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CivilComp Proceedings
ISSN 17593433 CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 106
A Mixed Finite Element Model based on LeastSquares Formulation for the Static Analysis of Laminated Composite Plates F. Moleiro^{1}, C.M. Mota Soares^{1}, C.A. Mota Soares^{1} and J.N. Reddy^{2}
^{1}IDMEC/IST, Department of Mechanical Engineering, Instituto Superior Técnico, Technical University of Lisbon, Portugal
F. Moleiro, C.M. Mota Soares, C.A. Mota Soares, J.N. Reddy, "A Mixed Finite Element Model based on LeastSquares Formulation for the Static Analysis of Laminated Composite Plates", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 106, 2006. doi:10.4203/ccp.83.106
Keywords: mixed finite elements, leastsquares, laminated composite plates.
Summary
This paper presents a mixed finite element model based on leastsquares variational
principles, as an alternative approach to the mixed weak form finite element models.
The mixed leastsquares model considers the static analysis of laminated composite
plates using the firstorder shear deformation theory, with generalized displacements
and stress resultants as independent variables. In fact, the mixed model is developed
using equalorder Lagrange interpolation functions with high levels, along with
full integration.
The motivation for this finite element model comes from the previous works of Pontaza and Reddy [1], Pontaza [2] and also, Duan and Lin [3]. Their recent mixed finite element formulations based on leastsquares variational principles have shown promising theoretical and computational advantages, altogether in solid and in fluid mechanics. Actually, Pontaza and Reddy [1] developed a mixed element based on leastsquares formulation for the bending of singlelayered isotropic plates, using the classical plate theory (CLT) and firstorder shear deformation theory (FSDT). As an extension of that work to the static analysis of laminated composite plates, a mixed leastsquares FSDT finite element model is here introduced. The real benefit of leastsquares variational principles combined with mixed formulations is that it leads to a variational unconstrained minimization problem, where the finite element approximation spaces can be chosen independently. Therefore, no restrictive compatibility conditions on the discrete spaces arises, contrary to the stability requirements intrinsic to mixed weak form finite element models. In addition, the mixed leastsquaresbased discrete model, once the boundary conditions are duly imposed, yields a symmetric positivedefinite system of algebraic equations, instead of the indefinite system in mixed weak form models. In overview, this paper addresses first the governing equations consistent with the mixed formulation of the proposed leastsquares finite element model. Basically, the governing equations consist of the plate equilibrium equations along with the laminate constitutive equations, as detailed in Reddy [4]. Secondly, the subsequent leastsquares formulation is presented and the finite element model derived. In essence, the leastsquares functional is defined by measuring the residuals of the governing equations in terms of suitable norms. Then, the finite element model is developed by minimizing the leastsquares functional with respect to the chosen approximation spaces. In fact, it should be noted that the use of highorder interpolation functions and full integration are the appropriate way to truly minimize the leastsquares functional. Pontaza and Reddy [1] and later Pontaza [2] demonstrated the exponential fast decay of the leastsquares functional with increasing order of the element. Lastly, numerical examples to assess the predictive capabilities of the proposed mixed leastsquares model are considered. Static analysis results for four laminated composite plates with different boundary conditions and a series of sidetothickness ratios, ranging from thick to thin laminates, are thoroughly examined. For comparison, analytical solutions using FSTD by the wellknown Navier and Lévy methods described in Reddy [4] are presented alongside the numerical results. Each laminate considered, is systematically modelled using either 4th, 6th or 8thorder elements in uniform meshes of , or elements. Overall, the numerical results show excellent agreement with the analytical solutions, for the entire sidetothickness ratios investigated. In fact, the proposed mixed leastsquares model with highorder interpolation functions is shown to be insensitive to shearlocking. Most especially, convergence of the computed results towards the analytical solutions is verified for both  and refinements. References
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