A Newton-Type Method and Applications
The framework of asymptotic analysis in singularly perturbed geometrical domains can be employed to produce two-term asymptotic expansions for a class of shape functionals. In Chap. 6 one-term expansions of functionals are required for algorithms of shape-topological optimization. Such an approach corresponds to the simple gradient method in shape optimization. The Newton method of shape optimization can be replaced, for shape-topology optimization, by two-term expansions of shape functionals. Thus, the resulting approximations are more precise and the associated numerical methods are much more complex compared to one-term expansion topological derivative algorithms. In particular, numerical algorithms associated with first order topological derivatives of shape functionals have been presented in Chap. 6, together with an account of their applications currently found in the literature, with emphasis on shape and topology optimization. In this chapter second order topological derivatives are introduced. Second order algorithms of shape-topological optimization are used for numerical solution of representative examples of inverse reconstruction problems. The main feature of these algorithms is that the method is non-iterative and thus very robust with respect to noisy data as well as independent of initial guesses.