In most shape optimization problems, the optimal solution does not belong to the set of genuine shapes but is a composite structure. the homogenization method consists in relaxing the original problem thereby extendin...
详细信息
In most shape optimization problems, the optimal solution does not belong to the set of genuine shapes but is a composite structure. the homogenization method consists in relaxing the original problem thereby extending the set of admissible structures to composite shapes. From the numerical viewpoint, an important asset of the homogenization method with respect to traditional geometrical optimization is that the computed optimal shape is quite independent from the initial guess (at least for the compliance minimization problem). Nevertheless, the optimal shape being a composite, a post-treatement is needed in order to produce an almost optimal non-composite (i.e. workable) shape. the classical approach consists in penalizing the intermediate densities of material, but the obtained result deeply depends on the underlying mesh used and the level of details is not controllable. In a previous work, we proposed a new post-treatement method for the compliance minimization problem of an elastic structure. the main idea is to approximate the optimal composite shape with a locally periodic composite and to build a sequence of genuine shapes converging toward this composite structure. this method allows us to balance the level of details of the final shape and its optimality. Nevertheless, it was restricted to particular optimal shapes, depending on the topological structure of the lattice describing the arrangement of the holes of the composite. In this article, we lift this restriction in order to extend our method to any optimal composite structure for the compliance minimization problem.
the 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 S...
详细信息
the 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. Sokolowski, SIAM J. Math. Anal. 40:1152–1200, 2008;P.I. Plotnikov, E.V. Ruban, J. Sokolowski, J. Math. Pures Appl. 92:113–162, 2009;P.I. Plotnikov, J. Sokolowski, Dokl. Akad. Nauk 397:166–169, 2004;P.I. Plotnikov, J. Sokolowski, J. Math. Fluid Mech. 7:529–573, 2005;P.I. Plotnikov, J. Sokolowski, Comm. Math. Phys. 258:567– 608, 2005;P.I. Plotnikov, J. Sokolowski, SIAM J. Control Optim. 45:1165–1197, 2006;P.I. Plotnikov, J. Sokolowski, Uspekhi Mat. Nauk 62:117–148, 2007;P.I. Plotnikov, J. Sokolowski, 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. Sokolowski, SIAM J. Control Optim. 48:4680–4706, 2010;P.I. Plotnikov, J. Sokolowski, J. Math. Sci. 170:34–130, 2010). the shape derivatives (J. Sokolowski, 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. Sokolowski, 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. Sokolowski, A. Żochowski, Numerical experiments in drag minimization for compressible Navier-Stokes flows in bounded domains, in Proceedings of the 14thinternational IEEE/IFAC conference on methods and models in automation and robotics, mmar’09, 2009, 4 pp;
暂无评论