Analysis and numeric of mixed approach for frictional contact problem in electro-elasticity

1. Introduction

The numerical study of the piezoelectric [1, 2] contact problems [3, 4, 5, 6, 7] presents a great challenges, because the non coercivity and non differentiability of some terms. The linear term coupling the mechanical field and electric potential is non coercive and non symmetric. The term corresponding to the friction is convex and non differentiable, in the variational formulation in one hand. In the other hand, the non-linear coupling mechanical field and frictional contact, it can be shown in the norm of the tangential component of the mechanical field present in the frictional function.

To overcome these difficulties, the authors develop some methods like finite elements method [8, 9], penalty method and fixed point method [4]. The most and sufficient method for this type of problem is the ones based on convex duality [10] and the introduction of Lagrange multipliers [11, 12, 13]. Primal-dual active sets strategy, which is an equivalent to infinite dimensional semismooth Newton method, is applied in [14] while in [15] the author propose a numerical approximation and based Uzawa block relaxation method. Alternating directions method of multipliers (ADMM) is based in [16].

In mechanic of structures one is always interested in the determination of the stress tensor \(\sigma\) more than the mechanical displacement itself. Methods have been developed for calculate an approximation of \(\sigma\) from \(u\) and the drawback is not easy to build an approximation space of tensors satisfying the equilibrium relations and required regularity. A mixed variational formulation have been developed to handle this difficulty. Concerning the piezoelectric contact problems, mixed formulation were developed in [17, 18, 19].

In this paper, we introduce a mixed variational approach based Lagrange multipliers which describes the static frictional contact between a piezoelectric body and non-conductive foundation. The standard mixed variational formulation of contact problems is formally in the following form [17, 18, 20]:

The idea consists in decoupling the contact from the friction by introducing two convex sets, one is reserved to the contact multiplier and the second is stated for the friction. This approach leads to decoupled inequalities in mixed variational problem. Since \(a(\cdot,\cdot)\) is not symmetric we follow the standards techniques and steps based on [17,18 ] but with different fixed point map (\(Step\, 3\) in the proof) and hence more analysis is needed to get the existence and uniqueness result. The resulting problem is an system by blocks of two unknowns (displacement and potential), regarding its structure and the form of blocks we use Gauss elimination technique and then we obtain a Schur complement in the matrix corresponding to displacement subproblem. It is well known (see [16, 21]) that this technique allows a suitable preconditioner for conjugate gradient method employed to solve the resulting symmetric and positive definite system.

To prove the efficiency of this approach, we state suitable numerical fixed point scheme. The convergence is proved basing abstract perturbed problem and fixed point process, Banach fixed point is hence applied. For details concerning the mathematical tools we refer to [22, 23, 24, 25].

The paper is structured as: In Section 2, we present the model of equilibrium process of the elastic piezoelectric body in frictional contact with a non-conductive foundation,we introduce the functional spaces for various quantities, list the assumptions on given data and derive the weak formulation of the problem. In Section, 2.2 we state and prove our main existence and uniqueness result, Theorem 1. The proof of these theorem are carried out in several steps and are based on an abstract result in the study of elliptic variational inequalities and Banach fixed point technique. The successive iterative method is detailed followed by the convergence result in Section 4. In Section 5, we conclude with finite element discretization and we give some numerical experiments by simple example .

2. Problem setting and main results

2.1. Problem setting

The piezoelectric body occupies in its reference configuration (initial configuration) the domain \(\Omega\subset\mathbb{R}^{d}\), \(d=2,3\). We suppose that \(\Omega\) is bounded with a smooth (enough) boundary \(\partial \Omega=\Gamma\). We denote by \(n\) be the outer normal to \(\Gamma\) and summation over repeated indices is implied and the index that follows a comma represents the partial derivative with respect to the corresponding component of the variable. The indices take values in \(\{1,\,\cdots,\,d\}\) the summation convention over repeated indices is used.

