Numerical analysis of a frictionless viscoelastic piezoelectric contact problem

DOI:10.1051/m2an:2008022 www.esaim-m2an.org

NUMERICAL ANALYSIS OF A FRICTIONLESS VISCOELASTIC PIEZOELECTRIC CONTACT PROBLEM^∗

Mikael Barboteu

, Jose Ramon Fern´ andez

and Youssef Ouafik

Abstract. In this work, we consider the quasistatic frictionless contact problem between a viscoelastic piezoelectric body and a deformable obstacle. The linear electro-viscoelastic constitutive law is em- ployed to model the piezoelectric material and the normal compliance condition is used to model the contact. The variational formulation is derived in a form of a coupled system for the displacement and electric potential fields. An existence and uniqueness result is recalled. Then, a fully discrete scheme is introduced based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximative solutions and, as a consequence, the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, some two-dimensional examples are presented to demonstrate the performance of the algorithm.

Mathematics Subject Classification. 65N15, 65N30, 74D10, 74M15, 74S05, 74S20.

Received July 23, 2007. Revised January 18, 2008.

Published online June 5, 2008.

1. Introduction

In this work, we study, from the numerical point of view, a frictionless contact problem between a viscoelastic piezoelectric body and a deformable obstacle.

Piezoelectricity is the ability of certain crystals, like the quartz (also ceramics (BaTiO₃, KNbO₃, LiNbO₃, etc.) and even the human mandible or the human bone), to produce a voltage when they are subjected to mecha- nical stress. On a nanoscopic scale, the piezoelectric phenomenon arises from a nonuniform charge distribution within a crystal unit cells. When such a crystal is mechanically deformed, the positive and negative charge centers displace by differing amounts. Thus, while the overall crystal remains electrically neutral, the difference in charge center displacement results in an electric polarization within the crystal. Electric polarization due to mechanical input is perceived as piezoelectricity.

The piezoelectric effect is characterized by the coupling between the mechanical and the electrical properties of the material: it was observed that the appearance of electric charges on some crystals was due to the action

Keywords and phrases. Piezoelectricity, viscoelasticity, normal compliance, error estimates, numerical simulations.

∗The work of the second author was partially supported by the Ministerio de Educaci´on y Ciencia (Project MTM2006-13981).

1Laboratoire de Math´ematiques et Physique pour les Syst`emes (MEPS), Bˆatiment B3, case courrier 12, 52 Avenue Paul Alduy, 66860 Perpignan, France. [email protected]; [email protected]

2Departamento de Matem´atica Aplicada, Facultade de Matem´aticas, Campus Sur s/n, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain. [email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2008

f

Foundation Γ

Γ Γ

Γ

Γ Ω

A B

f

C D

F

q

ν g u = 0

F

Figure 1. A viscoelastic piezoelectric body in contact with a foundation.

of body forces and surface fractions and, conversely, the action of the electric field generated strain or stress in the body. This kind of materials appears usually in the industry as switches in radiotronics, electroacoustics or measuring equipments.

Different models have been developed early to describe the interaction between the electric and mechanical fields (see, e.g., [2,4,13,17–20,25–28] and the references therein). Recently, contact problems involving elastic- piezoelectric materials [3,5,12,15,21,23,24] or viscoelastic piezoelectric materials [22] have been studied.

In this paper, we consider a viscoelastic piezoelectric body which may become in contact with a deformable obstacle, the so-called foundation. The contact is assumed frictionless and a normal compliance condition is employed to model it (see [14,16]). This paper continues [22], providing the numerical analysis of the variational problem and some numerical results which exhibit its behaviour, and extends the results of [3,12] to the case of viscoelastic materials.

The rest of the paper is structured as follows. In Section 2 we present the mechanical model, provide its variational formulation and state an existence and uniqueness result, Theorem2.1. Then, a fully discrete scheme is introduced in Section3based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. A main error estimates result is proved, Theorem3.3, from which the linear convergence of the algorithm is deduced under suitable regularity conditions (see Cor.3.4). Finally, some numerical examples are presented in Section4in order to show the performance of the method.

2. Mechanical problem and its variational formulation

Denote byS^d the space of second order symmetric tensors onR^d and by “·” and · the inner product and the Euclidean norms onR^d andS^d.

Let Ω ⊂ R^d, d = 1,2,3, denote a domain occupied by a viscoelastic piezoelectric body with a smooth boundary Γ =∂Ω. We denote byν the unit outer normal vector to Γ and we assume that this boundary is decomposed into three measurable parts Γ_D, Γ_F, Γ_C, on one hand, and on two measurable parts Γ_A and Γ_B, on the other hand, such that meas (Γ_D)>0, meas (Γ_A)>0, and Γ_C ⊆Γ_B. Finally, let [0, T], T >0, be the time interval of interest (see Fig.1).

