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Daology:
Complementarity
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Complementarity(Under construction)Philosophy | Physics | Mechanics | Mathematics | Biblic Approach | Home
Definition Two opposite (conjugate) elements which are intergrated into an elementary system by certain conservation lawRelated concepts: Daology, Duality, Symmetry, Triality |
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Physical Explanation complementarity principle in physics, tenet that a complete knowledge of phenomena on atomic dimensions requires a description of both wave and particle properties. The principle was announced in 1928 by the Danish physicist Niels Bohr. Depending on the experimental arrangement, the behaviour of such phenomena as light and electrons is sometimes wavelike and sometimes particle-like; i.e., such things have a wave-particle duality (q.v.). It is cmpossible to observe both the wave and particle aspects simultaneously. Together, however, they present a fuller description than either of the two taken alone. In effect, the complementarity principle implies that phenomena on the atomic and subatomic scale are not strictly like large-scale particles or waves (e.g., billiard balls and water waves). Such particle and wave characteristics in the same large-scale phenomenon are incompatible rather than complementary. Knowledge of a small-scale phenomenon, however, is essentially incomplete until both aspects are known. Click here for a list of other articles that contain information on this subject Web stis for Complementarity in physics:
Mechanics Complementarity in mechanics is regarded as the (Lagrange) dual formulations of the free energy principles. The study of the complementary variational methods in mechanics has a very long history. In solid mechanics, the well-known Hellinger-Reissner principle was first proposed in 1914. Since the boundary condition was clarified by Reissner in 1953, the complementary energy principles and mixed variational methods (i.e. Lagrange duality) in mechanics have been studied extensively during the last fifty years (cf. e.g. Koiter, 1976; Nemat-Nasser, 1977; Stumpf, 1978; Lee \& Shield, 1980; Bufler, 1983; Atluri, 1984; Ogden, 1984 and much more). Many monographs have been published in applied mathematics (Arthurs, 1980; Sewell, 1987), continuum mechanics (Washizu, 1968; Oden-Reddy, 1983) and linear dynamics (Tabarrok-Rimrott, 1994). Since Hellinger-Reissner principle involves both the second Piola-Kirchhoff stress $\bT$ and the displacement $\bu$, it is not considered as a pure complementary energy principle. In order to apply this principle, one needs to understand the structure of the critical points of the Hellinger-Reissner energy. But until recently this structure was not adequately understood. In finite deformation theory, the existence of a pure complementary energy principle dual to the potential variational problem has been argued for many years. In 1989, Gao and Strang discovered that in geometrically nonlinear systems, there exists a so-called complementary gap function between the total potential energy and the Fenchel-Rockafellar dual functional. They proved that if this gap function is positive on equilibrium admissible space, the generalized Hellinger-Reissner energy $L(\bu, \bT)$ is a saddle functional. In this case, the total potential is convex and there exists only one dual problem. However, if this gap function is negative, the total potential is nonconvex and the system has two dual problems (Gao, 1997a). A triality extremum theory in nonconvex problems was discovered recently and a pure complementary energy in finite elasticity is proposed recently (Gao, 1997a,b, 1998a,b, 2000). References: Arthurs, AM (1980),
Atluri, SN (1984),
Bufler, H (1983), On the work theorems for finite and incremental
elastic deformations with discontinuous
Gallagher, RH (1993),
Gao, DY (1992), Global extremum criteria for finite elasticity, {\em ZAMP}, {\bf 43}, 924-937. Gao, DY (1997a), Dual extremum principles in finite deformation theory
with applications in post-buckling analysis of extended nonlinear
beam model, {\em Appl. Mech. Reviews, ASME},
Gao, DY (1998a),
Gao, DY (1998b),
Gao, DY and Strang, G (1989a),
Koiter, WT (1976),
Nemat-Nasser, S (1972),
Oden, JT and Reddy, JN (1983), {\em Variational Methods in Theoretical Mechanics}, Springer-Verlag. Ogden, RW (1984),
Pian, THH and Tong, P (1980),
Reddy, BD (1992),
Reissner, E. (1996),
Sewell, MJ (1987), {\em Maximum and Minimum Principles},
Strang, G (1986),
Stumpf, H (1978),
Tonti, E (1972),
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that contain information on this subject
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Mathematics Complementarity in optimization and mathematical
programming is essentially equivalent to the criticality conditions (KKT)
of certain variational problems with inequality (unilateral) constraints.
CPNET: Complementarity Problem Net Complementarity problem
network for Mathematical programming
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