Contact with friction using the augmented Lagrangian Method: a conditional constrained minimization problem

Contact with Friction using the Augmented Lagrangian Method: a Conditional Constrained Minimization Problem

Alberto Luiz Serpa

Eaton Limited. Transmission Division. Rua Clark, 2061. CEP 13279-400. Valinhos. SP. Brazil

[e-mail]

Department of Computational Mechanics. Faculty of Mechanical Engineering. State University of Campinas.Unicamp. 13083-970. P.O. Box 6122.Campinas. SP. Brazil

[e-mail]

Fernando Iguti

Department of Computational Mechanics. Faculty of Mechanical Engineering. State University of Campinas.Unicamp. 13083-970. P.O. Box 6122. Campina. SP. Brazil

This work presents a formulation of the contact with friction between elastic bodies. This is a non linear problem due to unilateral constraints (inter-penetration of bodies) and friction. The solution of this problem can be found using optimization concepts, modelling the problem as a constrained minimization problem. The Finite Element Method is used to construct approximation spaces. The minimization problem has the total potential energy of the elastic bodies as the objective function, the non-inter-penetration conditions are represented by inequality constraints, and equality constraints are used to deal with the friction. Due to the presence of two friction conditions (stick and slip), specific equality constraints are present or not according to the current condition. Since the Coulomb friction condition depends on the normal and tangential contact stresses related to the constraints of the problem, it is devised a conditional dependent constrained minimization problem. An Augmented Lagrangian Method for constrained minimization is employed to solve this problem. This method, when applied to a contact problem, presents Lagrange Multipliers which have the physical meaning of contact forces. This fact allows to check the friction condition at each iteration. These concepts make possible to devise a computational scheme which lead to good numerical results.

Keywords: Finite Elements, Contact Problem, Friction, Mathematical Programming, Augmented Lagrangian.

Introduction

The contact problem is a subject of interest because usually forces transmission in mechanical systems occurs through a contact between two bodies. In several applications the stresses knowledge in the specific contact region is important.

The main difficulties of contact problems are: analytical solutions are not available, except for particular geometries; how to deal with unknown boundary conditions (contact region and contact forces); and the friction effect. These two last determine a non linear feature to the problem.

The solution of the contact problem using the Finite Element Method has been following three general approaches: adaptations of usual formulations using incremental and iterative schemes in order to take into account the contact conditions; formulation of variational inequalities (Duvaut and Lions, 1972; Glowinski, Lions and Trémolières, 1976), which can represent the contact problem, and the relationship of these inequalities with a mathematical programming method (Panagiotopoulos, 1975; Panagiotopoulos and Lazaridis, 1987); and the direct application of the mechanics minimum principles, and the solution through a constrained minimization problem (Haug, Chand and Pan, 1977; Klarbring and Björkman, 1988).

In this work the contact problem between elastic bodies, including the friction effect (Coulomb classical law), is formulated as a conditional dependent constrained minimization problem. This minimization problem presents inequality constraint equations which represent the non-inter-penetration condition of the bodies, and equality constraint equations are used to represent the friction effect. There are two friction conditions, adhesion and sliding, depending on the contact forces. Thus, the selection of the equality constraint equations set depends on the friction condition.

This work shows a detailed formulation of the contact problem with friction as a conditional dependent constrained minimization problem. The satisfaction of the correct friction condition is achieved simultaneously with the progress through the iterations. Distinction between the static and kinetic friction coefficients are allowed in this formulation. The elasticity equations were dealt with using the usual concepts of the Finite Element Method, see Hughes (1987).

An Augmented Lagrangian Method is employed to solve this minimization problem. This method is used to solve the constrained minimization problem through successive unconstrained minimizations (Luenberger, 1989). The Lagrange multipliers have the physical meaning of the contact forces, allowing the verification of the friction conditions. Besides this, the Augmented Lagrangian Method has a general feature which allows an extension of this work to other non linear effects.

