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Intro to Shape Optimization

Shape Optimization Introductory Course

This course was given by Dorin Bucur in November 2010 in Timisoara, and was my first contact with the field of shape optimization. As an outsider, at that time, I was really impressed by this course and by the problems presented here. The course does not go into deep details, and it is aimed to be a presentation of the field.

1. The Dido Problem. This is an ancient problem. The hystorical sources say that queen Dido was allowed to have as much land as can be encompassed by an oxhide. Being very smart, she cut the oxhide into a very thin and long strip and encompassed a semicircle having the sea as the straight boundary.

Therefore, the problem was to maximize the area enclosed by a fixed perimeter, (a similar problem is to minimize the perimeter which encloses a fixed area).

2. The honey comb problem. When bees construct their combs, they have to do a lot of work to build up the walls of the cells in which they deposit the honey. It is well known that these cells have hexagonal shape. This inspired the following conjecture, namely that the configuration of the combs in a bee hive is optimal, as a consequence of the fact that bees are very efficient workers and do not waste time and material in building their combs.

This ca be translated in a mathematical language in the following way: consider a large region ${\Omega}$ and a partition of ${\Omega}$ in ${N}$ parts of equal areas ${\Omega_1,...,\Omega_N}$. Our goal is to prove that the problem

$\displaystyle \min_{|\Omega_i|=|\Omega|/N} \sum_{i=1}^N \text{Per}\; (\Omega_i)$

has a solution and that the desired solution is formed of a hexagonal partition. The conjecture was proved by Hales in the sense that the hexagonal partition is the best one asymptotically as ${N}$ becomes very large. Some numerical results which show what happens for some concrete cases were developed by Edouard Oudet.

3. The lamp Problem Some lamps have their upper part that covers the bulb in the shape that can be seen in the figure. A constructor of such lamps can be interested in finding the shape of the upper part which costs the least in the sense that it has the least area among the shapes which connect the upper and lower circle.

A mathematical model can be given in the following way: Consider ${u: B(0,2)\setminus B(0,1) \rightarrow \Bbb{R}_+}$ with ${u(1)=c,u(2)=0}$. Find the function ${u}$ that minimizes the functional

$\displaystyle \mathcal{A}(u)=\int_{B(0,2)\setminus B(0,1)}\sqrt{1+|\nabla u|^2}dxdy$

It is interesting to note that the optimal shape is not a truncated cylinder as one may think. More details can be found in the post about the Catenary.

4. The drum problem (Rayleigh conjecture)

Consider a set ${\Omega}$ in the plane, which is open, convex and bounded. This models the membrane of a drum. The principal tone of the drum is given by the first eigenvalue of the Laplacian operator with Dirichled boundary condition in ${\Omega}$, i.e. the smallest non-zero ${\lambda}$ for which there exists a non-zero solution ${u}$ of the problem

$\displaystyle \begin{cases} -\Delta u=\lambda u & \text{ in } \Omega \\ u=0 & \text{ on }\partial \Omega \end{cases}$

Rayleigh conjectured that the shape which minimizes the first eigenvalue of the Laplacian, when ${\Omega}$ has constant area, is the disk. The conjecture was proved by Faber and Krahn.

5. The Munford-Shah functional (1986) This functional proposed a way to enhance the quality of the photos taken from satellites, which were often blurry and had imperfections. The idea was to study the shades of grey and the shape of the interface between different levels of grey. Consider an rectangular set ${D}$. We have an observed image ${f}$ which can be considered as a function ${f:D \rightarrow \Bbb{R}_+}$ or ${f: D \rightarrow [0,255]}$. We have two unknowns ${\Gamma}$ (the contours which sepparate the levels of grey) and ${u}$, the new image. The proposed method to enhance the image is to solve the following problem

$\displaystyle \min \int_D (u-f)^2 +\int_{D\setminus \Gamma} |\nabla u|^2dx +\ell(\Gamma).$

The first term asks that the distance between the observed image and the new image is small. The second term says that there aren’t any great jumps of the new image outside the contours which sepparate the levels of grey. The last term denotes the length of ${\Gamma}$, and we expect in a clear image that ${\Gamma}$ does not oscillate very much, and so its length is not expected to be very large.

A proof of the existence of a solution for the Dido Problem is given in the next part, proof which resembles the proof given in this post to the existence of the solution to the Isoperimetric inequality.

6. Swimming suits that decrease the drag As a consequence of the fall of many swimming world records in the years 2008-2009, swimming suits that cover all the body and that are made of some special materials were banned from swimming competitions, being considered as technological doping. It was discovered that there are textures which can highly minimize the drag, and therefore give decissive advantages to swimmers that wear those suits. In his phd thesis, Matthieu Bonnivard, under the coordination of Dorin Bucur, has discovered that sometimes adding certain roughness to an object can decrease the drag of an object moving throgh a fluid. Again, looking at some natural facts, the skin of the shark is far from being smooth, but we all know that the shark is some of the fastest creatures that move through water.