Original articlesA fictitious domain decomposition method for a nonlinear bonded structure
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 , a system of two isotropic (linear) elastic bodies and , bonded along their common boundary by a thin adhesive layer, see Fig. 1. The common boundary is assumed to be nonempty surface of positive measure. Let be the displacement field of the body . We set the displacement field of the bonded structure and the displacement jump across . Under small deformations assumption, constitutive equations are, for each body ,
Constrained optimization formulation
To transform the unconstrained minimization problem (2.8) into a constrained minimization problem we introduce the (interface) auxiliary unknown . We then obtain
Find such that where the interface functional is now
With (3.1)–(3.2), we associate the Lagrangian functional such that the solution to (3.1)–(3.2) reduces to the following saddle-point problem
Find
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 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 . For or for large values of , the number of iterations required for convergence is relatively small. For other values of , 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.
References (22)
- et al.
A Comment on the article A finite element method for the simulation of strong and weak discontinuities in solid mechanics by A. Hansbo and P. Hansbo [Comput. Methods Appl. Mech. Engrg 193 (2004) 3523-2540]
Comput. Methods Appl. Mech. Engrg.
(2006) - et al.
The finite element method with Lagrange multipliers on the boundary: circumventing the Babuška—Brezzi condition
Comput. Methods Appl. Mech. Engrg.
(1991) - et al.
A finite element method for the simulation of strong and weak discontinuities in solid mechanics
Comput. Methods Appl. Mech. Engrg.
(2004) - et al.
Boundary Lagrange multipliers in finite element methods: error analysis in natural norms
Numer. Math.
(1992) - et al.
Adaptive optimization of convex functionals in banach spaces
SIAM J. Numer. Anal.
(2005) A direct asymptotic analysis on a nonlinear model with thin layers
Ann. Univ. Ferrara - Sez. VII - Sec. Mat.
(2003)- et al.
An optimization-based domain decomposition method for nonlinear wall laws in coupled systems
Math. Models Methods Appl. Sci.
(2004) - et al.
Fictitious domain finite element methods using cut elements: I a stabilized Lagrange multiplier method
Comput. Methods Appl. Mech. Engrg.
(2010) - et al.
A domain decomposition method for bonded structures
Math. Models Methods Appl. Sci.
(1998) An optimization based domain decomposition method for a bonded structure
Math. Models Methods Appl. Sci.
(2002)
Convergence analysis of optimization-based domain decomposition methods for a bonded structure
Appl. Numer. Math.
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