Simo and Laursen (1992) applied the Augmented Lagrangian Method to solve the contact with friction between an elastic body and a rigid one without differentiating the static and kinetic friction coefficients. Their algorithm is quite different from this presented here since the tangential forces are not updated simultaneously with the normal ones.

A formulation of the Augmented Lagrangian Method and its relationship with a contact problem is presented in detail. The constraint equations are formulated usind a 2D quadrilateral finite element. An algorithm to solve the contact with friction is presented and tested using some examples.

Nomenclature

a: defines relative positioning of i, j and k nodes.

A, B: coefficient matrices of the constraint equations h_j(u) and c_i(u) respectively.

cond: matrix condition number.

c(u), h(u): vectors of equality and inequality constraint equations.

D, U: diagonal and upper triangular matrices resulting from K factorization.

f: external load vector.

f_c: stiffness and load contact terms in the non linear equations system.

f_d: unbalanced forces.

H: Hessian matrix.

i, j, k: typical nodes to formulate a constraint equation.

k (in the algorithm): Augmented Lagrangian iteration.

K: stiffness matrix.

l, m: numbers of equality and inequality constraint equations respectively.

ndof: number of degrees of freedom (number of optimization variables).

n, t: vectors of normal and tangential forces.

P: matrix related to the partition of U for static condensation.

(P₁): frictionless contact minimization problem.(P₂): contact with friction minimization problem.

(P₃): generic minimization problem.

q_h, q_v : horizontal and vertical distributed loads respectively.

r: penalty parameters vector (set of r_i e r_j).

r₀: initial penalty parameters vector (components r₀).

R_i, R_j: diagonal matrices of r_i and r_j respectively.

t: increasing rate of penalty parameters.

t : trial tangential force vector.

T: transformation matrix between local and global coordinate systems.

u: displacement vector (optimization variables vector).

x: nodal coordinates.

[a]₊= maximum[0,a].

| |: absolute value.

|| ||: Euclidian norm.

overlined symbols (example u ): related to local coordinate system.

Greek symbols

a , b ,: vectors of independent terms of linear constraints h_j(u) and c_i(u) respectively.

d l , d n : parameters to control maximum variation of l _j and n _jrespectively.

e ₁, e ₂: convergence criteria parameters.

F (u,n ,l ,r): Augmented Lagrangian function.

l , n : vectors of the Lagrange multipliers.

m _d, m _e: static and kinetic friction coefficients respectively.

Ñ : denotes gradient.

P : total potential energy.

q : rotation angle between local and global coordinate systems.

Subscripts

A, B: related to bodies A and B respectively.

a, b: related to vector and matrix partitions for static condensation.

i, j: related to constraint equations c_i(u) and h_j(u).

n, t: related to normal and tangential contact directions.

1, 2: related to coordinates components 1 or 2 respectively, except for e .

Superscripts

*: related to optimum solution.

i, j, k: related to typical nodes i, j and k.

t: denote matrix or vector transposition.

-1: denote matrix inversion.

Contact as a Minimization Problem

The total potential energy of a discretized system of elastic bodies A and B in quasi-static conditions, see Zienkiewicz and Taylor (1991), can be written in the form

(1)

where u_Aand u_B are the displacement vectors of the bodies A and B respectively, K_A and K_B are the stiffness matrices of the bodies A and B respectively, and f_A and f_B are the equivalent load vectors of the bodies A and B respectively. The sub-indexes A and B are omitted in order to simplify the notation.

The frictionless contact problem can be represented by the minimization problem (P₁):

(2)

where h_j(u) are inequalities which represent the inter-penetration of the bodies (if h_j(u)£ 0 there is no inter-penetration, and when h_j(u)> 0 there is inter-penetration of the bodies).

The classical Coulomb law of friction considers the existence of the kinetic friction coefficient m _d and the static friction coefficient m _e, and states that m _d£ m _e.

Defining n and t, normal and tangential forces vector respectively, and t the trial tangential force, it is possible to write the friction conditions (adhesion and sliding) in the form:

(3)

(4)

where u_t/||u_t|| is a relative tangential displacement unit vector. There is no relative displacements in the adhesion situation, and there is relative displacements in case of sliding. The tangential force related to the sliding condition is redefined according to the kinetic friction coefficient and this force is opposite to the relative displacement.

