Computational and Structural Optimization

  • Georgios E. Stavroulakis
Part of the Applied Optimization book series (APOP, volume 46)

Abstract

Optimization deals with the determination of the extremum or the extrema of a given function over the space where the function is defined or over a subset of it. Several optimization problems arise in nature and they are known, mainly for historical reasons, as principles. The principles of minimum potential energy in statics, the maximum dissipation principle in dissipative media and the least action principle in dynamics are some examples (see, among others, [Hamel, 1949], [Lippmann, 1972], [Cohn and Maier, 1979], [de Freitas, 1984], [de Freitas and Smith, 1985], [Panagiotopoulos, 1985], [Hartmann, 1985], [Sewell, 1987], [Bazant and Cedolin, 1991]). Furthermore, mathematical optimization is tightly connected with optimal structural design, control and identification. Applications include contemporary questions in biomechanics, like the understanding of the inner structure in bones [Wainwright and et.al., 1982] or of the shape in trees [Mattheck, 1997].

Keywords

Assure Liner Lution Excavation 

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References

  1. Aubin, J. P. (1993). Optima and equilibria Springer Verlag, Berlin.Google Scholar
  2. Aubin, P. and Frankowska, H. (1991). Set—valued analysis Birkhäuser, Berlin-Heidelberg.Google Scholar
  3. Baiant, Z. P. and Cedolin, L. (1991). Stability of structures. Elastic, inelastic, fracture and damage theories Oxford University Press, New York, Oxford.Google Scholar
  4. Bertsekas, D. P. (1982). Constrained optimization and Lagrange multiplier methods Academic Press, New York.Google Scholar
  5. Bolzon, G., Ghilotti, D., and Maier, G. (1997). Parameter identification of the cohesive crack model. In Sol, H. and Oomens, C., editors, Material identification using mixed numerical experimental methods, pages 213-222, Dordrecht. Kluwer Academic Publishers.Google Scholar
  6. Brousse, P. (1991). Optimization in mechanics: problems and methods Elsevier Science Publications, Amsterdam.Google Scholar
  7. Ciarlet, P. G. (1989). Introduction to numerical linear algebra and optimization Cambridge University Press, Cambridge.Google Scholar
  8. Clarke, F. H. (1983). Optimization and nonsmooth analysis J. Wiley, New York.Google Scholar
  9. Cohn, M. Z. and Maier, G., editors (1979). Engineering plasticity by mathematical programming Pergamon Press, Oxford.Google Scholar
  10. Cottle, R. W., Pang, J. S., and Stone, R. E. (1992). The linear complementarity problem Academic Press, Boston.Google Scholar
  11. de Freitas, J. A. T. (1984). A general methodology for nonlinear structuralanalysis by mathematical programming. Engineering Structures, 6:52–60.CrossRefGoogle Scholar
  12. de Freitas, J. A. T. and Smith, D. L. (1985). Energy theorems for elastoplastic structures in a regime of large displacements. J. de mecanique theorique et ppliqee, 4(6):769-784.MATHGoogle Scholar
  13. Dem’ yanov, V. F. and Rubinov, A. M. (1995). Introduction to constructive nonsmooth analysis Peter Lang Verlag, Frankfurt-Bern-New York.Google Scholar
  14. Dem’yanov, V. F., Stavroulakis, G. E., Polyakova, L. N., and Panagiotopoulos, P. D. (1996) Quasidifferentiability and nonsmooth modelling in mechanics, engineering and economics. Kluwer Academic, Dordrecht CrossRefMATHGoogle Scholar
  15. Dem’yanov, V. F. and Vasiliev, L. N. (1985) Nondifferentiable optimization. Optimization Software, New York CrossRefGoogle Scholar
  16. Ekeland, I. and Temam, R. (1976) Convex analysis and variational problems. North-Holland, Amsterdam MATHGoogle Scholar
  17. Elster, K.-H., Reinhardt, R., Schäuble, M., and Donath, G. (1977). Einführung in die nichtlineare Optimierung BSB B.G. Teubner Verlagsgesellschaft, Leipzig.Google Scholar
  18. Facchinei, F., Jiang, H., and Qi, L. (1999). A smoothing method for mathematical programs with equilibrium constraints. Mathematical Programming, 85:107-134.MathSciNetCrossRefMATHGoogle Scholar
