The purpose of this paper is to show equivalence between continuum and discrete formulations in sensitivity analysis when a linear velocity field is used. 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.

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