Letx∈Ω andt∈[0, T] be the spatial and time variables, respectively, and, in order to simplify the writing, we do not indicate the dependence of the functions onxandt. Moreover, a dot above a variable represents the derivative with respect to the time variable.

Let us denote by u the displacement field, σ the stress tensor, ε(u) = (εij(u))^d_i,j=1 the linearized strain tensor given by

ε_ij(u) = 1 2

∂u_i

∂x_j +∂u_j

∂x_i

, andϕthe electric potential.

The body is assumed viscoelastic piezoelectric and satisfying the following constitutive law (see [7,22]), σ=Aε( ˙u) +Bε(u)− E^∗E(ϕ),

whereAandBare the fourth-order viscosity and elastic tensors, respectively,E(ϕ) = (E_i(ϕ))^d_i=1 represents the electric field defined by

E_i(ϕ) =−∂ϕ

∂x_i, i= 1, . . . , d,

and E^∗= (e^∗_ijk)^d_i,j,k=1 denotes the transpose of the third-order piezoelectric tensorE = (e_ijk)^d_i,j,k=1. We recall that

e^∗_ijk =e_kij, for all i, j, k= 1, . . . , d.

Following [4] the following constitutive law is satisfied for the electric potential, D=Eε(u) +βE(ϕ),

whereD is the electric displacement field andβ is the electric permittivity tensor.

Since the process is assumed quasistatic, the inertia effects are negligible and therefore, Divσ+f₀=0 in Ω×(0, T),

divD=q₀ in Ω×(0, T),

where f₀ is the density of the body forces acting in Ω andq₀ is the volume density of free electric charges.

Moreover, Div and div represent the divergence operators for tensor and vector functions, respectively.

We turn now to describe the boundary conditions.

On the boundary part Γ_D we assume that the body is clamped and thus the displacement field neglects there, that isu=0on Γ_D×(0, T). Moreover, we assume that a density of traction forces, denoted byf_F, acts on the boundary part Γ_F,i.e.,

σν =f_F on Γ_F×(0, T).

On the part Γ_C the body can become in contact with a deformable insulator obstacle, the so-called foundation.

According to [14] the following normal compliance contact condition is employed,

−σν=p(u_ν−g) on Γ_C×(0, T),

whereσ_ν=σν·νis the normal stress,u_ν=u·νdenotes the normal displacement,grepresents the gap between the body and the obstacle measured along the normal directionν andpis a given function whose properties will be described below. Finally, we assume that the contact is frictionless and therefore,στ=σν−σ_νν=0.

Let Ω be subject to a prescribed electric potentialϕ_A on Γ_A and to a density of surface electric chargesq_F on Γ_B, that is,

ϕ=ϕ_A on Γ_A×(0, T), D·ν=q_F on Γ_B×(0, T).

We assume that q_F = 0 on Γ_C, that is, the foundation is supposed to be insulator. We note that it is straightforward to extend the results presented below to more general situations by decomposing Γ in a different way.

The mechanical problem of the quasistatic contact of a viscoelastic piezoelectric body with a deformable obstacle is then written as follows.

Problem P. Find a displacement field u: Ω×(0, T)→R^d, a stress field σ : Ω×(0, T)→S^d, an electric potential fieldϕ: Ω×(0, T)→Rand an electric displacement field D: Ω×(0, T)→R^d such that,

σ=Aε( ˙u) +Bε(u)− E^∗E(ϕ) in Ω×(0, T), (2.1)

D=Eε(u) +βE(ϕ) in Ω×(0, T), (2.2)

Divσ+f₀=0 in Ω×(0, T), (2.3)

divD=q₀ in Ω×(0, T), (2.4)

u=0 on Γ_D×(0, T), (2.5)

σν=f_F on Γ_F×(0, T), (2.6)

στ=0, −σ_ν =p(u_ν−g) on Γ_C×(0, T), (2.7)

ϕ=ϕ_A on Γ_A×(0, T), (2.8)

D·ν=q_F on Γ_B×(0, T), (2.9)

u(0) =u₀ in Ω. (2.10)

Here,u0represents an initial condition for the displacement field.

In order to obtain the variational formulation of Problem P, let us introduce the variational spaces V, Q andW, and the convex set W_Aas follows,

V ={v∈[H¹(Ω)]^d;v =0 on Γ_D},

Q={τ = (τ_ij)^d_i,j=1∈[L²(Ω)]^d×d ; τ_ij =τ_ji, i, j= 1, . . . , d}, W ={ψ∈H¹(Ω) ;ψ= 0 on Γ_A},

W_A={ψ∈H¹(Ω) ;ψ=ϕ_A on Γ_A},

and denote byH = [L²(Ω)]^d.

