# Application of Advanced Optimization Techniques to Parameter and Damage Identification Problems

## Abstract

A general formulation of an inverse problem of structural mechanics as an optimization problem is presented. The following features of a typical problem are accentuated: a large computational effort needed to evaluate the function values multiplied by the number of calls for the numerical simulation of the process under consideration, and that the function values often present some level of numerical noise. The main features of the Multipoint Approximation technique based on the Response Surface methodology (MARS) are presented with the emphasis on the choice of approximation functions. As a promising way of selection of the structure of approximations, the Genetic Programming methodology is presented. The use of optimization techniques for the solution of inverse problems of structural mechanics is illustrated by examples of damage recognition in steel structures and identification of parameters in various constitutive models.

## Keywords

Material Parameter Constitutive Model Sheet Metal Uniaxial Tension Bauschinger Effect## Preview

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## References

- Armstrong, P. J. and Frederick, C. O. (1966) A mathematical representation of the multiaxial Bauschinger effect.,
*GEGB report RD/B/N731*, Berkeley Nuclear Laboratories.Google Scholar - Audze, P. and Eglais, V. (1977). New approach for planing out of experiments, Problems of Dynamics and Strengths, 35, pp. 104–107, Riga, Zinatne Publishing House (in Russian).Google Scholar
- Baruch, M. (1982). 15 Optimal correction of mass and stiffness matrices using measured modes.
*AIAA J.***20**, 441.Google Scholar - Bates, S.J., Sienz, J. and Toropov, V.V. (2004) Formulation of the optimal Latin Hypercube design of experiments using a permutation genetic algorithm. Paper AIAA-2 004-2011, 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, CA, April 19–22Google Scholar
- Britanti, S., Maier, G. and Nappi, A. (1984) Inverse problems in structural elastoplasticity: a Kalman filter approach, A. Sawczuk and G. Bianchi, (eds.).
*Plasticity today. Modelling, methods and applications*, 311–329. Elsevier.Google Scholar - Box, G.E.P. and Draper, N.R. (1987). Empirical model-building and response surfaces, Wiley.Google Scholar
- Cawley, P. and Adams, R. D. (1979). The location of defects in structures from measurements of natural frequencies.
*J. Strain Analysis***14**, 49.Google Scholar - Chaboche, J. L. and Rousselier, G. (1983) On the plastic and viscoplastic constitutive equations-Part I: Rules developed with internal variable concept.
*Trans AS ME, J. Press. Vessel Technol.*,**105**, 153–158CrossRefGoogle Scholar - Chun, B. K., Jinn, J. T. and Lee, J. K. (2002a) Modeling the Bauschinger effect for sheet metals, part I: theory,
*Int. J. Plasticity*, 18, 571–595.MATHCrossRefGoogle Scholar - Chun, B. K., Jinn, J. T. and Lee, J. K. (2002b) Modeling the Bauschinger effect for sheet metals, part II: applications,
*Int. J. Plasticity*, 18, 597–616.MATHCrossRefGoogle Scholar - Distefano, N. (1970) On the identification problem in linear viscoelasticity,
*ZAMM*,**50**, 683–690.Google Scholar - Geng, L., Shen, Y. and Wagoner, R. H. (2002) Anisotropic hardening equations derived from reverse-bending testing,
*Int. J. Plasticity*, 18, 743–767.MATHCrossRefGoogle Scholar - Gioda, G. and Maier, G. (1980) Direct search solution of an inverse problem in elastoplasticity: identification of cohesion, friction angle and in situ stress by pressure tunnel tests,
*Int. J. Num. Meth. Engng.*,**15**, 1823–1848.MATHCrossRefGoogle Scholar - Gioda, G. and Sakurai, S. (1987) Back analysis procedures for the interpretation of field measurements in geomechanics.
*Int. J. for Numerical and Analytical Methods in Geomechanics*,**11**, 555–583.MATHCrossRefGoogle Scholar - Goldberg, D.E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley Publishing Company, Inc.Google Scholar
- Guccione, J. M., McCulloch, A. D. and Waldman, L. K. (1991) Passive material properties of intact ventricular myocardium determined from a cylindrical model.
*Transact. of the ASME, J. of Biomechanical Engng.*,**113**-51, 42–45.Google Scholar - Hajela, P. and Soeiro, F. J. (1990a) Structural damage detection based on static and modal analysis.
