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PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Shape Optimization of Thin-walled Structures Based on a New Shell Element and Uniform Strain Energy Density Criterion
P. Khosravi, R. Sedaghati and R. Ganesan
Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec, Canada
P. Khosravi, R. Sedaghati, R. Ganesan, "Shape Optimization of Thin-walled Structures Based on a New Shell Element and Uniform Strain Energy Density Criterion", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 216, 2006. doi:10.4203/ccp.83.216
Keywords: shape optimization, geometric nonlinearity, thin-walled structures, shell element, strain energy density.
This paper presents a methodology for design optimization of thin-walled structures undergoing large deflections subject to displacement and system stability constraints. Thin-walled structures used in aerospace industry and transportation systems consist of components in the form of plates or shells with small thickness compared to other dimensions. These structures are the most efficient among all of the structural systems, but they display nonlinear behavior under applied loads. This kind of nonlinearity which is due to the large structural deformations is called "geometric nonlinearity".
In practical applications, structures have to be optimized to have minimum mass or maximum load capacity. Developing design optimization methodologies that combine efficiently the iterative optimization process and iterative nonlinear analysis is a challenging and complex task which has not received appropriate attention. The major problem in the design optimization of nonlinear structures is that the combination of iterative optimization process and iterative nonlinear analysis makes the procedure computationally very expensive and extremely difficult. Some designers use the linear analysis instead of iterative nonlinear analysis in order to increase the efficiency of the optimization process, however it has been shown that it may lead to infeasible design which may cause structural failure . Considering this, it is required to develop efficient, accurate and robust methodologies for design optimization of nonlinear thin- walled structures. From a literature review, it has been noted that the optimization algorithms for nonlinear structures were mainly based on the optimality criterion technique because of its computational efficiency. For example the optimality criterion method has been employed to minimize the weight of the truss structures under the stability constraint .
In this study an optimality criterion based on Karush-Kuhn-Tucker conditions  is developed for mass minimization problems. This criterion, called in this study "uniform strain energy density criterion", states that at the optimal structural shape, the strain energy density in all elements affected by the shape design variables are equal. Shape design variables are the variables which control the shape of the structure. The proposed optimality criterion is then used to optimize different thin-walled structures through a recurrence relation.
In the finite element analysis of the structure the geometric nonlinearity is considered, and the co-rotational method is used for this purpose . The membrane element used in the shell formulation is a recently developed optimal membrane element  which has the least error in strain energy computation. The proposed optimization algorithm for the minimization of the total mass is combined with the nonlinear co-rotational analysis to design optimized plate and shell structures with geometric nonlinearity. The efficiency of the developed optimality criterions is also compared with that of sequential quadratic programming (SQP) method. This study concludes that the complex task of shape optimization in nonlinear thin-walled structures can be performed by the proposed method in a more efficient way.
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