The viscosity tensorA(x) = (aijkl(x))^d_i,j,k,l=1:τ ∈S^d→ A(x)(τ)∈S^d satisfies:

(a)a_ijkl=a_klij =a_jikl for i, j, k, l= 1, . . . , d.

(b)a_ijkl∈L^∞(Ω) for i, j, k, l= 1, . . . , d.

(c) There existsm_A>0 such thatA(x)τ·τ ≥m_Aτ²

∀τ ∈S^d, a.e.x∈Ω.

(2.11)

The elastic tensor B(x) = (bijkl(x))^d_i,j,k,l=1:τ ∈S^d→ B(x)(τ)∈S^d verifies:

(a)b_ijkl=b_klij =b_jikl for i, j, k, l= 1, . . . , d.

(b)b_ijkl∈L^∞(Ω) for i, j, k, l= 1, . . . , d. (2.12) The piezoelectric tensorE(x) = (eijk(x))^d_i,j,k=1 :τ ∈S^d→ E(x)(τ)∈R^d satisfies:

(a)e_ijk=e_ikj for i, j, k= 1, . . . , d.

(b)e_ijk∈L^∞(Ω) for i, j, k= 1, . . . , d. (2.13) The permittivity tensorβ(x) = (β_ij(x))^d_i,j,k,l=1 :w∈R^d→β(x)(w)∈R^d verifies:

(a)β_ij=β_ji for i, j= 1, . . . , d.

(b)β_ij∈L^∞(Ω) for i, j= 1, . . . , d.

(c) There existsm_β>0 such thatβ(x)w·w≥m_βw²

∀w∈R^d, a.e.x∈Ω.

(2.14)

The normal compliance functionp(x) :r∈R→p(x, r)∈[0,∞) satisfies:

(a) There existsm_p>0 such that

|p(x, r₁)−p(x, r₂)| ≤m_p|r₁−r₂|

∀r₁, r₂∈R, a.e.x∈Γ_C.

(b) (p(x, r₁)−p(x, r₂))(r₁−r₂)≥0 ∀r₁, r₂∈R, a.e.x∈Γ_C. (c) The mappingx∈Γ_C→p(x, r) is measurable on Γ_C,

for allr∈R.

(d)p(x, r) = 0 for allr≤0.

(2.15)

The following regularity is assumed on the density of volume forces, tractions, volume electric charges and surface electric charges:

f₀∈C([0, T];H), f_F ∈C([0, T]; [L²(Γ_F)]^d),

q₀∈C([0, T];L²(Ω)), q_F ∈C([0, T];L²(Γ_B)). (2.16) Finally, we assume that the gap function, the initial displacements and the boundary conditionϕ_A satisfy

g∈L²(Γ_C), g(x)≥0 a.e. x∈Γ_C, u0∈V, ϕ_A∈C([0, T], C(Γ_A)). (2.17) Using the Riesz’ Theorem, we define the linear mappings f : [0, T]→V and q: [0, T]→W as follows,

(f(t),w)V =

Ωf₀(t)·wdx+

ΓF

f_F(t)·wdΓ, ∀w∈V, (q(t), ψ)_W =

Ωq₀(t)ψdx+

ΓB

q_F(t)ψdΓ, ∀ψ∈W.

We notice that regularity assumptions (2.16) imply thatf ∈C([0, T];V) andq∈C([0, T];W).

Let us denote byj:V ×V →Rthe normal compliance functional given by j(u,v) =

ΓC

p(u_ν−g)v_νdΓ, ∀u,v ∈V, where, for allv∈V, we letv_ν =v·ν.

Plugging (2.1) into (2.3) and (2.2) into (2.4), keeping in mind that E(ϕ) = −∇ϕand using the boundary conditions (2.5)–(2.9), applying a Green’s formula we derive the following variational formulation of Problem P.

Problem VP. Find a displacement fieldu: [0, T]→V and an electric potential fieldϕ: [0, T]→W_A such that u(0) =u0 and for allt∈[0, T],

(Aε( ˙u(t)),ε(w))Q+ (Bε(u(t)),ε(w))Q+ (E^∗∇ϕ(t),ε(w))Q+j(u(t),w)

= (f(t),w)_V, ∀w∈V, (2.18)

(β∇ϕ(t),∇ψ)H−(Eε(u(t)),∇ψ)H = (q(t), ψ)_W, ∀ψ∈W. (2.19) Using analogous ideas to those employed in [21] for a normal compliance elastic problem or in [22] for the case of viscoelastic materials, we obtain the following theorem which states the existence of a unique weak solution to Problem VP.

