ( λ n, λ t1, λ t2 ) t. These forces are exressed using two locally orthonormal base systems connected to the frame; B = [n, a 1, a 2 ] used to describe λ and B = [ n, ā 1, ā 2 ] used for λ. These local systems for describing the frame kinematics are defined in the following way; a 1 and a 2 are the orthogonal vectors contained in the frame tangent lane at the considered oint and vector n oints towards solid Ω. The barred base system B at the same osition, will be defined in oosite direction to B. In order to describe the motion of the two solids we use the dislacement fields u and ū defined on Ω and Ω resectively, which, when added to the reference configurations X and X, rovide us with the current ositions x and x: x = X + u x = X + ū The motion of the contact frame is described using its dislacements v from its initial configuration Y, roviding a current osition (1) y = Y + v (2) however, this motion will be restricted to ermanently maintain the frame just in the middle between the two contact interfaces. To do that, the relative frame-contact interface distance Preared using cnmauth.cls

3 PARTITIONED FORMULATION OF FRICTIONAL CONTACT PROBLEMS 3 Ω Ω Contact Zone Γ C λ λ n Λ a 2 Ω Contact Frame Λ a 1 a 2 λ n a 1 Ω λ Figure 1. Left: Contact interfaces with intercalated frame. Rigth: Exloded view with localized Lagrange multiliers. vectors k = (k n, k t1, k t2 ) t and k = ( k n, k t1, k t2 ) t are exressed in a local coordinate system of the frame by k = B t (x y) k = B t ( x y) k k (3) = 0 Previous definitions[5] make the contact frame to coincide with the contact zone where k n is equal to zero. Also k t will give us the direction of the relative motion of the contact interfaces, a variable needed to formulate the frictional behavior. 3. ENFORCEMENT OF THE CONTACT CONDITIONS The behavior of the contact interface is governed by the non-enetration condition and the Coulomb friction law. To formulate the contact conditions, let s introduce the augmented Lagrange multilier variable λ(r) = λ + rk with a enalty arameter r > 0, and the Coulomb disk of radius g rojection oerator P Cg (.) : R 2 R 2 { [x, y] t if x 2 + y 2 g 2 P Cg (x, y) = g [x, y]t otherwise (4) 2 x2 +y then, the comlete fulfilment of contact conditions can be assured using the cone rojection oerator P (.) : R 3 R 3 alied to the augmented Lagrangian multiliers in the following Preared using cnmauth.cls

4 4 J. A. GONZÁLEZ, K. C. PARK AND C. A. FELIPPA way [ P (λ(r)) = max(0, λ n (r)) P Cµ max(0,λn) (λ t (r)) This definition allows us to exress the contact conditions with the final exression ] (5) P (λ(r)) = λ (6) 4. VARIATIONAL FORMULATION To derive the equilibrium equations of the constrained system we use the variational formulation roosed by Park and Felia [8, 9] where the roblem is treated as if all bodies were entirely free, formulating the virtual work by summing u the contributions of each body. The localized constraint conditions are constructed and then multilied by unknown coefficients, denoted herein as the localized Lagrange multiliers. The resulting constraint functional is added to the virtual work of the free substructures to yield the total work of the system. The variational functional that reresents the total energy of the system δπ, is then comosed by the energy of the two comletely free substructures lus the interface constraint functional associated with the contact henomena where the contact interface otential δπ total i δπ = δπ free + δ π free + δπ total i (7) consists of contributions from both sides δπ total i = δπ i + δ π i (8) which will be derived in this section. To do that, let us decomose each one of the two interface functionals into two terms δπ i = δπ k + δπ c δ π i = δ π k + δ π c (9) the first one is related with the kinematic ositioning of the frame, equation (3), that is enforced in a weak sense using the variational form δπ k = (δ{λ [B t (x y) k]})d (10) and the second one reresents the virtual work of the contact forces, contribution to the weak form that can be exressed in the following way δπ c = (λ δk)d (11) where the contact forces λ have to satisfy the unilateral contact law and the frictional law. These restrictions can be automatically satisfied relacing the Lagrange multiliers by the rojection oerator (5) obtaining δπ c = (P (λ (r)) δk)d (12) Preared using cnmauth.cls