Representing the adhesion condition as c_i(u) = 0 (equality constraint equations) one can write a generalized minimization problem in the form:

(5)

It can be noted in problem (P₂) the presence of c_i(u) = 0 as a function of the friction condition in terms of the normal and the tangential forces n_iand t_i.

The sliding condition occurs when the adhesion capacity is overcome (relative displacements). In this case, the constraint c_i(u) = 0 does not appear, and the corresponding sliding friction force is applied. In the case of adhesion, the constraint c_i(u) = 0 must be present, and it ensures that there is no relative displacement in the considered point.

An Augmented Lagrangian Method for the Contact Problem

Consider the generic minimization problem (P₃):

(6)

The Augmented Lagrangian function related to (P₃), see Bertsekas (1976), Powell (1978) and Luenberger (1989), can be written as

(7)

where [a]₊=max[0,a]; n _i and l _j are the Lagrange Multipliers (also represented by the vectors n and l ); and r_i and r_j are the penalty parameters (vector r).

The null gradient condition of the Augmented Lagrangian function of Eq.(7) is

(8)

To satisfy the Kuhn-Tucker condition, see Luenberger (1989) and Bazaraa (1993), at each iteration, one should have the Lagrange multipliers updated by

(9)

The Augmented Lagrangian algorithm is based on the unconstrained minimization (Wolfe, 1978) of F (u,n ^k,l ^k,r^k) and on updating the Lagrange multipliers and penalty parameters. This process is repeated until the convergence is reached. See Luenberger (1989) and Bazaraa (1993) for details.

It is convenient to denote the contact terms as

(10)

and Eq.(8) can be written as

(11)

The set of the constraint equations can be represented as vectors, i.e., h(u)=[h₁(u) h₂(u) ... h_m(u)]^t and c(u)=[c₁(u) c₂(u) ... c_l(u)]^t.

An interesting particular situation occurs when the constraint equations are linear functions, i.e.,

(12)

where A is a m x ndof matrix, a is a m x 1 vector, B is a l x ndof matrix, b is a l x 1 vector, and ndof is the number of degrees of freedom; and when

(13)

where K is the stiffness matrix and f is the load vector. In this case, the Augmented Lagrangian function gradient using Eq.(12) and Eq.(13) becomes

(14)

where R_i and R_j are diagonal matrices of r_i and r_j respectively, and [ ]_* denotes the presence of the term inside the brackets according to the value of the term [ ]₊ in the Augmented Lagrangian function of Eq.(7).

The Hessian matrix of the Augmented Lagrangian function is

(15)

One can notice that the constraint equations generate stiffness and forces terms. These stiffness terms affect directly the condition number of the Hessian matrix, Eq.(15). The numerical ill conditioning in Penalty Methods and the convergence effects are widely discussed in the literature, see Luenberger (1989).

Constraint Equations Formulation

In the case of a 2D finite element discretization problem using linear quadrilateral isoparametric finite elements (4 nodes), see Hughes (1987), the side of the element is a line segment and the displacement field in each side is a linear one.

Figure 1 shows the notation used to establish the constraint equations. In this figure one can see the definition of the local coordinate system (x₁,x₂) and the global coordinate system (x₁,x₂). The typical nodes i, j and k are used to establish each constraint of the problem.

The formulation of this section is related to a typical inequality constraint equation and to a typical equality constraint equation.

The local coordinate system is defined based on the tangential and normal contact directions. Thus, x₁ is oriented in the jk segment direction, and x₂ is oriented in the jk segment outer normal direction.

