Quasidifferentiability in nonsmooth, nonconvex mechanics
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Abstract
Nonconvex and nonsmooth optimization problems arise in advanced engineering analysis and structural analysis applications. In fact the set of inequality and complementarity relations that describe the structural analysis problem are generated as optimality conditions by the quasidifferential potential energy optimization problem. Thus new kind of variational expressions arise for these problems, which generalize the classical variational equations of smooth mechanics, the variational inequalities of convex, nonsmooth mechanics and give a solid, computationally efficient explication of hemivariational inequalities of nonconvex, nonsmooth mechanics. Moreover quasidifferential calculus and optimization software make this approach applicable for a large number of problems. The connection of quasidifferential optimization and nonsmooth, nonconvex mechanics is discussed in this paper. A number of representative examples from elastostatic analysis applications are treated in details. Numerical examples illustrate the theory.
Key words
quasidifferentiability codifferentiability d.c. optimization nonconvex energy nonsmooth mechanics variational inequalities hemivariational inequalitiesPreview
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References
- 1.Curnier, A. (1993),Méthodes numériques en mécanique des solides, Presses polytechniques et universitaires romandes, Lausanne.Google Scholar
- 2.Demyanov, V.F. (1989), Codifferentiability and codifferentials of nonsmooth functions,Soviet Math. Dokl.,38(3), 631–634.Google Scholar
- 3.Demyanov, V.F. and Dixon L.C.W. (eds.) (1986),Quasidifferential calculus, Mathematical Programming Study 29, North-Holland, AmsterdamGoogle Scholar
- 4.Demyanov, V.F. and Vasiliev, L.N. (1985),Nondifferentiable Optimization, Optimization Software, New York.Google Scholar
- 5.Demyanov, V.F. and Rubinov, A.M. (1983), On quasidifferentiable mappings,Math. Operationsforsch. u. Statistik. Serie Optimization 14, 3–21.Google Scholar
- 6.Demyanov, V.F. and Rubinov, A.M. (1985),Quasidifferentiable Calculus, Optimization Software, New York.Google Scholar
- 7.Demyanov, V.F. and Rubinov, A.M. (1990),Foundations of Nonsmooth Analysis. Quasidifferential Calculus, (in Russian), Nauka, Moscow,431p.Google Scholar
- 8.Demyanov, V.F and Rubinov, A.M. (1995),Introduction to Constructive Nonsmooth Analysis, Peter Lang Verlag, Frankfurt a.M.-Bern-New York, 414p.Google Scholar
- 9.Demyanov, V.F., Stavroulakis, G.E., Polyakova, L.N. and Panagiotopoulos, P.D. (1995),Quasidifferentiability and nonsmooth modelling in Mechanics, Engineering And Economics, Kluwer Academic Press (to appear).Google Scholar
- 10.Duvaut, G. and Lions, J.L. (1972),Les inequations en mechanique et en physique, Dunod, Paris.Google Scholar
- 11.Horst, R. and Tuy, H. (1990),Global Optimization, Springer Verlag, Berlin-Heidelberg.Google Scholar
- 12.Moreau, J.J. (1968), La notion de sur-potentiel et les liaisons unilaterales en elastostatique,C.R. Acad. Sc. Paris 267A, 954–957.Google Scholar
- 13.Pallaschke, D. and Urbanski, R. (1994), Reduction of quasidifferentials and minimal representations,Mathematical Programming 66, 161–180.Google Scholar
- 14.Panagiotopoulos, P.D. (1985),Inequality problems in mechanics and applications. Convex and nonconvex energy functions, Birkhäuser Verlag, Basel-Boston-Stuttgart.Google Scholar
- 15.Panagiotopoulos, P.D. (1988), Nonconvex superpotentials and hemivariational inequalities. Quasidiferentiability in mechanics, in Moreau, J.J. and Panagiotopoulos, P.D. (eds.),Nonsmooth Mechanics and Applications, CISM Lect. Nr. 302, Springer Verlag, Wien-New York.Google Scholar
- 16.Panagiotopoulos, P.D. and Stavroulakis, G.E. (1992), New types of variational principles based on the notion of quasidifferentiability,Acta Mechanica 94, 171–194.Google Scholar
- 17.Panagiotopoulos, P.D. (1993),Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer Verlag, Berlin-Heidelberg-New York.Google Scholar
- 18.Polak, E. (1989), Basics of minimax algorithms, in Clarke, F.H., Demyanov, V.F. and Giannessi, F. (eds.),Nonsmooth Optimization and Related Topics, Plenum Publ., New York.Google Scholar
- 19.Polyakova, L.N. (1981), Necessary conditions for an extremum of quasidifferentiable functions,Vestnik Leningrad Univ. Math. 13, 241–249.Google Scholar
- 20.Polyakova, L.N. (1986), On minimizing the sum of a convex function and a concave function,Mathematical Programming Study 19, 69–73.Google Scholar
- 21.Rockafellar, R.T. (1970),Convex Analysis, Princeton University Press, Princeton.Google Scholar
- 22.Rubinov, A.M. and Akhundov, I.S. (1992), Differences of compact sets in the sense of Demyanov and its application to non-smooth analysis, itOptimization23, 179–189.Google Scholar
- 23.Rubinov, A.M. and Yagubov, A.A. (1986), The space of star-shaped sets and its aplications in nonsmooth optimization,Mathematical Programming Study 29, 176–202.Google Scholar
- 24.Stavroulakis, G.E. (1993), Convex decomposition for nonconvex energy problems in elastostatics and applications,European Journal of Mechanics A/Solids 12(1), 1–20.Google Scholar
- 25.Stavroulakis, G.E. and Panagiotopoulos, P.D. (1993), Convex multilevel decomposition algorithms for non-monotone problems,Int. J. Num. Meth. Engng. 36, 1945–1966.Google Scholar
- 26.Stavroulakis, G.E. and Panagiotopoulos, P.D. (1994), A new class of multilevel decomposition algorithms for nonmonotone problems based on the quasidifferentiability concept,Comp. Meth. Appl. Mech. Engng. 117, 289–307.Google Scholar
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