Compliance minimization of thin plates made of material with predefined Kelvin moduli. Part I. Solving the local optimization problem

• Autorzy: Dzierżanowski G. , Lewiński T.
• Języki publikacji: EN

Abstrakty

EN

The paper deals with compliance minimization of a transversely homogeneous plate, subjected to the in-plane and transverse loadings acting simultaneously. The set of design variables includes the eigenstates of Hooke’s tensor whose eigenvalues, i.e. Kelvin moduli fields, are assumed to be fixed on the middle plane of the plate, but no isoperimetric condition is imposed. The optimization task reduces to an equilibrium problem of an effective hyperelastic plate. The effective potential is explicitly expressed in terms of the invariants of both the strain fields involved.

Słowa kluczowe

EN

free material optimization; compliance minimization; anisotropic plates

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Archives of Mechanics
• Rocznik: 2012
• Tom: Vol. 64, nr 1
• Strony: 21--21
• Opis fizyczny: –40, Bibliogr. 40 poz.

Twórcy

autor

Dzierżanowski G.

autor

Lewiński T.

• Warsaw University of Technology Faculty of Civil Engineering, al. Armii Ludowej 16 00-637 Warszawa, Poland,

Bibliografia

• 1. N.V. Banichuk, Optimization of anisotropic properties for continuum bodies and structural elements using spectral methods of tensor analysis, Mech. Struct. Mach. 24, 71–87, 1996.
• 2. M.P. Bendsøe, J.M. Guedes, R.B. Haber, P. Pedersen, J.E. Taylor, An Analytical Model to Predict Optimal Material Properties in the Context of Optimal Structural Design, ASME J. Appl. Mech. 61, 930–937, 1994.
• 3. M.P. Bendsøe, A.R. Diaz, R. Lipton, J.E. Taylor, Optimal design of material properties and material distribution for multiple loading conditions, Int. J. Numer. Meth. Eng. 38, 1149–1170, 1995.
• 4. M.P. Bendsøe, O. Sigmund, Topology Optimization. Theory, Methods and Applications, Springer, Berlin, 2003.
• 5. A. Blinowski, On the decomposition of the isotropic tensorial function in orthogonal bases, Bull. Polish Acad. Sci. Tech. Sci., 27, 11–16, 1980.
• 6. A. Blinowski, J. Ostrowska-Maciejewska, On the elastic orthotropy, Arch. Mech., 48, 129–141, 1996.
• 7. A. Blinowski, J. Ostrowska-Maciejewska, J. Rychlewski, Two-dimensional Hooke’s tensors-isotropic decomposition, effective symmetry criteria, Arch. Mech., 48, 325–345, 1996.
• 8. I.N. Bronstein, K.A. Siemiendjajew, G. Musiol, H. Mühlig, Taschenbuch der Mathematik, Verlag Harri Deutsch, Thun, Frankfurt am Main, 2001.
• 9. S. Czarnecki, T. Lewiński, Free material design formulation controlling Kelvin’s moduli and stiffness distributors, [in:] 7th World Congress on Structural and Multidisciplinary Optimization WCSMO-7, B.M. Kwak [Ed.], (CD-ROM), ISSMO, Seoul, Korea, 2007.
• 10. S. Czarnecki, T. Lewiński, The free material design revisited, [in:] 23rd IFIP TC7 Conference on System Modelling and Optimization, Book of Abstracts, IFIP, A. Korytowski, W. Mitkowski, M. Szymkat [Eds.], pp. 75–76, AGH University of Science and Technology, Cracow, Poland, 2007.
• 11. S. Czarnecki, T. Lewiński, On free material design problem for two loading conditions, [in:] Eighth World Congress of Structural and Multidisciplinary Optimization WCSMO- 8, H.C. Rodrigues, J.M. Guedes, P.R. Fernandes, J.O. Folgado, M.M. Neves [Eds.], (CD-ROM), ISSMO, Lisboa, Portugal, 2009.
• 12. S. Czarnecki, T. Lewiński, The stiffest designs of elastic plates. Vector optimization for two loading conditions. Comp. Meth. Appl. Mech. Engrg., 200, 1708–1728, 2011.
• 13. S. Czarnecki, T. Lewiński, The free material optimization of elastic plates and membrane shells. The case of two loading conditions, [in:] Shell Structures, Theory and Applications vol. 2, W. Pietraszkiewicz, I. Kreja [Eds.], pp. 59–62, CRC Press, Taylor and Francis Group, Balkema, London, 2010.
• 14. J. Du, J.E. Taylor, Application of an energy-based model for the optimal design of structural materials and topology, Struct. Multidiscip. Optimiz., 24, 277–292, 2002.
• 15. G. Dzierżanowski, T. Lewiński, Maximization of stiffness of elastic plates and shells of the Kelvin moduli distributed according to a given pattern, [in:] Shell Structures. Theory and Applications vol. 2, W. Pietraszkiewicz, I. Kreja [Eds.], pp. 79–81, CRC Press, Taylor and Francis Group, Balkema, London, 2010.