Considering the nodes i, j and k, their related degrees of freedom , (local coordinate system) and according to Fig.1 it is possible to write:

(16)

(17)

Thus the coefficient matrix of h_j(u) is

(18)

The other terms that appear in the expression of the gradient of the Augmented Lagrangian function, Eq.(14), for a typical inequality constraint equation, are:

(19)

(20)

Similarly it is possible to define an equality constraint equation:

(21)

The coefficient matrix of c_i(u) is

(22)

The other terms in the gradient of the Augmented Lagrangian function, Eq.(14), related to an equality constraint equation, i.e., B^t R_i B , B^t R_i b and B^t v and can be similarly calculated.

According to the definition of the equality constraint equations, the sign of these constraint equations can be used to determine the relative tangential displacement direction in the contact region.

The terms previously presented were formulated in the local coordinate system and they can be rewritten in the global coordinate system by means of a transformation of coordinates.

Consider matrix T defined as

(23)

The relation between the displacements in the local coordinate system and in the global coordinate system is

(24)

Thus, an inequality constraint equation and the related terms in the global coordinate system are:

(25)

(26)

In the same way for an equality constraint equation:

(27)

(28)

It is possible to see that the constraint equations, in the way they were formulated, have the same matrix structure of a standard finite element scheme. It generates a symmetric "stiffness" matrix and a "load" vector.

It can be verified that the constraint equations established according to the present node-to-segment scheme can generate an incompatible displacement field like the one illustrated in the Fig.2.

Numerical Schemes

One can consider a corresponding normal and tangential force (and related constraint equations) for each point of contact. Thus, the indexes i and j can be the same.

The following algorithm, using the previous formulation, is proposed to solve the contact with friction problem:

1. Initial definitions: friction coefficients m _e and m _d ; starting point u⁰= 0 (in general); Lagrange multipliers (l⁰, n ⁰); and penalty parameters (r_i⁰, r_j⁰).

2. k=0 (iteration counter).

3. While the convergence criteria is not satisfied, repeat item 3.1 to 3.4:

3.1) Contact forces definition (through the Lagrange multipliers):

n = l^k (normal forces) and t = n^k (tangential forces).

3.2) Solve the non linear system of Eq.(11) in order to obtain u^k+1. Consider u^k as starting point, and

(assembly).

with the following verification of the friction conditions for each inequality and equality couple:

3.3) Update Lagrange multipliers and penalty parameters.

3.4) k=k+1.

4. u^* =u^k+1 is the obtained solution. End.

In the case of linear constraints the non linearity of this problem is due to [ ]₊ term and due to changes in the equality constraint set due to the friction condition test. If the active constraint set does not change during the iterative procedure a particular situation of a linear system of equations appears. If the active constraint set changes one can use a line search procedure to enlarge the convergence region. The secant line search (Zienkiewicz and Taylor, 1991) testing the new interval at each iteration, can be a suitable procedure for this case because does not require the objective function evaluation. This line search scheme is employed in this work.

The null gradient condition of the Augmented Lagrangian function can be rewritten as

(29)

The static condensation according to Guyan (1965) can be an interesting scheme because the number of contact degrees of freedom is in general much smaller than the system total number of degrees of freedom. The static condensation is very suitable in this case because the terms [A[^t R_j A]_* and B^t R_i B affect only the degrees of freedom related to the contact region. So, the equation system to be solved at each iteration has the same order of the partition matrix of the contact degrees of freedom.

A generic system can be partitioned in the form

(30)

By the use of Eq.(30) it is possible to write

(31)

(32)

Through Eq.(31) one has

(33)

Substituting Eq.(33) in Eq.(32) it follows that

(34)

The solution, using the static condensation, consists in obtaining u_b using Eq.(34), and then u_a by the use of Eq.(33). This is a suitable process when it is necessary to solve the u_b system several times and then once for u_a. This is a typical situation of a non linear system where the non linearity is restricted to the u_b degrees of freedom.

The usual concept of the Gauss factorization consists in decomposing the stiffness matrix in the form

(35)

where U is an upper triangular matrix with unit diagonal (sometimes referred to as unit upper triangular matrix), and D is a diagonal matrix. The matrix U has the same storage profile as K, and the fact that U has diagonal entries equal to one allow to save storage by placing the diagonal entries of D in the diagonal entries of the array that storages U ("skyline" storage), see Hughes (1987).

