Keywords: mortar method, composite plates, optimization.

A mortar method is introduced for the analysis of L-shaped generally-orthotropic thin composite plates. The underlying differential operator is a fourth-order one that calls for interface conditions on the normal displacement as well as on its normal derivative. We therefore move from the few existing approaches for the bi-Laplacian operator and extend it to handle the fourth-order laminated plate operator. However, we do not resort to Lagrange polynomials and relevant quadrature points as frequently done in the field of mortar methods but make use of standard polynomials with exact (symbolic) integration. A few numerical examples concerning the analysis and optimization of laminated composite plates are presented to validate the proposed approach.

The design of laminated composite plates and shells is an extremely challenging task mainly because of the large number of design variables, i.e. laminae thicknesses and ply angles. Furthermore, one may show that the design process is non-convex so that peculiar numerical schemes are to be developed that are able to escape local minima of the objective function in the design variable space. In fact, the huge variety of design alternatives often advice to cast the design problem as an optimal design one. Thousands of structural and sensitivity analysis are then to be performed for the optimal design to be achieved. The finite-element method, though far more general than spectral approaches as to the geometry of the structure, may however lead to very cumbersome if not prohibitive design sessions. Therefore, spectral methods may represent a sound alternative, at least in the case of rather regular design domains. In fact, standard spectral methods are basically limited to convex quadrilateral domains whereas domain decomposition approaches, [1], and mortar methods in paricular, [2], represent a recent powerful tool that, among others, present the following beneficial features:

• the possibility of adopting non-conforming schemes whereby one may approximate unknown fields in adjacent sub-domains with polynomials of different order or even couple spectral and finite element approaches;
• the possibility of using non-matching grids on adjacent sub-domains;
• the inherent parallel nature of the approach according to which each sub-domain may be analyzed with its own processor and some a-posteriori condition used to correctly glue all the computed solutions.

Coming to laminated composite plates, several theories are at disposal that range from the classical lamination theory to sophisticated higher-order models for which reference is made to [3]. In this respect, we shall be using the classical lamination theory since the focus of the paper is on the discretization scheme rather than on the mechanical model. However, the method is in principle applicable to higher-order models as well as will be elucidated in some forthcoming contribution. The differential operator that governs the laminated plate under investigation is therefore a "full" fourth-order differential operator that encompasses as particular cases the bi-Laplacian (isotropic plates) and the anisotropic bi-Laplacian (orthotropic plates). Unlike what is frequently done, we do not use Legendre polynomials in conjunction with Gauss-Lobatto or generalized Gauss schemes, [5], but symbolic polynomials with exact closed-form integration. From a procedural standpoint, our method finds its root in the works [4,5] that, though limited to the bi-Laplacian case, provide general convergence theorems with a-priori error estimates that put the approach on sound mathematical bases. Within the rather limited existing literature dealing with fourth-order problems, the recent contribution [6] is worth mentioning that uses mortars in a finite-element framework.

1

P. Le Tallec, "Domain decomposition methods in computational mechanics", Comput. Mech. Adv., vol. 1, pp. 121-220, 1994.

2

C. Bernardi, Y. Maday and A.T. Patera, "A new nonconforming approach to domain decomposition: the mortar element method", in Nonlinear Partial Differential Equations and Theit Applications, Collège de France Seminar XI, H. Brézis and J.-L. Lions, eds., Pitman, Boston, MA, 1992.

3

J.N. Reddy, "Mechanics of laminated composite plates", CRC Press, Boca Raton, 1997.

4

Z. Belhachmi, "Nonconforming mortar element methods for the spectral discretization of two-dimensional fourth-order problems", SIAM J. Numer. Anal., vol. 34, no. 4, pp. 1545-1573, 1997. doi:10.1137/S0036142995284363

5

Z. Belhachmi, "Methodes d'elements spectraux avec joints por la resolution de problemes d'ordre quatre", These de Doctorat de l'Universite Pierre et Marie Curie, Paros, 1994.

6

L. Marcinkowski, "Domain decomposition methods for mortar finite element discretization of plate problems", SIAM J. Numer. Anal., vol. 39, no. 4, pp. 1097-1114, 2001. doi:10.1137/S0036142900371192

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