Abstract
The application of mathematical programming methods in a variety of practically motivated engineering mechanics problems provides a fertile field for interdisciplinary interaction between the mathematical programming and engineering communities. This paper briefly outlines several topical problems in engineering mechanics involving the use of mathematical programming techniques. The intention is to attract the attention of mathematical programming experts to some of the still open questions in the intersection of the two fields.
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References
M.Z. Cohn and G. Maier (eds.), Engineering Plasticity by Mathematical Programming, NATO Advanced Study Institute, University of Waterloo 1977, Pergamon Press, New York, 1979.
G. Maier and J. Munro, Mathematical programming applications to engineering plastic analysis, Applied Mechanics Reviews 35, 1982, 1631–1643.
G. Maier and D. Lloyd Smith, Update to “Mathematical programming applications to engineering plastic analysis”, ASME Applied Mechanics Update, 1985, 377–383.
M.C. Ferris and J.S. Pang, Engineering and economic applications of complementarity problems, SIAM Review 39, 1997, 669–713.
M.C. Ferris and J.S. Pang (eds.), Complementarity and Variational Problems: State of the Art, SIAM, Philadelphia, Pennsylvania, 1997.
V. Carvelli, Limit and Shakedown Analysis of Three-Dimensional Structures and Periodic Heterogeneous Materials, Ph.D. Thesis, Po-litecnico di Milano, 1999.
Y.G. Zhang and M.W. Lu, An algorithm for plastic limit analysis, Computer Methods in Applied Mechanics and Engineering 126, 1995, 333–341.
J. Lubliner, Plasticity Theory, Macmillan Publishing Company, 1990.
A. Capsoni and L. Corradi, A finite element formulation of the rigid-plastic limit analysis problem, International Journal for Numerical Methods in Engineering 40, 1997, 2063–2086.
A. Corigliano, G. Maier and S. Pycko, Dynamic shakedown analysis and bounds for elastoplastic structures with nonassociative, internal variable constitutive laws, International Journal of Solids and Structures 32, 1995, 3145–3166.
C. Polizzotto, A unified treatment of shakedown theory and related bounding techniques, Solid Mechanics Archives 7, 1982, 19–75.
F. Tin-Loi, Deflection bounding at shakedown, Proceedings ASCE, Journal of Structural Division 106, 1980, 1209–1215.
C. Comi, G. Maier and U. Perego, Generalized variable finite element modeling and extremum theorems in stepwise holonomic elastoplasticity with internal variables, Computer Methods in Applied Mechanics and Engineering 96, 1992, 213–237.
I. Kaneko, On some recent engineering applications of complementarity problems, Mathematical Programming Study 17, 1982, 111–125.
G. Maier, A matrix structural theory of piecewise linear plasticity with interacting yield planes, Meccanica 5, 1970, 55–66.
G. Cocchetti and G. Maier, Static shakedown theorems in piecewise linearized poroplasticity, Archive of Applied Mechanics 68, 1998, 651–661.
R.W. Lewis and B.A. Schrefler, The Finite Element Method in the Deformation and Consolidation of Porous Media, John Wiley & Sons, Chichester, UK, 1998.
G. Bolzon, D. Ghilotti and G. Maier, Parameter identification of the cohesive crack model, in Material Identification Using Mixed Numerical Experimental Methods (H. Sol and C.W.J. Oomens, eds.), Dordrecht, Kluwer Academic Publishers, 1997, pp. 213–222.
G. Bolzon, G. Maier and F. Tin-Loi, On multiplicity of solutions in quasi-brittle fracture computations, Computational Mechanics 19, 1997, 511–516.
G. Bolzon, G. Maier and G. Novati, Some aspects of quasi-brittle fracture analysis as a linear complementarity problem, in Fracture and Damage in Quasibrittle Structures (Z.P. Bažant, Z. Bittnar, M. Jirásek and J. Mazars, eds.), E&FN Spon, London, 1994, pp. 159–174.
G. Bolzon, G. Maier and F. Tin-Loi, F. Holonomic and nonholo-nomic simulations of quasi-brittle fracture: a comparative study of mathematical programming approaches, in Fracture Mechanics of Concrete Structures (F.H. Wittmann, ed.), Aedificatio Publishers, Freiburg, 1995, pp. 885–898.
F. Tin-Loi and M.C. Ferris, Holonomic analysis of quasibrittle fracture with nonlinear softening, in Advances in Fracture Research (B.L. Karihaloo, Y.W. Mai, M.I. Ripley and R.O. Ritchie, eds.), Pergamon Press, 1997, pp. 2183–2190.
G. Bolzon and G. Maier, Identification of cohesive crack models for concrete on the basis of three-point-bending tests, in Computational Modelling of Concrete Structures (R. de Borst, N. Bicanic, H. Mang and G. Meschke, eds.), Balkema, Rotterdam, 1998, pp. 301–310
R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem, Academic Press, 1992.
