Structural and Multidisciplinary Optimization

, Volume 40, Issue 1, pp 353–364

Shape sensitivity analysis with design-dependent loadings—equivalence between continuum and discrete derivatives


DOI: 10.1007/s00158-009-0374-4

Cite this article as:
Akbari, J., Kim, N.H. & Ahmadi, M.T. Struct Multidisc Optim (2010) 40: 353. doi:10.1007/s00158-009-0374-4


The purpose of this paper is twofold: (1) showing equivalence between continuum and discrete formulations in sensitivity analysis when a linear velocity field is used and (2) presenting shape sensitivity formulations for design-dependent loadings. The equations for structural analysis are often composed of the stiffness part and the applied loading part. The shape sensitivity formulations for the stiffness part were well-developed in the literature, but not for the loading part, especially for body forces and surface tractions. The applied loads are often assumed to be conservative or design-independent. In shape design problems, however, the applied loads are often functions of design variables. In this paper, shape sensitivity formulations are presented when the body forces and surface tractions depend on shape design variables. Especially, the continuum–discrete (C–D) and discrete–discrete (D–D) approaches are compared in detail. It is shown that the two methods are theoretically and numerically equivalent when the same discretization, numerical integration, and linear design velocity fields are used. The accuracy of sensitivity calculation is demonstrated using a cantilevered beam under uniform pressure and an arch dam crown cantilever under gravity and hydrostatic loading at the upstream face of the structure. It is shown that the sensitivity results are consistent with finite difference results, but different from the analytical sensitivity due to discretization and approximation errors of numerical analysis.


Shape optimization Shape sensitivity Continuum sensitivity Discrete sensitivity Design-dependent loading 

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.University of FloridaGainesvilleUSA
  2. 2.Department of Mechanical & Aerospace EngineeringUniversity of FloridaGainesvilleUSA
  3. 3.Department of Civil EngineeringTarbiat Modares UniversityTehranIran
  4. 4.Faculty of Civil Engineering, Department of EngineeringMalayer UniversityMalayerIran

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