Abstract
Shape optimization of the fine scale geometry of elastic objects is investigated under stochastic loading. Thus, the object geometry is described via parametrized geometric details placed on a regular lattice. Here, in a two dimensional set up we focus on ellipsoidal holes as the fine scale geometric details described by the semiaxes and their orientation. Optimization of a deterministic cost functional as well as stochastic loading with risk neutral and risk averse stochastic cost functionals are discussed. Under the assumption of linear elasticity and quadratic objective functions the computational cost scales linearly in the number of basis loads spanning the possibly large set of all realizations of the stochastic loading. The resulting shape optimization algorithm consists of a finite dimensional, constraint optimization scheme where the cost functional and its gradient are evaluated applying a boundary element method on the fine scale geometry. Various numerical results show the spatial variation of the geometric domain structures and the appearance of strongly anisotropic patterns.
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References
Allaire G.: Shape Optimization by the Homogenization Method, vol. 146. Springer Applied Mathematical Sciences, Berlin (2002)
Allaire G., Bonnetier E., Francfort G., Jouve F.: Shape optimization by the homogenization method. Numerische Mathematik 76, 27–68 (1997)
Allaire G., Jouve F.: A level-set method for vibration and multiple loads structural optimization. Comput. Methods Appl. Mech. Eng. 194(30–33), 3269–3290 (2005)
Allaire G., Jouve F., Maillot H.: Topology optimization for minimum stress design with the homogenization method. Struct. Multidisc. Opt. 28, 87–98 (2004)
Allaire G., Kohn R.V.: Optimal design for minimum weight and compliance in plane stress using extremal microstructures. Euro. J. Mech. Solids 12(6), 839–878 (1993)
Ambrosio L., Buttazzo G.: An optimal design problem with perimeter penalization. Calculus Variations Partial Differ. Equ. 1, 55–69 (1993)
Bebendorf, M.: Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems, vol. 63 of Lecture Notes in Computational Science and Engineering (LNCSE). Springer, Berlin, 2008. ISBN 978-3-540-77146-3
Ben-Tal A., El-Ghaoui L., Nemirovski A.: Robust Optimization. Princeton University Press, Princeton (2009)
Bendsøe M.P.: Optimization of structural topology, shape, and material. Springer, Berlin (1995)
Chambolle A.: A density result in two-dimensional linearized elasticity, and applications. Arch. Ration. Mech. Anal. 167(3), 211–233 (2003)
Cherkaev, A., Cherkaeva, E.: Optimal design for uncertain loading conditions. In: V. Berdichevsky, V. Jikov, G. Papanicolaou (eds.) “Homogenization”, vol. 50 of Series on Advances in Mathematics for Applied Sciences, pp. 193–213 World Scientific, Singapore (1999)
Cherkaev A., Cherkaeva E.: Principal compliance and robust optimal design. J. Elast. 72, 71–98 (2003)
Ciarlet, P.G.: Mathematical Elasticity Volume I: Three-Dimensional Elasticity, vol. 20. Studies in Mathematics and its Applications, North-Holland (1988)
Clements D., Rizzo F.: A method for the numerical solution of boundary value problems governed by second-order elliptic systems. IMA J. Appl. Math. 22, 197–202 (1978)
Conti S., Held H., Pach M., Rumpf M., Schultz R.: Shape optimization under uncertainty—a stochastic programming perspective. SIAM J. Optim. 19, 1610–1632 (2009)
Conti S., Held H., Pach M., Rumpf M., Schultz R.: Risk averse shape optimization. SIAM J. Cont. Optim. 49(3), 927–947 (2011)
de Gournay F., Allaire G., Jouve F.: Shape and topology optimization of the robust compliance via the level set method. ESAIM: Cont. Optim. Calculus Var. 14, 43–70 (2007)
Delfour, M.C., Zolésio, J.: Shapes and geometries: analysis, differential calculus and optimization. Adv. Des. Control 4. SIAM, Philadelphia (2001)
Evans, L.C.: Partial Differential Equations, vol. 19. AMS Graduate Studies in Mathematics (2002)
Guedes J.M., Rodrigues H.C., Bendsøe M.P.: A material optimization model to approximate energy bounds for cellular materials under multiload conditions. Struct. Multidiscip. Optim. 25, 446–452 (2003)
Hackbusch W.: Integral Equations, vol. 120 of International Series of Numerical Mathematics. Birkhäuser Verlag, Basel (1995)
Huyse, L.: Free-form airfoil shape optimization under uncertainty using maximum expected value and second-order second-moment strategies. ICASE report ; no. 2001-18. ICASE, NASA Langley Research Center Available from NASA Center for Aerospace Information, Hampton, VA (2001)
Lenz, M.: Modellierung und Simulation des effektiven Verhaltens von Grenzflächen in Metalllegierungen. Dissertation, Universität Bonn (2007)
Marsden J., Hughes T.: Mathematical Foundations of Elasticity. Dover Publications Inc., New York (1993)
Michell, A.: LVIII. The limits of economy of material in frame-structures. Philosophical Magazine Series 6 8, 47, 589–597 (1904)
Nocedal J., Wright S.J.: Numerical Optimization. Springer Series in Operations Research, Berlin (1999)
Penzler P., Rumpf M., Wirth B.: A phase-field model for compliance shape optimization in nonlinear elasticity. ESAIM: COCV 18(1), 229–258 (2012)
Pflug G.C., Römisch W.: Modeling, Measuring and Managing Risk. World Scientific, Singapore (2007)
Schultz R., Tiedemann S.: Conditional value-at-risk in stochastic programs with mixed-integer recourse. Math. Program. 105(2–3), 365–386 (2006)
Shapiro A., Dentcheva D., Ruszczyński A.: Lectures on Stochastic Programming. SIAM-MPS, Philadelphia (2009)
Shewchuk J.: Triangle: engineering a 2d quality mesh generator and delaunay triangulator. Appl. Comput. Geom. Towards Geom. Eng. 1148, 203–222 (1996)
Shewchuk J.: Delaunay refinement algorithms for triangular mesh generation. Comput. Geom. Theory Appl. 22, 21–74 (2002)
Sokołowski J., Zolésio J.-P.: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, Berlin (1992)
Suzuki K., Kikuchi N.: A homogenization method for shape and topology optimization. Comput. Methods Appl. Mech. Eng. 93(3), 291–318 (1991)
Wächter, A.: An Interior Point Algorithm for Large-Scale Nonlinear Optimization with Applications in Process Engineering. Phd thesis, Carnegie Mellon University (2002)
Wächter A., Biegler L.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)
Zhuang C., Xiong Z., Ding H.: A level set method for topology optimization of heat conduction problem under multiple load cases. Comput. Methods Appl. Mech. Eng. 196, 1074–1084 (2007)
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Geihe, B., Lenz, M., Rumpf, M. et al. Risk averse elastic shape optimization with parametrized fine scale geometry. Math. Program. 141, 383–403 (2013). https://doi.org/10.1007/s10107-012-0531-1
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DOI: https://doi.org/10.1007/s10107-012-0531-1