*AIAA J.***28**, 1110.Google Scholar - Hajela, P. and Soeiro, F. J. (1990b) Recent Developments in Damage Detection Based on System Identification Methods.
*Structural Optimization***2**, 1.CrossRefGoogle Scholar - Hassiotis, S. and Jeong, G. D. (1993) Assessment of structural damage from natural frequency measurements.
*J. Computers & Structures***49**, 679.CrossRefGoogle Scholar - Hamasaki, H., Yoshida, F., Shinbata, K., Toropov, V. V. (2003) Identification of material properties for lead-free solder using micro-indentation experiments, FE simulation and optimization. In Short Paper of
*5th World Congress of Structural and Multidisciplinary Optimization*, Lido di Jesolo, Italy, May 2003, 101–102Google Scholar - Hawkins, R., Wright, J.C. (1971) Mechanical properties and press-formability of copper/mild steel sandwich sheet materials,
*J. Inst. Metal.*99, 357–371.Google Scholar - Hendriks, M. A. N. (1991)
*Identification of the mechanical behaviour of solid materials*, Ph.D. Thesis, Eindhoven University of Technology, The Netherlands.Google Scholar - Iding, R. H., Pister, K. S. and Taylor, R. L. (1974) Identification of nonlinear elastic solids,
*Comput. Meth. Appl. Mech. Engng*,**5**, 121–142.CrossRefGoogle Scholar - Kanetake, K., Tozawa, Y, Kato, T. and Aiba, S. (1981) Effect of texture on deformation behavior of aluminum sheets,
*J. Jpn Inst. Light Metals*, 31, 307–312Google Scholar - Kavanagh, K. T. and Clough, R. W. (1971) Finite element applications in the characterization of elastic solids,
*Int. J. Solids & Structures*,**7**, 11–23.MATHCrossRefGoogle Scholar - Kavanagh, K. T. (1972) Extension of classical experimental techniques for characterizing composite-material behavior,
*Experimental mechanics*,**12**, 50–56.CrossRefGoogle Scholar - Kim, J.-K., Yu, T.-X. (1997) Forming and failure behaviour of coated, laminated and sandwiched sheet metals: a review, J. Materials Processing Technology 63, 33–42.CrossRefGoogle Scholar
- Koza, J.R. (1992)
*Genetic Programming: On the Programming of Computers by Means of Natural Selection*. MIT Press.Google Scholar - Kristen, H. A. D. (1976) Determination of rock mass elastic moduli by back analysis of deformation measurements,
*Proc. Symp. on Exploration for Rock Engineering*, Johannesburg, 165–172.Google Scholar - Lapierre, H. and Ostiguy, G. (1990) Structural model verification with linear quadratic optimization theory.
*AIAA J.***28**, 1497.Google Scholar - Lead-Free Solder Project. (1997) NCMS Report 0401 RE96.Google Scholar
- Lin, E. I.-H. and Sackman, J. L. (1975) Identification of dynamic properties of nonlinear viscoelastic materials and the associated wave propagation problem,
*Int. J. Solids & Structures*, 11, 1145–1159.CrossRefMATHGoogle Scholar - Ma, X. and Yoshida, F. (2003) Rate-dependent indentation hardness of a power-law creep solder alloy,
*Appl. Phys. Lett*. 82, 188–190.CrossRefGoogle Scholar - Madsen, K. and Hegelund, P. (1991)
*Non-gradient Subroutines for Non-linear Optimization*. Institute for Numerical Analysis, Technical University of Denmark, Report NI-91-05.Google Scholar - Maier, G. (1981) Inverse problem in engineering plasticity: a quadratic programming approach.
*Rendiconti dell’ Accademia Nazionale dei Lincei*, Serie VIII,**LXX**-4, 203–209.Google Scholar - Maier, G., Giannessi, F. and Nappi, (1982) A. Indirect identification of yield limits by mathematical programming.
