The Relationship between the Variational Approach and the Implicit Differentiation Approach to Shape Design Sensitivities

Abstract

The most commonly used approach to the design sensitivity problem results from the implicit differentiation of the discretized equilibrium equations. The most general implementation of this technique requires that finite differences be used to differentiate the element stiffness matrices. Proper choice of the step size is necessary to obtain high levels of accuracy and to avoid round-off errors. Furthermore, since it is necessary to operate on the element matrices, this method is difficult to implement into a general purpose finite element program. A more recent shape design sensitivity formulation, based upon variational calculus, avoids having to differentiate the discretized equations and results in an analytical expression for the derivative. This approach is based upon the total derivative of the variational state equation and uses an adjoint variable technique for design sensitivity analysis. Only structural response data on the boundary of the structure are necessary, thereby making implementation into a general purpose program less difficult. This paper attempts to compare the two different techniques and point out the similarities. Two test problems demonstrate the accuracy of the variational approach.