A method for shape design sensitivity analysis of thin shell structures is developed. The curvature distribution on the middle surface of thin shell structure is the design to be determined. The thin shell theory employed is the general Koiter model in the Cartesian coordinates. In order to obtain the explicit formulas for the sensitivity, both the direct variational method and the indirect one based on the material derivative concept are studied. A general integral form in terms of various responses is considered. The results obtained from the two formulations, are shown to be equivalent. Numerical implementation based on these formulations is carried out and the results are compared with those by finite differencing. Through an iterative optimization process with the numerical sensitivity calculation, the optimum shapes of several problems are obtained.
For the convenience of shape sensitivity analysis it is assumed that the middle surface of the shell can be projected onto a plane. This provides that the variational equation of the static problem of a thin shell structure depend explicitly on the function $Z=Z(x^1,x^2)$ of the middle surface which is the shape to be designed. In the direct variational method, the desired shape sensitivity information is obtained by taking derivatives of the variational equation directly with repect to the shape function z. The other method is developed by using the material derivative concept in which the velocity field corresponds to a shape design change. Both methods use the adjoint system in order to express the variations of the state variables as an explicit function of the shape design change or the velocity field. Applying the material derivative concept over the mapped surface which is constant for any perturbed configuration, a general shape sensitivty formula is obtained. Restricting the velocity field to only z component in the Cartesian coordinates, this formula is shown equivalent to that derived by the direct variational method.
Shape design parameterization of the middle surface is illustrated using a natural bicubic spline surface which is controlled by design shape variables z defined on the mapped surface. The velocity fileds are also represented by bicubic splines. Finite elements used to find the solution of the variational equilibrium equation are triangular of class $C^1$, having 18 degrees of freedom for each node. Two kinds of velocity fileds are considered for numerical implementation of design sensitivity analysis, and their results are compared with those obtained by finite differencing. Accurate design sensitivity predictions are obtained. Two problems, a doubly curved square shell problem and a triangular shell problem, are analyzed and optimized to demonstrate the accuracy and versatility of the shape design of a thin shell structure.
