Skip to main content
SpringerLink
Log in

A continuum mechanical-based formulation of the variational sensitivity analysis in structural optimization. Part I: analysis

  • Research Papers
  • Published:
Structural optimization Aims and scope Submit manuscript

Abstract

A variational formulation of shape design sensitivity analysis is outlined, starting from a differential geometry-based representation of continuum mechanics.

A rigorous analysis using convected curvilinear coordinates yields a decomposition of all continuum mechanical functions into independent geometry and displacement mappings. Using this representation of geometry and displacements defined on a fixed parameter space, their influence on physical quantities can easily be separated. Consequently, the variations of continuum mechanical quantities with respect to either geometry or displacements can be performed similarly using the well-known linearization techniques in nonlinear mechanics.

The proposed methodology for performing variational design sensitivity analysis is formulated for general nonlinear hyperelastic material behaviour using either the Lagrangian or Eulerian description. The differences and similarities of the formulation presented compared with the material derivative approach and the domain parametrization approach are highlighted and discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abraham, R.; Marsden, J.E.; Ratiu, T. 1983:Manifolds, tensor analysis and applications. Reading, MA: Addison-Wesley

    Google Scholar 

  • Adelman, H.M.; Haftka, R.T. 1986: Sensitivity analysis of discrete structural systems.AIAA J. 24, 823–832

    Google Scholar 

  • Arora, J.S. 1993: An exposition of the material derivative approach for structural shape sensitivity analysis.Comp. Meth. Appl. Mech. Engng. 105, 41–62

    Google Scholar 

  • Arora, J.S.; Cardoso, J.E.B. 1989: A design sensitivity analysis principle and its implementation into ADINA.Comp. & Struct. 32, 691–705

    Google Scholar 

  • Arora, J.S.; Cardoso, J.B. 1992: Variational principle for shape design sensitivity analysis.AIAA J. 30, 538–547

    Google Scholar 

  • Arora, J.S.; Lee, T.H.; Cardoso, J.B. 1992: Structural shape sensitivity analysis: relationship between material derivative and control volume approaches.AIAA J. 30, 1638–1648

    Google Scholar 

  • Barthold, F.J. 1993: Theorie und Numerik zur Berechnung und Optimierung von Strukturen aus isotropen, hyperelastischen Materialien.Technical Report F93/2, Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover

  • Barthold, F.J. 1994: Berechnung und Optimierung von Strukturen aus isotropen hyperelastischen Materialien.ZAMM 74, T300-T302

    Google Scholar 

  • Barthold, F.-J.; Falk, A. 1993: INAOPT; Inelastic ANnalysis and OPTimization; Programmdokumentation (Eingabeschreibung, Elementbibliothek, Algorithmen, Optimale Formgebung).Technical Report F94/1, Institut für Baumechanik und Numerische Mechanik, Universität Hannover

  • Barthold, F.J.; Stein, E. 1994: A continuum mechanical approach for analytical sensitivity analysis in structural optimization.Technical Report 94/7, Institut für Baumechanik und Numerische Mechanik, Universität Hannover

  • Barthold, F.J.; Falk, A.; Stein, E. 1994: Structural optimization for rubberlike materials using analytical sensitivity analysis. In: Gilmore, B.J.; Hoeltzel, D.A.; Dutta, D.; Eschenauer, H.A. (eds.)Advances in design automation, pp. 43–50. New York: ASME

    Google Scholar 

  • Becker, A. 1992: Strukturoptimierung stabilitätsgefährdeter Systeme mittels analytischer Gradientenermittlung.Technical Report F92/1, Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover

  • Belegundu, A.D.; Rajan, S.D. 1988: A shape optimization approach based on natural design variables and shape functions.Comp. Meth. Appl. Mech. Engng. 66, 87–106

    Google Scholar 

  • Cardoso, J.B.; Arora, J.S. 1988: Variational method for design sensitivity analysis in nonlinear structural mechanics.AIAA J. 26, 595–603

    Google Scholar 

  • Céa, J. 1981a: Problems of shape optimal design. In: Haug and Céa (eds.)Optimization of distributed parameter structures, parts I & II, pp. 1005–1048. Alphen an den Rijn, The Netherlands: Sijthoff & Noordhoff

    Google Scholar 

  • Céa, J. 1981b: Numerical methods of shape optimal design. In: Haug and Céa (eds.)Optimization of distributed parameter structures, parts I & II, pp. 1049–1087. Alphen an den Rijn, The Netherlands: Sijthoff & Noordhoff

    Google Scholar 

  • Choi, K.K.; Chang, K.-H. 1993:Shape design sensitivity analysis and optimization of elastic solids. In: Kamat, M.P. (ed.)Structural optimization: status and promise, Volume 150, pp. 569–609

  • Choi, K.K.; Haug, E.J. 1983: Shape design sensitivity analysis of elastic structures.J. Struct. Mech. 11, 231–269

    Google Scholar 

  • Choi, K.K.; Santos, J.L.T. 1987: Design sensitivity analysis of non-linear structural systems. Part I: theory.Int. J. Num. Meth. Engng. 24, 2039–2055

