Research Paper
Struct Multidisc Optim 28, 99–112 (2004) DOI 10.1007/s00158-004-0423-y
An augmented Lagrangian optimization method for contact analysis problems, 1: formulation and algorithm A.R. Mijar and J.S. Arora
Abstract A review of existing augmented Lagrangian methods (ALM) for contact analysis problems reveals that they have not been implemented with automatic penalty updates as intended in their original development. Therefore, although the methods are an improvement over the penalty methods, solution with them still depends on the user-specified penalty values for the contact constraints. To overcome this drawback, an ALM is developed and discussed for contact analysis problems that automatically update the user-specified penalty values to obtain the final appropriate values. Further, to solve the frictional contact analysis problem accurately, a two-phase formulation is proposed. Solution of the Phase 1 problem removes penetration of the contacting nodes and brings them exactly to their initial contact points. In addition, a new contact constraint is introduced which allows determination of the precise friction force at the contacting nodes. Phase 2 of the formulation checks the friction conditions and solves the friction problem to bring the structure to an equilibrium state. Phases 1 and 2 are then combined to provide a general algorithm for multi-node frictional contact problems. The two-phase procedure also removes dependence of the contact solution on the number of load steps for the elastostatic problem. Numerical evaluation of the formulation and the algorithm is presented in Part 2 of the paper.
Key words frictional contact, optimization, augmented Lagrangian, elastostatic, finite elements Received: 4 August 2003 Revised manuscript received: 28 February 2004 Published online: 4 August 2004 Springer-Verlag 2004 A.R. Mijar and J.S. Arorau Optimal Design Laboratory, College of Engineering, The University of Iowa, Iowa City, IA 52242, USA e-mail:
[email protected]
1 Introduction A two-phase formulation and associated numerical algorithm for solution of frictional contact problems is proposed and evaluated using simple numerical examples. The procedure overcomes the difficulty of dependence of the solution process on two user-specified parameters: (i) the penalty parameter for contact constraints, and (ii) the number of load steps. With the existing procedures, the penalty parameter and the number of load steps are problem dependent and require some numerical experimentation by the user for their specification. Furthermore, the specified number of load steps may yield convergence of a Newton-like iterative procedure but may fail to predict the friction force accurately. The user may be unaware of such an inaccurate solution, which is not a desirable situation. In addition, such procedures require more computations as well as user’s time. The paper is divided into two parts; Part 1 presents formulations and the augmented Lagrangian method (ALM), and Part 2 presents their numerical evaluations using simple test problems. In this section, we present some background material related to the proposed formulation and the solution algorithm, a review of existing implementations of ALM for frictional contact problems, and motivation and objectives of the present research. In the remaining sections, the frictional contact problem is defined, basic concepts of ALM are discussed, and a detailed algorithm is presented for numerical implementation. It is noted that frictional contact problems are nonlinear, requiring an incremental solution process coupled with equilibrium iterations. For dynamic problems, the total time of interest is divided into smaller increments for integration purposes. For quasi-static problems, a timelike parameter representing the load is divided into load increments. Throughout the paper, we refer to these as “load” increments since only quasi-static problems are considered.
100 1.1 Background Contact problems in computational solid mechanics have been an active research area in recent years due to their importance in engineering design and analysis. Two basic formulations have been studied in the literature to solve such problems (Kikuchi and Oden 1988): (1) Variational inequality formulation (VI); and (2) Variational equality formulation (VE). Since a detailed review of the formulations for frictionless and frictional contact problems has been presented elsewhere (Mijar and Arora 2000a, 2000b), it is not included here. However a review of the literature related to the augmented Lagrangian approach for contact analysis is presented later in this section. Most implementations for solution of practical frictional contact problems are based on VE formulation (e.g. in ANSYS, ADINA, ABAQUS, DYNA3D). In this formulation, the contact surface between the bodies is assumed to be known at the start of the load increment, which is updated iteratively. Iterations within the load step are used to advance the solution from the nth step to the (n + 1)th step. The Lagrange multiplier method (LMM) (Bathe and Chaudhary 1985) and the penalty method (PM) (Zhong 1993; Ju and Rowlands 1999) have traditionally been used to impose contact constraints. Implementation of the Lagrange multiplier method comes at a cost of an additional variable for each constraint and indefinite structure of the resulting matrix problem while the penalty method suffers from ill-conditioning since the constraints are satisfied only in the limit of infinite penalty values. Treatment of the frictional and smooth contact problems with the penalty method has been analysed recently with respect to the penalty value and the number of load steps. A simple test problem that has an analytical solution is used in that study (Mijar 2000; Mijar and Arora 2000b). For the penalty method, it is concluded that (i)
Final satisfaction of the contact constraints depends heavily on the value of the penalty parameter specified by the user for both frictional and smooth contact problems. (ii) Solution of the smooth contact problem does not depend on the number of load steps used in calculations. However, the path to the final solution does depend on the number of load steps used. This brings out the path-dependent nature of the solution process for contact analysis problems. (iii) For frictional contact problems, the solution heavily depends on the number of load steps used, more so for the stick case than for the slip case. This is due to the path-dependent nature of the frictional contact problem. This difficulty is attributed to inaccurate prediction of frictional forces at the contact points, such as due to the use of a return-mapping-type procedure from plasticity.
