Solving Quasistatic Contact Problems Using Nonsmooth Optimization Approach

This paper is devoted to a study of time-dependent hemivariational inequality. We prove existence and uniqueness of its solution, provide fully discrete scheme and reformulate this scheme as a series of nonsmooth optimization problems. This theory is later applied to a sample quasistatic contact problem describing a viscoelastic body in frictional contact with a foundation. This contact is governed by a nonmonotone friction law with dependence on normal component of displacement and tangential component of velocity. Finally, computational simulations are performed to illustrate obtained results.



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1 Introduction

Currently various engineering applications require scientific analysis of contact phenomena. In order to study such problems a framework in the form of variational and hemivariational inequalities complemented by theory of Finite Element Method emerged. Literature in the field of Contact Mechanics grows rapidly. Main ideas and mathematical tools were introduced in monographs [6, 14, 17, 18, 20]. Other comprehensive studies include [10, 19, 21, 22]. Theory of nonsmooth optimization used in our approach was presented in [1] and applied to a sample static contact problem in [2]. Numerical analysis of contact problems can be found for instance in papers [3, 8, 9, 12, 13, 16] and for a recent study [11].

This paper is a continuation of our previous work [15], where we reformulated time-independent hemivariational inequality as optimization problem and used it to solve a static mechanical contact problem. Here, we consider time-dependent hemivariational inequality, and use similar idea to solve it numerically. In this case we prove existence and uniqueness of a solution using fixed point argument and apply this abstract framework to solve a quasistatic mechanical contact problem. This approach allows us to substitute Coulomb’s law of dry friction by more general, nonmonotone friction law with dependence on tangential component of velocity. Additionally, the friction bound is determined by normal component of displacement, describing penetration of the foundation.

The rest of this paper is organized as follows. In Section 2 we consider an abstract problem in the form of hemivariational inequality and show that under appropriate assumptions it has a unique solution. Section 3 contains a discrete scheme which approximates solution to introduced abstract problem. We also prove the theorem concerning numerical error estimate. An application of presented theory to a mechanical contact model is introduced in Section 4. Finally, in Section 5, we describe computational algorithm used to solve mechanical contact problem. Then we present results of simulations and empirical error estimation for a set of sample data.

2 Hemivariational inequality

We start with basic notation used in this paper. For a normed space , we denote by its norm, by its dual space and by the duality pairing of and . By we denote a generic constant (value of may differ in different equations).

Let us now assume that is locally Lipschitz continuous. The generalized directional derivative of at in the direction is defined by

The generalized subdifferential of at is a subset of the dual space given by

If is a locally Lipschitz function of variables, then we denote by and the Clarke subdifferential and generalized directional derivative with respect to -th variable of , respectively.

Let now be a reflexive Banach space and be a Banach space. Let be a linear and continuous operator from to , and . We denote by the adjoint operator to . Let be a time interval with . Let , , and let . The functional is assumed to be locally Lipschitz continuous with respect to its second argument. We formulate the considered hemivariational inequality as follows.

Problem :  Find such that for all ,


We introduce the following assumptions.

 Operator satisfies

  1. is linear and bounded,

  2. is symmetric, i.e., for all ,

  3. there exists such that for all .

 Operator is Lipschitz continuous, i.e., there exists such that
a   .


  1. is locally Lipschitz continuous for all ,

  2. there exist such that
    for all ,

  3. there exist such that

    for all .




We remark that assumption (c) is equivalent to the following condition

for all . In the special case when does not depend on its first variable, we obtain a relaxed monotonicity condition, i.e. for all


So, condition (c) is a generalization of .

Define an operator by the formula

We can reformulate Problem as the following one.

Problem :  Find such that for all ,

Let us now present the following theorem.

Theorem 1

Under assumptions , , , , and , Problem  has a unique solution .

Proof. We use a fixed point argument. Given , define

Then . Consider the auxiliary problem of finding a function such that


for all . Applying Theorem 4.2 in [11] with , we know that there exists a unique element which solves this inequality for each . Let us show that the function . For simplicity, with , denote , , for . We get that


Taking in (2.3), in (2.4) and adding, we get

Using strong monotonicity of the operator guaranteed by (c), assumptions , (c) and we obtain



By the smallness assumption , , and from continuity of and , we deduce that is a continuous function. This allows us to define an operator via the relation

Let us prove that the operator has a unique fixed point . For two arbitrary functions , let be the solution of (2.2) for , . Similarly to (2.5), we have


we derive from the previous inequality that

This shows that is a history dependent operator. Applying Theorem 3.20 in [11], we know that the operator has a unique fixed point . Moreover, by the definition of , is a solution to Problem . Uniqueness of a solution to Problem  is a consequence of uniqueness of fixed point.   

3 Numerical scheme

Let us now fix and let be a finite dimensional space with a discretization parameter . For a given , we introduce the time step and the temporal nodes , . Let be an approximation of element in space . We write and define

We present the following discretized version of Problem in the form of an operator inclusion problem.

Problem : Find such that for all

Now we introduce some preliminary material, namely we recall a special case of the Jensen inequality and the discrete version of Gronwall inequality. On several occasions, we will also apply the elementary inequality

Lemma 2

(the Jensen inequality) Let be a set of positive measure and let be an integrable function. Then

In particular, for , , , we have

Lemma 3

(the Gronwall inequality, [10, Lemma 7.25])  Let be given. For we define . Let , be two nonnegative sequences satisfying for and for all

Then there exists a constant such that

Let us now prove the following lemma.

Lemma 4

Under assumptions , , , , and if Problem  has a solution , then it is unique and satisfies


for all with a positive constant .

Proof. To simplify the notation in this proof, we write instead of . Let be a solution to Problem and let us fix . This means that there exists such that

From the definition of generalized subdifferential of we have for all ,

After reformulation, we obtain discretized version of hemivariational inequality ,


Let us now assume that Problem has two solutions and . We will prove inductively that these solutions are equal. For we get . In inequality (3.4) for a solution we set , and for a solution we set . Then adding these inequalities, we obtain

Hence, (c) and (c) yield

and consequently from we have . We now show that if for , then . Similarly, in (3.4) for a solution we set , and for a solution we set . Adding these inequalities we obtain

We observe that , hence

Again (c) and (c) yield

Under assumption , we obtain that . This equality holds for , hence a solution of  is unique.

Now, in order to prove , we set in (3.4) to obtain



Using (c), we get

From Proposition 3.23(ii) in [19] and assumption (b) we have



From ,


Using (c), (3.6) and (3.7) in (3.5), we get


Taking sufficiently small , using () and inequality (3.1), we obtain

From discrete version of the Gronwall inequality (Lemma 3) with

we get

and then

which concludes the proof of (3.3).   

We now consider an optimization problem, which is equivalent to Problem  under the stated assumptions. To this end, let us fix and assume to be known, hence also is given. Let a functional be defined for all as follows


The next lemma collects some properties of the functional .

Lemma 5

Under assumptions , , , , and , the functional defined by (3.8) has the following properties

  1. is locally Lipschitz continuous,

  2. for all ,

  3. is strictly convex,

  4. is coercive.

Proof. The proof of (i) is immediate since for the functional is locally Lipschitz continuous as a sum of locally Lipschitz continuous functions with respect to .

For the proof of (ii), we observe that from , and , the functions

are strictly differentiable with

Now, using the sum and the chain rules for generalized subgradient (cf. Propositions 3.35 and 3.37 in

[19]), we obtain

which concludes (ii).

In order to prove (iii), let us fix . We take for . From (ii), there exists such that

From the equivalent condition to (c), and consequently from (2.1), we have

Hence and from (c), we obtain