Shape optimization

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Shape optimization, deals with problems of the form: find a shape (in two or three dimensions) which is optimal in a certain sense, while satisfying certain requirements. Topology optimization is, in addition, concerned with the number of connected components/boundaries within a structure.

Mathematically, shape optimization this can be posed as the problem of finding a bounded set D, minimizing a functional

$\mathcal{F}(D)$ ,

possibly subject to a constraint of the form

$\mathcal{G}(D)=0.$

Usually we are interested in sets D which are Lipschitz or C¹ boundary, which is a way of saying that we would like to find a rather pleasing shape as a solution, not some jumble of bits and pieces. Sometimes additional constraints should be imposed to that end.

Shape optimization is an infinite-dimensional optimization problem.

1 Examples
2 Techniques
- 2.1 Keeping track of the shape
- 2.2 Iterative methods
3 References
4 External links

[edit] Examples

Among all three-dimensional shapes of given volume, find the one which has minimal surface area. Here:

$\mathcal{F}(D)=\mbox{Area}(\partial D)$ ,

with

$\mathcal{G}(D)=\mbox{Volume}(D)-\mbox{const.}$

The answer is the inside of a sphere.
Find the shape of an airplane wing which minimizes drag. Here the constraints could be the wing strength, or the wing dimensions.
Find the shape of various mechanical structures, which can resist a given stress while having a minimal mass/volume.

[edit] Techniques

Shape optimization problems are usually solved numerically, by using iterative methods. That is, one starts with an initial guess for a shape, and then gradually evolves it, until it morphs into the optimal shape.

[edit] 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 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 $Ω 0$ is the initial shape, and $Ω t$ is the shape at time t, one considers the diffeomorphisms

$f_t:\Omega_0\to \Omega_t, \mbox{ for } 0\le t\le t_0.$

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

[edit] Iterative methods

If one represents a shape by means of a function, shape optimization can be considered a calculus of variations problem. One can differentiate the objective functional $\mathcal{F}$ , and use gradient descent, Newton's method, or some other iterative method to evolve towards the optimal solution. Typically, gradient descent is preferred, even if requires a large number of iterations, because, it can be extremely hard to find the second-order derivative (that is, the Hessian) of the objective functional $\mathcal{F}$ .

If the shape optimization problem has constraints, that is, the functional $\mathcal{G}$ is present, one has to find ways to convert the constrained problem into an unconstrained one. Sometimes ideas based on Lagrange multipliers can work.

[edit] References

Mohammadi, B.; Pironneau, O. (2001) Applied Shape Optimization for Fluids. Oxford University Press. ISBN 0-19-850743-7.

Haslinger, J.; Mäkinen, R. (2003) Introduction to Shape Optimization: Theory, Approximation and Computation. Society for Industrial and Applied Mathematic. ISBN 0-89871-536-9.

Laporte, E.; Le Tallec, P. (2003) Numerical Methods in Sensitivity Analysis and Shape Optimization. Birkhäuser. ISBN 0-8176-4322-2.

[edit] External links

TopOpt Group — Free interactive programs for 2D and 3D compliance optimization, free MATLAB programme and more information on theory and applications.

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Category: Optimization