Editing Shape optimization (section)

=== Keeping track of the shape ===
To solve a shape optimization problem, one needs to find a way to represent a shape in the [[computer memory]], and follow its evolution. Several approaches are usually used.

One approach is to follow the boundary of the shape. For that, one can 
sample the shape boundary in a relatively dense and uniform manner, that is, to consider enough points to get a sufficiently accurate outline of the shape. Then, one can evolve the shape by gradually moving the boundary points. This is called the ''Lagrangian approach''.

Another approach is to consider a [[function (mathematics)|function]] defined on a rectangular box around the shape, which is positive inside of the shape, zero on the boundary of the shape, and negative outside of the shape. One can then evolve this function instead of the shape itself. One can consider a rectangular grid on the box and sample the function at the grid points. As the shape evolves, the grid points do not change; only the function values at the grid points change. This approach, of using a fixed grid, is called the ''Eulerian approach''. The idea of using a function to represent the shape is at the basis of the [[level-set method]].

A third approach is to think of the shape evolution as of a flow problem. That is, one can imagine that the shape is made of a plastic material gradually deforming such that any point inside or on the boundary of the shape can be always traced back to a point of the original shape in a one-to-one fashion. Mathematically, if <math>\Omega_0</math> is the initial shape, and <math>\Omega_t</math> is the shape at time ''t'', one considers the [[diffeomorphism]]s

:<math>f_t:\Omega_0\to \Omega_t, \mbox{ for } 0\le t\le t_0.</math>

The idea is again that shapes are difficult entities to be dealt with directly, so manipulate them by means of a function.