Keywords: mononofin design, swimming propulsion, identification, optimization.

The models of 3D unsteady coupled fluid-structure systems such as swimming fins are numerically too expensive to be optimized. Designers must therefore make preliminary decisions based on simplified models. In particular, considering 2D cases for optimization is a standard practice. The 2D design then needs to be translated into its 3D counterpart.

The current work is an example of such a problem. The objective is to design a 3D swimming monofin using a 2D optimal flexural stiffness distribution determined in [1]. In this study, a 2D monofin was represented by rigid bars linked by torsional springs, in dynamic equilibrium with the fluid. The swimmer was composed of linear articulated segments, whose kinematics was imposed and identified from experimental data. The sheet vortex fluid model presented in [2] was used, which accounts for a two-dimensional unsteady, inviscid and incompressible fluid flow going past a thin obstacle. The propulsive power provided by the monofin has been maximized with an upper bound on the total power expended by the swimmer. The design variables were the spring rigidities.

The flexural stiffness distribution obtained from the two-dimensional optimization is now translated into a three-dimensional shape. The mapping can be seen as an identification procedure where the "experience" is a 2D bars system whose behavior is approximated by a 3D finite element model of the fin. In its most general statement, this identification problem is ill-posed since many combinations of shape and thickness distribution can represent the 2D monofin. In practice, however, the planform shape of the monofin is dictated by manufacturing (mold cost) and marketing considerations which yield forms that mimick marine mammals. Since the fin planform shape is given, the spring stiffnesses are mapped into the fin thickness distribution. The equivalence between the two models can be sought in terms of static behavior, modal behavior, or a mix of static and modal behaviors. The advantages of static equivalence is that the load cases can be taken from the 2D flow simulation and large displacements analyses are available. However, it neglects the fin inertia, which may be realistic in comparison to water inertia and fin flexural stiffness. On the contrary, the modal dynamic identification accounts for both flexural and inertial terms but it is, in essence, a small displacements analysis. Furthermore, 3D non-bending modes have no pendant in the 2D system. For this reason, only the first natural mode, which has empirically been found on the monofin to consistently be bending, is considered. The 3D thickness distribution is found by minimizing

where,

is a weight factor between static displacement and modal criteria, are the target displacements at the bar joints of the simplified model, are the displacements at the corresponding control points of the finite element model, is the target first natural frequency of the simplified model and is the first natural frequency of the finite element model.

The effects of the identification formulations on the final thickness distribution are described. Firstly, when (), the influence of the load cases and the small or large displacements analyses are described. Secondly, the difference between minimizing and is exhibited. Finally, the problem is solved for a mixed static and modal criterion ( ).

Based on the results, it is recommended to choose the mixed static/dynamic formulation when translating a 2D fin design into 3D. Indeed, the static and dynamic formulations do no yield the same designs. In particular, it was found that the 2D model, which has a constant chordwise mass density, has a dynamic behavior which cannot be reproduce by 3D fins. In terms of static analysis, a small displacements finite element model is sufficient. It is observed that small and large displacements formulations do not present significant design differences. Moreover, when mixed formulation is considered, a highly accurate static analysis is not essencial because static and dynamic equivalences between the 2D and 3D models are traded-off, so that there is no longer a precise deformed shapes match.

1

M.A. Luersen, R. Le Riche, D. Lemosse, "Swimming Monofin Optimization". Submited to the 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, New York, Aug. 30 - Sept. 1, 2004.

2

O. Le Maître, S. Huberson, E. Souza de Cursi, "Unsteady model of sail and flow interaction". Journal of Fluids and Structures, 13, 37-59, 1999. doi:10.1006/jfls.1998.0188

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