Theorem 2.1. Assume that (2.11)–(2.17)hold. Then there exists a unique solution to Problem VP with the following regularity

u∈C¹([0, T];V), ϕ∈C([0, T];W_A).

The proof of Theorem2.1is based on classical results of parabolic nonlinear variational equations and partial differential equations (see [22]).

3. Fully discrete approximations: error estimates

We now introduce a finite element algorithm to approximate solutions to Problem VP and we derive an error estimate on them. Moreover, in order to simplify the writing we assume, in this section, thatϕ_A= 0 (and then W_A=W). It is straightforward to extend the results presented below to a more general case.

The discretization of (2.18)–(2.19) is done as follows. First, we consider two finite dimensional spacesV^h⊂V and W^h ⊂ W approximating the spaces V and W, respectively. h > 0 denotes the spatial discretization parameter.

Remark 3.1. In the numerical simulations presented in the next section, V^h and W^h consist of continuous and piecewise affine functions, that is,

V^h={w^h∈[C(Ω)]^d; w^h_{|T r} ∈[P₁(T r)]^d T r∈ T^h, w^h=0 on Γ_D}, (3.1) W^h={ψ^h∈C(Ω) ; ψ_|^h

T r ∈P₁(T r) T r∈ T^h, ψ^h= 0 on Γ_A}, (3.2) where Ω is assumed to be a polygonal domain, T^h denotes a finite element triangulation of Ω, and P₁(T r) represents the space of polynomials of global degree less or equal to one inT r.

To discretize the time derivatives, we use a uniform partition of [0, T], denoted by 0 =t₀< t₁< . . . < t_N =T, and letk be the time step size, k =T /N. For a continuous function f(t) letf_n =f(t_n), and for a sequence {wn}^N_n=0 we letδw_n = (w_n−w_n−1)/k denote the divided differences.

In this section, no summation is assumed over a repeated index, and c denotes a positive constant which depends on the problem data and the continuous solution, but it is independent of the discretization parametersh andk.

Thus, using the backward Euler scheme, the fully discrete approximation of Problem VP is the following.

Problem VP^hk. Find a discrete displacement field u^hk ={u^hk_n }^N_n=0⊂V^h and a discrete electric potential fieldϕ^hk={ϕ^hk_n }^N_n=0⊂W^h such thatu^hk₀ =u^h₀ and for alln= 1, . . . , N,

(Aε(δu^hk_n ),ε(w^h))_Q+ (Bε(u^hk_n ),ε(w^h))_Q+ (E^∗∇ϕ^hk_n ,ε(w^h))_Q

+j(u^hk_n ,w^h) = (f_n,w^h)_V, ∀w^h∈V^h, (3.3) (β∇ϕ^hk_n ,∇ψ^h)_H−(Eε(u^hk_n ),∇ψ^h)_H = (q_n, ψ^h)_W, ∀ψ^h∈W^h, (3.4) whereu^h₀ is an appropriate approximation of the initial condition u0.

We notice that the fully discrete problem VP^hk can be seen as a coupled system of variational equations.

Using classical results of nonlinear variational equations (see [9]) we obtain that Problem VP^hkadmits a unique solutionu^hk⊂V^h andϕ^hk⊂W^h, which we summarize in the following.

Theorem 3.2. Assume that (2.11)–(2.17)hold. Then there exists a unique solution to ProblemVP^hk. Our interest in this section lies in estimating the numerical errorsun−u^hk_n V andϕn−ϕ^hk_n W. We have the following main error estimates result.

Theorem 3.3. Assume that(2.11)–(2.17)hold. Let(u, ϕ)and(u^hk, ϕ^hk)denote the solutions to ProblemsVP and VP^hk, respectively. Then, the following error estimates hold for all w^h = {w^h_j}^N_j=1 ⊂ V^h and ψ^h = {ψ_j^h}^N_j=1⊂ W^h,

1≤n≤Nmax {un−u^hk_n ²_V +ϕn−ϕ^hk_n ²_W} ≤c

1≤n≤Nmax ϕn−ψ^h_n²_W +

N j=1

k

u˙j−δuj²_V +uj−w^h_j²_V

+ max

1≤n≤Nun−w^h_n²_V +u₀−u^h₀²_V + 1

k

N−1

j=1

uj−w^h_j −(u_j+1−w^h_j+1)²_V .