1.2 Overview of existing augmented Lagrangian approaches The augmented Lagrangian-type algorithms have been used to solve contact problems. Therefore it is important to review these implementations and relate them to the proposed approach. The augmented Lagrangian method (ALM), proposed originally by Hestenes (1969) and Powell (1969) for solution of general optimization problems, is known to provide significant advantages over the more traditional Lagrange multiplier and penalty methods (Gill et al. 1981; Luenberger 1984; Arora et al. 1991). Through use of the ALM, constraints are satisfied exactly using finite penalty values. The initial value of the penalty parameter is updated automatically inside the algorithm without user interference in order to determine its final finite value. The augmented Lagrangian (AL) functional can be interpreted as the ordinary Lagrangian augmented with a penalty term. In all of the existing ALM-based contact solution procedures presented in the following paragraphs, the AL functional is treated exactly as the penalty function in the penalty method. That is, the initial value of the penalty parameter defined by the user is not changed during the solution process. To specify appropriate penalty values for these implementations, a trial-and-error procedure is required, which is not desirable. Since these approaches are similar to the penalty method, we refer to them as penalty plus Lagrange multiplier methods, P+LMM. Although the approaches are an improvement over the penalty method, they have the same drawbacks as the penalty method. This will be demonstrated in Part 2 of the paper with examples. Therefore, as noted by Laursen (1992), it remains to automate fully the ALM for frictional contact problems, i.e. to incorporate the automatic penalty updates. This is one of the purposes of the present paper – to show how the ALM from the optimization theory with automatic penalty updates can be implemented for contact analysis problems. Using P+LMM, Simo and Laursen (1992) formulated and solved the frictional contact problem. An adaptation of the return-mapping algorithm from computational plasticity was used to handle Coulomb’s friction law. The approach led to decreased ill-conditioning of the governing equations and satisfied the contact constraints with reasonable values for the penalties. Examples of an elastic block against rigid foundation and extrusion of aluminum cylinder were solved. Wriggers and Zavarise (1993) employed an approach similar to the one by Simo and Laursen (1992) to accommodate the micromechanically based constitutive equations for contact interfaces. Attention was restricted to the case of small strains and displacements. Numerical problems of two elastic blocks in contact with each other and contact between a ring and a plane surface were solved. Zavarise et al. (1995) treated thermo-mechanical frictional contact problems using a similar approach. The
101 frictional contact problem of an elastic block and rigid foundation was studied using the penalty method and P+LMM. Auricchio and Sacco (1996) used two ALMlike procedures based on the works of Alart and Curnier (1991) and Simo and Laursen (1992) to treat frictionless contact between a plate and a rigid obstacle. The P+LMM have also been implemented in commercial finite-element programs like ANSYS. Contact load/time prediction schemes are also devised to improve contact solution accuracy and efficiency. Although these schemes improve the solution process, they do not eliminate the basic difficulties. This will be investigated in Part 2 of the paper. 1.3 Motivation Some of the existing frictional contact formulations and solution methods have limitations in the sense that their solution depends heavily on the user-defined parameters: (i) the penalty values and (ii) the number of load steps. Therefore, convergence of the incremental/iterative solution process does not guarantee accuracy of solution of the contact problem. The P+LMM is an improvement over the penalty method; however the method can still converge to inaccurate solutions of frictional contact problems. One may argue that, with a large enough value for the penalty parameter, P+LMM should work well. This may be true in some cases; however, in general, the procedure has the same drawbacks as the penalty method: (i)
(ii) (iii)
(iv)
What is a large enough value for the penalty parameter for a given problem? Some numerical experimentation is needed to determine such a value. This process needs more computations, and more importantly, additional human resources. Too large a value for the penalty parameter causes ill-conditioning in the numerical calculations. Too small a value for the penalty parameter leads to slower convergence or no convergence of the solution process. The contact constraints may not be satisfied to the desired accuracy leading to inaccurate solutions.
Thus uncertainty about the penalty parameter value is not overcome with the existing P+LMM implementations. In this paper, it will be shown that the ALM can be implemented in such a way that an appropriate finite value for the penalty parameter is determined automatically and the contact constraints can be satisfied to the desired accuracy. Dependence of the solution on the number of load steps with the penalty method or P+LMM is troubling because accuracy of the solution cannot be guaranteed in general. The main reason is that most existing implementations and formulations have difficulty in calculating the friction force accurately. A return-mapping-like
procedure from computational plasticity has been commonly used to handle friction conditions and calculate the friction force. However, it has been shown that the procedure has difficulty in simulating contact stick /slip phenomena (Mijar 2000; Mijar and Arora 2000c). Automatic load stepping procedures have been developed and implemented to improve the quality of frictional contact solution. Such solution enhancement schemes are geared more towards obtaining a converged contact solution using less computational effort; they do not eliminate solution dependence on the number of load steps (Mijar and Arora 2000c, 2002). Thus, it remains to investigate alternate frictional contact formulations that yield more accurate solutions without depending on the number of load steps.