*Eng. Struct.*,**4**, 86–98CrossRefGoogle Scholar - Majlessi, S. A., Dadras, P. (1983) Pure plastic bending of sheet laminates under plane strain condition,
*Int. J. Mech. Sciences*. 25, 1–14.MATHCrossRefGoogle Scholar - Mackay, M. D., Beckman, R. J., and Conover, W. J. (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, 21, 239–245.CrossRefMathSciNetGoogle Scholar
- Marti, K. (1991) Stochastic programming: Numerical solution techniques by semi-stochastic approximation methods, in Slowinski, R. and Teghem, J. (eds.),
*Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty*, Kluwer, 23–43.Google Scholar - Oomens, C. W. J., van Ratingen, M. R., Janssen, J. D., Kok, J. J. and Hendriks, M. A. N. (1993) A numerical-experimental method for a mechanical characterization of biological materials,
*J. Biomechanics*,**26**-4/5, 617–621.CrossRefGoogle Scholar - Pedersen, P. (1989) Optimization methods applied to identification of material parameters.
*Proc. of the GAMM seminar on Discretization methods and structural optimization, Lecture Notes in Engineering*,**42**. Springer.Google Scholar - Pedersen, P. and Frederiksen, P. S. (1990) Sensitivity analysis for identification of material parameters,
*Proc. 9th Int. Conf. on Experimental Mechanics*, Copenhagen, 545–551.Google Scholar - Pedersen, P. and Frederiksen, P. S. (1992) Identification of orthotropic material moduli by a combined experimental/numerical approach,
*Measurement*,**10**, 113–118.CrossRefGoogle Scholar - Pharr, G. M. and Oliver, W. C. (1989) Nanoindentation of silver-relations between hardness and dislocation structure,
*J. Mater. Res.*, Vol. 4, No. 1, 94–101.Google Scholar - Prager, W. (1956) A new method of analyzing stresses and strains in work-hardening plastic solids,
*Appl. Mech.*, 23, 493–496.MATHMathSciNetGoogle Scholar - Ravaii, H., Toropov, V.V. and Horoshenkov, K.V. (1997) Structural damage recognition based on optimization techniques. In Gutkowski, W., Mroz, Z., eds.,
*Proc. of 2nd World Congress of Structural and Multidisciplinary Optimization*, Zakopane, Poland, May 1997, vol. 1, pp. 299–304, Polish Academy of Sciences, 1997.Google Scholar - Ravaii, H., Toropov, V.V. and Horoshenkov, K.V. (1998a) Mixed numerical-experimental damage recognition in steel structures. In Allison, I.M. (ed.),
*Experimental Mechanics. Advances in Design, Testing and Analysis-Proc. 11th Int. Conf. on Experimental Mechanics*, Oxford, August 24–28, 1998, vol. 1, A.A. Balkema, Rotterdam, 77–82.Google Scholar - Ravaii, H., Toropov, V.V. and Mahfouz, S.Y. (1998b) Application of a genetic algorithm and derivative-based techniques to identification of damage in steel structures. In: M. Tanaka, G.S. Dulikravich, eds.,
*Inverse Problems in Engineering Mechanics-Proc. Int. Symposium*, Nagano City, Japan, March 24–27, 1998, Elsevier, 571–580.Google Scholar - Rikards, R. (1993) Elaboration of optimal design models for objects from data of experiments. In Pedersen, P., ed., Optimal design with advanced materials, The Frithiof Niordson volume. Proceedings of the IUTAM Symposium, Lyngby, Denmark, Elsevier, 149–162.Google Scholar
- Semiatin, N. L. and Piehler, H. R. (1979a) Deformation of sandwich sheet materials in uniaxial tension,
*Metal. Trans*. 10A, 85–96.Google Scholar - Semiatin, N. L. and Piehler, H. R. (1979b) Formability of sandwich sheet materials in plane strain compression and rolling,
*Metal. Trans.*10A, 97–107.Google Scholar - Semiatin, N. L. and Piehler, H. R. (1979c) Forming limits of sandwich sheet materials,
*Metal. Trans.*10A, 1107–1118.Google Scholar - Shiratori, E., Ikegami, K. and Yoshida, F. (1976a) The subsequent yield Surfaces after proportional preloading of the Tresca-type material,
*Bulletin of the JSME*, 19, 1122–1128.Google Scholar - Shiratori, E., Ikegami, K., Yoshida, F., Kaneko, K. and Koike, S., (1976b) The subsequent yield surfaces after preloading under combined axial load and torsion,
*Bulletin of the JSME*, 19–134, 877–883.Google Scholar - Sol, H., De Visscher, J. and De Wilde, W. P. (1993) Identification of the viscoelastic material properties of orthotropic plates using a mixed numerical/experimental technique, C. A. Brebbia and G. M. Carlomagno, eds.,
*Computational methods and Experimental Measurements VI*, vol. 2, Comp. Mech. Publications, 131–142.Google Scholar - Sol, H. and Oomens, C. W. J., Eds. (1997)