5 PARTITIONED FORMULATION OF FRICTIONAL CONTACT PROBLEMS 5 Substituting (12) and (10) into the first of (9) rovides the exression for the total variation of the interface otential at the non barred side δπ i = (λ δ{b t (x y)} + δλ {B t (x y) k} + δk { λ + P (λ (r))})d (13) Similarly, one can obtain δ π i as δ π i = ( λ δ{ B t ( x y)} + δ λ { B t ) ( x y) k} + δk { λ + P ( λ (r) })dγc (14) for the barred side. To deal with the first terms of equations (13) and (14) we decomose them in the following way λ δ{b t (x y) = (δu δv) {Bλ} + λ n δn (x y) + λ tα δa α (x y) (15) λ δ{ B t ( x y) = (δū δv) { B λ} + λ n δ n ( x y) + λ tα δā α ( x y) (16) where the variation of the normal and tangential frame unitary vectors can be written, see [5], as δa α = Q α δv,α, δā α = Q α δv,α (17) δn = e βα (a β I)Q α δv,α, δ n = e βα (ā β I) Q α δv,α (18) Substituting into (13) and (14) leads to the final exression for the interface otentials: δπ i = (δλ {B t (x y) k} + (δu δv) {Bλ} + δv,α {Q α Φ α (x y)} + δk { λ + P (λ (r))})d (19) with Φ α = λ tα I + λ n e αβ (a β I), and for the barred: side δ π i = (δ λ { B t ( x y) k} + (δū δv) { B λ} + δv,α { Q α Φ α ( x y)} + δk { λ + P ( λ (r) ) })d (20) with Φ α = λ tα I λ n e αβ (a β I). It is imortant to mention that receding interface constraint functional will not lead to a standard minimization roblem, for the Coulomb disk C g inside the cone rojection oerator (5) is a function of the normal contact through the friction limit g which deends on the solution u. To overcome this difficulty and following to Alart and Curnier [10] we have obtained a articular form of quasi-variational functional by substituting g by the convex set µ max(0, λ n ); for this reason, the minimization roblem associated with (7) is considered as a quasi-variational roblem. 5. DISCRETE EQUATIONS The discrete contact roblem will be defined in terms of contact airs, i.e. coules formed by a set of interface nodes and its associated frame element. Those contact airs are established Preared using cnmauth.cls

6 6 J. A. GONZÁLEZ, K. C. PARK AND C. A. FELIPPA Solid Ω ( v, k ) Λ ( v, k ) ξ a 2 1 ξ 1 ξ n ( u, λ ) 2 a 2 3 ( v, k ) 2 2 ( v, k ) 3 3 Figure 2. Contact airs between the barred contact interface node and the frame. Contact airs between the non-barred contact interface node and the frame element is indeendently defined. before starting each time ste, calculating for every otentially contacting boundary node its nearest frame element and rojecting geometrically on it, see Figure 2. The frame dislacements v(ξ 1, ξ 2 ) and its average distance to the solids k(ξ 1, ξ 2 ) are interolated using isoarametric finite elements in the following form v(ξ 1, ξ 2 ) = N(ξ 1, ξ 2 ) v 1 :, k(ξ 1, ξ 2 ) = N(ξ 1, ξ 2 ) k 1 : (21) where n f is the number of nodes in the frame element, and N N nf 0 0 N(ξ 1, ξ 2 ) = 0 N N nf 0 v nf k nf (22) 0 0 N N nf is the shae functions aroximation matrix. The localized Lagrange multiliers are collocated with the contacting interface nodes so that they can be exressed in terms of Dirac s delta functions λ(ξ 1, ξ 2 ) = λ δ(ξ ξ ) (23) with ξ = (ξ 1, ξ 2 ) and ξ the frame coordinates of the node rojection, Figure 2. This definition of contact forces will reduce integrations over the contact zone to summations over the contact airs, i.e. n λ(ξ 1, ξ 2 ) f(ξ 1, ξ 2 )d = λ f(ξ ) (24) Preared using cnmauth.cls =1