1.4 Objectives of the paper The purpose of this paper is to propose and evaluate algorithms to eliminate the aforementioned two difficulties in existing formulations and solution procedures for the frictional contact problem. The difficulties with the penalty parameter selection can be avoided if the original intent of the ALM of automatic updating of the penalty parameter is implemented for contact analysis problems. The difficulty of solution dependence on the number of load steps can be overcome if alternate formulations and solution procedures are used to calculate the friction force more accurately. An alternative formulation is presented that allows calculation of an accurate friction force without use of the return-mapping procedure from computational plasticity. An associated numerical algorithm to solve the problem is proposed and evaluated. To illustrate and evaluate the basic ideas more clearly, a simpler problem of two-dimensional deformable-to-rigid-body contact under quasi-static loading is considered. Thus, the objectives of the paper are: (1) To propose, develop and evaluate a frictional contact formulation that does not require the use of the return-mapping procedure to determine the frictional force. Whether or not the formulation leads to solutions that are independent of the number of load steps will be investigated. (2) To propose and develop an algorithm that solves the frictional contact problem without dependence of the solution on the penalty values and the number of load steps. Since ALM from the optimization literature is deemed to be quite suitable to achieve these objectives, this method will be studied and developed for the frictional contact problem. The aim here is to develop a numerical algorithm such that material nonlinearity and geometrical nonlinearity for the problem would alone determine the number of load steps needed to solve the problem, and not the presence of frictional contact phenomena. Successful implementation of the algorithm will also show how ALM
102 from the optimization literature can be used such that the penalty values for the contact constraints are determined automatically without user interference. 2 Contact problem definition 2.1 Kinematics and statics Fig. 1 Contact configuration
Details of the problem formulation have been presented in Mijar and Arora (2000a, b) and other references cited therein. For completeness and convenience of reading, the material from these references is summarized here. A deformable-to-rigid body contact in two dimensions is shown in Fig. 1. The deformable body (contactor) is discretized into NOE finite elements consisting of NON nodes. Contact and friction effects are taken into account at discrete nodes on the surface of the contactor. The contact surface identified with a portion Γ2c on the boundary of the contactor contains NOC potential contact nodes (NOC ≤ NON). We consider a node P on the contactor with coordinate vector xP and points I and J on the edge of the target with coordinate vectors xI and xJ , respectively, in the reference configuration. Unit vectors n and t define the local coordinate system on the target. The initial normal gap dP 0 for the Pth node is defined J P as dP = (x − x ) · n. The current normal gap after some 0 P P movement is given as dP := dP n 0 − un , where un is the normal displacement at node P. Two bodies under consideration should satisfy the unilateral contact law, which consists of impenetrability, compression and complementary conditions, gP n
≤0,
RP n
≥0,
P RP n gn
P P P gP n := − dn = un − d0
=0,
where
for all P ∈ Γ2c
(1)
where the normal gap constraint is written in the standard ≤ form, gP n ≤ 0. These conditions are a part of the Karush–Kuhn–Tucker (KKT) necessary conditions of optimality for inequality constraints. The conditions are P shown in the RP n − gn plane in Fig. 2(a). Once the node P on the contactor surface comes into contact with the target edge, it exerts a contact force ¯f P = [RP fP ]T on the target. The friction force magnic n f
Fig. 2 Contact and friction laws
tude fP f at the point of contact depends on the normal contact force magnitude RP n . In this paper, Coulomb’s classical (rigid) and modified (elastic) dry friction laws are considered. For the classical law, depending on the friction force fP f at the point of contact, node P will either stick to the wall or slip along the wall. Note that the friction laws are stated for a static problem while considering the total tangential displacement at a given node. For dynamic problems, the sign of the friction force depends on the sign of the velocity of the node. Such velocity dependence (or, for nonlinear quasi-static problems, dependence on the incremental displacement) of the friction force is not considered in the current paper. Thus reversal of the friction force sign in a load increment is not allowed. The example problems used in the present study are developed to satisfy this assumption. For the classical friction law, stick and slip conditions are stated as [Fig. 2(b)]: P P ff − µRP (2) n < 0 ⇒ ut = 0 (Stick condition) P P ff − µRP n = 0 ⇒ ut > 0 (Slip condition)
(3)
where µ is the coefficient of friction and uP t is the magnitude of the tangential displacement along t. Note that the friction force acts in a direction opposed to the relative motion uP t of the contactor node P. The classical friction law has been modified to an elastic friction law to overcome computational difficulties of the VE formulation, as [ Fig. 2(c)]: P ff = Ks uP ⇒ uP (4) t t ≤ e (Stick condition) P f = µRP ⇒ uP > e (Slip condition) f n t
(5)
103 where Ks (called the sticking stiffness) is the slope of the part of the curve shown in Fig. 2(c) that corresponds to an elastic stick e. 2.2 Frictionless contact problem If we assume frictionless contact between the bodies shown in Fig. 1, then the following optimization problem gives the contact solution (note that a quantity in the nt coordinate system has an over-bar which is omitted for the vector quantities in the x1 − x2 coordinate system): 1 ¯T ¯ ¯ ¯T ¯ min Π := U KU − U F ¯ 2 U subject to
P P 2 gP n := un − d0 ≤ 0 for all P ∈ Γc
3 Augmented Lagrangian method (ALM) (6)
¯ is the [2NON × where Π is the total potential energy, K ¯ 2NON] global stiffness matrix, U is the [2NON × 1] global ¯ is the [2NON × 1] global external displacement vector, F P force vector and gn is the contact constraint for node P on the contactor. The displacement and external force vectors are written as 1 T ¯ = u ¯ ¯2 . . . u ¯ NON ; U u T ¯ P = uP u uP for P = 1, 2, . . ., NON (7) n t ¯ = ¯f 1 ¯f 2 . . . ¯f NON T ; F ¯f P = fP fP T for P = 1, 2, . . ., NON n t
(8)
¯ P and ¯f P are the displacement and external force where u vectors at node P in the n − t coordinate system, respectively. Note that lowercase letters are used to represent quantities associated with one node. It has been shown that the solution of the optimization problem in (6) is equivalent to the solution of the variational inequality for frictionless contact (Kikuchi and Oden 1988). Standard optimization techniques can ¯ be used to solve this problem for displacement vector U P and Lagrange multipliers Rn (i.e. contact forces) associated with NOC contacting nodes. 2.3 Frictional contact problem The finite-element discretization of the frictional contact variational equality leads to the following equilibrium equation (Mijar and Arora 2000b): ¯U ¯ −F ¯ +F ¯c = 0 K
¯ and the conIt is observed that the displacement U ¯ tact force Fc in (9) are both unknowns. The unilateral contact constraint in (1) and Coulomb’s friction conditions in (4) and (5) need to be enforced along with the other boundary conditions during the solution process. The initial contacting point of node P and the subsequent movement of this node on the target are functions of the ¯ Therefore, the normal contact forces RP applied force F. n and tangential friction forces fP f change during loading. Thus, (9) is nonlinear and needs to be solved using an incremental/iterative solution procedure.
