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 welldeveloped 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 J. Akbari · N. H. Kim University of Florida, Gainesville, FL 32611, USA e-mail: jalal.akbari@gmail.com N. H. Kim (B) Department of Mechanical & Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA e-mail: nkim@ufl.edu M. T. Ahmadi Department of Civil Engineering, Tarbiat Modares University, Tehran, Iran e-mail: mahmadi@modares.ac.ir Present Address: J. Akbari Malayer University, Hamadan, Iran 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.
