Kirchhoff's theorem: Difference between revisions

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'''Shape optimization''' is part of the field of [[optimal control]] theory. The typical problem is to find the [[shape]] which is optimal in that it minimizes a certain cost [[functional (mathematics)|functional]] while satisfying given [[constraint (mathematics)|constraints]]. In many cases, the functional being solved depends on the solution of a given partial differential equation defined on the variable domain.
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[[Topology optimization]] is, in addition, concerned with the number of connected components/boundaries belonging to the domain. Such methods are needed since typically shape optimization methods work in a subset of allowable shapes which have fixed topological properties, such as having a fixed number of holes in them. Topological optimization techniques can then help work around the limitations of pure shape optimization.
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==Definition==
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[[Mathematics|Mathematically]], shape optimization can be posed as the problem of finding a [[bounded set]] <math>\Omega</math>, [[maxima and minima|minimizing]] a [[functional (mathematics)|functional]]
:<math>\mathcal{F}(\Omega)</math>,
possibly subject to a [[constraint (mathematics)|constraint]] of the form
:<math>\mathcal{G}(\Omega)=0.</math>
Usually we are interested in sets <math>\Omega</math> which are [[Lipschitz continuity|Lipschitz]] or C<sup>1</sup> [[Boundary (topology)|boundary]] and consist of finitely many [[connected component (analysis)|components]], which is a way of saying that we would like to find a rather pleasing shape as a solution, not some jumble of rough bits and pieces. Sometimes additional constraints need to be imposed to that end to ensure well-posedness of the problem and uniqueness of the solution.


Shape optimization is an [[infinite-dimensional optimization]] problem. Furthermore, the space of allowable shapes over which the optimization is performed does not admit a [[vector space]] structure, making application of traditional optimization methods more difficult.
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==Examples==
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<ul>
<li>Among all three-dimensional shapes of given volume, find the one which has minimal surface area. Here:
:<math>\mathcal{F}(\Omega)=\mbox{Area}(\partial \Omega)</math>,
with
:<math>\mathcal{G}(\Omega)=\mbox{Volume}(\Omega)-\mbox{const.}</math>
The answer is the inside of a [[sphere]].
</li>
<li>Find the shape of an airplane wing which minimizes [[Drag (physics)|drag]]. Here the constraints could be
the wing strength, or the wing dimensions.
</li>
<li>Find the shape of various mechanical structures, which can resist a given [[Stress (physics)|stress]] while having a minimal mass/volume.
</li>
<li>Given a known three-dimensional object with a fixed radiation source inside, deduce the shape and size of the source based on measurements done on part of the boundary of the object. A formulation of this [[inverse problem]] using [[least-squares]] fit leads to a shape optimization problem.
</li>
</ul>


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To solve a shape optimization problem, one needs to find a way to
 
represent a shape in the [[Computer storage|computer memory]], and follow its evolution. Several approaches are usually used.  
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One approach is to follow the boundary of the shape. For that, one can
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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
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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
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one considers the [[diffeomorphism]]s


:<math>f_t:\Omega_0\to \Omega_t, \mbox{ for } 0\le t\le t_0.</math>
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The idea is again that shapes are difficult entities to be dealt with directly, so manipulate them by means of a function.
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=== Iterative methods using shape gradients ===
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:<math>x(0) = x_0 \in \Omega_0, \quad x'(s) = V(x(s)), \quad T_s(x_0) = x(s), \quad s \geq 0 </math>,
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where <math>\nabla \mathcal{F}</math> is called the shape gradient. This gives a natural idea of [[gradient descent]], where the boundary <math>\partial \Omega</math> 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.
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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 complex functions evaluated using numerical models (CFD, FEA,...). A convenient approach, suitable for a wide class of problems, consists in the parametrization of the CAD model coupled with a full automation of all the process required for function evaluation (meshing, solving and result processing). [[Mesh morphing]] is a valid choice for complex problems. In this case the parametrization is defined after the meshing stage acting directly on the numerical model used for calculation that is changed using mesh updating methods. There are several algorithms available for mesh morphing ([[deforming volume]]s, [[pseudosolid]]s, [[radial basis function]]s).
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== References ==
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* Allaire, G. (2002) ''Shape optimization by the homogenization method''. Applied Mathematical Sciences 146, Springer Verlag. ISBN 0-387-95298-5


* Ashok D. Belegundu, Tirupathi R. Chandrupatla. (2003) ''Optimization Concepts and applications in Engineering'' Prentice Hall. ISBN 0-13-031279-7.
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* Bendsøe M. P.; Sigmund O. (2003) ''Topology Optimization: Theory, Methods and Applications''. Springer. ISBN 3-540-42992-1.
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* Burger, M.; Osher, S.L. (2005) ''A Survey on Level Set Methods for Inverse Problems and Optimal Design''. European Journal of Applied Mathematics, vol.16 pp.263-301.
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* Delfour, M.C.; Zolesio, J.-P. (2001) ''Shapes and Geometries - Analysis, Differential Calculus, and Optimization''. SIAM. ISBN 0-89871-489-3.
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* Haslinger, J.; Mäkinen, R. (2003) ''Introduction to Shape Optimization: Theory, Approximation and Computation''. Society for Industrial and Applied Mathematic. ISBN 0-89871-536-9.
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* Laporte, E.; Le Tallec, P. (2003) ''Numerical Methods in Sensitivity Analysis and Shape Optimization. Birkhäuser. ISBN 0-8176-4322-2.
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*[http://www.topopt.dtu.dk 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:Mathematical optimization]]

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