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].
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References
Aubin, J. P. (1993). Optima and equilibria Springer Verlag, Berlin.
Aubin, P. and Frankowska, H. (1991). Set—valued analysis Birkhäuser, Berlin-Heidelberg.
Baiant, Z. P. and Cedolin, L. (1991). Stability of structures. Elastic, inelastic, fracture and damage theories Oxford University Press, New York, Oxford.
Bertsekas, D. P. (1982). Constrained optimization and Lagrange multiplier methods Academic Press, New York.
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.
Brousse, P. (1991). Optimization in mechanics: problems and methods Elsevier Science Publications, Amsterdam.
Ciarlet, P. G. (1989). Introduction to numerical linear algebra and optimization Cambridge University Press, Cambridge.
Clarke, F. H. (1983). Optimization and nonsmooth analysis J. Wiley, New York.
Cohn, M. Z. and Maier, G., editors (1979). Engineering plasticity by mathematical programming Pergamon Press, Oxford.
Cottle, R. W., Pang, J. S., and Stone, R. E. (1992). The linear complementarity problem Academic Press, Boston.
de Freitas, J. A. T. (1984). A general methodology for nonlinear structuralanalysis by mathematical programming. Engineering Structures, 6:52–60.
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.
Dem’ yanov, V. F. and Rubinov, A. M. (1995). Introduction to constructive nonsmooth analysis Peter Lang Verlag, Frankfurt-Bern-New York.
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
Dem’yanov, V. F. and Vasiliev, L. N. (1985) Nondifferentiable optimization. Optimization Software, New York
Ekeland, I. and Temam, R. (1976) Convex analysis and variational problems. North-Holland, Amsterdam
Elster, K.-H., Reinhardt, R., Schäuble, M., and Donath, G. (1977). Einführung in die nichtlineare Optimierung BSB B.G. Teubner Verlagsgesellschaft, Leipzig.
Facchinei, F., Jiang, H., and Qi, L. (1999). A smoothing method for mathematical programs with equilibrium constraints. Mathematical Programming, 85:107-134.
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.
Fletcher, R. (1990). Practical methods of optimization J. Wiley, Chichester. Friedman, A. (1982). Variational principles and free boundary problems. JWiley, New York.
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.
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
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.
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.
Gill, P. E., Murray, W., and Wright, M. H. (1981). Practical optimization Academic Press, New York.
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
Hamel, G. (1949). Theoretische Mechanik Springer Verlag, Berlin. Hartmann, F. (1985). The mathematical foundation of structural mechanics Springer Verlag, Berlin.
Haslinger, J., Miettinen, M., and Panagiotopoulos, P. (1999). Finite Element Approximation of Hemivariational Inequalities: Theory, Numerical Methods and Applications Kluwer Academic Publishers, Dordrecht.
Haslinger, J. and Neittaanmäki, P. (1996). Finite element approximation for optimal shape, material and topology design J. Wiley and Sons, Chichester. (2nd edition).
Hilding, D., Klarbring, A., and Pang, J.-S. (1999a) Minimization of maximum unilateral force. Computer Methods in Applied Mechanics and Engineering, 177:215–234.
Hilding, D., Klarbring, A., and Petersson, J. (1999b). Optimization of structures in unilateral contact. ASME Applied Mechanics Review, 52(4):139–160.
Hiriart-Urruty, J. B. and Lemaréchal, C. (1993). Convex analysis and minimization algorithms I Springer, Berlin-Heidelberg.
Hlavaèek, I., Haslinger, J., Necas, J., and Lovisek, J. (1988). Solution of variational inequalities in mechanics, volume 66 of Appl. Math. Sci Springer.
Kiwiel, K. C. (1985). Methods of descent for nondifferentiable optimizationSpringer, Berlin. Lecture notes in mathematics No. 1133.
Kornhuber, R. (1997). Adaptive monotone multigrid methods for nonlinear variational problems B.G. Teubner, Stuttgart.
Lippmann, H. (1972). Extremum and variational principles in mechanics Springer CISM Courses and Lectures 54, Wien.
Luo, Z. Q., Pang, J. S., and Ralph, D. (1996). Mathematical programs with equilibrium constraints Cambridge University Press, Cambridge.
Maier, G. (1982). Inverse problem in engineering plasticity: a quadratic programming approach. Accademia Nazionale die Lincei, Serie VII Volume LXX:203–209.
Maier, G., Giannessi, F., and Nappi, A. (1982). Indirect identification of yield limits by mathematical programming. Engineering Structures, 4:86-99.
Mattheck, C. (1997). Design in der Natur: der Baum als Lehrmeister Rombach,Freiburg im Breisgau.
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.
Migdalas, A., Pardalos, P., and Värbrand, P. (1997). Multilevel optimization:algorithms and applications Kluwer Academic Publishers, Dordrecht.
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.
Moreau, J. J. (1963). Fonctionnelles sous - différentiables. C R Acad. Sc. Paris, 257A:4117–4119.
Murty, K. G. (1988). Linear complementarity, linear and nonlinear programming Heldermann, Berlin.
Neittaanmäki, P., Rudnicki, M., and Savini, A. (1996). Inverse problems and optimal design in electricity and magnetism Oxford University Press, New York.
Oden, J. T. and Reddy, J. N. (1982). Variational methods in theoretical mechanics Springer Verlag, Berlin.
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.
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.
Panagiotopoulos, P. D. (1993). Hemivariational inequalities. Applications inmechanics and engineering Springer, Berlin —Heidelberg —NewYork.
Rockafellar, R. T. (1970). Convex analysis Princeton University Press, Princeton.
Rockafellar, R. T. (1982). Network flows and monotropic optimization J. Wiley, New York.
Rodrigues, J. F. (1987). Obstacle problems in mathematical physics North Holland, Amsterdam.
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.
Sewell, N. J. (1987). Maximum and minimum principles. A unified approach with applications Cambridge University Press, Cambridge.
Shimizu, K., Ishizuka, Y., and Bard, J. (1996). Nondifferentiable and two-level mathematical programming Kluwer Academic Publishers, Dordrecht.
Shor, N. Z. (1985). Minimization methods fornondifferentiable functions Sprin—ger, Berlin.
Stavroulakis, G. E. (1993). Convex decomposition for nonconvex energy problems in elastostatics and applications. European Journal of Mechanics A / Solids, 12(1):1–20.
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.
Stavroulakis, G. E. (1995b). Optimal prestress of structures with frictional unilateral contact interfaces. Archives of Mechanics (Ing. Archiv), 66:71–81.
Tin-Loi, B. (1999a). On the numerical solution of a class of unilateral contact structural optimization problems. Structural Optimization, 17:155–161.
Tin-Loi, B. (1999b). A smoothing scheme for a minimum weight problem in structural plasticity. Structural Optimization, 17:279–285.
Wainwright, S. and et.al. (1982). Mechanical design in organisms Princeton University Press, Princeton New Jersey.
Washizu, K. (1968).Variational methods in elasticity and plasticity Pergamon Press, Oxford.
Womersley, R. S. and Fletcher, R. (1986). An algorithm for composite nonsmooth optimization problems. Journal of Optimization Theory and Applications, 48:493–523.
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Stavroulakis, G.E. (2001). Computational and Structural Optimization. In: Inverse and Crack Identification Problems in Engineering Mechanics. Applied Optimization, vol 46. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0019-3_3
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