漂浮结构的动力学
On the dynamics of floating structures
AI总结：
本文研究了漂浮体与水面波的相互作用，提出了一种新的水波问题公式化方法，能够通过求解椭圆方程确定流体对漂浮体的内压，适用于一维和高维情况，并展示了其在非线性浅水方程和 Boussinesq 方程中的应用。
摘要：
本文针对漂浮体问题，研究水面波与漂浮体的相互作用。我们提出了一种新的水波问题公式化方法，能够方便地推广以考虑漂浮体的存在。所得方程具有压缩性-不可压缩性结构，其中流体对漂浮体内部施加的压力是拉格朗日乘数，可通过求解 $d$-维椭圆方程确定，其中 $d$ 为水平维度。在漂浮体自由漂浮的情况下，我们将流体对固体施加的水动力力和力矩分解，以展示附加质量效应；在 $d=1$ 的一维情况下，计算可显式进行。我们还证明了这种将内部压力作为拉格朗日乘数的方法可以应用于简化渐近模型，如非线性浅水方程和 Boussinesq 方程，并可以推广到这些简化模型的离散版本，提出了一维情况下的简单数值方案。最后，我们展示了基于这些数值方案的若干数值计算，并通过一些特定配置中的显式解来验证这些计算，例如在流体初始静止时，从非平衡位置释放物体返回平衡状态的问题。
英文摘要：
This paper addresses the floating body problem which consists in studying the interaction of surface water waves with a floating body. We propose a new formulation of the water waves problem that can easily be generalized in order to take into account the presence of a floating body. The resulting equations have a compressible-incompressible structure in which the interior pressure exerted by the fluid on the floating body is a Lagrange multiplier that can be determined through the resolution of a $d$-dimensional elliptic equation, where $d$ is the horizontal dimension. In the case where the object is freely floating, we decompose the hydrodynamic force and torque exerted by the fluid on the solid in order to exhibit an added mass effect; in the one dimensional case $d=1$, the computations can be carried out explicitly. We also show that this approach in which the interior pressure appears as a Lagrange multiplier can be implemented on reduced asymptotic models such as the nonlinear shallow water equations and the Boussinesq equations; we also show that it can be transposed to the discrete version of these reduced models and propose simple numerical schemes in the one dimensional case. We finally present several numerical computations based on these numerical schemes; in order to validate these computations we exhibit explicit solutions in some particular configurations such as the return to equilibrium problem in which an object is dropped from a non-equilibrium position in a fluid which is initially at rest.
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