*Material Identification using Mixed Numerical — Experimental Methods. Proc. Euromech 357 Colloquium*, Kerkrade, The Netherlands, April 1997, Kluwer.Google Scholar - Tanaka, K. (1986) A thermomechanical sketch of shape memory effect: One-dimensional tensile behavior,
*Res. Mech.*, 18, 251.Google Scholar - Tanaka, K., Kobayashi, S. and Sato, Y. (1986) Thermomechanics of transformation pseudoelasticity and shape memory effect in alloys,
*Int. J. Plasticity*, 2, 59.CrossRefGoogle Scholar - Tanaka, K., Tobushi, H. and Miyazaki, S. (1993)
*Mechanical Properties of Shape Memory Alloys*. Yokendo Ltd.Google Scholar - Tanaka, M. and Dulikravich, G.S., Eds. (1998)
*Inverse Problems in Engineering Mechanics. Proc. Int. Symposium*, Nagano City, Japan, March 1998, Elsevier.Google Scholar - Toropov, V. V. and Van der Giessen, E. (1993) Parameter identification for nonlinear constitutive models: Finite Element simulation-Optimization-Nontrivial experiments, In: Pedersen, P., ed.,
*Optimal Design with Advanced Materials*, Elsevier, 113–130.Google Scholar - Toropov, V.V., van der Giessen, E. and Yoshida, F. (1996) Material parameter identification based on nontrivial experiments, numerical simulation and optimization. In: Friswell, M.I.; Mottershead, J.E., eds.,
*Proceedings of 1st International Conf. on Identification in Engineering Systems*, Swansea, March 1996, The Cromwell Press Ltd., 328–337.Google Scholar - Toropov, V.V. and Markine, V.L. (1996) The use of simplified numerical models as mid-range approximations. 6th AIAA/NASA/ISSMO Symp. on Multidisciplinary Analysis and Optimization, Bellevue WA, Part 2, 952–958, AIAAGoogle Scholar
- Toropov, V. V. and Alvarez, L.F. (1998a) Application of genetic programming and response surface methodology to optimization and inverse problems. In: M. Tanaka, G.S. Dulikravich, eds.,
*Inverse Problems in Engineering Mechanics-Proc. Int. Symp.*, Nagano City, March 1998, Elsevier, 551–560.Google Scholar - Toropov, V. V. and Alvarez, L.F. (1998b) Approximation model building for design optimization using genetic programming methodology,
*Paper AIAA-98-4769*,*7th AIAA/USAF/NASA/ISSMO Symp. on Multidisciplinary Analysis and Optimization*, St. Louis, USA, September 1998, 1, AIAA, 490–498.Google Scholar - Toropov, V. V., Alvarez, L.F. and Ravaii, H. (1999a) Structural damage recognition using optimization techniques and genetic programming methodology,
*3rd ISSMO/UBCAD/UB/AIAA World Congress of Structural and Multidisciplinary Optimization*, Buffalo, NY, USA, May 17–21, 1999 (CD Proceedings).Google Scholar - Toropov, V. V., Alvarez, L.F. and Ravaii, H. (1999b) Recognition of damage in steel structures using genetic programming methodology. In: M.I. Friswell, J.E. Mottershead, A.W. Lees (eds.),
*Proc. 2nd International Conf. on Identification in Engineering Systems*, Swansea, March 29–31, 1999, The Cromwell Press Ltd., 382–391.Google Scholar - Toropov, V. V. (2001) Modelling and approximation strategies in optimization-global and mid-range approximations, response surface methods, genetic programming, low / high fidelity models. In: Blachut, J.; Eschenauer, H. A., eds.,
*Emerging Methods for Multidisciplinary Optimization*, CISM Courses and Lectures, No. 425, International Centre for Mechanical Sciences, 205–256, Springer-Verlag, 2001.Google Scholar - Venkataraman, S., Haftka, R. and Johnson, T. (1998) Design of shell structures for buckling using correction response surface approximations. Paper AIAA-98-4855,
*Proceedings of 7th AIAA/USAF/ NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization*, St. Louis: AIAA. 1131–1144.Google Scholar - Verguts, H., Sowerby, R. (1975) The pure plastic bending of laminated sheet metals, Int. J. Mech. Sciences, 17, 31–51.CrossRefGoogle Scholar
- Vestergaard, B. (1990) Experimentation and identification for models of visco-elastic materials,
*Proc. 9th Int. Conf on Experimental Mechanics*, Copenhagen, 601–608.Google Scholar - Wagoner, R. H., Geng, L. and Balakrishnan, V. (2000) Role of hardening law in springback, In Khan, A. S., Zhan, H. and Yuen Y., eds.,
*Proceedings of 8th Int. Symp. on Plasticity and Its Current Applications*, NEAT Press, 609–611Google Scholar - Wu, P. D. and Van der Giessen, E. (1991) Analysis of elastic-plastic torsion of circular bars at large strains,
*Arch. Appl Mech.*,**61**, 89–103.MATHGoogle Scholar - Yoshida, F., Hino, R. and Okada, T. (1990) Deformation and fracture of stainless-steel/aluminium sheet metal laminates in stretch bending.
