In this paper, two linite-element-based schemes for second-order shape sensitivity analysis are presented. In the first formulation, the AV-DD method, the first-order shape sensitivity equation is derived and expressed in terms of state and adjoint variables. The resultant equation is then directly differentiated to obtain the second-order shape sensitivity equation. In the second formulation, the DD-AV method, the functional of concern is differentiated twice to yield the second-order sensitivity equation in which the second-order shape derivatives can be eliminated by introducing a proper adjoint equation. A thermal fin problem and a thermal insulation layer problem have been studied to validate the proposed schemes. It is shown that both methods yield identical results, though the DD-AV method is computationally more efficient.G. I-W. HOU AND J. SHEEN by employing either of the above-mentioned methods. Thus, various combinations of AV and DD methods can provide four different methods to derive the second-order sensitivity equations, name 1 y :1. direct differentiation method in conjunction with direct differentiation method (DD-DD), 2. direct differentiation method in conjunction with adjoint variable method (DD-AV), 3. adjoint variable method in conjunction with direct differentiation method (AV DD), and 4. adjoint variable method in conjunction with adjoint variable method (AV-AV).Only limited research has paid attention to higher-order sensitivity analysis. Haug extended the adjoint variable method (AV-AV) to find out the first-and second-order design sensitivity equations of structural systemss and dynamic response.6 Haftka introduced an approach that combines the direct differentiation with the adjoint variable method (DD-AV) to obtain a more efficient method in terms of computation time.7 Haftka and Mroz applied the DD-AV method to calculate the first-and second-order design sensitivities of linear and non-linear structures.' The cited works5 -on higher-order sensitivity analysis deal with sizing variables in fixed configurations. D e r n~,~ however, extended the analysis presented in Reference 10 to derive general equations for calculating second-order shape sensitivities of a non-linear structure. No numerical example has yet been reported in Dems' work.The goal of this paper is, thus, to develop finite element method (FEM)-based schemes to calculate the second-order shape sensitivities. For a problem with N design variables, the number of analyses required for each of the above methods for finding second-order derivatives of M constraint functions is given as follows:DD-DD method-(I + N)(2 + N ) / 2 DD-AV method-1 + N + M AV-DD method-(1 + N)(1 + M ) AV-AV method-(1 + N)(1 + M )
Fiber-reinforced composite laminates are used in many aerospace and automobile applications. The magnitudes and durations of the cure temperature and the cure pressure applied during the curing process have significant consequences for the performance of the finished product. The objective of this study is to exploit the potential of applying the optimization technique to the cure cycle design. Using the compression molding of a filled polyester Sheet Molding Compound (SMC) as an example, a unified Computer Aided Design (CAD) methodology, consisting of three uncoupled modules, (i.e., optimization, analysis and sensitivity calculations), is developed to systematically generate optimal cure cycle designs. Various optimization formulations for the cure cycle design are investigated. The uniformities in the distributions of the temperature and the degree of cure within the specimen cured under the optimal cycles are compared with those resulting from conventional isothermal processing conditions with pre-warmed platens. Recommendations with regards to further research in the computerization of the cure cycle design are also addressed.
, P r i n c i p a l I n v e s t i g a t o r and Jeenson Sheen, Graduate Research A s s i s t a n t F i n a l Report
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