Comparison of two different approaches for making design sensitivity analysis an integrated part of finite element analysis

This paper presents a comparison of two approaches for incorporating design sensitivity calculations into finite element analysis. The formulations depend on the implicit differentiation method and require few additional calculations to obtain the design sensitivity derivatives. The first approach by-passes the direct calculation of the stiffness matrix derivatives by calculating the sensitivity derivatives at the stress recovery stage of the analysis. The second approach depends on the direct calculation of the derivatives of the stiffness matrix, which are stored for re-use in multiple load case analyses, and subsequent matrix multiplications to evaluate the design sensitivities. The two approaches are developed and implemented to calculate the design sensitivities for continuum and structural isoparametric elements. In addition, a criterion is developed to aid in deciding which approach is better for a given number of load cases. To demonstrate the validity of the developed criterion and to evaluate the relative merits of each approach, some sensitivity calculation test problems are solved with different numbers of load cases.