Motivation and Framework
It is not difficult to motivate, from a practical point of view, the kind of situations we would like to deal with and analyze. We have selected a typical example in heat conduction, but many other examples are as valid as this one. Suppose we have two ve
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Motivation and Framework
1.1 The Model Problem It is not difficult to motivate, from a practical point of view, the kind of situations we would like to deal with and analyze. We have selected a typical example in heat conduction, but many other examples are as valid as this one. Suppose we have two very different materials at our disposal: the first, with conductivity ˛1 D 1, is a good and expensive conductor; the other is a cheap material, almost an insulator with conductivity coefficient ˛0 D 0:001. These two materials are to be used to fill up a given design domain Q, which we assume to be a unit square for simplicity (Fig. 1.1), in given proportions t1 , t0 , with t1 C t0 D 1. Typically, t1 < t0 given that the first material is much more expensive than the second. We will take, for definiteness, t1 D 0:4, t0 D 0:6. The thermal device is isolated all over @Q, except for a small sink 0 at the middle of the left side where we normalize temperature to vanish, and there is a uniform source of heat all over Q of size unity. The mixture of the two materials is to be decided so that the dissipated energy is as small as possible. If we designate u.x; y/ as temperature, and use a characteristic function to indicate where to place the good conductor in Q, then we would like to find the optimal such distribution minimizing the cost functional Z u.x; y/ dx dy Q
that measures dissipated energy, among all those mixtures complying with divŒ.˛1 .x; y/ C ˛0 .1 .x; y///ru.x; y/ D 1 in Q; u D 0 on 0 ;
.˛1 .x; y/ C ˛0 .1 .x; y///ru.x; y/ n D 0 on @Q n 0 ; Z .x; y/ dx dy D 0:4: Q
© Springer International Publishing Switzerland 2016 P. Pedregal, Optimal Design through the Sub-Relaxation Method, SEMA SIMAI Springer Series 11, DOI 10.1007/978-3-319-41159-0_1
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1 Motivation and Framework
Fig. 1.1 Design domain for a thermal device
uy = 0 Distributed Heating
u=0
?
ux = 0
min: compliant
u =0 y
Fig. 1.2 Optimal distribution of the two materials
uy = 0
u=0
u =0 x
uy = 0
Vector n stands for the outer unit normal to @Q. Figure 1.2 shows the optimal design (mixture) of the two materials. Black indicates the good conductor. We notice that there are specific spots where the two materials tend to be mixed in finer and finer scales. By changing the various elements of this example (the cost functional, the boundary conditions, the total amount of the two materials, the conductivity constants, etc.), one can study many other situations. Understanding the structure of these optimal mixtures of two materials is the main, global aim of this book. Once we have a better perspective on the significance of these problems, we explain, in as clear terms as possible, what our objective is with this booklet. The following are the basic ingredients of our model problem. At this initial stage, we do not pay much attention to precise assumptions. 1. The design domain ˝ R2 is a bounded, Lipschitz domain. 2. There are two conducting, homogeneous, isotropic materials, with conductivity constants ˛1 > ˛0 > 0, at our dis
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