    Google Scholar 

  • Choi, K.K.; Seong, H.G. 1986: A domain method for shape design sensitivity analysis of built-up structures.Comp. Meth. Appl. Mech. Engng. 57, 1–15

    Google Scholar 

  • Dems, K.; Mróz, Z. 1984: Variational approach by means of adjoint systems to structural optimization and sensitivity analysis, structure shape variation.Int. J. Solids & Struct. 20, 527–552

    Google Scholar 

  • Dems, K.; Mróz, Z. 1985: Variational approach to first- and second-order sensitivity analysis of elastic structures.Int. J. Num. Meth. Engng. 21, 637–661

    Google Scholar 

  • Dems, K.; Mróz, Z. 1989: Shape sensitivity analysis and optimal design of physically nonlinear plates.Arch. Rat. Mech. 41, 481–501

    Google Scholar 

  • Falk, A. 1994: Adaptive Verfahren für die Formoptimierung flächiger Strukturen unter Berücksichtigung der CAD-FE-Kopplung.Technical Report F94/1, Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover (to appear)

  • Farin, G. 1993:Curves and surfaces for computer aided geometric design. (3rd edition) London: Academic Press

    Google Scholar 

  • Gopalakrishna, H.S.; Greimann, L.F. 1988: Newton-Raphson procedure for the sensitivity analysis of nonlinear structural behaviour.Comp. & Struct. 30, 1263–1273

    Google Scholar 

  • Haber, R.B. 1987: A new variational approach to structural shape design sensitivity analysis. In: Mota Soares, C.A. (ed.)Computer-aided optimal design, pp. 573–587. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Haftka, R.T.; Adelman, H.M. 1989: Recent developments in structural sensitivity analysis.Struct. Optim. 1, 137–151

    Google Scholar 

  • Haftka, R.T.; Grandhi, R.V. 1986: Structural shape optimization — a survey.Comp. Meth. Appl. Mech. Engng. 57, 91–106

    Google Scholar 

  • Haftka, R.T.; Mróz, Z. 1986: First- and second-order sensitivity analysis of linear and non-linear structures.AIAA J. 24, 1187–1192

    Google Scholar 

  • Haririan, M.; Cardoso, J.B.; Arora, J.S. 1987: Use of ADINA for design optimization of nonlinear structures.Comp. & Struct. 26, 123–133

    Google Scholar 

  • Haug, E.J.; Céa, J. (eds.) 1981:Optimization of distributed parameter structures, parts I & II. Alphen van den Rijn: Sijthoff & Noordhoff

    Google Scholar 

  • Haug, E.J.; Choi, K.K. 1986: Material derivative method for shape design sensitivity analysis. In: Bennet, J.A.; Botkin, M.E. (eds.)The optimum shape. New York: Plenum Press

    Google Scholar 

  • Haug, E.J.; Choi, K.K.; Komkov, V. 1986:Design sensitivity analysis of structural systems. Orlando: Academic Press

    Google Scholar 

  • Hou, J.W.; Chen, J.L.; Sheen, J.S. 1986: Computational method for optimization of structural shapes.AIAA J. 24, 1005–1012

    Google Scholar 

  • Hou, J.W.; Sheen, J.S.; Chuang, C.H. 1990: Shape sensitivity analysis and design optimization of linear thermoelastic solids.Proc. 31st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Material Conf. (held in Long Beach, CA)

  • Hughes, T.J.R.; Pister, K.S. 1978: Consistent linearization in mechanics of solids and structures.Comp. & Struct. 8, 391–397

    Google Scholar 

  • Kamat, M.P. (ed.) 1993: Structural optimization: status and promise.Progress in Astronautics and Aeronautics 150, Washington D.C.: AIAA

    Google Scholar 

  • Malvern, I.E. 1969:Introduction to the mechanics of a continuous medium. Englewood Cliffs: Prentice-Hall

    Google Scholar 

  • Marsden, J.E.; Hughes, T.J.R. 1983:Mathematical foundations of elasticity. Englewood Cliffs: Prentice-Hall

    Google Scholar 

  • Miehe, C. 1994: Apects of the formulation and finite element implementation of large strain isotropic elasticity.Int. J. Num. Meth. Engng. 37, 1981–2004

    Google Scholar 

  • Mróz, Z. 1986: Variational approach to sensitivity analysis and optimal design. In: Bennett, J.A.; Botkin, M.E. (eds.)The optimum shape. New York: Plenum Press

    Google Scholar 

  • Mróz, Z.; Kamat, M.P.; Plaut, R.H. 1985: Sensitivity analysis and optimal design of nonlinear beams and plates.Struct. Mech. 13, 245–266

    Google Scholar 

  • Ogden, R.W. 1984:Non-linear elastic deformations. Chichester: Ellis Horwood, John Wiley

    Google Scholar 

  • Phelan, D.G.; Haber, R.B. 1989: Sensitivity analysis of linear elastic systems using domain parametrization and mixed mutual energy principle.Comp. Meth. Appl. Mech. Engng. 77, 31–59