(9)
¯ c is the [2NON × 1] global contact force vector, where F ¯ c = ¯f 1 ¯f 2 . . . ¯f NON T for P = 1, 2, . . ., NON F c c c (10) T with ¯fcP = [RP fP n f ] as the contact force at the Pth contacting node (fP being the friction force). f
3.1 Basic idea of the augmented Lagrangian method The augmented Lagrangian method is one of the wellestablished multiplier methods in the optimization literature (Arora et al. 1991). To present the basic idea of the method more clearly, we consider the frictionless contact problem defined in (6). The frictional contact problem is treated in later sections. Since the frictionless contact problem involves only inequality constraints, equality constraints are not considered in the algorithm (although they can be treated routinely). The augmented Lagrangian functional Φ for the inequality-constrained frictionless contact problem in (6) is defined as follows: NOC P 2 P P 2 ¯ θ , Kn = Π + 1 Φ U, KP + θ − θ g (11) n n + 2 P=1
where θ is the vector containing parameters θP for the Pth contact constraint, Kn is the vector of positive penalty P P P parameters KP n > 0, (h)+ = max(0, h) and Kn θ = Rn is the Lagrange multiplier for the Pth constraint (i.e. Rn is the [NOC × 1] vector). Note that for the inequality constraints, θP ≥ 0. We define index sets for the active and inactive constraints as
P Active constraints: I1 := P : gP ≥ 0 , P ∈ Γ2c n +θ (12) P Inactive constraints: I2 := P : gP 0 is a step (j) size and λ(j) d(j) = ∆Rn is the change in the multiplier vector. To derive an expression for the search direction d, the gradient of the dual function Ψ(Rn ) of (15) with respect to the Lagrange multiplier RP n for the Pth node is written as: ¯ dΨ ∂Ψ T dU = + ∇Ψ ; dRP ∂RP dRP n n n
P ∈ Γ2c
(18)
¯ where ∇ represents the gradient with respect to U. P P T ¯ Equation (9) for frictionless contact (i.e. fc = [Rn 0] ) and (16) give ∇Ψ = 0. Therefore, using (12), (13) and (16) in (18), the gradient of the dual function Ψ(Rn ) with respect to the Lagrange multiplier RP n for the Pth node is given as
dΨ RP n P P = c = max g , − ; P ∈ Γ2c (19) n P dRP K n n where cP = ∂Ψ/∂RP n . It is noted that the direction of steepest ascent is d = c (or in component form dP = cP ,
105 for P ∈ Γ2c ). Therefore, using (17), the jth steepest ascent iteration to maximize the dual function is given as follows: P (j+1) P (j) P (j) P (j) Rn = Rn + Kn c ; P ∈ Γ2c (20) Using (19), we obtain
(j) P (j+1) P (j) P (j) P (j) (RP n) Rn = Rn + Kn max gn , − P (j) ; (Kn ) P ∈ Γ2c
(21)
Alternatively, the iteration given above can also be writ(j) (j) P (j) ten using (RP = (KP n) n ) (θ ) , as (j) P (j+1) P (j) = θ + max gP , − (θP )(j) ; P ∈ Γ2c θ n (22) Note that, for the steepest ascent iterations in (21) and (j) (22), the step sizes are observed to be λ(j) = (KP and n) (j) λ = 1, respectively. The method of steepest ascent is simple and robust and it is convergent when a line search is used along the search direction. However, a large number of iterations may be required for unconstrained maximization using this first-order update scheme. Other Lagrange multiplier update schemes [such as the conjugate gradient method (first-order update), and the Newton’s method and Tapia’s formula (second-order updates)] are available and explained elsewhere (Tapia 1974; Arora et al. 1991; Mijar 2000). Thus the ALM automatically determines appropriate values for the penalty parameters for the constraints. Also, the Lagrange multipliers for the constraints are determined. For the friction contact analysis problem, these multipliers represent the contact forces – frictional and normal forces at the contact points.
Fig. 3 ALM2 formulation
The basic idea of the formulation is generalized into a two-phase numerical algorithm to solve general (multinode) frictional contact problems in the next section. The proposed procedure does not use the return-mapping algorithm to calculate the friction force and the solution is expected to be independent of the number of load steps as well as penalty values. The proposed formulation and algorithm will be demonstrated and evaluated in Part 2 of the paper. 4.1 Phase 1: determination of initial contact point The objective in Phase 1 is to capture the initial contact point accurately and to determine the forces of constraints required in keeping the contacting node at that point. The following problem is defined for this phase (note that a prime indicates quantities associated with Phase 1): min Π (¯ u ) st
4 A two-phase frictional contact formulation To present basic ideas behind the two-phase formulation, a two-bar structure – rigid wall problem having one contacting node (P = 1) shown in Fig. 3 is considered. The scope of the two-phase formulation is restricted to quasistatic deformable-to-rigid-body frictional contact problems. Phase 1 of the formulation finds the initial contact point for node P and contact force at that point while Phase 2 checks the friction conditions and solves the friction problem. Phase 2 of the formulation does not have a potential energy functional associated with it (unlike Phase 1) due to the presence of path-dependent friction forces. Therefore, the load increment scheme is employed in the formulation, which naturally accommodates linear as well as nonlinear problems (geometric or material nonlinear effects).
gn gt
(23)
:=
un −
:=
ξ(gn )(ut −
d0 ≤ 0 dt ) = 0 with
ξ(gn )
0 = 1
gn < 0 gn ≥ 0
where Π is the total potential energy of the structure, un and ut are the Phase 1 normal and tangential displacements of the contacting node, gt is the tangential penetration distance of the contacting node inside the wall (Fig. 3), dt is the tangential displacement of the contacting node (at the instant of its initial contact with the wall) relative to its original position and ξ(gn ) is the unit step function. Note that the tangential distance constraint gt is an equality and it becomes active only if gn ≥ 0, i.e. when the contacting node penetrates the wall. The basic idea of this new constraint is to capture the initial contact point precisely and force the contact node to stick there through the loading process in Phase 1. The Lagrange multipliers for the constraints give the normal and tangential forces precisely, and thus the slip/stick condition
106 can be checked there. The constraint is ignored in Phase 2 to allow the contact node to slip depending on the friction condition. The penalty method or other optimization methods can be used to solve the problem defined in (23). However, due to drawbacks of these methods, we use the augmented Lagrangian method from optimization theory (Powell 1969; Hestenes 1969; Chahande and Arora 1993). Let us define the augmented Lagrangian functional for the Phase 1 problem as: Φ(¯ u , θn , Kn , θt , Kt ) = Π(¯ u ) + 1 Kn θn2 2 1 2 Kn θn gn + 1 Kn g2 n + Kt θt gt + Kt gt 2 2
