Optimal strokes at low Reynolds number: a geometric and numerical study of Copepod and Purcell swimmers

Abstract : In this article, we provide a comparative geometric and numerical analysis of optimal strokes for two different rigid links swimmer models at low Reynolds number: the Copepod swimmer (a symmetric swimmer recently introduced by [31]) and the long-standing three-link Purcell swimmer [29]. The design of strokes satisfying some performance criteria leads to investigate optimal control problems which can be analyzed in the framework of sub-Riemannian geometry. In this context nilpotent approximations allow to compute strokes with small amplitudes, which in turn can be used numerically to obtain general strokes. A concept of geometric efficiency (corresponding to the ratio between the displacement and the length of the stroke) is introduced to deduce global optimality properties, in particular for the Copepod case. For this model a detailed analysis of both abnormal and normal strokes is also described. First and second order optimality conditions, combined with numerical analysis, allow us to detect optimal strokes for both the Copepod and the Purcell swimmers. C 1-optimality is investigated using the concept of conjugate point. Direct and indirect numerical schemes are implemented in Bocop [7] and HamPath [16] software to perform numerical simulations, which are crucial to complete the theoretical study and evaluate the optimal solutions.
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Submitted on : Monday, January 30, 2017 - 8:25:56 AM
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Piernicola Bettiol, Bernard Bonnard, Jérémy Rouot. Optimal strokes at low Reynolds number: a geometric and numerical study of Copepod and Purcell swimmers. 2017. ⟨hal-01326790v3⟩

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