Shape design sensitivity analysis of eigenvalues using "exact" numerical differentiation of finite element matrices

As has been shown in recent years, the approximate numerical differentiation of element stiffness matrices which is inherent in the semi-analytical method of finite element based design sensitivity analysis, may give rise to severely erroneous shape design sensitivities in static problems involving linearly elastic bending of beam, plate and shell structures. This paper demonstrates that the error problem also manifests itself in semi-analytical sensitivity analyses of eigenvalues of such structures and presents a method for complete elimination of the error problem. The method, which yields "exact" numerical sensitivities on the basis of simple first-order numerical differentiation, is computationally inexpensive and easy to implement as an integral part of the finite element analysis. The method is presented in terms of semi-analytical shape design sensitivity analysis of eigenvalues in the form of frequencies of free transverse vibrations of plates modelled by isoparametric Mindlin finite elements. Finally, the development is illustrated via two examples of occurrence of the error phenomenon when the traditional method is used and it is shown that the problem is completely eliminated by the application of the new method.