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Structural design optimization, homogenization and relaxation of variational problems

  • Robert V. Kohn
  • Gilbert Strang
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 154)

Keywords

Effective Equation Torsional Rigidity Antiplane Shear Shape Optimization Problem Measurable Partition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1).
    Lagrange, J.L., “Sur la figure des colonnes”, Miscellanea Taurinensia V, 1770-1773, pg. 123.Google Scholar
  2. 2).
    Michell, A.G.M., “The limits of economy of material in frame structures”, Phil. Mag. S6, Vol 8, No. 47, pp. 589–597.Google Scholar
  3. 3).
    Haug, E.J., and Cea, J., Proceedings of NATO ASI on optimization of distributed parameter structures, Iowa City, 1980. Sithoff and Nordhoff, to appear.Google Scholar
  4. 4).
    Sawzuk, A. and Mroz, Z., Optimization in structural design (Proceedings of 1973 IUTAM symposium in Warsaw, Poland), Springer-Verlag, 1975.Google Scholar
  5. 5).
    Prager, W., Introduction to structural optimization, International Centre for Mechanical Science, Udine, Courses and Lectures no. 212, Springer-Verlag, 1974.Google Scholar
  6. 6).
    Rozvany, G.I.N., Optimal design of flexural systems, Pergamon Press, 1976.Google Scholar
  7. 7).
    Cea, J. and Malanowski, K., “An example of a max-min problem in partial differential equations”, SIAM J. Control vol 8, 1970, pp. 305–316.Google Scholar
  8. 8).
    Klosowicz, B. and Lurie, K.A., “On the optimal nonhomogeneity of a torsional elastic bar”, Arch. of Mechanics, vol 24, 1971, pp 239–249.Google Scholar
  9. 9).
    Mroz, Z., “Limit analysis of plastic structures subject to boundary variations”, Arch. Mech. Stos. vol 15, 1963, pp. 63–75.Google Scholar
  10. 10).
    Zavelani, A., Maier, G., and Binda, L., “Shape optimization of plastic structures by zero-one programming”, in IUTAM Warsaw Symposium, 1973, see reference 3.Google Scholar
  11. 11).
    Lurie, K.A., Fedorov, A.V., and Cherkaev, A.V., “Regularization of optimal design problems for bars and plates and elimination of contradictions within the necessary conditions of optimality”, Journal of Opt. Th. and Appl., to appear, 1982Google Scholar
  12. 12).
    Cea, J., “Shape optimal design: problems and numerical methods”, in Proc. of NATO-ASI on optimization of Distributed Parameter Structures, Iowa City, 1980Google Scholar
  13. 13).
    Olhoff, N., “Optimization of columns against buckling“ in Proceedings of NATO-ASI on optimization of distributed parameter structures, 1980.Google Scholar
  14. 14).
    Cinquini, C. and Mercier, B., “Minimal cost and elastoplastic structures”, Meccanica vol 11, no 4, 1976, pp 219–226.Google Scholar
  15. 15).
    Wheeler, L., “On the role of constant-stress surfaces in the problem of minimizing elastic stress concentration.”, International J. of Solids and Structures vol 12, 1967, pp 779–789.Google Scholar
  16. 16).
    Olhoff, N. and Taylor, J.E., J. Opt. Theory and Applic. Vol 27, 1979, pp 571–582.Google Scholar
  17. 17).
    Fleury, C. and Schmit, L.A. “Primal and dual methods in structural optimization,” J. Structural Div. ASCE vol 106, 1980, pp 1117–1133.Google Scholar
  18. 18).
    Cheng, K.T. and Olhoff, N., “An investigation concerning the optimal design of solid elastic plates”, Int. J. of Solids and Structures, to appear.Google Scholar
  19. 19).
    Cheng, K.T. and Olhoff, N., “Regularized formulation for optimal design of axisymmetric plates”, to appear.Google Scholar
  20. 20).
    Murat, F., “Contre-examples pour divers problemes ou le control intervient dans les coefficients”, Annali di Mat. Pura ed Appl. Ser 4, vol 112–113, 1977.Google Scholar
  21. 21).
    Simon, L., “On G-convergence of elliptic operators,” Indiana Univ. Math. Journal vol 28, pp 587–594.Google Scholar
