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Computational Structures Technology
Edited by: B.H.V. Topping and Z. Bittnar
C. Cinquini and P. Venini
Department of Structural Mechanics, University of Pavia, Italy
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.
- 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.
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