Springer Series in Computational Mathematics

Transcription

1 Springer Series in Computational Mathematics 16 Editorial Board R. L. Graham, Murray Hill J. Stoer, WOrzburg R. Varga, Kent (Ohio)

2 Jan Sokolowski Jean-Paul Zolesio Introduction to Shape Optimization Shape Sensitivity Analysis Springer-Verlag Berlin Heidelberg GmbH

3 Jan Sokolowski Systems Research Institute Polish Academy of Sciences ul. Newelska Warszawa Poland Jean-Paul Zolesio Centre National de la Recherche SCientifique et Institut non Lineaire de Nice B. P. 71 Faculte des Sciences Nice Cedex 2 France Mathematics Subject Classification (1991): 35B30, 49B50, 73C60, 73K40, 73T05 Ubrary 01 Congress Calaloging-in-Publicalion Dala Sokorowski, Jan, Inlroduction 10 shape optlmization: shape sensilivity analysis I J.Sokotowski, J. P. Zolesio. p. cm. - (Springer se ries In computational mathematics; 16) Includes blbllographlcal relerences and index.c ISBN ISBN (ebook) DOI /

4 Contents Chapter 1 Introduction to shape optimization 1.1. Preface Chapter 2 Preliminaries and the material derivative method 2.1. Domains in IRN of class C k 2.2. Surface measures on r 2.3. Functional spaces 2.4. Linear elliptic boundary value problems 2.5. Shape functionals 2.6. Shape functionals for problems governed by linear elliptic boundary value problems Shape functionals for transmission problems Approximation of homogenuous Dirichlet problems 2.7. Convergence of domains 2.8. Transformations Tt of domains 2.9. The speed method Admissible speed vector fields Vk(D) Eulerian derivatives of shape functionals Non-differentiable shape functionals Properties of Tt transformations Differentiability of transported functions Derivatives for t > Derivatives of domain integrals Change of variables in boundary integrals Derivatives of boundary integrals The tangential divergence of the field V on r Tangential gradients and Laplace-Beltrami operators on r Variational problems on r The transport of differential operators Integration by parts on r The transport of Laplace-Beltrami operators Material derivatives

5 2 Contents Material derivatives on r The material derivative of a solution to the Laplace equation with Dirichlet boundary conditions Strong material derivatives for Dirichlet problems The material derivative of a solution to the Laplace equation with Neumann boundary conditions Shape derivatives Derivatives of domain integrals (II) Shape derivatives on r Derivatives of boundary integrals 115 Chapter 3 Shape derivatives for linear problems The shape derivative for the Dirichlet boundary value problem The shape derivative for the Neumann boundary value problem Necessary optimality conditions Parabolic equations Neumann boundary conditions Dirichlet boundary conditions Shape sensitivity in elasticity Shape sensitivity analysis of the smallest eigenvalue Shape sensitivity analysis of the Kirchhoff plate Shape derivatives of boundary integrals: the non-smooth case in m? Shape sensitivity analysis of boundary value problems with singularities Hyperbolic initial boundary value problems 157 Chapter 4 Shape sensitivity analysis of variational inequalities Differential stability of the metric projection in Hilbert spaces Sensitivity analysis of variational inequalities in Hilbert spaces The obstacle problem in HI (il) Differentiability of the Newtonian capacity The shape controlability of the free boundary The Signorini problem Variational inequalities of the second kind Sensitivity analysis of the Signorini problem in elasticity Differential stability of solutions to variational inequalities in Hilbert spaces Shape sensitivity analysis The Signorini problem with given friction Shape sensitivity analysis Elasto-Plastic torsion problems Elasto-Visco-Plastic problems 234 References 240

6 Notation D domain in IRN with piecewise smooth boundary ad fl measurable set in IRN or in D, or domain of class C k r = afl boundary of fl n unit normal vector field on r, outward to fl No unitary extension of n to an open neighbourhood of r in IRN y(fl) function in WS,P(fl) X{} or X characteristic function of fl flc =D\fl(orIRN\fl) X{}C or XC characteristic function of flc Ifll N-dimensional measure of fl K mean curvature of r Char(D) = {X E L2(D) such that (1 - X)X = 0 a.e. on D} B unit ball in IRN Bo unit ball in IRN-l, Bo C B J(fl) domain functional (or cost functional) PD( fl) perimeter of fl in D A(x),B(x),C(x) continuous matrix functions on D * A( x ) transpose of A( x) Tt transformation of IR N or of D into IR N V( t)( ) = V( t, x) speed vector field DTt Jacobian of Tt 'Yt = 'Y(t) = det(dtt) M(Tt) M(Tt) = 'Y(t)* DTt- 1 Wt = w(t) = IIM(Tt}.nIIIRN on r (V) = H* DV + DV) dj(fl; V) Eulerian derivative G( fl) shape gradient G density of the shape gradient g(r) the density gradient g E Ll(IRN) a (non unique) distributed representation of g(r) 'Yr trace operator on r, e.g. 'Yr E C(Hl(fl); H~ (r)) obstacle function t/j

7 4 Notation 'Vr a an a ana divr.dr ij(nj V) yl(nj V) y * Tt yl(rj V) co'(n), CO'(nj1R N ) tangential gradient on r nonnal derivative on r cononnal derivative on r associated to the matrix A tangential divergence on r Laplace-Beltrami operator on r material derivative of y( n) at n in direction of the speed field. shape (domain) derivative of y( n) at n in direction of the speed field V transported distribution boundary shape derivative of y(r) at r in direction of the speed field V space of smooth functions (or of vector smooth functions) with compact supports in n