if gn + θn < 0 if gn + θn ≥ 0 (24)
where Kn > 0, Kt > 0, θn and θt are the ALM parameters associated with the normal and tangential constraints in (23). Parameters Kn and Kt are also called the penalty parameters for the respective constraints. The idea of ALM is to transform a given constrained problem into a sequence of unconstrained problems minimizing the ¯ for the given augmented Lagrangian Φ with respect to u estimates of Kn , Kt , θn and θt (Arora et al. 1991). It will be shown next that when the normal and tangential constraints in (23) are active, the products Kn θn = r fn and Kt θt = r ft yield the exact Lagrange multipliers (i.e. normal and tangential contact forces). The existing ALMbased contact formulations (Simo and Laursen, 1992, Alart and Curnier 1991) do not use parameters like θn and θt given above. We use such parameters to obtain the exact solution to the original constrained problem while using finite values of the penalty parameters Kn and Kt . In this methodology, inspired from duality theory, con∗ ∗ vergence to the optimal point (¯ u∗ , r fn , r ft ) is achieved by adjusting Kn , Kt , θn and θt iteratively. The idea is to increase the penalty parameters Kn and Kt to force the iterates of Lagrange multipliers (r fn , r ft ) into the re∗ ∗ gion about (r fn , r ft ) in which convergence is guaranteed by local duality theory (Luenberger 1984; Arora et al. 1991). Once in this region, the penalty parameters Kn and Kt are fixed and only the parameters θn and θt are varied (i.e. Lagrange multipliers are updated) such that ∗ ∗ (r fn , r ft ) → (r fn , r ft ). This approach facilitates automatic update of penalty values without user interference and determination of Lagrange multipliers for the constraints. In the event of no penetration, i.e. gn + θn < 0, the first-order optimality conditions for Φ in (24) simply yield the equilibrium equations for the structure. In the event of penetration, i.e. gn + θn ≥ 0, the first-order optimality ¯ give, conditions with respect to u ¯ u − ¯f + f¯ c = 0 ∇Φ(¯ u , θn , Kn , θt , Kt ) := k¯
(25)
where f¯ c = [Kn (gn + θn ) Kt (gt + θt )]T is the vector containing forces associated with the constraints on normal
penetration gn and tangential penetration gt . The ALM with automatic update of parameters Kn , Kt , θn and θt can be employed to obtain the solution of the problem ¯ and ¯fc (Powell 1969; Mijar 2000), i.e. displacement u at the initial contact point D shown in Fig. 3. This is discussed later in more detail while presenting the algorithm. At point D, the constraints are active (i.e. gn = gt = 0) and their exact Lagrange multipliers Kn θn = r fn and Kt θt = r ft are obtained. Note that θn is always ≥ 0 but θt is unrestricted in sign. This is due to the fact that θn is associated with the inequality constraint gn , the Lagrange multiplier of which is ≥ 0 while θt is associated with the equality constraint gt , the Lagrange multiplier of which can be positive, negative or zero (Arora et al. 1991). The Lagrange multipliers r fn and r ft are physically interpreted as normal and tangential reaction forces at the initial contact point D, respectively. Note that (25) is linear in terms of displacements since it is derived for linear elastic problems assuming the existence of the potential energy functional. For nonlinear problems (i.e. considering material and/or geometric nonlinear effects), the following incremental equilibrium equation can be written: ¯ (i−1) + k ¯ (i−1) ∆¯ k u(i) − ¯f + ¯r(i−1) + ¯fc(i−1) = 0 (26) c (i−1) ¯ (i−1) + k ¯ (i−1) ∆¯ k u(i) = ¯f ; c
ub
¯f (i−1) = ¯f − ¯r(i−1) − ¯f (i−1) = 0 c ub
(27)
¯ (i−1) = ∇¯r(i−1) is the structural stiffness mawhere k trix that corresponds to geometrical and material con¯ c(i−1) = ∇¯fc(i−1) is the contact stiffness maditions, k (i) trix , ∆¯ u is the vector of incremental displacements, ¯ (i−1) u ¯r(i−1) = k ¯ (i−1) is the nodal force vector corres(i−1) ponding to element stresses. ¯fub in (27) is the unbalanced force vector that needs to be driven to zero iteratively. Solving (27) for ∆¯ u(i) , the displacements at (i) ¯ =u ¯ (i−1) + ∆¯ the ith iteration are given as u u(i) . (i−1) For the non-penetration case (i.e. gn + θn < 0), the (i−1) ¯ c(i−1) = 0 contact force ¯fc = 0 and contact stiffness k in the above equilibrium equation. For the penetration (i−1) case (i.e. gn + θn ≥ 0), using (24) and (25), the contact force and contact stiffness terms required to solve the above equilibrium equation are given as follows: (i−1) K g + θ n n n ¯f (i−1) = c K g(i−1) + θ ; t t t ¯ (i−1) = ∇¯f (i−1) = k c c
Kn 0 0 Kt
(28)
Note that when the normal and tangential constraints are active, the contact force vector in (28) retrieves the exact (i−1) reaction forces at the initial contact point, i.e. ¯fc = T [Kn θn Kt θt ] . At the end of Phase 1, the contacting
107 node is at the initial contact point D (Fig. 3) and dis¯ ; reaction forces r fn and r ft at that point are placement u known. Due to these forces, the structure may not be in equilibrium, however. Phase 2 determines the equilibrium state. Note that the full external load is applied in Phase 1 in the foregoing development. This can be done for linearly elastic problems under the small displacement assumption. For nonlinear problems, the load is applied in increments as explained later. 4.2 Phase 2: solution of the friction problem In Phase 2, the objective is to find the nodal displace¯ considering the frictional contact effects (note ment u that double prime indicates variables associated with Phase 2). In other words, Phase 2 determines whether the contacting node is going to slip or stick at the current load level. The constraint gt on the tangential displacement of the node is now released and the node is allowed to slip if needed. If the stick condition with the classical Coulomb’s law is determined, the solution process is complete at the current load level. For the modified Coulomb’s friction law, the elastic stick is determined. If the contacting node is determined to slip, then the friction problem is solved to calculate the slip displacement at the current load level. At point D in Fig. 3, the friction force ff is determined using Coulomb’s classical/modified law, as |r ft | < µr fn
| ft | ≥ µ fn r
r
⇒ (Stick condition) ⇒ ff = r ft (Classical law)
(29)
⇒ ff = Ks ut (Modified law)