  19. Ferris, M. and Tin-Loi, F. (1999). On the solution of a minimum weight elastoplastic problem involving displacement and complementarity constraints.Computer Methods in Applied Mechanics and Engineering, 174:107–120.CrossRefMATHGoogle Scholar
  20. Fletcher, R. (1990). Practical methods of optimization J. Wiley, Chichester. Friedman, A. (1982). Variational principles and free boundary problems. JWiley, New York.Google Scholar
  21. Galka, A. and Telega, J. (1995). Duality and the complementary energy principle for a class of nonlinear structures. part is Five-parameter shell model. part ii: Anomalous dual variational principles for compressed elastic beams. Archives of Mechanics, 47(4):677–724.Google Scholar
  22. Gao, D. (1998). Duality, triality and complementary extremum principles in non-convex parametric variational problems with applications IMA Journal of Applied Mathematics, 61(3):199–236 MathSciNetCrossRefMATHGoogle Scholar
  23. Giannessi, F., Jurina, L., and Maier, G. (1978). Optimal excavation profile or a pipeline freely resting on a sea floor. In 4 Congresso Nationale di Meccanica Teorica ed Applicata, pages 281–296 AIMETA.Google Scholar
  24. Giannessi, F., Jurina, L., and Maier, G. (1982). A quadratic complementarity problem related to the optimal design of a pipeline freely resting on a rough sea bottom. Engineering Structures, 4:186–196.Google Scholar
  25. Gill, P. E., Murray, W., and Wright, M. H. (1981). Practical optimization Academic Press, New York.Google Scholar
  26. Givoli, D. (1999). A direct approach to the finite element solution of elliptic optimal control problems Numerical Methods for Partial Differential Equations, 15(3):371–388 MathSciNetCrossRefMATHGoogle Scholar
  27. Hamel, G. (1949). Theoretische Mechanik Springer Verlag, Berlin. Hartmann, F. (1985). The mathematical foundation of structural mechanics Springer Verlag, Berlin.Google Scholar
  28. Haslinger, J., Miettinen, M., and Panagiotopoulos, P. (1999). Finite Element Approximation of Hemivariational Inequalities: Theory, Numerical Methods and Applications Kluwer Academic Publishers, Dordrecht.Google Scholar
  29. Haslinger, J. and Neittaanmäki, P. (1996). Finite element approximation for optimal shape, material and topology design J. Wiley and Sons, Chichester. (2nd edition).Google Scholar
  30. Hilding, D., Klarbring, A., and Pang, J.-S. (1999a) Minimization of maximum unilateral force. Computer Methods in Applied Mechanics and Engineering, 177:215–234.MathSciNetCrossRefMATHGoogle Scholar
  31. Hilding, D., Klarbring, A., and Petersson, J. (1999b). Optimization of structures in unilateral contact. ASME Applied Mechanics Review, 52(4):139–160.CrossRefGoogle Scholar
  32. Hiriart-Urruty, J. B. and Lemaréchal, C. (1993). Convex analysis and minimization algorithms I Springer, Berlin-Heidelberg.Google Scholar
  33. Hlavaèek, I., Haslinger, J., Necas, J., and Lovisek, J. (1988). Solution of variational inequalities in mechanics, volume 66 of Appl. Math. Sci Springer.Google Scholar
  34. Kiwiel, K. C. (1985). Methods of descent for nondifferentiable optimizationSpringer, Berlin. Lecture notes in mathematics No. 1133.Google Scholar
  35. Kornhuber, R. (1997). Adaptive monotone multigrid methods for nonlinear variational problems B.G. Teubner, Stuttgart.Google Scholar
  36. Lippmann, H. (1972). Extremum and variational principles in mechanics Springer CISM Courses and Lectures 54, Wien.Google Scholar
  37. Luo, Z. Q., Pang, J. S., and Ralph, D. (1996). Mathematical programs with equilibrium constraints Cambridge University Press, Cambridge.Google Scholar
  38. Maier, G. (1982). Inverse problem in engineering plasticity: a quadratic programming approach. Accademia Nazionale die Lincei, Serie VII Volume LXX:203–209.Google Scholar
  39. Maier, G., Giannessi, F., and Nappi, A. (1982). Indirect identification of yield limits by mathematical programming. Engineering Structures, 4:86-99.CrossRefGoogle Scholar
  40. Mattheck, C. (1997). Design in der Natur: der Baum als Lehrmeister Rombach,Freiburg im Breisgau.Google Scholar
  41. Matthies, H. G., Strang, G., and Christiansen, E. (1979). The saddle point of a differential problem. In Glowinski, R., Rodin, E., and Zienkiewicz, O., editors, Energy methods in finite element analysis, New York. J. Wiley and Sons.Google Scholar
