Original articles
A fictitious domain decomposition method for a nonlinear bonded structure

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Abstract

We study a fictitious domain decomposition method for a nonlinearly bonded structure. Starting with a strongly convex unconstrained minimization problem, we introduce an interface unknown such that the displacement problems on each subdomain become uncoupled in the saddle-point equations. The interface unknown is eliminated and a Uzawa conjugate gradient domain decomposition method is derived from the saddle-point equations of the stabilized Lagrangian functional. To avoid interface fitted meshes we use a fictitious domain approach, inspired by XFEM, which consists in cutting the finite element basis functions around the interface. Some numerical experiments are proposed to illustrate the efficiency of the proposed method.

Introduction

A bonded structure consists of two elastic adherents bonded by a thin adhesive layer. In real world applications, bonded structures can be encountered in aircraft industry (composite structure, e.g., metal/polymer); civil engineering (concrete block/mortar); geology (granite/clay or quartzite/magma). To avoid numerical difficulties due the geometrical and mechanical differences between the adhesive and the adherents, simplified models obtained by asymptotic analysis are used. In such models, the adhesive layer disappears, replaced by a transmission condition. We refer to [6], [19] and references therein for asymptotic analysis of bonded structures. With simplified models, the domain decomposition approach applies in a natural way since the subdomains and the interface are already defined [9], [11].

In this paper we study a bonded structure composed of linear elastic adherents and a nonlinear adhesive layer. Our model generalizes the linear model studied in [9], [20] and the nonlinear model studied in [6], [7]. Starting with a strongly convex unconstrained minimization problem, we introduce an interface unknown such that the displacement problems on each subdomain become uncoupled if the multiplier is known. Then we eliminate the auxiliary unknown to obtain a reduced Lagrangian functional. We derive a Uzawa conjugate gradient algorithm based on the saddle-point equations of the stabilized Lagrangian functional. For our domain decomposition method, we use a fictitious domain approach to avoid the generation of compatible meshes. The fictitious domain method we use was initiated by [14] for the Poisson equation (see also [1], [8], [18]) and allows the use of different degrees of freedom for the interface multiplier and the subdomains. The approach is inspired by XFEM [16] since it consists partially in cutting the finite element basis function around the interface. But unlike XFEM, the finite element function spaces are not enriched with singular functions, even though [2] shows an equivalence between both methods.

The paper is organized as follows. In Section 2, we described the model problem in a continuous level. In Section 3 we introduce a standard finite element discretization and the Uzawa conjugate gradient algorithm. In Section 4 we present the stabilized fictitious domain discretization and the corresponding Uzawa conjugate gradient algorithm. Finally in Section 5, we carry out some numerical experiments to illustrate the behavior of the proposed method.

Section snippets

Model problem

We consider, in R2, a system of two isotropic (linear) elastic bodies Ω1 and Ω2, bonded along their common boundary Γ by a thin adhesive layer, see Fig. 1. The common boundary Γ is assumed to be nonempty surface of positive measure. Let ui be the displacement field of the body Ωi. We set u=(u1,u2) the displacement field of the bonded structure and [u]=(u1−u2)|Γ the displacement jump across Γ. Under small deformations assumption, constitutive equations are, for each body Ωi, σi(ui)=2μiεi(ui)+λi(

Constrained optimization formulation

To transform the unconstrained minimization problem (2.8) into a constrained minimization problem we introduce the (interface) auxiliary unknown ϕ=K[u]. We then obtain

Find (u,ϕ)∈V×Lp(Γ) such that J(u)+JΓ(ϕ)≤J(v)+JΓ(φ),∀(v,φ)∈V×Lp(Γ)K[u]−ϕ=0, where the interface functional is now JΓ(ϕ)=1p∫Γ|ϕ|pdΓ.

With (3.1)–(3.2), we associate the Lagrangian functional L(u,ϕ;λ)=J(v)+JΓ(ϕ)+(K[u]−ϕ,λ)Γ.such that the solution to (3.1)–(3.2) reduces to the following saddle-point problem

Find (u,ϕ;λ)∈V×Lp(Γ)×Lq(Γ)

Fictitious domain approach

To avoid the generation of meshes which coincide on the interface, we chose the fictitious domain approach in which the choice of the degrees of freedom for the multiplier on Γ is made independently of the mesh. The fictitious domain used here has been first introduced by Haslinger and Renard [14] for the Poisson problem.

Numerical experiments

We now study the numerical behavior of Algorithm 2. We have implemented Algorithm 2 in MATLAB, using vectorized assembling functions and the mesh generator provided in [12], [13], on a computer running Linux (Ubuntu 16.04) with 3.00 GHz clock frequency and 32GB RAM. The test problem used is designed in order to illustrate the numerical behavior of the algorithms more than to model actual bonded structures.

The test problem used is illustrated in Fig. 2, derived from [11]. The adherents and the

Conclusion

We have studied a fictitious domain decomposition method for a nonlinearly bonded structure. Numerical experiments show that the proposed method is numerically scalable, with respect to the mesh size for a fixed value of p. For p=2 or for large values of p, the number of iterations required for convergence is relatively small. For other values of p, the number of iterations required for convergence can become large, since the convergence of the conjugate gradient method in a finite number of

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This work was supported by the Conseil Régional d’Auvergne-Rhone-Alpes and by the European Regional Development Fund MMaSyF allocation doctorale CPER 2015-15CEB069.

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