Shape sensitivity analysis of the work functional for the compressible Navier–Stokes equations

International audienceThe compressible Navier–Stokes equations with nonhomogeneous Dirichlet conditions in a bounded domain with an obstacle are considered (P.I. Plotnikov, J. Sokolowski, Compressible Navier-Stokes Equations. Theory and Shape Optimization, Birkhäuser, Basel, 2012). The dependence of local solutions on the shape of an obstacle is analyzed (P.I. Plotnikov, E.V. Ruban, J. Sokołowski, SIAM J. Math. Anal. 40:1152–1200, 2008; P.I. Plotnikov, E.V. Ruban, J. Sokołowski, J. Math. Pures Appl. 92:113–162, 2009; P.I. Plotnikov, J. Sokołowski, Dokl. Akad. Nauk 397:166–169, 2004; P.I. Plotnikov, J. Sokołowski, J. Math. Fluid Mech. 7:529–573, 2005; P.I. Plotnikov, J. Sokołowski, Comm. Math. Phys. 258:567–608, 2005; P.I. Plotnikov, J. Sokołowski, SIAM J. Control Optim. 45:1165–1197, 2006; P.I. Plotnikov, J. Sokołowski, Uspekhi Mat. Nauk 62:117–148, 2007; P.I. Plotnikov, J. Sokołowski, Stationary boundary value problems for compressible Navier-Stokes equations, in Handbook of Differential Equations: Stationary Partial Differential Equations, vol. VI, Elsevier/North-Holland, Amsterdam, 2008, pp. 313–410; P.I. Plotnikov, J. Sokołowski, SIAM J. Control Optim. 48:4680–4706, 2010; P.I. Plotnikov, J. Sokołowski, J. Math. Sci. 170:34–130, 2010). The shape derivatives (J. Sokołowski, J.-P. Zolésio, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992) of solutions to the compressible Navier–Stokes equations are derived. The shape gradient (J. Sokołowski, J.-P. Zolésio, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992) of the work functional is obtained. In this way the framework for numerical methods of shape optimization (P. Plotnikov, J. Sokołowski, A. Żochowski, Numerical experiments in drag minimization for compressible Navier-Stokes flows in bounded domains, in Proceedings of the 14th International IEEE/IFAC Conference on Methods and Models in Automation and Robotics, MMAR’09, 2009, 4 pp; A. Kaźmierczak, P.I. Plotnikov, J. Sokołowski, A. Żochowski, Numerical method for drag minimization in compressible flows, in 15th International Conference on Methods and Models in Automation and Robotics (MMAR), pp. 97–101, 2009

Shape-topological differentiability of energy functionals for unilateral problems in domains with cracks and applications

International audienceA review of results on first order shape-topological differentiability of energy functionals for a class of variational inequalities of elliptic type is presented.The velocity method in shape sensitivity analysis for solutions of elliptic unilateral problems is established in the monograph (Sokołowski and Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992). The shape and material derivatives of solutions to frictionless contact problems in solid mechanics are obtained. In this way the shape gradients of the associated integral functionals are derived within the framework of nonsmooth analysis. In the case of the energy type functionals classical differentiability results can be obtained, because the shape differentiability of solutions is not required to obtain the shape gradient of the shape functional (Sokołowski and Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992). Therefore, for cracks the strong continuity of solutions with respect to boundary variations is sufficient in order to obtain first order shape differentiability of the associated energy functional. This simple observation which is used in Sokołowski and Zolésio (Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992) for the shape differentiability of multiple eigenvalues is further applied in Khludnev and Sokołowski (Eur. J. Appl. Math. 10:379–394, 1999; Eur. J. Mech. A Solids 19:105–120, 2000) to derive the first order shape gradient of the energy functional with respect to perturbations of the crack tip. A domain decomposition technique in shape-topology sensitivity analysis for problems with unilateral constraints on the crack faces (lips) is presented for the shape functionals.We introduce the Griffith shape functional as the distributed shape derivative of the elastic energy evaluated in a domain with a crack, with respect to the crack length. We are interested in the dependence of this functional on domain perturbations far from the crack. As a result, the directional shape and topological derivatives of the nonsmooth Griffith shape functional are obtained with respect to boundary variations of an inclusion

Passive control of singularities by topological optimization. The second order mixed shape derivatives of energy functionals for variational inequalities

The class of nonsmooth shape optimization problems for variational inequalities is con-sidered. The variational inequalities model elliptic boundary value problems with the Signorini type unilateral boundary conditions. The shape functionals are given by the first order shape derivatives of the elastic energy. In such a way the singularities of weak solutions to elliptic boundary value problems can be characterized. An example in solid mechanics is given by the Griffith's functional, which is defined in the plane elasticity to measure SIF, the so-called stress intensity factor, at the crack tips. Thus, the topological optimization can be used for passive control of singularities of weak solutions to variational inequalities. The Hadamard directional differentiability of metric projection onto the positive cone in fractional Sobolev spaces is employed to the topological sensitivity analysis of weak solutions of nonlinear elliptic boundary value problems. The first order shape derivatives of energy functionals in the direction of specific velocity fields depend on the solutions to variational inequalities in a subdomain. The domain decomposition technique is used in order to separate the unilateral boundary conditions and the energy asymptotic analysis. The topological derivatives of nonsmooth integral shape functionals for variational inequalities are derived. The singular geometrical doamin perturbations in an elastic body Ω are approximated by the regular perturbations of bilinear forms in variational inequality, without any loss of precision for the purposes of the second order shape-topological sensitivity analysis. The second-order shape-topological directional derivatives are obtained for the Laplacian and for the linear elasticity in two and three spatial dimensions. In the proposed method of sensitivity analysis, the singular geometrical perturbations ǫ → ωǫ ⊂ Ω centered at x ∈ Ω are replaced by regular perturbations of bilinear forms supported on the manifold ΓR = {|x − x| = R} in an elastic body, with R > ǫ > 0. The obtained expressions for topological derivatives are easy to compute and therefore useful in numerical methods of topological optimization for contact problems