(3.5)

Proof. First, let us obtain an error estimation on the electric potential. Then, taking (2.19) at timet=t_n for ψ=ψ^h∈W^h and subtracting it to (3.4) we obtain that

(β∇(ϕn−ϕ^hk_n ),∇ψ^h)_H−(Eε(un−u^hk_n ),∇ψ^h)_H = 0, ∀ψ^h∈W^h. Thus, we have

(β∇(ϕn−ϕ^hk_n ),∇(ϕn−ϕ^hk_n ))_H−(Eε(un−u^hk_n ),∇(ϕn−ϕ^hk_n ))_H

= (β∇(ϕn−ϕ^hk_n ),∇(ϕn−ψ^h))_H−(Eε(un−u^hk_n ),∇(ϕn−ψ^h))_H, for allψ^h∈W^h.

Applying the Cauchy’s inequality

ab≤a²+ 1

4b², a, b, ∈R, >0, (3.6)

and using properties (2.13) and (2.14), after some algebra we find that

ϕn−ϕ^hk_n ²_V ≤c(un−u^hk_n ²_V +ϕn−ψ^h2_W), ∀ψ^h∈W^h. (3.7) Secondly, we proceed now to estimate the numerical errors on the displacement field. We rewrite variational equation (2.18) at timet=t_n forw=w^h∈V^h and we subtract it to variational equation (3.3) to obtain

(Aε( ˙un−δu^hk_n ),ε(w^h))_Q+ (Bε(un−u^hk_n ) +E^∗∇(ϕn−ϕ^hk_n ),ε(w^h))_Q +j(un,w^h)−j(u^hk_n ,w^h) = 0, ∀w^h∈V^h.

Therefore,

(Aε( ˙un−δu^hk_n ),ε(un−u^hk_n ))_Q+j(un,un−u^hk_n )−j(u^hk_n ,un−u^hk_n ) + (Bε(un−u^hk_n ) +E^∗∇(ϕ_n−ϕ^hk_n ),ε(un−u^hk_n ))_Q

= (Aε( ˙un−δu^hk_n ),ε(un−w^h))_Q+j(un,un−w^h)−j(u^hk_n ,un−w^h) + (Bε(un−u^hk_n ) +E^∗∇(ϕn−ϕ^hk_n ),ε(un−w^h))_Q, ∀w^h∈V^h.

From properties (2.15), we have (see [8] for details),

j(un,un−u^hk_n )−j(u^hk_n ,un−u^hk_n )≥0,

j(un,un−w^h)−j(u^hk_n ,un−w^h)≤cun−u^hk_n Vun−w^hV.

Using repeatedly inequality (3.6) and properties (2.11), (2.12), (2.13) and (2.15), after easy calculations it follows that

(Aε(δun−δu^hk_n ),ε(un−u^hk_n ))_Q ≤c

u˙n−δun²_V +un−u^hk_n ²_V +ϕn−ϕ^hk_n ²_W +un−w^h2_V +un−u^hk_n ²_V

+ (Aε(δun−δu^hk_n ),ε(un−w^h))_Q , ∀w^h∈V^h,

where >0, here and below, is assumed small enough and we denote δun= (un−u_n−1)/k.

Keeping in mind that

(Aε(δun−δu^hk_n ),ε(un−u^hk_n ))_Q≥ c 2k

un−u^hk_n ²_V − un−1−u^hk_n−1²_V ,

proceeding by induction it leads to the following inequality, un−u^hk_n ²_V ≤c

n j=1

k

u˙j−δuj²_V +ϕj−ϕ^hk_j ²_W +uj−w^h_j²_V

+uj−u^hk_j ²_V + (Aε(δuj−δu^hk_j ),ε(uj−w^h_j))_Q +u0−u^h₀²_V,

(3.8)

for allw^h={w^h_j}ⁿ_j=0⊂V^h. Since (see [11])

n j=1

k(Aε(δuj−δu^hk_j ),ε(uj−w^h_j))_Q

= n j=1

(Aε(uj−u^hk_j −(uj−1−u^hk_j−1)),ε(uj−w^h_j))_Q

≤un−u^hk_n ²_V +cun−w^h_n²_V +cu₀−u^h₀²_V +cu₁−w^h₁²_V +c

n−1

j=1

uj−u^hk_j Vuj−w^h_j −(uj+1−w^h_j+1)V,

combining (3.7) and (3.8) and applying a discrete version of Gronwall’s inequality (see [10] for details), we

obtain (3.5).

We notice that the above error estimates are the basis for the analysis of the convergence rate of the algorithm.

