Sensitivity Analysis - Mäkinen R. A. E; Haslinger J (Introduction to Shape Optimization, 2003)
From the previous chapter we already know that a continuous dependence of solutions to state problems on design variations is a fundamental property ensuring the existence of optimal solutions. Continuity is important but not enough. To better understand the problem, other properties are needed. Differentiability is one of the most important of these. The need to deal with such information gave rise to a special discipline in optimization called sensitivity analysis. Sensitivity analysis develops appropriate tools and concepts enabling us to analyze the differentiability of various objects, such as solutions to state problems, cost and constraint functionals, etc., with respect to control variables, and in particular with respect to design variables in sizing and shape optimization (SSO). On the basis of these results one can derive necessary optimality conditions satisfied by solutions to optimal control problems. The interpretation of optimality conditions reveals important properties of optimal solutions that are not usually directly seen from the original setting of the problem. Sensitivity analysis also plays an important role in computations: it provides us with gradient information required by the gradient type methods most frequently used for the numerical minimization of discretized problems. This chapter deals with sensitivity analysis in SSO. The basic ideas are the same. Each branch of structural optimization, however, develops its own techniques, taking into account its features. This will be seen, in particular, in the case of shape optimization, for which appropriate tools enabling us to describe the change in geometry have to be introduced. This chapter starts with sensitivity analysis in the algebraic setting of problems because of its simplicity. Here we explain common ideas. These results will then be adapted following the specific needs of SSO.