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带到了从波浪中获得清洁能源的问题上。海浪是清洁能源的永久来源。通过波浪能转换器（WEC）收集这种能量是可持续能源议程的重大挑战之一。概念验证已经实现，目前的首要目标是实现具有商业效率的动力输出（PTO），这涉及新的几何建模。我们提出的贡献，这一议程是发展的基础数学的一类下一代浮动WEC，特别是管道波能converterswith灵活的底部地形，命名为FlexSlosh WEC。FlexSlosh WEC是一个自由浮动的刚体与外部海洋表面波的流体动力学相互作用，提取能量从其内部流体晃动在一个灵活的底部地形。柔性底部可以利用可变形材料的新用途，使得能够使用利用分布式波纹管动作的新的分布式嵌入式能量转换器技术（DEEC-Tec）。FlexSlosh WEC的基础数学是基于两个水平空间维度中刚体运动及其内部耗散浅水晃动之间“动态耦合”的非线性偏微分方程的广义Lie-Poisson括号公式，并且在计算上基于几何和结构保持数值分析。几何和结构保持方法尊重基础数学结构，即特定的几何或拓扑，和守恒定律的偏微分方程（PDE），他们解决，是新一代先进的数值模拟技术的发展偏微分方程。它们的优点是对于高度耦合的非线性系统的“长时间”计算建模来说是鲁棒的、稳定的、快速的和精确的，这对于开发用于具有商业效率的波能提取的几何优化工具是如此重要。内部流体运动、WEC的刚体运动、模拟柔性底部地形的弹性体以及外部波浪运动。本项目将集中在前三个方面。泊松括号，拉格朗日，哈密顿，和结构保持数值方案已被推导出这些组件作为独立的系统。动态耦合带来了新的挑战，特别是在长时间的积分过程中保持各分量之间正确的能量和动量分配是非常重要的，本文的研究目标是（1）为FlexSlosh WEC动力学发展新的广义Poisson括号和Casimir不变量，即刚体运动与其内部浅水晃动的动力耦合和柔性底地形与边界的耦合;（2）发展了新的能量和位势拟能守恒的有限差分辛格式，用于耦合非线性系统的长时间积分，这些数学进展将在严格的理论和数值框架内发展波能模拟所需的方面。通过开发新的连续和离散微分几何路径来转换海浪能量，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.