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 is the initial shape, and is the shape at time t, one considers the diffeomorphisms The idea is again that shapes are difficult entities to be dealt with directly, so manipulate them by means of a function. Iterative methods using shape gradients Consider a smooth velocity field and the family of transformations of the initial domain under the velocity field : , and denote Then the Gâteaux or shape derivative of at with respect to the shape is the limit of if this limit exists. If in addition the derivative is linear with respect to , there is a unique element of and where is called the shape gradient. This gives a natural idea of gradient descent, where the boundary is evolved in the direction of negative shape gradient in order to reduce the value of the cost functional. Higher order derivatives can be similarly defined, leading to Newtonlike methods. Typically, gradient descent is preferred, even if requires a large number of iterations, because, it can be hard to compute the second-order derivative (that is, the Hessian) of the objective functional . If the shape optimization problem has constraints, that is, the functional is present, one has to find ways to convert the constrained problem into an unconstrained one. Sometimes ideas based on Lagrange multipliers, like the adjoint state method, can work. Geometry parametrization Shape optimization can be faced using standard optimization methods if a parametrization of the geometry is defined. Such parametrization is very important in CAE field where goal functions are usually