(30)
⇒ (Slip condition) ⇒ ff = µr fn sgnr ft (Classical/Modified law) (31)
Note that, due to the presence of friction force, Phase 2 problem does not have a potential energy function associated with it. The virtual work equation for the structure leads to the equation: ¯ (¯ ¯ ) − ¯f + ¯fc = 0 k u + u
⇒
¯ u − ¯f + ¯f = 0 k¯ c c
(32)
where ¯fc is the contact force vector (containing reaction forces at the initial contact point) from Phase 1 (refer to (25)) and ¯fc = [Rn ff ]T is the contact force vector (containing the normal contact force and tangential friction force) of Phase 2. The non-penetration constraint for Phase 2 is given as follows: gn := un ≤ 0
(33)
Note that the Lagrange multiplier Rn for the nonpenetration constraint in (33) representing the normal contact force is given as
Rn =
0
if gn + θn < 0
Kn (gn + θn )
if gn + θn ≥ 0
(34)
where Kn > 0 and θn ≥ 0 are the ALM parameters associated with the constraint in (33). The final value of the Lagrange multiplier Kn θn = Rn is obtained when the unilateral contact constraint is satisfied exactly, i.e. gn = 0. ¯ , (32) is nonSince ¯fc is dependent on displacement u linear and the following incremental equation for Phase 2 can be written: ¯ (i−1) + k ¯ (i−1) ∆¯ k u(i) − ¯fc + ¯r(i−1) + ¯fc(i−1) = 0 (35) c ¯ (i−1) u ¯ (i−1) is the nodal force vector where ¯r(i−1) = k corresponding to element stresses. The required contact force vector and contact stiffness matrix terms to solve the above equilibrium equation are given as (1) Stick case: classical Coulomb’s law (i−1) g K + θ n ¯f (i−1) = n n ; c r ft ¯ (i−1) = ∇¯f (i−1) = k c c
Kn
0
0
0
(36)
(2) Stick case: modified Coulomb’s law (i−1) + θ K g n n n ¯f (i−1) = ; c (i−1) Ks ut Kn 0 (i−1) (i−1) ¯ k = ∇¯fc = c 0 Ks
(37)
(3) Slip case: classical/modified Coulomb’s law (i−1) Kn gn + θn ¯f (i−1) = ; c (i−1) (i−1) Kn gn + θn µ sgn ut ¯ (i−1) = ∇¯f (i−1) = k c c
Kn
0 (i−1)
µ Kn sgn ut
0
(38)
¯ is found At the end of Phase 2, the nodal displacement u such that the structure is in equilibrium while satisfying contact and friction conditions. The foregoing two-phase ALM-based treatment for the frictional contact problem is designed to obtain exact solutions for frictional contact problems. The sum of the displacements from Phase 1 and Phase 2 yields the final exact displacement ¯=u ¯ + u ¯ ). (i.e. u 4.3 Unified treatment of Phases 1 and 2 Note that the structure of Phase 1 and Phase 2 equilibrium equations (25) and (32) or (26) and (35) is similar.
108 We drop the prime and double prime from these equilibrium equations and write the following general equation: ¯ (i−1) + k ¯ (i−1) ∆¯ k u(i) − ¯f + ¯r(i−1) + ¯fc(i−1) = 0 ⇒ c
(i−1) ¯ (i−1) + k ¯ (i−1) ∆¯ k u(i) = ¯fub c
(39)
¯ (i−1) u ¯ (i−1) is the nodal force vector corwhere ¯r(i−1) = k (i−1) responding to element stresses and ¯fub is the unbalanced force vector that needs to be driven to zero iteratively. Solving (39) for ∆¯ u(i) , the displacements at the (i) ¯ (i−1) + ∆¯ ¯ =u u(i) . At the beith iteration are given as u ginning of incremental analysis, Phase 1 terms, i.e. contact force vector and contact stiffness matrix of (28) are used in (39). Contacting node at the initial contact point (i.e. point D in Fig. 3) marks the end of Phase 1. At this point, friction conditions are checked and Phase 2 terms, i.e. contact force vector and contact stiffness matrix of (36)–(38) are incorporated in (39). Note that for simplicity of expression, the full external load is applied to the structure as shown in (39). This is not necessary since the external load can be applied incrementally as well. In that case, all the terms of (39) will belong to load step t + ∆t. This general approach, which allows for the inclusion of material and geometric nonlinearity, is adopted while explaining the ALM2 algorithm in the next section.
amounts to fixing the step size of the steepest ascent update). (iv) If the contact constraints for nodes undergoing Phase 1 are satisfied to the desired accuracy, these nodes are exactly at their initial contact points. Check the friction conditions for such nodes. If the contact constraints for nodes undergoing Phases 1 and 2 are satisfied to the desired accuracy, break the augmentation loop and go to Step (i) to increment the load parameter. Otherwise, continue. (v) Update the ALM parameters Kn , Kt , θn and θt for nodes whose contact constraints are not satisfied to the desired accuracy, using an automatic procedure (this step amounts to solving the dual problem). (vi) End augmentation loop. (vii) End time step loop. It is noted that a precursor to ALM2 is ALM1, which has been developed and evaluated (Mijar 2000). In that algorithm, the friction force was calculated using the return-mapping procedure; however, the penalty parameters were determined automatically in the algorithm. Thus the solution was independent of the user-specified penalty values, but it still depended on the load step size. Therefore ALM2 was developed to overcome both of these difficulties. 5.2 ALM2 algorithm
5 Augmented Lagrangian algorithm Phases 1 and 2 described in the last section for one potential contacting node need to be unified algorithmically in order to solve the multi-node frictional contact problem. This will be called the ALM2 algorithm. Also, prime and double prime over the quantities (used in Sect. 4.1) are no longer used since Phases 1 and 2 are unified in the algorithm. It is noted here that another algorithm where Phases 1 and 2 are treated separately has also been developed and demonstrated (Mijar and Arora 2001). That algorithm, called ALM3, has also worked quite well. 5.1 Outline of the ALM2 algorithm (i) (ii)
Begin the load increment loop. Begin the augmentation loop, which means to update the ALM parameters Kn , Kt , θn and θt as discussed in the detailed algorithm later. (iii) Solve the incremental equation (39) for the displacements iteratively until convergence is achieved. This step is equivalent to minimizing the augmented Lagrangian in (24) for nodes undergoing Phase 1 and solving equilibrium equation (35) for nodes undergoing Phase 2. Note that the ALM parameters Kn , Kt , θn and θt are held fixed during this step (which