  22. 22).
    De Giorgi, E., “Convergence problems for functionals and operators”, in Proc. of the International Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978; De Giorgi, Magenes, & Mosco editors, Pitagora Editrice, Bologna, 1980.Google Scholar
  23. 23).
    Duvaut, G., “Comportement microscopique d'une plaque perforée périodiquement”, to appear.Google Scholar
  24. 24).
    Cioranescu, D. and Saint Jean Paulin, J., “Homogenization dans des ouverts à cavités,” C.R. Acad. Sci. Paris A, vol 284 (1977) pp 857–860.Google Scholar
  25. 25).
    Carbone, L., “Sur un problème d'homogénéisation avec des constraints sur le gradient”, J. Math. Pures et Appl. 58, 1979, pp 275–297.Google Scholar
  26. 26).
    Strang, G. and Kohn, R., “Optimal design of cylinders in shear”, to appear in proceedings of the 1981 MAFELAP Conference, Brunel University.Google Scholar
  27. 27).
    Kohn, R. and Strang, G., “Optimal design for torsional rigidity”, to appear in proceedings of the conference on Mixed and Hybrid Methods in Finite Element Methods, Atlanta, 1981.Google Scholar
  28. 28).
    Cherepanov, G.P., “Inverse problems of the plane theory of elasticity”, P.M.M. vol 38, no 6, 1974, pp 963–979.Google Scholar
  29. 29).
    Acker, A., “Interior free boundary problems for the Laplace equation”, Arch. Rat. Mech. Anal. vol 75, 1981, pp 157–168.Google Scholar
  30. 30).
    Alt, H.W. and Caffarelli, L.A., “Existence and regularity for a minimum problem with free boundary”, J. Riene Angew. Math. 325 (1981) pg 105.Google Scholar
  31. 31).
    Banichuk, N., Doklady Akad. Nauk USSR vol 242, pp 1042–1045, 1978.Google Scholar
  32. 32).
    Jouron, C., “Sur un problème d'optimisation où la constrainte porte sur la fréquence fondamentale”, RAIRO Analyse Numerique vol 12, 1978, pp 349–376.Google Scholar
  33. 33).
    Murat, F., “Compacite par compensation”, Ann. Scuola Norm.Sup Pisa vol 5, 1978, pp. 489–507.Google Scholar
  34. 34).
    Murat, F., “Compacite par compensation II”, in proc. of the International Meeting on Recent Methods in Nonlinear Analysis, De Giorgi, Magenes, Mosco editors, Pitagora Editrice, Bologna, 1980.Google Scholar
  35. 35).
    Tartar, L., “Estimation de coefficients homogeneises”, Springer Lect. Notes in Math. vol 704, pp. 364–373.Google Scholar
  36. 36).
    Benedict, R.L., “Optimal design for elastic bodies in contact”, in proceedings of the NATO-ASI on optimization of distributed parameter structures, (ref.3).Google Scholar
  37. 37).
    Banichuk, N.V.; Kartvelishvili, V.M.; and Mironov, A.A., “Optimization problems with local performance criteria in the theory of plate bending”, Mechanics of Solids, 1978, no 1.Google Scholar
  38. 38).
    Ekeland, I. and Temam, R., Convex Analysis and Variational Problems, North-Holland, 1976.Google Scholar
  39. 39).
    Vogelius, M.; Kohn, R.; Papanicolaou, G., “Effective equations for plates and beams with rapidly varying thickness”, to appear.Google Scholar
  40. 40).
    Olhoff, N.; Lurie, K.A.; Cherkaev, A.V.; Fedorov, A.V.; “Sliding Regimes and Anisotropy in Optimal Design of Vibrating Axisymmetric Plates”, Int. J. Solids Structures, to appear.Google Scholar
  41. 41).
    Raitum, U.E., “On optimal control problems for linear elliptic equations”, Soviet Math. Dokl. Vol 20, pp 129–132, 1979.Google Scholar
  42. 42).
    Raitum, U.E., “The extension of extremal problems connected with a linear elliptic equation”, Soviet Math. Dokl. Vol 19, pp 1342–1345, 1978.Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Robert V. Kohn
    • 1
  • Gilbert Strang
    • 2
  1. 1.Courant Institute of Mathematical SciencesUSA
  2. 2.Massachusetts Institute of TechnologyUSA

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