Path-following and semismooth Newton methods for the variational inequality arising from two membranes problem

A semismooth Newton method, based on variational inequalities and generalized derivative, is designed and analysed for unilateral contact problem between two membranes. The problem is first formulated as a corresponding regularized problem with a nonlinear function, which can be solved by the semismooth Newton method. We prove the convergence of the method in the function space. To improve the performance of the semismooth Newton method, we use the path-following method to adjust the parameter automatically. Finally, some numerical results are presented to illustrate the performance of the proposed method.


Introduction
Contact problems appear in many applications in industry and engineering, such as the contact between two elastic membranes [1][2][3][4][5][6][7]. This problem describes the equilibrium position of two membranes under the action between them. The membranes cannot interpenetrate and are fixed on the boundary. In this system, there are three unknowns: the position of each membrane and the action of each membrane on the other one [2]. One of the main challenges is the fact that the contact zone is not known in advance and has to be identified. Although the main results on existence and uniqueness can be found in the recent literature [1][2][3][4], little attention has been paid to methods for the numerical solution. Therefore, the accurate and efficient numerical simulation of the contact problem is necessary.
We note that different Newton methods have been successfully applied to constrained problems such as complementary problems and variational inequalities in finite or infinite dimensional space [8][9][10][11][12][13][14][15][16][17]. Motivated by theoretical and numerical results obtained in recent years, we develop a coupling procedure with combination of semismooth Newton methods (SSNMs) and path-following methods (PFMs) in function space [10,15,18]. The essence of the procedure is to reduce the problem to a regularized problem which can be solved by SSNM. The main advantage of SSNM is that the inequality constraints are formulated as a nonlinear system which is equivalent to a sequence of linear systems. However, the convergence speed of SSNM is sensitive to a parameter. To make SSNM more efficient, we propose a path-following strategy to update the parameter automatically for numerical implementation.
The paper is organized as follows: In the next section, we start with the formulation of the contact problem between two elastic membranes and recall some basic results. In Sect. 3, we give a regularized problem and its convergence. The semismooth Newton method is proposed in Sect. 4. A path-following method, based on the semismooth Newton method, is presented in Sect. 5. Finally, in Sect. 6 some numerical results are given to show the performance of our method.

Problem setting and main results
We consider two elastic membranes in unilateral contact. Throughout the paper, let Ω be the bounded and convex domain in R 2 with a Lipschitz boundary Γ . For given functions f 1 , f 2 and nonnegative g, the problem is to determine the displacements u 1 , where the tension coefficients μ 1 > 0 and μ 2 > 0. The solution (u 1 , u 2 ) of (2.1)-(2.5) may be interpreted as a vertical displacement of two membranes stretched by different horizontal heights and pressed together by vertical forces with two densities. In this system, λ represents the action of the second membrane on the first one and -λ is the reaction. The contact condition (2.3) describes the non-interpenetration of two membranes in Ω, and the first membrane can press the second one in the domain that is in contact, i.e. u 1u 2 = 0. If there is no contact, i.e. u 1u 2 > 0, then the action vanishes with λ = 0. The boundary conditions (2.4) and (2.5) mean that the first membrane is fixed on Γ at the height g which is a nonnegative function, and the second one is fixed at zero, respectively. (More details are given in [2][3][4].) To give the weak formulation of the problem (2.1)-(2.5), we introduce the following space of functions: and the convex subsets For given (f 1 , f 2 ) in L 2 (Ω)×L 2 (Ω) and g in H 1 2 + (∂Ω), we consider the following variational problem: Find (u 1 or a variational inequality: For the above problems, we summarize the main conclusions for the existence and uniqueness as follows (see Proposition 1, Lemma 2 and Proposition 3 in [3]).
is a weak solution of (2.1)-(2.5) if and only if it is a solution of (2.6).

Proposition 2.2 For any solution
In this paper, we consider the numerical method of the unilateral contact problem.
We also require that the bilinear form a(·, ·) satisfies the weak maximum principle, i.e. for all v ∈ H 1 0 (Ω), where v + = max(0, v). This property can easily be proved by using the bilinear form a(·, ·).
We note that the condition (2.3) can be rewritten as for any γ > 0 [11,15]. If we replace (3.3) by whereλ ∈ L 2 (Ω) is given, then problem (2.1)-(2.5) can be expressed as Consequently, the optimization problem for system (3.5) is It follows from the uniform convexity of J(γ , ·, ·) that the system (3.5) admits a unique solution (u 1γ , u 2γ , λ γ ) for every γ > 0 [11,15]. To highlight the dependence on γ , the solution is denoted by (u 1γ , u 2γ ) and the corresponding multiplier by λ γ . In the following theorem, we can show that the problem (2.6) can be approximately formulated as the optimization problem (3.6) with γ → ∞.

Semismooth Newton method
This section is devoted to the discussion of an iterative algorithm for solving (3.5). Note that the direct application of a Newton algorithm is impeded by the fact that the maxfunction is not differentiable. Alternatively we shall apply a semismooth Newton method to the mapping F : L 2 (Ω) → L 2 (Ω) defined by We now briefly recall those facts on semismooth Newton methods which are relevant for the present context [11,13,15].
for every x ∈ U.

Theorem 4.1 Suppose that x * ∈ D is a solution to F(x) = 0 and that F is Newtondifferentiable in an open neighborhood U containing x * and that G(x)
Let us consider Newton-differentiability of the max-operation. We introduce candidates for the generalized derivatives of the form where y ∈ X.

Proposition 4.1
The mapping max(0, ·) with 1 ≤ p < q < ∞ is Newton-differentiable on L q (Ω) and G m is a generalized derivative.
Now we can describe our semismooth Newton method for the problem (3.5) as follows. (4.1) (5) Stop or set k := k + 1 and go to (2).
Following the analysis in [11,15], we have the same results.

Proposition 4.2 If
is the solution to (4.1).

Path-following method
As in Theorem 3.1, the solution converge only if γ → ∞. If the parameter γ is too small, the SSNM converges slowly. On the contrary, if the γ is too big, it may result in a badly conditioned problem. Therefore, the SSNM needs a continuous procedure with respect to γ . We mention that path-following schemes for problems posed in function space have become popular in recent years. Such a procedure has already been applied to obstacles and contact problems in linear elasticity [10,15,18].
In this section, we give a brief review of path-following method for treating semismooth Newton methods, which can be applied to the unilateral contact problem between membranes. We introduce the primal infeasibility measure ρ F , and the complementarity measure ρ C for the (k + 1)th iterate as follows: Then we can update the parameter γ by where τ > 1 and q ≥ 1. So we obtain the following path-following method.  Evolution of E k with respect to k for different N Next, we investigate the convergence behavior of our method. In Fig. 2 we provide the evolution of the relative error, , with respect to the iteration index k for N = 200 × 200, N = 400 × 400 and N = 800 × 800. We note that our method converges for different mesh sizes. Although the number of iterations increases for increasing number of elements, the finer grid yields the smaller relative error.