Thus, let Ω be a polyhedral domain and denote by T^h a triangulation of Ω compatible with the partition of the boundary Γ =∂Ω into Γ_D, Γ_F, Γ_C on one hand, and on Γ_A and Γ_B, on the other hand. Let V^h andW^h be defined by (3.1) and (3.2), respectively, and assume that the discrete initial conditionu^h₀ is obtained by

u^h₀ = Π^hu0, (3.9)

where Π^h= (π^h)^d_i=1: [C(Ω)]^d→V^h, andπ^h:C(Ω)→W^his the standard finite element interpolation operator (see,e.g., [6]).

Then, we have the following corollary which states the linear convergence of the algorithm under suitable regularity conditions.

Corollary 3.4. Assume that(2.11)–(2.17)hold. Let(u, ϕ)and(u^hk, ϕ^hk)denote the solutions to ProblemsVP and VP^hk, respectively, and let the discrete initial condition be given by (3.9). Under the following regularity conditions

u∈H²(0, T;V)∩H¹([0, T]; [H²(Ω)]^d), ϕ∈C([0, T];H²(Ω)), (3.10) the linear convergence of the algorithm is achieved, that is, there exists a positive constant c >0, independent of the discretization parameters handk, such that

1≤n≤Nmax {un−u^hk_n V +ϕn−ϕ^hk_n W} ≤c(h+k). (3.11) Proof. We have the following approximation properties of the finite element spacesV^h andW^h (see [6]),

1≤n≤Nmax inf

ψ_n^h∈W^hϕn−ψ_n^hW ≤chϕ_C([0,T];H²_(Ω)),

1≤n≤Nmax inf

w_n^h∈V^hun−w^h_nV ≤chu_C([0,T];[H²_(Ω)]^d₎.

Moreover, from the definition of the finite element interpolation operator Π^h it follows that u₀−u^h₀V ≤chu_C([0,T];[H²_(Ω)]^d₎.

It is easy to check that

N j=1

ku˙j−δuj²_V ≤ck²u²_H2(0,T;V).

Finally, we find that (see [11]), 1 k

N−1 j=1

uj−w^h_j −(u_j+1−w^h_j+1)²_V ≤ch²u²_H1(0,T;[H²(Ω)]^d).

Combining the previous estimates and (3.5) it leads to (3.11).

We notice that the regularity conditions (3.10) have not been proved yet. However, since the problem is quasistatic and the contact is produced with a deformable obstacle, it seems reasonable to assume them.

Anyway, this is an open and interesting problem that we hope to address in the near future.

4. Numerical results

In order to recover the convergence results of the fully discrete method discussed in the previous section, some experiments have been done in the study of two two-dimensional test problems. First, in Section 4.1we describe the algorithm used to solve Problem VP^hk and, secondly, in Sections 4.2 and 4.3, we consider two two-dimensional test problems in order to highlight the linear convergence obtained in Corollary 3.4, but also to describe some mechanical aspects of the frictionless viscoelastic piezoelectric contact behaviour.

4.1. Numerical algorithm

The algorithm, used in solving the fully discrete frictionless contact problem VP^hk, is based on a backward Euler difference for the time derivatives and on a penalty approach (see [29] for more details) to simulate the normal compliance law. In order to give the solution algorithm, we have to introduce the expressions of

the functions w^h, u^h and δu^h(resp. ϕ^h and ψ^h) by considering theirs values at the i^th nodes of T^h and the basis functionsαⁱ (resp. γⁱ) of the spaceV^h(resp. W^h) fori= 1, . . . , N_tot(N_tot is the total number of nodes),

w^h=

Ntot

i=1

wⁱαⁱ, u^h=

Ntot

i=1

uⁱαⁱ, δu^h=

Ntot

i=1

δuⁱαⁱ, (4.1)

and ψ^h=

Ntot

i=1

ψⁱγⁱ, ϕ^h=

Ntot

i=1

ϕⁱγⁱ. (4.2)

The penalty approach shows us that Problem VP^hk can be governed by the following system of nonlinear equations

A(˜ δun) +G(un, ϕ_n) + ˜F(un) =0. (4.3) The vectorsun,δun andϕ_n represent respectively the generalized vectors defined as follows

un ={uⁱ_n}^N_i=1^tot, δun={δuⁱ_n}^N_i=1^tot and ϕ_n={ϕⁱ_n}^N_i=1^tot. (4.4) The generalized contact operator ˜F(un) ∈ R^d×Ntot×R^Ntot is defined by ˜F(un) = (F(un),0_ψ), where the penalized contact operatorF(un) =c_p dist(u_ν(t_n)−g,R⁺)ν^t∈R^d×Ntot denotes the gradient of the penalized contact functionalL^hk(un) =^c₂^p

