Abstract
The complementary finite-element programming method and algorithm for solving finite deformation problems in nonsmooth mechanics are presented. This method provides a dual approach for the numerical solutions of the mixed boundary-value problem governed by nonsmooth physical laws. Application to non-smooth plastic flow is illustrated.
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References
P. D. Panagiotopoulos,Inequality Problems in Mechanics and Applications, Birkháuser, Boston/ Basel/Stuttgart (1985).
J. E. Taylor, Distributed parameter optimal structural design: some basic problem formulations and their applications. In: C. A. Mota Soares (eds.).Computer Aided Optimal Design: Structural and Mechanical Systems. Springer-Verlag, Berlin/Heidelberg (1987) pp. 3–85.
E. Reissner, see the list of Publications of Eric Reissner. In: S. Nemat-Nasser (eds.),Mechanics Today Vol. 5, Pergamon Press, Oxford (1980) 561–569.
B. M. Fraeijis de Veubeke, Displacement and equilibrium methods in the finite element methods. In: O. C. Zienkiewicz and G. S. Holister (eds.).Stress Analysis, Wiley, London (1965) 145.
J. T. Oden and J. N. Reddy,Variational Methods in Theoretical Mechanics, Springer-Verlag, Berlin/Heidelberg/New York (1983).
T. H. H. Pian and T. Tong, Reissner's principle in finite element formulations. In: S. Nemat-Nasser (ed.),Mechanics Today, Vol. 5, Pergamon Press (1980) 377–395.
A. M. Arthurs,Complementary Variational Principles, Second edition. Clarendon Press (1980).
I. Ekeland and R. Temam,Convex Analysis and Variational Problems, North-Holland (1976).
M. J. Sewell,Maximum and minimal principles, Cambridge University Press, Cambridge (1987).
Y. Gao and G. Strang, Geometric nonlinearity: Potential energy, complementary energy, and the gap function.Quart. Appl. Math., 47 (1989) 487–504.
Y. Gao and G. Strang, Dual extremum principles in finite deformation elastoplastic analysis.Acta Appl. Math., 17 (1989) 257–267.
Y. Gao, Dynamically loaded rigid-plastic analysis under large deformation.Quart. Appl. Math., 48 (1990) 731–739.
D. Y. Gao, Global extremum criteria for nonlinear elasticity.Z. Angew Math. Phys. (ZAMP), 43 (1992) 742–755.
Y. Gao and Y. K. Cheung, On the extremum complementary energy principles for nonlinear elastic shells.Int. J. Solids & Struct., 26 (1990) 683–693.
S.-T. Yau and Y. Gao, Obstacle problems for von Kármán equations.Adv. Appl. Math., 12 (1992) 123–141.
D. Y. Gao, Duality theory in nonlinear buckling analysis for von Karman equations.Studies in Applied Mathematics 94 (1995) 423–444.
C. Felippa, Will the force method come back?Trans. ASME, J. Appl. Mech., 54 (1987) 726–728.
R. H. Gallagher, Finite element structural analysis and complementary energy.Finite Element in Analysis and Design, 13 (1993) 115–126.
Y. Gao, On the complementary bounding theorems for limit analysis.Int. J. Solids Structures, 24 (1988) 545–556.
Y. Gao, Panpenalty finite element programming for plastic limit analysis.Compute & Structures, 28 (1988) 749–755.
T. Belytschko and P. G. HodgeJr., Plane stress limit analysis by finite element,J. Eng. Mech. Div., ACE, 96 (1970) 913–944.
M. R. Ranaweera and F. A. Leckie, Bound methods for limit analysis. In: H. Tottenham and C. A. Brebbia (eds.),Finite Element Techniques in Structural Mechanics, Southampton Univ. Press, Southampton (1970).
R. Casciaro and L. Cascini, A mixed formulation and mixed finite elements for limit analysis.Int. J. Num. Meth. Eng., 18 (1982) 211–243.
E. Christiansen and S. Larsen, Computations in limit analysis for plastic plates.Int. J. Nem. Meth. Eng., 19 (1983) 169–184.
C. Lemaréchal, J. J. Strodiot and A. Bihain, On a bundle algorithm for nonsmooth optimization. In: O. Mangasarian, R. Meyer and S. Robinson (eds.),Nonlinear Programming 4, Academic Press, New York (1981) 245–282.
K. Kiwiel, Methods of descent for nondifferentiable optimization. In:Lecture Notes in Mathematics. Springer-Verlag, Berlin/Heidelberg (1985).
J. J. Strodiot and V. H. Nguyen, On the numerical treatment of the inclusion 0 ∈ ∂ % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadAgaaaa!3D1A!\[f\] (x). In: J. J. Moreau, P. D. Panagiotopoulos and G. Strang (eds.),Topics in nonsmooth mechanics. Birkhauser Verlag, Basel (1988) 267–294.
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Gao, D.Y. Complementary finite-element method for finite deformation nonsmooth mechanics. J Eng Math 30, 339–353 (1996). https://doi.org/10.1007/BF00042755
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DOI: https://doi.org/10.1007/BF00042755