Sensitivity analysis. Finite difference or direct approach.

To calculate the sensitivity of an objective function F to a modification of the design parameters s, we start from

where u are the state variables that depend on s and S is the set of state equations of the system. Applying the chain rule of differentiation and operating, this yields:

The evaluation of the partial derivatives is problem-specific. In most cases, they are calculated following a finite differences scheme:

This method is extremely inefficient when the number of degrees of freedom greatly exceeds the number of functions to be evaluated, as is usually the case in shape optimization problems. Therefore, this approach is not recommended except for simple cases.