7 PARTITIONED FORMULATION OF FRICTIONAL CONTACT PROBLEMS 7 with n the total number of contact airs. Exression (24) is useful in order to maintain the contact interface generic, leading to modular couling software. To manage the interface discrete variables we introduce the substructural interface nodal indicator L, the well known Boolean finite element assembling oerator defined in the following way u = L u u, λ = L λ λ (25) v = L v v, k = L k k where L is used to extract the variable associated with a contact interface node from the global unknowns vector. Using the receding definitions, the discrete variational form of the total energy functional can be finally written δπ = δu f + δū f + δλ g + δ λ ḡ δv (q + q) + δk ( + ) (26) where f = n =1 n =1 L t uf, g = n =1 n L t λ g q = L t vq, = L t k =1 with the corresonding exressions for the barred quantities. For each contact airs, the following equations hold (27) f = {f int f ext + Bλ}, f = { f int f ext + B λ} (28) g = {B t (x y) Nk}, ḡ = { B t ( x y) Nk} (29) q = { N t Bλ + N ț αq α Φ α (x y)}, q = { N t B λ + N ț α Q α Φ α ( x y)} (30) = {N t ( λ + P (λ(r)))}, = { N t ( λ + P ( λ(r) ) )} (31) where equation (28) reresents the equilibrium equation of each substructure (term Bλ are the contact forces exressed in the global system), equation (29) governs the relative motion between the frame and the substructures, equation (30) evaluates the forces acting on the frame and equation (31) imoses the fulfilment of the contact conditions. The stationary of the above variational equation, viz., δπ = 0, is obtained by solving for the unknown z = ( u, ū, λ, λ, v, k ) the following B-differentiable system F(z) = f f g ḡ q q + = 0 (32) where non F-differentiability occurs because equation + can resent non-linear directional derivatives. The artitioned equations of motion for the frame-based contact roblem are obtained from the Jacobian of system (32), that can be exressed as Preared using cnmauth.cls

8 8 J. A. GONZÁLEZ, K. C. PARK AND C. A. FELIPPA K 0 B 0 L v 0 u f 0 K 0 B L v 0 ū f B t L b N k λ 0 B t 0 0 L b N k λ = g ḡ L t v L t v L t b L t b D v D v 0 v q + q 0 0 P λ P λ 0 P k + P k k (33) where only non-differentiable terms P λ, P λ, P k and P k need a secial treatment. For examle, comonents of the receding equation for the non barred quantities are obtained by assembling contributions of each contact airs : B = n =1 n =1 L t ub L λ, L v = n =1 n L t ul v L v L b = L t λ L bl v, N k = L t λ N kl k P λ = n =1 D v = n =1 =1 L t vd v L v L t k P λl λ, P k = n =1 L t k P kl k (34) with sums extended to the n contact airs from that side and where each term comes from differentiation of equations (28), (29), (30) and (31) L v = { Φ t αq α } N,α (35) (x y) t (a 2 I) (x y) t (a 1 I) L b = Bt N (x y) t Q 1 N,1 0 Q 2 N,2 0 (x y) t D v = +N ț α (36) N k = {N} (37) N t [Φ t αq α ]N,α + N ț α[q α Φ α ]N 1 y,α {Qα Φ α (x y) a α + a α Q α Φ α (x y) +[Φ α (x y)] a α Q α }N,α } Exressions for the barred side can be similarly obtained. On the other hand, terms P λ and P λ are decomosed into their normal and tangential arts: P λ = P λn + P λt and P k = P kn + P λt (39) with the following definition for the normal art: (38) 1. Case λ n (r) < 0 P λn = Nt Preared using cnmauth.cls

9 PARTITIONED FORMULATION OF FRICTIONAL CONTACT PROBLEMS 9 2. Case λ n (r) 0 P kn = Nt r N In the receding equations one can see that instead of calculating a comlicated non-linear directional derivative for the non-differentiable case λ n (r) = 0, it has been relaced by a more simle linear derivative coming from the right side. A similar technique is used for nondifferentiable oints of the tangential contribution: 1. Case λ n 0 2. Case λ n > 0 (a) When λ t (r) µλ n P λt = Nt P kt = Nt r 0 N 0 0 r (b) otherwise, if λ t (r) > µλ n with α = P λt = P kt = µλ n, β = λ 2 t1 (r)+λ 2 t (r) 2 and R = λ t (r) λ t (r). Nt Nt α λt 1 (r) λ n Ψ 11 Ψ 12 α λt 2 (r) λ n Ψ 21 Ψ rθ 11 rθ 12 N 0 rθ 21 rθ 22 µλ n (λ 2 t 1 (r)+λ 2 t 2 (r)) 3 2, Ψ = (α 1)I βr, Θ = αi βr In the above exressions we are using a linearized substitute for the non-linear directional derivative aearing when any of these equalities hold λ n = 0, λ n (r) = 0 or λ t (r) = µλ n, although the function can be exected to be normally differentiable in the large majority of ractical cases. 6. SOLVING THE NON-LINEAR SYSTEM To solve system (32) the Generalized Newton s Method with Line Search (GNMLS) has been used. GNMLS is an effective extension of the Newton s Method for B-differentiable functions roosed by Pang [11] in a general context and articularized by Alart [12] and Christensen [13] for the contact case. This method is based on the comutation of the non-linear directional Preared using cnmauth.cls