Z. Cen and G. Maier, Bifurcations and instabilities in fracture of cohesive-softening structures: a boundary element analysis, Fatigue and Fracture of Engineering Materials and Structures 15, 1992, 911–928.
G. Maier, G. Novati and Z. Cen, Symmetric Galerkin boundary element method for quasi-brittle fracture and frictional contact problems, Computational Mechanics 13, 1993, 74–89.
G. Bolzon, Hybrid finite element approach to quasi-brittle fracture, Computers and Structures 60, 1996, 733–741.
J.J. Judice and G. Mitra, An enumerative method for the solution of linear complementarity problems, European Journal of Operations Research 36, 1988, 122–128.
S.P. Dirkse and M.C. Ferris, The PATH solver: a non-monotone stabilization scheme for mixed complementarity problems, Optimization Methods & Soßware 5, 1995, 123–156.
A. Brooke, D. Kendrick, A. Meeraus and R. Raman, GAMS: A User’s Guide, Gams Development Corporation, Washington, DC 20007, 1998.
M.C. Ferris and T.S. Munson, Complementarity problems in GAMS and the PATH solver, Journal of Economic Dynamics and Control 24, 2000, 165–188.
Z.Q. Luo, J.S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, 1996.
D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, 1989.
G. Maier, Inverse problem in engineering plasticity: a quadratic programming approach, Academia Nazionale dei Lincei, Serie VIII, vol LXX, 1981, 203–209.
G. Maier, F. Giannessi and A. Nappi, Indirect identification of yield limits by mathematical programming, Engineering Structures 4, 1982, 86–98.
A. Nappi, System identification for yield limits and hardening moduli in discrete elastic-plastic structures by nonlinear programming, Applied Mathematical Modelling 6, 1982, 441–448.
H. Jiang, D. Ralph and F. Tin-Loi, Identification of yield limits as a mathematical program with equilibrium constraints, in Proceedings, 15th Australasian Conference on the Mechanics of Structures and Materials (ACMSM15) (R.H. Grzebieta, R. Al-Mahaidi and J.L. Wilson, eds.), Balkema, 1997, pp. 399–404.
D. Ralph, A piecewise sequential quadratic programming method for mathematical programs with linear complementarity constraints, in Proceedings, 7th Conference on Computational Tech-niques and Applications (CTAC95) (A. Easton and R. May, eds.), World Scientific Press, 1995, pp. 663–668.
F. Tin-Loi and M.C. Ferris, A simple mathematical programming method to a structural identification problem, in Proceedings, 7th International Conference on Computing in Civil and Building Engineering (ICCCBE-VII) (C.K. Choi, C.B. Yun and H.G. Kwak, eds.), Techno-Press, 1997, pp. 511–518.
F. Facchinei, H. Jiang and L. Qi, A smoothing method for mathematical programs with equilibrium constraints, Mathematical Programming 85, 1999, 107–134.
C. Kanzow, Some noninterior continuation methods for linear complementarity problems, SIAM Journal on Matrix Analysis and Applications 17, 1996, 851–868.
M.C. Ferris and F. Tin-Loi, Nonlinear programming approach for a class of inverse problems in elastoplasticity, Structural Engineering and Mechanics 6, 1998, 857–870.
H. Jiang and D. Ralph, Smooth SQP methods for mathematical programs with nonlinear complementarity constraints, Research Report, Department of Mathematics and Statistics, The University of Melbourne, 1998.
L.D. Davis, Handbook of Genetic Algorithms, Van Nostrand Rein-hold, New York, 1991.
S. Bittanti, G. Maier and Nappi, Inverse problems in structural elastoplasticity: a Kaiman filter approach, in Plasticity Today (A. Sawczuk and G. Bianchi, eds.), Elsevier Applied Science Publishers, London, 1993, pp. 311–329.
S.P. Dirkse and M.C. Ferris, Modeling and solution environments for MPEC: GAMS & MATLAB, in Reformulation: Nons-mooth, Piecewise Smooth, Semismooth and Smoothing Methods (M. Fukushima and L. Qi, eds.), Kluwer Academic Publishers, 1999, pp. 127–148.
A. Drud, CONOPT — a large-scale GRG code, ORSA Journal on Computing 6, 1994, 207–216.
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Maier, G., Bolzon, G., Tin-Loi, F. (2001). Mathematical Programming in Engineering Mechanics: Some Current Problems. In: Ferris, M.C., Mangasarian, O.L., Pang, JS. (eds) Complementarity: Applications, Algorithms and Extensions. Applied Optimization, vol 50. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3279-5_10
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DOI: https://doi.org/10.1007/978-1-4757-3279-5_10
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