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In this lecture which is the first in a series of four lectures we will
lay down the foundation for the ideas presented later. We are concerned with
mathematical tools that will enable us the analysis and the construction
of efficient algorithms for the solution of shape optimization problems
governed by fluid dynamics models, ranging from the full potential
equation to the full compressible Navier-Stokes equations. We restrict all
our discussions to gradient based methods. We begin this
lecture with a short review of basic ideas in optimization where we start
with algebraic problems and constraints. We derive the optimality conditions
for such cases as an introduction to our problems of interest which
include optimal control and
shape design. We demonstrate the derivation of the optimality conditions
for a control problem and discuss the case of finite dimensional control as
well as the infinite dimensional control case. Our last topic is the
derivation of optimality conditions for shape design problem. We derive the
variation of functionals with respect to the domain (shape) of integration.
Examples
for optimal shape design problems which minimize the deviation of the pressure
from a given pressure distribution, are given
for the full potential equation and the Euler equation.

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Shlomo Ta'asan
2001-08-22