水下结构水弹性分析的不连续有限元方法

中文摘要 AI 辅助

本文针对水下弹性板水弹性建模中出现的波-结构相互作用问题，开发了一种有限元方法。该方法适用于二维设置和三维通道。对于问题的流体部分，我们使用对称不连续伽辽金格式，以捕捉穿过板的势的不连续性。为了施加索末菲辐射条件，我们借鉴狄利克雷-诺依曼映射方法，利用半无限域中的解析解构造适当的非局部边界条件。对于板，我们使用连续/不连续伽辽金方法求解四阶算子，而无需使用连续可微的有限元。我们证明了该方法的适定性，表明所得半双线性形式有界且满足Gårding不等式，从而保证解的唯一性和稳定性。我们使用Julia中的Gridap包将我们的公式实现为开源工具。我们针对超奇异边界积分方法验证了我们的方法，发现结果高度一致。通过二维斜板和三维板几何形状（包括中心矩形板、偏移矩形板和环形板）的模拟，展示了该方法的灵活性。本文提出的有限元方法可直接应用于压电双晶片波能转换的建模。此外，该方法可轻松扩展以考虑固定或刚性结构或可变底部地形的波散射，且不限于水弹性应用。

英文摘要

This paper develops a finite element method for wave-structure interaction problems arising in the hydroelastic modelling of submerged elastic plates. The approach is formulated for both two-dimensional settings and three-dimensional channels. For the fluid part of the problem, we use a symmetric discontinuous Galerkin scheme that captures the discontinuity in the potential across the plate. To impose the Sommerfeld radiation conditions, we take inspiration from the Dirichlet-to-Neumann map approach and use the analytic solution in the semi-infinite domains to construct appropriate non-local boundary conditions. For the plate, we use a continuous/discontinuous Galerkin method to resolve the 4$^{\rm th}$ order operator without using continuously differentiable finite elements. We prove well-posedness of the method, showing that the resulting sesquilinear form is bounded and satisfies the Gårding inequality, leading to a unique and stable solution. We implement our formulation as an open-source tool using the Gridap package in Julia. We validate our method against a hypersingular boundary integral method, finding excellent agreement. The flexibility of the methodology is demonstrated through simulations of a two-dimensional slanted plate and three-dimensional plate geometries including centred rectangular plates, offset rectangular plates, and annulus-shaped plates. The finite element method presented here can be readily applied to model wave energy conversion of piezoelectric bimorphs. Furthermore, the method could easily be extended to consider wave scattering from fixed or rigid structures or from variable bottom topography, and is not limited to hydroelastic applications.

发表机构

• Queensland University of Technology（昆士兰科技大学）
• Delft University of Technology（代尔夫特理工大学）
• University of Newcastle（纽卡斯尔大学）

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