Control of Moving Domains, Shape Stabilization and Variational Tube Formulations
This paper deals with the control of a moving dynamical domain in which a non cylindrical dynamical boundary value problem is considered. We consider weak Eulerian evolution of domains through the convection of a measurable set by (non necessarily smooth)
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ontrol of Moving Domains, Shape Stabilization and Variational Tube Formulations Jean-Paul Zol´esio Abstract. This paper deals with the control of a moving dynamical domain in which a non cylindrical dynamical boundary value problem is considered. We consider weak Eulerian evolution of domains through the convection of a measurable set by (non necessarily smooth) vector field V . We introduce the concept of tubes by “product space” and we show a closure result leading to existence results for a variational shape principle. We illustrate this by new results: heat equation and wave equation in moving domains with various boundary conditions and also the geodesic characterisation for two Eulerian shape metrics leading to the Euler equation through the transverse field considerations. We consider the non linear Hamilton-Jacobi like equation associated with level set parametrization of the moving domain and give new existence result of possible topological change in finite time in the solution.
1. The use of BV perimeter in shape optimization In order to derive existence results in classical shape optimization (see for example [9]) I introduced after 1984 ([38], [39]) the concept of a functional regularized with the perimeter: Jσ (Ω) = J(Ω) + σ PD (Ω)
(1.1)
This has been in the context of large water wave modelling (non shallow water free boundary) in which the “small” parameter σ turns to be the surface tension. That result was emphasized in [36] after having been presented to a large audience, then in [37] (that paper was kept two years before being accepted without changes for publication). For dynamical modelling (artery [11], fluid structure interaction [26], . . . ) the concept of a tube (ζ, V ) and a tube functional to be extremized with respect to
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the tube was introduced in [40], [13], . . . in the following form: τ PD (Ωt (V )) dt. J(ζ, V ) = j(ζ, V ) + σ
(1.2)
0
Following that idea we consider in this paper new control functionals for non cylindrical heat equation, non cylindrical wave equations and shape metrics. Many results are completely new: the optimality conditions and existence results for both heat and wave problems as well as the existence of solution to the level set equation with possible topological change in finite time. The Euler equation for the shape geodesic generalizes in some sense the variational formulation for the incompressible Euler equation. In deriving the new geodesic conditions for the ¯ 1 , Ω2 ) and dE (Ω1 , Ω2 ) we introduce new technical results new shape metrics δ(Ω such as the expression for the boundary shape derivative vΓ t and the new weak form for the transverse vector field evolution equation. Also the structure of the adjoint field Λ is clarified when the right-hand side is a shape gradient like measure γt∗ (g nt ), then Λ =!∇λ∇χΩt . The cubic energy expression for the wave equation with homogeneous Dirichlet condition derived in 1984 is also generalized for the ∂y first time to the “co-normal Neumann” condition ∂y ∂t y + ∂ν ∗ y = 0 on Γt . t
2. Sha
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