  42. Migdalas, A., Pardalos, P., and Värbrand, P. (1997). Multilevel optimization:algorithms and applications Kluwer Academic Publishers, Dordrecht.Google Scholar
  43. Mistakidis, E. and Stavroulakis, G. (1998). Nonconvex optimization in mechanics. Algorithms, heuristics and engineering applications by the F.E.M Kluwer Academic, Dordrecht and Boston and London.Google Scholar
  44. Moreau, J. J. (1963). Fonctionnelles sous - différentiables. C R Acad. Sc. Paris, 257A:4117–4119.Google Scholar
  45. Murty, K. G. (1988). Linear complementarity, linear and nonlinear programming Heldermann, Berlin.Google Scholar
  46. Neittaanmäki, P., Rudnicki, M., and Savini, A. (1996). Inverse problems and optimal design in electricity and magnetism Oxford University Press, New York.Google Scholar
  47. Oden, J. T. and Reddy, J. N. (1982). Variational methods in theoretical mechanics Springer Verlag, Berlin.Google Scholar
  48. Outrata, J., Kocvara, M., and Zowe, J. (1998). Nonsmooth approach to optimization problems with equilibrium constraints: theory, applications, and numerical results Kluwer Academic Publishers, Dordrecht.Google Scholar
  49. Panagiotopoulos, P. D. (1985). Inequality problems in mechanics and applications. Convex and nonconvex energy functions Birkhäuser, Basel - Boston - Stuttgart. Russian translation, MIR Publ., Moscow 1988.Google Scholar
  50. Panagiotopoulos, P. D. (1993). Hemivariational inequalities. Applications inmechanics and engineering Springer, Berlin —Heidelberg —NewYork.Google Scholar
  51. Rockafellar, R. T. (1970). Convex analysis Princeton University Press, Princeton.Google Scholar
  52. Rockafellar, R. T. (1982). Network flows and monotropic optimization J. Wiley, New York.Google Scholar
  53. Rodrigues, J. F. (1987). Obstacle problems in mathematical physics North Holland, Amsterdam.Google Scholar
  54. Schramm, H. and Zowe, J. (1992). A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results. SIAM J. Optimization, 2:121–152.MathSciNetCrossRefMATHGoogle Scholar
  55. Sewell, N. J. (1987). Maximum and minimum principles. A unified approach with applications Cambridge University Press, Cambridge.Google Scholar
  56. Shimizu, K., Ishizuka, Y., and Bard, J. (1996). Nondifferentiable and two-level mathematical programming Kluwer Academic Publishers, Dordrecht.Google Scholar
  57. Shor, N. Z. (1985). Minimization methods fornondifferentiable functions Sprin—ger, Berlin.Google Scholar
  58. Stavroulakis, G. E. (1993). Convex decomposition for nonconvex energy problems in elastostatics and applications. European Journal of Mechanics A / Solids, 12(1):1–20.MathSciNetMATHGoogle Scholar
  59. Stavroulakis, G. E. (1995a). Optimal prestress of cracked unilateral structures: finite element analysis of an optimal control problem for variational inequalities. Computer Methods in Applied Mechanics and Engineering, 123:231–246.MathSciNetCrossRefMATHGoogle Scholar
  60. Stavroulakis, G. E. (1995b). Optimal prestress of structures with frictional unilateral contact interfaces. Archives of Mechanics (Ing. Archiv), 66:71–81.Google Scholar
  61. Tin-Loi, B. (1999a). On the numerical solution of a class of unilateral contact structural optimization problems. Structural Optimization, 17:155–161.Google Scholar
  62. Tin-Loi, B. (1999b). A smoothing scheme for a minimum weight problem in structural plasticity. Structural Optimization, 17:279–285.CrossRefGoogle Scholar
  63. Wainwright, S. and et.al. (1982). Mechanical design in organisms Princeton University Press, Princeton New Jersey.Google Scholar
  64. Washizu, K. (1968).Variational methods in elasticity and plasticity Pergamon Press, Oxford.Google Scholar
  65. Womersley, R. S. and Fletcher, R. (1986). An algorithm for composite nonsmooth optimization problems. Journal of Optimization Theory and Applications, 48:493–523.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Georgios E. Stavroulakis
    • 1
  1. 1.Institute of Applied Mathematics, Department of Civil EngineeringTechnical University Carolo WilhelminaBraunschweigGermany

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