Using the partition concept it is possible to write

(36)

Through this partition it is possible to rewrite Eq.(34) in the form

(37)

and Eq.(31) as

(38)

This formulation of static condensation allows to apply the usual computational routines of the Gauss elimination.

In the proposed algorithms the Lagrange Multipliers are updated according to a sequence based in the Kuhn-Tucker conditions, Luenberger (1989) and Bazaraa (1993), Eq.(9). The penalty parameters affect the algorithm convergence. They influence the numerical conditioning of the Hessian matrix, and also affect the Lagrange Multipliers variation rate.

An empirical penalty parameters update scheme is used in this work, i.e.,

(39)

where t e r₀ are determined by computational tests for each specific problem. A bad choice of parameters can generate an oscillatory behaviour of the algorithm and can cause divergence.

A scheme to control the variation of the penalty parameters and the Lagrange multipliers was employed using the value limits given by d l and d n . This scheme is:

if r_jh_j(u^k) < d l then

and r_j is updated,

else

and r_j is not updated;

if then

and r_i is updated,

else

and r_i is not updated.

The matrix condition number is also monitored in order to avoid ill conditioned problems. In this work, the system of equations condition number was estimated using the estimator of Gill, Murray and Wright (1991). This estimator uses matrices U and D which are the results of the K matrix factorization. If the problem is ill conditioned, another values of r₀ and t are used to define a new penalty parameters update sequence.

Examples and Numerical Results

In this section it is presented some examples and numerical results related to the proposed algorithms. Microsoft FORTRAN Power Station version 1.0 using double precision was used to develop the computer codes. A Hewlett Packard Vectra 486/33VL computer was used to process the examples.

The convergence criteria used in this work are based in a feasible point and in a maximum point variation, i.e.,

(41)

The unbalanced nodal forces f_d, which is a measure of the nodes equilibrium in terms of forces, see Irons and Ahmad (1980), are also employed as a parameter to check the convergence of the algorithms, i.e.,

(42)

It is usual to compare max |f_d| to the external applied forces norm ||f||.

Example 1: Rectangular block in contact with a rigid plane

This example refers to a block in contact with a rigid plane surface according to Fig.3. It is assumed the plane strain hypothesis. The mesh employed in the solution of this problem presents 379 degrees of freedom and 33 couple of constraints (inequalities and equalities). The nodes are numbered from the left to the right in the contact region.

Two cases in terms of loads and friction coefficients were studied: case 1 - q_h=15N/mm², q_v=5N/mm², m _e=m _d=1.0; and case 2 - q_h=10N/mm², q_v=15N/mm², m _e=m _d=0.2.

The results in terms of displacements and forces are shown in table 1 when e ₁=10^-8. It is possible to observe for case 1 an adhesion contact region corresponding to the nodes 25 to 33, a sliding contact region defined by the nodes 4 to 24, and a separation of the bodies in the region defined by the nodes 1 to 3. The condition number estimated in the solution is cond=20.69. The maximum unbalanced nodal force is max|f_d|=3.74x10^-2 compared to the external forces norm ||f||=207.57. One can observe for case 2 an adhesion contact region corresponding to the nodes 20 to 33, and a sliding contact region defined by the nodes 1 to 19. There is not separation in this case, so that u_n=0 for all contact nodes. The condition number estimated in the solution is cond=17.03. The maximum unbalanced force is max|f_d|=7.62x10^-5 compared to the external forces norm ||f||=222.25.

The obtained values present a good correlation with the results obtained by Barbosa (1986), and also a good result in terms of unbalanced forces.

Table 2 shows the penalty parameters effect on the convergence behaviour for this example in terms of the Augmented Lagrangian number of iterations and in terms of the processing time (in seconds).

Example 2: Dovetail (two elastic bodies)

This example is illustrated in Fig.4. It is assumed a plane stress situation. This problem presents 507 degrees of freedom and 17 couple of constraints.

