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Section: Scientific Foundations

Geometrical optimization

In view of enhancing the robustness of algorithms in shape optimization or shape evolution, modeling the moving geometry is a challenging issue. The main obstacle between the geometrical viewpoint and the numerical implementation lies in the basic fact that the shape gradients are distributions and measures lying in the dual spaces of the shape and geometrical parameters. These dual spaces are usually very large since they contain very irregular elements. Obviously, any finite dimensional approach pertains to the Hilbert framework where dual spaces are identified implicitly to the shape parameter spaces. But these finite-dimensional spaces sometimes mask their origin as discretized Sobolev spaces, and ignoring this question leads to well-known instabilities; appropriate smoothing procedures are necessary to stabilize the shape large evolution. This point is sharp in the “narrow band” techniques where the lack of stability requires to reinitialize the underlying level equation at each step.

The mathematical understanding of these questions is sought via the full analysis of the continuous modeling of the evolution. How can we “displace” a smooth geometry in the direction opposite to a non smooth field, that is going to destroy the boundary itself, or its smoothness, curvature, and at least generate oscillations.

The notion of Shape Differential Equation is an answer to this basic question and it arises from the functional analysis framework to be developed in order to manage the lack of duality in a quantitative form. These theoretical complications are simplified when we return to a Hilbert framework, which in some sense, is possible, but to the undue expense of a large order of the differential operator implied as duality operator. This operator can always be chosen as an ad hoc power of an elliptic system. In this direction, the key point is the optimal regularity of the solution to the considered system (aerodynamical flow, electromagnetic field, etc.) up to the moving boundary whose regularity is itself governed by the evolution process.

We are driven to analyse the fine properties concerning the minimal regularity of the solution. We make intensive use of the “extractor method” that we developed in order to extend the I. Lasiecka and R. Triggiani “hidden regularity theory”. For example, it was well known (before this theory) that when a domain $ \upper_omega$ has a boundary with continuous curvatures and if a “right hand side” f has finite energy, then the solution u to the potential problem -$ \upper_delta$u = f is itself in the Sobolev space H2($ \upper_omega$)$ \cap$H01($ \upper_omega$) so that the normal derivative of u at the boundary is itself square integrable. But what does this result become when the domain boundary is not smooth? Their theory permitted for example to establish that if the open set $ \upper_omega$ is convex, the regularity property as well as its consequences still hold. When the boundary is only a Lipschitzian continuous manifold the solution u loses the previous regularity. But the “hidden regularity” results developed in the 80's for hyperbolic problems, in which the H2($ \upper_omega$) type regularity is never achieved by the solution (regardless the boundary regularity), do apply. Indeed without regularity assumption on the solution u , we proved that its normal derivative has finite energy.

In view of algorithms for shape optimization, we consider the continuous evolution $ \upper_omega$t of a geometry where t may be the time (governing the evolution of a PDE modeling the continuous problem); in this case, we consider a problem with dynamical geometry (non cylindrical problem) including the dynamical free boundaries. But t may also be the continuous version for the discrete iterations in some gradient algorithm. Then t is the continuous parameter for the continuous virtual domain deformation. The main issue is the validity of such a large evolution when t is large, and when t$ \rightarrow$$ \infty$ . A numerical challenge is to avoid the use of any “smoother” process and also to develop “shape-Newton” methods [65] . Our evolution field approaches permit to extend this viewpoint to the topological shape optimization ( [67] ).

We denote G($ \upper_omega$) the shape gradient of a functional J at $ \upper_omega$ . There exists Im1 ${s\#8712 \#8477 ^+}$ such that Im2 ${G{(\#937 )}\#8712 H^{-s}{(D,\#8477 ^N)}}$ , where D is the universe (or “hold all”) for the analysis. For example Im3 ${D=\#8477 ^N}$ . The regularity of the domains which are solution to the shape differential equation is related to the smoothness of the oriented distance function Im4 $b_\#937 $ which turns out to be the basic tool for intrinsic geometry. The limit case Im5 ${b_\#937 \#8712 C^{1,1}{(\#119984 )}}$ (where Im6 $\#119984 $ is a tubular neighborhood of the boundary $ \upper_gamma$ ) is the important case.

If the domains are Sobolev domains, that is if Im7 ${b_\#937 \#8712 H^r{(\#119984 )}}$ , then we consider a duality operator, Im8 ${\#119964 \#8712 \#8466 (H^r,H^{-s})}$ satisfying: Im9 ${\lt \#119964 \#966 ,\#966 \gt \#8805 {|\#966 |}_H^2}$ where H denotes a root space. We consider the following problem: given $ \upper_omega$0 , find a non autonomous vector field Im10 ${V\#8712 C^0{([0,~\#8734 }{[,H^r{(D,\#8477 ^N)})}\#8745 C{([}0,~,\#8734 {[~,L^\#8734 {(D,\#8477 ^N)})}}$ such that, Tt(V) being the flow mapping of V ,

Im11 ${\#8704 t\gt 0,~~~\#119964 .V{(t)}+G(T_t{(V)}{(\#937 _0)})=0}$

Several different results have been derived for this equation under boundedness assumptions of the following kind:

Im12 ${\mtext there~\mtext exists~~~M\gt 0~~~~~\mtext so~\mtext that,~~\#8704 \#937 ,~~~||G(\#937 )||\#8804 ~M}$

The existence of such bound has first been proved for the problem of best location of actuators and sensors, and have since been extended to a large class of boundary value problems. The asymptotic analysis (in time t$ \rightarrow$$ \infty$ ) is now complete for a 2D problem with help of V. Sverak continuity results (and extended versions with D. Bucur). These developments necessitate an intrinsic framework in order to avoid the use of Christoffel symbols and local mappings, and to work at minimal regularity for the geometries.

The intrinsic geometry is the main ingredient to treat convection by a vector field V . Such a non autonomous vector field builds up a tube. The use of BV topology permits these concepts to be extended to non smooth vector fields V , thus modeling the possible topological changes. The transverse field concept Z has been developed in that direction and is now being applied to fluid-structure coupled problems. The most recent results have been published in three books [2] , [13] , [1] .


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