Below we use \(\mathbb{S}^{d}\) to denote the space of second order symmetric tensors on \(\mathbb{R}^{d}\) while "\(.\)" and \(\| \cdot \|\) will denote the inner product and the Euclidean norm on \(\mathbb{S}^{d}\) and \(\mathbb{R}^{d}\), that is \begin{eqnarray*}u. v=u_{i}v_{i}, \hspace{1cm}\|v\|=(v. v)^{\frac{1}{2}},\hspace{0.5cm}\forall\;\; u,v\in \mathbb{R}^{d},\\ \sigma. \tau=\sigma_{ij}\tau_{ij}, \hspace{1cm}\|\tau\|=(\tau.\tau)^{\frac{1}{2}},\hspace{0.5cm}\forall\;\; \sigma,\tau\in \mathbb{S}^{d}.\end{eqnarray*}

We also use the notations \(u_{n}\) and \(u_\tau\) for the normal and tangential displacement, that is \(u_{n}=u. n\) and \(u_\tau=u-u_{n}n\). Similarly we denote by \(\sigma_{n}\) and \(\sigma_\tau\) the normal and tangential stress tensor given by \(\sigma_{n}=\sigma n. n\), \(\sigma_\tau=\sigma n-\sigma_{n}n\).\\ We introduce the following functional spaces on \(\Omega\); \begin{eqnarray*} &&H= L^{2}(\Omega)^{d}=\left\lbrace u=(u_{i})\; | \;u_{i}\in L^{2}(\Omega)\right\rbrace ,\;\;\; \mathcal{H}= \left\lbrace \sigma = \sigma_{ij},\; \sigma_{ij}=\sigma_{ji}\in L^{2}(\Omega)\right\rbrace, \\ &&H_{1}=\left\lbrace u \in H\; | \;\epsilon(u) \in \mathcal{H}\right\rbrace ,\;\;\;\mathcal{H}_1= \left\lbrace \sigma \in \mathcal{H}\;|\; Div \;\sigma \in H \right\rbrace \end{eqnarray*} endowed with the inner products \begin{eqnarray*}&&(u,v)_{H}=\int_{\Omega} u_{i}v_{i}dx,\hspace{0.5cm} (\sigma,\tau)_{\mathcal{H}}=\int_{\Omega} \sigma_{ij}\tau_{ij}dx.\\ &&(u,v)_{H_{1}}= (u,v)_{H}+(\varepsilon(u),\varepsilon(v))_{\mathcal{H}},\hspace{0.5cm} (\sigma,\; \tau)_{\mathcal{H}_{1}}=(\sigma,\; \tau)_{\mathcal{H}}+(Div\; \sigma,\; Div\; \tau)_{H} .\end{eqnarray*}

Problem 1. Find the displacement field \(u:\Omega\longrightarrow \mathbb{R}^{d}\) and the electric potential field \(\varphi:\Omega\longrightarrow \mathbb{R}\) such that (2)-(12) hold.

Let us introduce the following Hilbert spaces: $$V=\left\lbrace v\in [H^{1}(\Omega)]^{d}/v=0\text{ on }\Gamma_{1}\right\rbrace, $$ $$W=\left\lbrace \varphi\in H^{1}(\Omega)/\varphi =0\text{ on }\Gamma_{a}\right\rbrace, $$ $$K=\left\lbrace v\in [H^{\frac{1}{2}}(\Gamma_{3})]^{d}/v_{n}\leq g\text{ on }\Gamma_{3}\right\rbrace.$$

If \(u\) and \(\varphi\) are regular functions which satisfy (2)-(10), then we find: $$\int_{\Omega}\mathcal{A}\varepsilon(u)\varepsilon(v)dx+\int_{\Omega}\mathcal{B}^*\nabla \varphi\varepsilon(v) dx=\int_{\Omega}f_{0}vdx+\int_{\Gamma_{2}}f_{2}vd\Gamma+\int_{\Gamma_{3}}(\sigma n).vd\Gamma$$