    Google Scholar 

  • Phelan, D.G.; Vidal, C.; Haber, R.B. 1991: An adjoint variable method for sensitivity analysis of non-linear elastic systems.Int. J. Num. Meth. Engng. 31, 1649–1667

    Google Scholar 

  • Ryu, Y.S.; Haririan, M.; Wu, C.C.; Arora, J.S. 1985: Structural design sensitivity analysis of nonlinear response.Comp. & Struct. 21, 245–255

    Google Scholar 

  • Seong, H.G.; Choi, K.K. 1987: Boundary-layer approach to shape design sensitivity analysis.Mech. Struct. & Mach. 15, 241–267

    Google Scholar 

  • Simo, J.C.; Ortiz, M. 1985: A unified approach to finite deformation elastoplastic analysis based on a use of hyperelastic constitutive equations.Comp. Meth. Appl. Mech. Engng. 49, 221–245

    Google Scholar 

  • Simo, J.C.; Pister, K.S. 1984: Remarks on rate constitutive equations for finite deformation problems.Comp. Meth. Appl. Mech. Engng. 46, 201–215

    Google Scholar 

  • Simo, J.; Taylor, R.L. 1991: Quasi-incompressible finite elasticity in principal stretches continuum basis and numerical algorithms.Comp. Meth. Appl. Mech. Eng. 85, 273–310

    Google Scholar 

  • Tortorelli, D.A. 1992: Sensitivity analysis for nonlinear constrained elastostatic systems.Int. J. Num. Meth. Engng. 33, 1643–1660

    Google Scholar 

  • Tortorelli, D.A.; Haber, R.B. 1989: First order design sensitivities for transient conduction problems by an adjoint method.Int. J. Num. Meth. Engng. 28, 733–752

    Google Scholar 

  • Tortorelli, D.A.; Wang, Z. 1993: A systematic approach to shape sensitivity analysis.Int. J. Solids & Struct. 30, 1181–1212

    Google Scholar 

  • Tortorelli, D.A.; Haber, R.B.; Lu, S.C.-Y. 1989: Design sensitivity analysis for nonlinear thermal systems.Comp. Meth. Appl. Mech. Engng. 77, 61–77

    Google Scholar 

  • Tortorelli, D.A.; Lu, S.C.-Y.; Haber, R.B. 1990: Design sensitivities for elastodynamic systems.Mech. Struct. Mach. 18, 77–106

    Google Scholar 

  • Tortorelli, D.A.; Lu, S.C.-Y.; Haber, R.B. 1991: Adjoint sensitivity analysis for nonlinear dynamic thermoelastic systems.AIAA J. 29, 253–263

    Google Scholar 

  • Tortorelli, D.A.; Subramani, G.S.; Lu, S.C.-Y.; Haber, R.B. 1991: Sensitivity analysis for coupled thermoelastic systems.J. Solids & Struct. 27, 1477–1497

    Google Scholar 

  • Truesdell, C.; Noll, W. 1965: The nonlinear field theories of mechanics. In: Flügge, S. (ed.)Handbuch der Physik III/3. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Tsay, J.J.; Arora, J.S. 1990: Nonlinear structural design sensitivity analysis for path dependent problems. Part 1: general theory.Comp. Meth. Appl. Mech. Engng., 183–208

  • Wu, C.C.; Arora, J.S. 1987: Design sensitivity analysis and optimization of nonlinear structural response using incremental procedure.AIAA J. 25, 1118–1125

    Google Scholar 

  • Yang, R.J.; Botkin, M.E. 1986a: The relationship between the variational approach and the implicit differention approach to shape design sensitivities In: Bennent, J.A.; Botkin, M.E. (eds.)The optimum shape. New York: Plenum Press

    Google Scholar 

  • Yang, R.J.; Botkin, M.E. 1986b: Comparison between the variational and implicit differentiation approaches to shape design sensitivities.AIAA J. 24, 1027–1032

    Google Scholar 

  • Yang, R.J.; Choi, K.K. 1985: Accuracy of finite element based shape design sensitivity analysis.Struct. Mech. 13, 223–239

    Google Scholar 

  • Yao, T.M.; Choi, K.K. 1989: 3-D shape optimal design and automatic finite element regridding.Int. J. Num. Meth. Engng. 28, 369–384

    Google Scholar 

  • Zolesio, J.P.: The material derivative (or speed) method for shape optimization. In: Haug, E.J.; Céa, J. (eds.)Optimization of distributed parameter structures, parts I & II, pp. 1089–1151

Download references

Author information

Authors and Affiliations

  1. Institute for Structural Mechanics and Computational Mechanics, University of Hannover, Appelstraße 9A, D-30167, Hannover, Germany

    F. -J. Barthold & E. Stein

Authors
  1. F. -J. Barthold
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Stein
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barthold, F.J., Stein, E. A continuum mechanical-based formulation of the variational sensitivity analysis in structural optimization. Part I: analysis. Structural Optimization 11, 29–42 (1996). https://doi.org/10.1007/BF01279652

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01279652

Keywords

Search

Navigation