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We adapt, via asymptotic expansion, Kapitsa's formula for the effective potential of a pendulum with vibrating suspension to rapidly forced potential flows with free boundaries. Determination of time-averaged stationary states leads to an optimal shape design problem. Under periodic boundary conditions existence and uniqueness of smooth minimizers to the averaged energy is proved using local coerciveness. In the numerical part of the article, 2D and 3D finite element approximations including related error estimates are discussed. Some illustrating examples are sketched.

Variational and Finite Element Analysis of Vibroequilibria