The underpinning mathematics for a novel wave energy converter: the FlexSlosh WEC

这笔小额赠款的目的是完成一项重要研究的两个步骤，该研究将数学的力量引入波浪产生的清洁能源问题。海浪是清洁能源的永恒来源。通过波浪能转换器（WECs）收集这种能量是可持续能源议程的巨大挑战之一。概念验证已经完成，目前的首要目标是实现具有商业效率的功率起飞（PTO），这涉及到新的几何建模。我们对这一议程的贡献是为下一代浮动wecc开发基础数学，特别是具有柔性底部地形的导管波能转换器，称为FlexSlosh WEC。FlexSlosh WEC是一个自由浮动的刚体，它与外部海洋表面波浪相互作用，从其内部流体晃动中提取能量。柔性底部可以利用可变形材料的新用途，从而使用利用分布式波纹管作用的新型分布式嵌入式能量转换器技术（DEEC-Tec）。FlexSlosh WEC的基础数学是基于广义的李泊松支架公式的非线性偏微分方程的“动态耦合”之间的刚体运动和内部耗散浅水晃动在两个水平空间维度，并计算基于几何和结构保持数值分析。几何和结构保持方法是求解偏微分方程的新一代高级数值模拟技术，它尊重其潜在的数学结构，即特定的几何或拓扑以及守恒定律。它们的优点是鲁棒、稳定、快速和精确，可用于高耦合非线性系统的“长时间”计算建模，这对于开发具有商业效率的波浪能提取几何优化工具非常重要。完全耦合非线性系统包括四个子系统：内部流体运动、WEC的刚体运动、模拟柔性底部地形的弹性体运动和外部波动运动。本项目将集中于前三个方面。泊松括号、拉格朗日、哈密顿和保持结构的数值格式已被导出作为独立系统的这些分量。动态耦合带来了新的挑战，特别是在长时间集成过程中保持组件之间正确的能量和动量分配是非常重要的。本研究的目的是：(1)为FlexSlosh WEC动力学建立新的广义泊松支架和Casimir不变量，即刚体运动与其内部浅水晃动之间的动力耦合以及与柔性底部地形的边界耦合；(2)为耦合非线性系统的长时间积分建立了新的有限差分保能保势辛格式。提出的数学进展将在严格的理论和数值框架中发展波能建模所需的方面。通过开发新的连续和离散的微分几何途径来转换海浪能量，拟议的基础数学项目将有助于实现2050年的净零目标。

The purpose of this small grant is to complete two steps in an important study that brings the power of mathematics to the problem of clean energy derived from waves. Ocean waves are a perpetual source of clean energy. Harvesting of this energy via Wave Energy Convertors (WECs) is one of the great challenges of the sustainable energy agenda. The proof of concept has been achieved, and the current overarching aim is to achieve power take-off (PTO) with commercial efficiencies, and this involves new GEOMETRIC modelling. Our proposed contribution to this agenda is to develop the underpinning mathematics for a class of next-generation floating WECs, in particular ducted wave energy converterswith flexible bottom topography, named FlexSlosh WEC.The FlexSlosh WEC is a freely floating rigid body in hydrodynamic interaction with exterior ocean surface waves, which extracts energy from its interior fluid sloshing over a flexible bottom topography. The flexible bottom may exploit novel use of deformable materials enabling the use of new distributed embedded energy converter technologies (DEEC-Tec) utilising distributed bellows action. The underpinning mathematics of the FlexSlosh WEC is based on a generalised Lie-Poisson bracket formulation of nonlinear partial differential equations for `dynamic coupling' between rigid-body motion and its interior dissipative shallow-water sloshing in two horizontal space dimensions, and computationally based on geometric and structure-preserving numerical analysis.Geometric and structure-preserving methods, which respect the underlying mathematical structure, i.e. specific geometric or topological, and conservation laws of the partial differential equations (PDEs) they solve, are a new generation of advanced numerical simulation techniques for evolutionary PDEs. Their advantages are being robust, stable, fast and precise for `long-time' computational modelling of highly-coupled nonlinear systems, which is so important for the development of a geometric optimisation tool for wave energy extraction with commercial efficiency.The fully coupled nonlinear system involves four subsystems: the interior fluid motion, the rigid body motion of the WEC, the elastic body modelling the flexible bottom topography, and the exterior wave motion. This project will concentrate on the first three. Poisson brackets, Lagrangians, Hamiltonians, and structure-preserving numerical schemes have been derived for these components as independent systems. Dynamic coupling brings in new challenges, in particular it is important to maintain the correct energy and momentum partition between components over long-time integration.The aim of the proposed research is (1) to develop new generalised Poisson bracket and Casimir invariants for the FlexSlosh WEC dynamics, i.e. dynamic coupling between rigid-body motion and its interior shallow-water sloshing and boundary coupling with the flexible bottom topography; and (2) to develop new finite difference energy- and potential-enstrophy-conserving symplectic scheme for long-time integration of the coupled nonlinear system.The proposed mathematical advances will develop the needed aspects of wave energy modelling in rigorous theoretical and numerical frameworks. By developing new continuum and discrete differential geometric pathways to transformation of ocean wave energy, the proposed underpinning mathematics project will contribute to the 2050 net zero target.