The displacements and forces in the contact region are shown in table 3. The results are related to e ₁=10^-8. It can be noted in table 3 a separation region corresponding to nodes 1 and 2, and a sliding region corresponding to nodes 3 to 17. The contact nodes are numbered from the left to the right in the contact region.

The obtained results are very similar to those ones presented by Raous, Chabrand and Lebon (1988).

Table 4 shows some data related to the effect of the penalty parameters in the convergence properties for this example.

Example 3: Elastic blocks in contact

Figure 5 shows the problem scheme and the corresponding mesh. It is assumed a plane stress situation. This problem presents 258 degrees of freedom and 11 inequalities constraints and 11 equalities constraints. Three cases in terms of friction coefficients were analyzed (m ₁=0.002, m ₂=0.2 and m ₃=0.4). The nodes are numbered from the left to the right in the contact region.

The obtained results in terms of displacements and contact forces are shown in tables 5, 6, and ⁷ for e ₁=10^-8.

Figure 6 shows the contact region with 20000 times amplified displacements for m ₁=0.002. It is possible to verify the incompatibility of the displacement fields and the effect of the convergence criteria in terms of the feasibility of the constraints.

The obtained results present a good correlation with the work of Lee (1994).

Conclusions

In this work the contact with friction was modeled as a minimization problem. The total potential energy of the bodies is the objective function, the inter-penetration of the bodies are considered as inequality constraints, and the friction is considered through adhesion and sliding using equality constraints. An equality constraint is present in the adhesion situation, and not present in the sliding case when the dynamic friction forces are imposed. The scheme adopted in this work characterizes what it is called here the conditional dependent constrained minimization problem and can be a more general way of solving this class of problems. Other non linear effects can also be included in this formulation.

Using the Augmented Lagrangian Method, the constrained minimization problem is solved through successive unconstrained minimizations of the Augmented Lagrangian function. In the Augmented Lagrangian Method, the optimality conditions of the constrained minimization problem corresponds to the optimality conditions of the unconstrained minimization of the Augmented Lagrangian function.

The Augmented Lagrangian Method applied to the contact problem allows the employment of the Lagrange multipliers as the contact forces. This physical interpretation allows to test the friction condition, defining the presence or not of a specific set of equality constraints in the iteration.

The null gradient condition of the Augmented Lagrangian function represents a system of equations. In this system, the constraint equations have a similar matrix structure to the one in the Finite Element Method. They give way to a symmetric stiffness matrix and a load vector.

The numerical results obtained are satisfactory when compared with some references.

It can be noted that the penalty parameters influence directly the numerical conditioning of the system of equations. Some schemes to control the penalty parameters increasing rate were used, looking for a better convergence behaviour of the algorithm. The matrix condition number monitoring, the maximum variation of Lagrange multipliers, and the maximum penalty parameters variation were also used. The values of the maximum variation of the Lagrange multipliers can be defined through a comparison with the external applied forces.

The presented algorithm allow an extension of this work to other non linear situations using a proper definition of the gradient of the total potential energy (Zienkiewicz and Taylor, 1991) and of the constraint equations. This scheme can also be used incrementally and, in general, the convergence can be improved. A convenient choice of the penalty parameters, mild increasing rate for example, can avoid the uncontrolled variations in the active constraint equations set. In this case, the solution of problem (P₂) can be found without using load steps, particularly when the non linearity is uniquely due to the contact.

Some aspects that should be considered in order to improve the proposed algorithm are the development of some optimized criteria to update penalty parameters without user’s interference, see Serpa and Iguti (1998); the use of some adaptive mesh scheme to reduce the incompatibility of the displacement fields in the contact region. In the h refinement, new finite elements can be suitably created, and in the p refinement higher order interpolation terms can be conveniently created; and the application of other constraint formulation, which ensures a better contact compatibility, see Quiroz and Beckers (1995), also needs to be investigated.

Manuscript received: January 1999. Technical Editor: Hans Ingo Weber.

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Publication Dates

• Publication in this collection
  18 Dec 2000
• Date of issue
  2000

History

• Received
  Jan 1999