ΓC dist²(u_ν(t_n)−g,R⁺)dΓ in the directionu,

( ˜F(un)·(w, ψ))_Rd×Ntot×R^Ntot = ((F(un),0_ψ)·(w, ψ))_Rd×Ntot×R^Ntot = (∇unL^hk(un),w)_Rd×Ntot

and 0_ψ is the zero element of R^Ntot. c_p is the surface stiffness coefficient and so 1/c_p is the deformability coefficient. In addition, the generalized viscous term ˜A(δun) ∈ R^d×Ntot×R^Ntot is defined by ˜A(δun) = (A(δun),0_ψ). Thus, the termsA(δun)∈R^d×Ntot andG(un, ϕ_n) represent respectively the viscous term and the elastic-piezoelectric term given by

(( ˜A(δun)·(w, ψ))_R^d×Ntot×R^Ntot = ((A(δun),0_ψ)·(w, ψ))_R^d×Ntot×R^Ntot (4.5)

= (A(δun)·w)_R^d×Ntot = (Aε(δun),ε(w^h))_Q ∀w^h∈V^h,

(G(un, ϕ_n)·(w, ψ))_R^d×Ntot×R^Ntot = (Bε(un),ε(w^h))_Q+ (Eε(w^h),∇ϕn)_H−(f_n,w^h)_V

−(Eε(un),∇ψ^h)_H+ (β∇ϕ_n,∇ψ^h)_H−(q_n, ψ^h)_W ∀w^h∈V^h, ψ^h∈W^h,

where w (resp. ψ) denotes the generalized vector constituted by the values wⁱ (resp. ψⁱ) for i = 1, . . . , N_tot. We remark that the volume and surface forces are contained in the termG(un, ϕ_n).

The solution algorithm consists in a combination between the finite differences (backward Euler difference) and the linear iterations method (Newton method). To solve (4.3), at each time increment the variables (un, ϕ_n) are treated simultaneously through a Newton method and therefore in what follows we use x_n to denote the pair (un, ϕ_n). The algorithm that we used in the viscoelastic piezoelectric case can be developed in three steps which are the following:

•A prediction step

This step gives the initial displacement and the velocity by the following formula ϕ⁰_n+1=ϕ_n+1, u⁰_n+1=u_n+1 and δu⁰_n+1=δun.

• A Newton linearization step

At an iterationiof the Newton method, we have

xⁱ⁺¹_n+1=xⁱ_n+1−

Qⁱ_n+1

k +Kⁱ_n+1+Tⁱ_{n+1 −1}

×

A(δu˜ ⁱ_n+1) +G(uⁱ_n+1, ϕⁱ_n+1) + ˜F(uⁱ_n+1) ,

with Kⁱ_n+1=Du,ϕG(uⁱ_n+1, ϕⁱ_n+1),Qⁱ_n+1=Du,ϕA(˜ δuⁱ_n+1),Tⁱ_n+1=Du,ϕF˜(uⁱ_n+1),and wherexⁱ⁺¹_n+1 denotes the pair (uⁱ⁺¹_n+1, ϕⁱ⁺¹_n+1); i and n represent the Newton iteration index and the time index, respectively. Here, Du,ϕG,Du,ϕA˜ andDu,ϕF˜denote the differentials of the functionsG, ˜Aand ˜F with respect to the variablesu andϕ. This leads us to solve the resulting linear system

Qⁱ_n+1

k +Kⁱ_n+1+Tⁱ_n+1

Δxⁱ

=−A(˜ δuⁱ_n+1)−G(uⁱ_n+1, ϕⁱ_n+1)−F˜(δuⁱ_n+1),

(4.6)

where Δx = (Δuⁱ,Δϕⁱ) with Δuⁱ = uⁱ⁺¹_n+1 −uⁱ_n+1 and Δϕⁱ = ϕⁱ⁺¹_n+1 −ϕⁱ_n+1. We solve the linear system of equations (4.6) by using a Conjugate Gradient Method with efficient preconditioners to overcome the poor conditioning of the matrix due to the penalized contact terms. In particular, we used specific incomplete LU factorization methods like the Element-By-Element preconditioner (see [1]). We notice that, in the case where the operators ˜A and G are linear, the matrices Qⁱ_n+1 and Kⁱ_n+1 do not change during the Newton iterations.

• A correction step

Once the system (4.6) is resolved, we updatexⁱ⁺¹_n+1 and δuⁱ⁺¹_n+1 by

xⁱ⁺¹_n+1=xⁱ_n+1+ Δxⁱ and δuⁱ⁺¹_n+1=δuⁱ_n+1+Δuⁱ k ·

For more considerations about Computational Contact Mechanics, see the recent monograph [29].