It is straightforward that \(M_{T,N}\) are two closed convex sets of \(H_\Gamma^{*}\) and \(0_{H_\Gamma^{*}}\in M_{N,T}\). We introduce two dual Lagrange multipliers \(\lambda_N\) and \(\lambda_T \in M\) as follows: $$\left\langle \lambda_N, v\right\rangle_{\Gamma_{3}}:=-\int_{\Gamma_{3}}\sigma_{n}v_{n}d\Gamma ,\hspace{0.5cm}v\in V, \; \text{and}\;\left\langle \lambda_T, v\right\rangle_{\Gamma_{3}}:=-\int_{\Gamma_{3}}\sigma_\tau v_\tau d\Gamma , \hspace{0.5cm}v\in V.$$

We define two bi-linear and continuous forms \(b_{1}\) and \(b_{2}\) for all \(v \in V,\; \delta_{1}, \delta_{2} \in H_\Gamma^{*}\) as follows: $$b_{1}: \tilde{V} \times H_\Gamma^{*} \longrightarrow \mathbb{R},\hspace{1cm}b_{1}(\tilde{v},\delta_{1}):=\left\langle \delta_{1} ,v \right\rangle_{\Gamma_{3}},$$ $$b_{2}:\tilde{V} \times H_\Gamma^{*} \longrightarrow \mathbb{R},\hspace{1cm}b_{2}(\tilde{v},\delta_{2}):=\left\langle \delta_{2} ,v \right \rangle_{\Gamma_{3}}.$$

Problem 2.(Weak formulation of Problem Problem 1) Find \(\tilde{u}\in\tilde{V}\) and \(\lambda=(\lambda_N,\lambda_T)\in M_{N}\times M_{T}(\mathcal{F}\lambda_N)\) such that:

2.2. Main results

Theorem 1. Assume (13)-(17), then the Problem 2 has unique solution \((\tilde{u},\lambda)\in\tilde{V}\times M\). Moreover if \((\tilde{u}_{1},\lambda)\) and \((\tilde{u}_{2},\beta)\) are two solutions of Problem 2 for given data \(\tilde{f}_{1}\) and \(\tilde{f}_{2}\) respectively, then $$||\tilde{u}_{1}-\tilde{u}_{2}||_{\tilde{V}}+||\lambda -\beta||_{H_\Gamma^{*} \times H_\Gamma^{*}}\leq C(||\tilde{f}_{1}-\tilde{f}_{2}||).$$

Proposition 1. Let \((u^{\ell},\lambda^\ell)\) be the solution generated by the Algorithm 1, then

3. Proof of the main result

Problem 3. For given \(f\in X\), find \(u\in X\) and \(\lambda=(\lambda_N,\lambda_T)\in M\) such that:

Theorem 2. Let \(f \in X\) and assume that (30)-(34) hold. Then, there exists a unique solution \((u,\lambda)\) of \textbf{Problem \(AWF\)} Moreover, if \((u_{1},\lambda)\) and \((u_{2},\gamma)\) are two solutions of the \textbf{Problem \(AWF\)} for given data functions \(f_{1}\in X\) and \(f_{2}\in X\) respectively, then, there exists \(SC>0\) such that:

Proof. We consider the symmetric \(a_{0}(.,.)\) and anti-symmetric \(c(.,.)\) part of \(a(.,.)\) respectively, defined by $$a_{0}:X\times X\longrightarrow \mathbb{R} ,\;\;a_{0}(u,v):=(a(u,v)+a(v,u))/2,\hspace{1cm}u, \forall\;\; v\in X,$$ $$c:X\times X\longrightarrow \mathbb{R} , \;\;c(u,v):=(a(u,v)-a(v,u))/2,\hspace{1cm}u,\forall\;\; v\in X.$$ For given \(0

Remark 1. In practice, the Algorithm 2 is solved using Tresca friction with slip bound \(S^\ell\) and an fixed point iteration is used to compute the problem with Coulomb friction, i.e. \(S^{\ell+1}=-\mathcal{F}\lambda_N^{\ell}\).

4.2. Numerical example

5. Conclusion

Author Contributions

Competing Interests