The ALM2 algorithm can treat several potential contact nodes. Some of the nodes may be in frictional contact and some may have penetration while others may not be in contact yet. The detailed algorithm is given as follows: (1) Initialize: (1.1) Decide the number of load steps τ (i.e. choose the load increment ∆t). Set the load t = 0. ¯ (0) + Initialize the tangent stiffness matrix (K (0) ¯ Kc ). Set α > 1, β > 1, ε1 1, ε2 1, where parameter α is used to monitor the constraint violations, parameter β is used to increase the penalty parameter (if required) and ε1 , ε2 are convergence tolerances. Set the counters for the number of cumulative augmentations (jc = 0) and cumulative Newton–Raphson iterations (ic = 0). (0) (0) (1.2) Set the ALM parameters Kn > 0, Kt > 0, (0) (0) and θ n = θ t = 0. Note that Kn and Kt are [NOC × 1] vectors which contain penalty paP rameters KP n and Kt that correspond to conP straints gP , g and gP n n t at the Pth contacting node [refer to (23) and (33)], respectively. Also, θ n and θ t are [NOC × 1] vectors containing parameters θnP and θtP corresponding to these constraints. It is recalled that the surface Γ2c on the deformable body contains NOC potential contact nodes. Initialize vectors y and z
109 to 0 . Note that y and z are [NOC × 1] vectors containing parameters yP and zP (to be defined later) corresponding to the Pth contacting node. Parameter yP is used to keep track of the load step in which the Pth node penetrates the wall (i.e. no contact to contact step) while parameter zP is used to keep track of the friction slip/stick condition at the Pth contacting node. (1.3) Increment the load step to t + ∆t. Find the external load at step t + ∆t, ¯ = tF ¯ + ∆F ¯ F
(40)
¯ is the load at load step t and ∆F ¯ where t F is the load increment. Set parameters V1 1, V2 1 and the augmentation counter j = 0. Note that the parameter V1 is used to moniP tor the violation of constraints (gP n and gt ) at nodes that are undergoing Phase 1 while parameter V2 is used to monitor the violation of constraints (gP n ) at nodes that are undergoing Phase 2. (1.4) Set the following quantities from the previous converged load step t: ¯ (j=0) = t U ¯, K ¯ (j=0) = t K ¯, K ¯ (j=0) = t K ¯c, U c ¯ (j=0) = t F ¯c F c
(41)
(1.5) Initialize the following index sets: (j=0)
S
≡ S; t
(j=0) S1
≡∅;
(j=0) S2
≡∅
(42)
where set S contains nodes that were not in contact at the previous converged load step t while empty sets S1 and S2 are for nodes undergoing Phase 1 and Phase 2, respectively. These sets are defined and updated in later steps. (2) Start the augmentation loop: (2.1) Set j = j + 1, jc = jc + 1 and the iteration counter i = 0. (2.2) Provide initial conditions for the current augmentation step, ¯ (j) = U ¯ (j−1) , U
¯ (j) = K ¯ (j−1) , K
¯ (j) = K ¯ (j−1) , K c c
¯ (j) = F ¯ (j−1) , F c c (j)
(j−1)
S(j) ≡ S(j−1) , S1 ≡ S1
(j)
(j−1)
, S2 ≡ S2
(43) (3) Minimize the augmented Lagrangian function: This step solves the Newton–Raphson equation [similar to (39)] iteratively while keeping the ALM (j) (j) (j) (j) parameters Kn , Kt , θ n and θ t fixed at their cur¯ (j) . rent values to obtain U (3.1) Set i = i + 1 and ic = ic + 1.
(3.2) Solve the Newton–Raphson equation (39), to ¯ (i) . obtain incremental displacement ∆U (i) (3.3) Calculate the normal penetration (gP = n) P (i) P 2 (un ) − d0 for all nodes P ∈ Γc . ¯ (i−1) ≤ ε1 (i.e. the (3.4) If the unbalanced force F ub current load step has converged), then break the iterations loop and go to Step 4; otherwise ¯ (i) = U ¯ (i−1) + update the displacement to U (i) ¯ and continue with the iterations and go ∆U to Step 3.5. (3.5) Calculate the contact force vector and the contact stiffness matrices for nodes P ∈ S(j) and (j) P ∈ S1 (i.e. nodes that are not in contact and nodes that are undergoing Phase 1, respectively): (3.5.1) (i) (j) P gP + θ < 0 n n ⇒ No penetration condition ⇒ Set yP = 0 P (i) (i) ¯ =0; k =0 (44) ⇒ ¯fcP c
(i) P (j) (3.5.2) (gP ≥0 ⇒ Penetran ) + (θn ) tion condition ⇒ Set yP = yP + 1 ⇒ Find ¯ P )(i) using (28). (gP )(i) = (¯fcP )(i) and (k c t P (i) P (ut ) − dt in (28) is the tangential penetration of Pth node with distance dP t (refer to Fig. 3) as dP t
=
P (i) ut (i) (uP n)
dP only for yP = 1 0
(45)
Note that distance dP t is calculated only when parameter yP = 1 indicating that node P penetrated the wall during the analysis (i.e. no contact to contact condition). When the parameter yP > 1, the Pth node is either on the target or inside it. (3.6) Calculate the contact force vector and the con(j) tact stiffness matrix for nodes P ∈ S2 (i.e. nodes that are undergoing Phase 2): (3.6.1) Calculate the normal contact force at node P as follows: P (i) P (j) P (i) P (j) Rn = Kn gn + θn (46) (3.6.2) If zP = 1 (stick condition indicator to be defined later), contact force vector (¯fcP )(i) ¯ P )(i) are given and contact stiffness matrix (k c using (36) and (37). (3.6.3) If zP = 2 (slip condition indicator to ¯ P )(i) are given be defined later), (¯fcP )(i) and (k c using (38). ¯ (i) + (3.7) Compute the tangent stiffness matrix (K (i) (i) ¯ ¯ Kc ) and the internal force vector FI = (i) (i) (i) ¯ U ¯ +F ¯ c using the contact force vectors K
110 and the contact stiffness matrices obtained in Steps (3.5) and (3.6). Compute the residual ¯ (i) = F ¯ −F ¯ (i) , and go to Step 3.1 force vector F I ub (i.e. continue with the iterations). (4) Determine the maximum constraint violation: (4.1) Define the following index sets: (j) (j) S(j) = P: gP + θnP < 0 , zP = 0 ; n for P ∈ Γ2c (47) (j) (j) (j) S1 = P: gP + θnP ≥ 0 , zP = 0 ; n (48) for P ∈ Γ2c (j) (j) = P: gP + θnP ≥ 0 , zP = 0 ; n (49) for P ∈ Γ2c
(j) S2
where set S contains nodes that are not in contact with the target, set S1 contains nodes that are undergoing Phase 1 and set S2 contains nodes that are undergoing Phase 2. ¯1 (4.2) Define the constraint violation parameter V for nodes undergoing Phase 1, ¯ 1 = max max gP (j) , − θP (j) ; V n n P (j) (j) max gt (50) for P ∈ S1 ¯2 (4.3) Define the constraint violation parameter V for nodes undergoing Phase 2, (j) (j) ¯ 2 = max max gP , − θnP V n (j) for P ∈ S2 (51) (j) S1 :
(5) Perform augmentation for nodes P ∈ ¯ 1 ≤ ε2 is satisfied, (5.1) If the convergence criterion V (j) nodes in set S1 are exactly at the initial contact point on the target. This is equivalent to completion of Phase 1 for the current load step, as explained in Sect. 4.1. Therefore, we have P P r P P P the reaction forces r fP n = Kn θn and ft = Kt θt (j) for nodes P ∈ S1 . We check the friction conditions for these nodes: r P P ft < µr fP n ⇒ (Stick condition) ⇒ z = 1 (52) r P P ft ≥ µr fP n ⇒ (Slip condition) ⇒ z = 2 (53) P
Note that z = 1 indicates node P is sticking, zP = 2 indicates node P is slipping while zP = 0 indicates node P is not in contact. (j+1) (j) Set (KP = (KP and (θnP )(j+1) = (θnP )(j) n) n) (j)
(j)
for nodes P ∈ S1 . Update the set S1 to (j+1) S1 = ∅, i.e. it is an empty set indicating completion of Phase 1 for the current load step. Go to Step 6.