• 16. G. Dzierżanowski, T. Lewiński, Optimal orientation of anisotropic material with Niven Kelvin moduli in FMO problems for plates and shells, [in:] 2nd International Conference on Engineering Optimization EngOpt 2010, H. Rodrigues, J. Herskovits, C. Mota Soares [Eds.], Lisboa, Portugal, 2010.
• 17. S. Gaile, G. Leugering, M. Stingl, Free material optimization for plates and shells, [in:] 23rd IFIP TC7 Conference on System Modelling and Optimization, Book of Abstracts, IFIP, A. Korytowski, W. Mitkowski, M. Szymkat [Eds.], pp. 62–63, AGH University of Science and Technology, Cracow, Poland, 2007.
• 18. J.M. Guedes, H.C. Rodrigues, M.P. Bendsøe, A material optimization model to approximate energy bounds for cellular materials under multiload conditions, Struct. Multidiscip. Optimiz., 25, 446–452, 2003.
• 19. J.M. Guedes, E. Lubrano, H.C. Rodrigues, S. Turteltaub, Hierarchical optimization of material and structure for thermal transient problems, [in:] 6th World Congress on Structural and Multidisciplinary Optimization WCSMO-6, J. Herskovits, S. Mazorche, A. Canelas [Eds.], (CD-ROM), ISSMO, Rio de Janeiro, 2005.
• 20. J. Haslinger, M. Kočvara, G. Leugering, M. Stingl, Multidisciplinary Free Material Optimization, SIAM J. Appl. Math., 70, 2709–2728, 2010.
• 21. H.R.E.M. Hörnlein, M. Kočvara, R. Werner, Material optimization: bridging the gap between conceptual and preliminary design, Aeorosp. Sci. Technol., 5, 541–554, 2001.
• 22. M. Kočvara, M. Stingl, Free material optimization for stress constraints, Struct. Multidiscip. Optimiz., 33, 323–335, 2007.
• 23. M. Kočvara, M. Stingl, J. Zowe, Free material optimization: recent progress, Optimization, 57, 79–100, 2008.
• 24. T. Lewiński, The stiffest plates and shells of the uniformly distributed Kelvin moduli, [in:] 8th World Congress on Computational Mechanics (WCCM8), 5th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2008), B.A. Schrefler, U. Perego [Eds.], (CD-ROM), International Center for Numerical Methods in Engineering (CINME), Barcelona, Spain, 2008.
• 25. M.M. Mehrabadi, S.C. Cowin, Eigentensors of linear anistropic elastic materials, Q. J. Mech. Appl. Math., 43, 15–41, 1990.
• 26. M. Moakher, A.N. Norris, The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry, J. Elast., 85, 215–263, 2006.
• 27. A.N. Norris, The isotropic material closest to a given anisotropic material, J. Mech. Mat. Struct., 1, 223–238, 2006.
• 28. U.T. Ringertz, On finding the optimal distribution of material properties, Struct. Optimiz., 5, 265–267, 1993.
• 29. R.T. Rockafellar, Integral functionals, normal integrands and measurable selections, [in:] Nonlinear Operators and the Calculus of Variations, L. Waelbroeck (ed.), Lecture Notes in Mathematics (543), pp. 157-207, Springer, 1976.
• 30. H.C. Rodrigues, J.M. Guedes, M.P. Bendsøe, Hierarchical optimization of material and structure, Struct. Multidiscip. Optimiz., 24, 1–10, 2002.
• 31. J. Rychlewski, On Hooke’s Law, Prikl. Mat. Mekh., 48, 420–435, 1984 [in Russian].
• 32. J. Rychlewski, On thermoelastic constants, Arch. Mech., 36, 77–95, 1984.
• 33. J. Rychlewski, Unconventional approach to linear elasticity, Arch. Mech., 47, 2, 149–171, 1995.
• 34. J. Rychlewski, A qualitative approach to Hooke’s tensor, Arch. Mech., 52, 737–759 (Part I), ibid. 53, 45–63 (Part II), 2000.
• 35. M. Stingl, M. Kočvara, G. Leugering, Theory of multidisciplinary free material optimisation, [in:] Eighth World Congress of Structural and Multidisciplinary Optimization WCSMO-8, H.C. Rodrigues, J.M. Guedes, P.R. Fernandes, J.O. Folgado, M.M. Neves [Eds.], (CD-ROM), ISSMO, Lisboa, Portugal, 2009.
• 36. M. Stingl, M. Kočvara, G. Leugering, A Sequential Convex Semidefinite Programming Algorithm with an Application to Multiple-Load Free Material Optimization, SIAM J.Optim., 20, 130–155, 2009.
• 37. M. Stingl, M. Kočvara, G. Leugering, Free Material Optimization with Fundamental Eigenfrequency Constraints, SIAM J. Optim., 20, 524–547, 2009.
• 38. S. Sutcliffe, Spectral decomposition of the elasticity tensor, ASME, J. Appl. Mech., 59, 762–773, 1992.
• 39. J.E. Taylor, An energy model for the optimal design of linear continuum structures, Struct. Optimiz., 16, 116–127, 1998.
• 40. P.S. Theocaris, D.P. Sokolis, Spectral decomposition of the compliance tensor for anisotropic plates, J. Elast., 51, 89–103, 1998.