A Mathematical Introduction to Sizing and Shape Optimization - Mäkinen R. A. E; Haslinger J (Introduction to Shape Optimization, 2003)
The aim of this chapter is to present ideas that are used in existence and convergence analysis in sizing and shape optimization (SSO). As we shall see, the basic ideas are more or less the same: we first prove that solutions of state problems depend continuously on design variables (and we shall specify in which sense). Then, imposing appropriate continuity (or better lower semicontinuity) assumptions on a cost functional, we immediately arrive at an existence result. The same scheme remains more or less true when doing the convergence analysis. Before we give an abstract setting for optimal sizing and shape design problems and prove abstract existence and convergence results, we show how to proceed in particular model examples. The same ideas will be used later on in the abstract form. 2.1 Thickness optimization of an elastic beam: Existence and convergence analysis Let us consider a clamped elastic beam of variable thickness e subject to a vertical load f. The beam is represented by an interval I = [0, ℓ], ℓ > 0. We want to find the thickness distribution in I minimizing the compliance of the beam, given by the value J (u (e)), where J ( y ) = ∫ I ƒ y d x 2.1 and u (e) is the solution of the following boundary value problem: { ( β e 3 u ″ ) ″ ( x ) = ƒ ( x ) ∀ x ∈ ] 0 , ℓ [ , u ( 0 ) = u ′ ( 0 ) = u ( ℓ ) = u ′ ( ℓ ) = 0 . 2.2 Here β ∈ L∞ (I), β ≥ β 0 = const. > 0, is a function depending on material properties and on the shape of the cross-sectional area of the beam. The solution of (2.2) is assumed to be a function of e, playing the role of a control variable. To define an optimization problem, one has to specify a class Uad of admissible thicknesses. As we already know, the result of the optimization process depends on, among other factors, how large Uad is.