¯ 1 ≤ ε2 is not sat(5.2) If the convergence criterion V (j) isfied, nodes in set S1 are not at the initial contact point on the wall, i.e. normal and tangential penetrations have not been removed completely. Establish an index set of constraints the violation of which did not improve by the factor α: (j) P (j) I3 = P: max gP , − θ > V1 /α , n (j) (j) and/or gP > V /α P ∈ S 1 t 1
(54)
¯ 1 ≥ V1 , for nodes P ∈ I3 set (5.3) If V P (j+1) (j) P (j+1) P (j) Kn = β KP ; θn = θn /β n (55) P (j+1) (j) P (j+1) P (j) = β KP ; θt = θt /β Kt t (56) That is, increase the penalty parameters KP n and KP t by the factor β and reduce the parameters θnP and θtP by the same factor, thus P P keeping the Lagrange multipliers r fP n = Kn θn r P P P and ft = Kt θt unchanged in this step. Continue with Step 6. ¯ 1 < V1 , update the parameters θP and θP (5.4) If V n t by employing, say, the steepest ascent update scheme [refer to (22)] and continue to Step 5.5: (j) (j) P (j+1) P (j) = θn + max gnP , − θnP θn (57) P (j+1) P (j) P (j) θt = θt + gt
(58)
¯ 1 ≤ V1 /α, set V1 = V ¯ 1 and go to Step 6. (5.5) If V Otherwise, continue. ¯ 1 < V1 , for nodes P ∈ I3 set (5.6) If V1 /α < V P (j+1) (j) Kn = β KP ; n P (j+1) P (j+1) θn = θn /β
(59)
P (j+1) (j) = β KP ; Kt t P (j+1) P (j+1) θt = θt /β
(60)
¯ 1 , and go to the next augmentation and V1 = V Step 6. (j) (6) Perform augmentation for nodes P ∈ S2 : ¯ 2 ≤ ε2 is satisfied, (6.1) If the convergence criterion V (j+1) P (j) P (j+1) set (KP ) = (K ) , (θ ) = (θnP )(j) and n n n (j) zP = 0 for nodes P ∈ S2 . Break the augmentation loop and go to the next load step, i.e. Step 1.2.
111 ¯ 2 ≤ ε2 is not sat(6.2) If the convergence criterion V isfied, establish the constraint index set the violation of which did not improve by the factor α: (j) (j) I4 = P: max gnP , − θnP > V2 /α , (j) (61) P ∈ S2 ¯ 2 ≥ V2 , for nodes P ∈ I4 set (6.3) If V P (j+1) (j) P (j+1) P (j) Kn = β KP ; θn = θn /β n (62) and go to Step 2.1. ¯ 2 < V2 , update the parameter θP employ(6.4) If V n ing the steepest ascent update scheme [refer to (57)] and continue to Step 6.5. ¯ 2 and go to Step 2.1, ¯ 2 ≤ V2 /α, set V2 = V (6.5) If V i.e. the next augmentation step. Otherwise, continue. ¯ 2 < V2 , for nodes P ∈ I4 set (6.6) If V2 /α < V
Acknowledgements We would like to acknowledge support of Ford Motor Company for this research under URP, with Dr. C.C. Wu as the project monitor.
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(j) P (j+1) = β KP ; Kn n P (j+1) P (j+1) θn = θn /β
augmented Lagrangian method by introducing the penalty update schemes into the incremental/iterative solution process for contact analysis problems. This is designed to make the contact solution independent of the penalty parameter value (demonstrated in Part 2 of the paper). 2. The proposed use of an equality constraint on the tangential displacement is expected to alleviate the difficulty of accurate determination of the friction force at the contact points. The proposed two-phase ALM2 solution process is expected to obtain accurate frictional contact solutions irrespective of the number of load steps and penalty values (investigated in Part 2 of the paper).
(63)
¯ 2 , and go to Step 2.1. and V2 = V 6 Concluding remarks An overview of the existing ALM-based frictional contact problem formulations and solution methods was presented. It was noted that these methods did not follow the intent of ALM from the optimization literature. Therefore the contact solution with the methods still depended on the user-specified penalty value, as with the penalty method. It was also noted that solution with existing ALM-based formulations depended on the userdefined number of load steps due to the difficulty of accurately predicting the friction force at the contact nodes. Two-dimensional deformable-to-rigid-body frictionless and frictional contact problems were defined. The basic idea of the augmented Lagrangian method (ALM) from optimization literature was presented in the context of contact problems. A new formulation and the corresponding algorithm ALM2 were developed in order to eliminate the contact solution dependence on both the penalty values and the number of load steps. The following conclusions are drawn based on the current investigation: 1. The augmented Lagrangian method from optimization theory is appropriate to solve contact problems due to its advantages over the more traditional penalty and Lagrange multiplier methods. The proposed ALM2 contact algorithm fully automates the
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