*Proceedings of 5th Int. Symp. on Plasticity and Its Current Applications*, Elsevier, 869–872.Google Scholar - Yoshida, F., Hino, R. and Okada, T. (1995) Deformation and fracture of stainless-steel/aluminium sheet metal laminates in stretch bending/unbending. In Tanimura, S. and Khan, A. S. eds.,
*Proceedings of the 5th International Symposium on Plasticity and Its Current Applications*, Gordon and Breach Publ., 869–872.Google Scholar - Yoshida, F. (1995) Ratchetting behaviour of 304 stainless steel at 650°C under multiaxially strain-controlled and uniaxially/multiaxially stress-controlled conditions,
*European J. Mechanics, A/Solids*, 14, 97–117.Google Scholar - Yoshida, F. (1997) Deformation and fracture of sheet metal laminates in plastic forming, In Hui, D., ed.,
*Proceedings of 4th International Conference on Composite Engineering*, 61–64.Google Scholar - Yoshida, F., Urabe, M. and Toropov, V. V. (1998) Identification of material parameters in constitutive model for sheet metals from cyclic bending tests,
*Int. J. Mechanical Sciences*, 40–2, 237–249.CrossRefGoogle Scholar - Yoshida, F. and Hino, R. (1997) Forming limit of stainless steel-clad aluminium sheets under plane stress condition,
*J. Materials Processing Technology*, 63, 66–71.CrossRefGoogle Scholar - Yoshida, F. and Urabe, M. (1999) Computer-aided process design for the tension levelling of metallic strips,
*J. Materials Processing Technology*, 89/90, 218–223.CrossRefGoogle Scholar - Yoshida, F., Toropov, V.V., Kyogoku, H. and Sakuma, T. (1999) Identification of material parameters of Ti-Ni shape memory alloy. In: M.I. Friswell, J.E. Mottershead, A.W. Lees, eds.,
*Proceedings of 2nd International Conf. on Identification in Engineering Systems*, Swansea, March 29–31, 1999, The Cromwell Press Ltd., 150–159.Google Scholar - Yoshida, F. and Urabe, M. (2000) Computer-aided process design for tension levelling of clad sheet metal,
*Key Engineering Materials*, 177–180, 503–508.CrossRefGoogle Scholar - Yoshida, F. (2000) A constitutive model of cyclic plasticity,
*Int. J. Plasticity*, 16, 359–380.MATHCrossRefGoogle Scholar - Yoshida, F., Uemori, T. and Fujiwara, K. (2002) Elastic-plastic behavior of steel sheets under in-plane cyclic tension-compression at large strain,
*Int. J. Plasticity*18, 633–659.MATHCrossRefGoogle Scholar - Yoshida, F. and Uemori, T. (2002) A model of large-strain cyclic plasticity describing the Bauschinger effect and workhardening stagnation,
*Int. J. Plasticity*18, 661–689.MATHCrossRefGoogle Scholar - Zhao, K. M., Lee, J. K. (2001) Generation of cyclic stress-strain curves for sheet metals,
*ASME, J. Eng. Mater. Technology*, 123, 391–397.CrossRefGoogle Scholar