4.2. First two-dimensional example

In order to recover the theoretical numerical behaviour of the fully discrete scheme discussed in Section3, we carry out some numerical simulations based on an academic viscoelastic piezoelectric contact problem with normal compliance law. To do that, we consider a piezoelectric body extending indefinitely in the first direc- tion X₁ of a Cartesian coordinate frame (O, X₁, X₂, X₃). The material used was assumed to be an academic isotropic piezoceramic with hexagonal symmetry like zinc oxide material (class 6 mm in the international clas- sification [13]) but in which we introduce a viscous behaviour. In the crystallographic frame, theX₃-direction is a six-fold revolution symmetry axis and the (X₁OX₃) and (X₂OX₃) planes are mirrors. The electric and mechanical loads applied to the body are supposed to be constant along the X₁ direction. As a consequence, the fields E, D, ε and σ turn out to be constant along X₁. In addition, we suppose that: ε₁₁ = 0, ε₁₂ = 0, ε₁₃ = 0 and D₁ = 0; thus, we have to consider a plane problem. In the frame (O, X₂, X₃), constitutive equations (2.1) and (2.2) can be written by using a compressed matrix notation instead of the tensor notation

Table 1. Material viscoelastic constants of the considered piezoelectric body.

Elastic (GPa) Viscoelastic (GPa·s) b₂₂ b₂₃ b₃₃ b₄₄ a₂₂ a₂₃ a₃₃ a₄₄ 210 105 211 42.5 21 10.5 21.1 4.25

Table 2. Material electric constants of the considered piezoelectric body.

Piezoelectric (C·m⁻²) Permittivity (C²·N⁻¹·m⁻²) e₃₂ e₃₃ e₂₄ β₂₂/₀ β₃₃/₀

−0.61 1.14 −0.59 −8.3 −8.8 as follows,

⎡

⎢⎢

⎢⎢

⎣ σ₂₂ σ₃₃ σ₂₃ D₂ D₃

⎤

⎥⎥

⎥⎥

⎦=

⎡

⎢⎢

⎢⎢

⎣

b₂₂ b₂₃ 0 0 e₃₂ b₂₃ b₃₃ 0 0 e₃₃

0 0 b₄₄ e₂₄ 0

0 0 e₂₄ −β₂₂ 0 e₃₂ e₃₃ 0 0 −β₃₃

⎤

⎥⎥

⎥⎥

⎦

⎡

⎢⎢

⎢⎢

⎣ ε₂₂ ε₃₃ 2ε₂₃

−E₂

−E₃

⎤

⎥⎥

⎥⎥

⎦

+

⎡

⎢⎢

⎢⎢

⎣

a₂₂ a₂₃ 0 0 0 a₂₃ a₃₃ 0 0 0

0 0 a₄₄ 0 0

0 0 0 0 0

0 0 0 0 0

⎤

⎥⎥

⎥⎥

⎦

⎡

⎢⎢

⎢⎢

⎣

˙ ε₂₂

˙ ε₃₃ 2 ˙ε₂₃

−E₂

−E3

⎤

⎥⎥

⎥⎥

⎦ (4.7)

where ˙ε_ij =1 2

∂u˙_i

∂x_j +∂u˙_j

∂x_i

. In equation (4.7), taking advantage of the symmetries of the mechanical tensors, the passage of the fourth-order viscosity and elastic tensors (b_ijkl anda_ijkl) to the second-order tensors (with the matrix notation: b_pq anda_pq) is done by using the following identifications:

b_ijkl≡b_pq=

⎛

⎝ b₂₂ b₂₃ 0 b₂₃ b₃₃ 0 0 0 b₄₄

⎞

⎠ and a_ijkl≡a_pq=

⎛

⎝ a₂₂ a₂₃ 0 a₂₃ a₃₃ 0 0 0 a₄₄

⎞

⎠

with the following notation:

ij orkl= 22 −→ porq= 2, ij orkl= 33 −→ porq= 3, ij orkl= 23 or 32 −→ porq= 4.

In the same way for the third order piezoelectric tensor, we have,

e_ijk≡e_iq=

0 0 e₂₄ e₃₂ e₃₃ 0

with

jk= 22 −→ q= 2, jk= 33 −→ q= 3, jk= 23 or 32 −→ q= 4.

The coefficient values are given in Tables 1 and 2. The permittivity constant of the vacuum ₀ = 8.885× 10⁻¹²C²·N⁻¹·m⁻².

As a first two-dimensional example, we consider the problem depicted in Figure 2, where a square body is in contact with a foundation. The domain Ω = (0,1)×(0,1) is a cross-section of a three-dimensional square