10 10 J. A. GONZÁLEZ, K. C. PARK AND C. A. FELIPPA derivative of the objective function; however, it is well known that in contact roblems this non-linear directional derivative rarely needs to be comuted and can be substituted by a linearized version without affecting the algorithm convergence; this is the aroach adoted herein in obtaining (39). If we define the scalars β (0, 1), σ (0, 1/2), and ε > 0 but small, the alication of GNMLS algorithm to solve the non-linear equations F(z) = 0, can be summarized in the following stes: 1. Time integration t: Solve for z t+ t with the known state variable vector z t. (a) Inner GNMLS iterations, loo k. (b) Use GMRES to solve for z t k in the system F(zt k ; zt k ) = F(zt k ). (c) Obtain first integer m = 1, 2,... that fulfills H(z t k + βm z t k ) (1 2σβm )H(z t k ) with H(z) = 1 2 Ft (z)f(z). (d) Udate the solution z t k+1 = zt k + τ k z t k with τ k = β m. (e) If H(z t k+1 ) ɛ continue, else comute new GNMLS iteration k k Make z t+ t = z t k+1 and solve for next time ste t t + t. The algorithm uses two nested loos, the external one t cares for time marching and the internal one k does the Newton subiterations. On each one of this subiterations a linear system has to be solved to comute the search direction, where we use a sarse matrix storage scheme combined with the GMRES solver. When a direction z t k is obtained, it is scaled by a factor of τ k obtained from a decreasing error condition. 7. APPLICATIONS The roblem considered is the Hertzian contact of two geometrically identical elastic sheres of radius R 1 = R 2 = 8 mm indented alying an external load P = 1 N normal to the contact zone. The material roerties for the first shere are E 1 = 100 GPa, ν 1 = 0.2 and E 2 = 100 GPa, ν 2 = 0.4 for the second. This difference in the Poisson s ratio, together with a friction coefficient µ = 0.25 will roduce a distribution of tangential stresses in the contact zone that will not affect considerably the normal ressure, which will be very close to the Hertz solution that redicts a contact zone radius of mm. The meshes for both sheres are comletely identical, generated using 1280 hexahedral elements and 1569 nodes for each one of them, and refining the otentially contact zone (a square of 1mm 2 ) with 100 quadrilateral elements and 121 contact nodes, see Figure 3. The contact frame is a lane of 1mm 2 that uses the same discretization as the otential contact zone and the normal load is alied using ten time stes.it should be noted that because Poisson s ratios for the two contacting sheres are different, the initially matching interface meshes will lead to non-matching meshes. The convergence rate for one ste solution is resented in Figure 4 in terms of relative error H(z k )/H(z 1 ) for each GNMLS subiteration k together with the scaling arameter τ k. On the right side of the same Figure is shown the normal distance from the frame to the sheres in the deformed state where the contact zone is defined by k n 0 and with the mesh resolution used its radius is estimated to be around 0.35mm. Preared using cnmauth.cls

11 PARTITIONED FORMULATION OF FRICTIONAL CONTACT PROBLEMS 11 Figure 3. Mesh and frame used for the Hert roblem. -0.5mm Relative error H(z k )/H(z 1 ) E τ 1E-4 k E-5 H(z k )/H(z 1 ) 0.3 1E E E mm Iteration number (k) -0.5mm Solution advance a=0.35mm 0.5mm Disacement (mm) k n Figure 4. Convergence rate of the GNMLS algorithm and normal distance k n between the deformed sheres and the frame. Preared using cnmauth.cls

