Application of topological derivative to accelerate genetic algorithm in shape optimization of coupled models

Abstract

In the paper we consider a new method based on the genetic algorithm for finding the location and size of small holes in the domain, in which the coupled linear and nonlinear boundary value problems are defined. The linear and nonlinear components are connected by the transmission conditions on the interface boundary. The expansion with respect to small parameter of the shape functional for nonlinear component and the expansion of Steklov-Poincaré operator for linear component are derived in order to determine the form of topological derivative for shape functional defined for coupled model. Then the topological derivative is employed for evaluation of the probability density applied to generation of location of holes by the genetic algorithm.

On dual dynamic programming in shape optimization of coupled models

“…The approximate value of our functional J is similar like the result in the numerical example presented in Szulc and Zochowski (2015). However we know how far we are from the inf J (γ ), while in Szulc and Zochowski (2015) is only known some step of calculation.…”

“…The approximate verification theorem allows us to calculate approximate solution but also to state how far we are from the inf J (γ ). In Szulc and Zochowski (2015) the value of calculated numerically functional is almost the same (we took the same parameters in our example) but the authors in Szulc and Zochowski (2015) can only assert that it is some step of calculation of the minimal value. In a subsequent paper we are going to apply the dynamic programming technique to more general shape optimization problems.…”

“…In the interior subdomain ω the physical phenomenon are described by the linear PDE and in the exterior domain the processes are governed by nonlinear PDE subject to some external function. We took that problem from the paper (Szulc and Zochowski 2015) where more details is presented. The design problem can be considered as a two level optimization problem.…”

“…So, except for very particular cases, there is no optimal location in optimum design problem (see Delfour and Zolesio 2001). That is why many authors confine themself only to numerical analyzis and this is just done in Sokolowski and Zochowski (1999) applying topological derivative and in Szulc and Zochowski (2015). In the last decade some optimum design problems were investigated for time-dependent state equations, including the wave equation or the heat equation (see e.g.…”

Topological derivative and neural network for inverse problems of coupled models

“…According to [11,12] we can rewrite the condition in (1. The expantion of the Stieklov-Poincare operator was described in details in [14] The problem (l.5) can then be rewritten as follows:…”