12 12 J. A. GONZÁLEZ, K. C. PARK AND C. A. FELIPPA FEM Hertz λ n λ n y=0.0 y= y= x(mm) y(mm) Contact zone X (mm) Figure 5. Distribution of normal multiliers in the contact zone and comarison with Hertz solution. λt λt y=0.0 y=0.3 y=0.2 FEM y(mm) x(mm) Contact zone X (mm) Figure 6. Distribution of tangential multiliers with µ = 0.25 and the normal load N alied with ten time stes. The normal Lagrange multiliers are resented in Figure 5 comared with the Hertz solution (right) for three cross sections along the x direction. The maximum normal Lagrange multilier λ n max =0.035 N is obtained at the center of the contact zone and the differences between the numerical results and the Hertz solution can be attributed to a coarse discretization combined with the use of linear elements. It is well known that the couling between the normal and tangential stresses for this roblem is very low, meaning that normal tractions are not considerably affected by the frictional henomena. It was observed also that the number of stes used to aly the normal load didn t modify the normal solution but affected considerably the tangential multiliers inside the central stick zone. Finally, the tangential Lagrange multiliers in x direction can be seen in Figure 6 obtaining the same solution rotated 90 o for the orthogonal direction y. It reresents a comlex shae that have to satisfy the sli condition at the external annular region making the tangential forces equal to the friction limit. Preared using cnmauth.cls

13 PARTITIONED FORMULATION OF FRICTIONAL CONTACT PROBLEMS CONCLUSION A formulation of the frictional contact roblem using localized Lagrange multiliers to connect the contacting substructures to an adatative contact frame is resented. The contact conditions are mathematically formulated using rojection functions and the contact frame is allowed to move freely between the substructures maintaining the contact zone as an unknown. The GNMLS is alied to solve the non-smooth system of equations reresenting the equations of motion and different regularization techniques to imrove convergence are roosed and tested. The algorithm and introduced formulations rove to be very robust and efficient when solving 3D non-matching contact roblems. Suggested methodology simlifies the rocedure to solve this kind of roblems, minimizing geometrical knowledge of the contacting substructures needed to formulate the contact interface behavior. This fact can facilitate further extensions for contact roblems, like iterative arallel comutations or connection between different numerical techniques like the FEM and the Boundary Element Method. REFERENCES 1. Sim, J. C., Wriggers, P. and Taylor, R. L. A erturbed lagrangian formulation for the finite element solution of contact roblems. Comuter Methods in Alied Mechanics and Engineering (50), (1985). 2. Paadooulos, P. and Taylor, R. L. A mixed formulation for the finite element solution of contact roblems. Comuter Methods in Alied Mechanics and Engineering (94), (1992). 3. Sim, J. C. and Laursen, T.A. An augmented lagragian treatment of contact roblems involving friction. Comuters & Structures 42, (1992). 4. Puso, M. A. and Laursen, T. A. A mortar segment-to-segment contact method for large deformation solid mechanics. Comuter Methods in Alied Mechanics and Engineering (193), (2004). 5. Rebel G, Park KC, Felia CA. A contact formulation based on localized Lagrange multiliers: formulation and alication to two-dimensional roblems. International Journal for Numerical Methods in Engineering 2002; 54(2): Park KC, Felia CA, Gumaste UA. A localized version of the metod of Lagrange multiliers and its alications. Comutational Mechanics 2000; 24: Park KC, Felia CA, Rebel G. A simle algorithm for localized construction of non-matching structural interfaces. International Journal for Numerical Methods in Engineering 2002; 53: Park KC, Felia CA. A variational framework for solution method develoments in structural mechanics. Journal of Alied Mechanics 1998; 65: Park KC, Felia CA. A localized version of the metod of Lagrange multiliers and its alications. Comutational Mechanics 2000; 24: Alart P, Curnier A. A mixed formulation for frictional contact roblems rone to Newton like solution methods. Com. Al. Mech. Engng. 1991; 32: Pang JS. Newton s method for B-differentiable equations. Math. Oer. Research 1990; 15(2): Alart P. Méthode de Newton généralisée en mécanique du contact. J. Math. Pures Al. 1997; 76: Christensen PW, Klarbring A, Pang JS, Strömberg N. Algorithms for frictional contact roblems based on mathematical rogramming. Linköing Univ., Sweden 1998; Preared using cnmauth.cls