Numerical algorithms for higher-order accurate discretizations of flows on deforming domains

加拿大是世界上最大的水电和风能生产国之一。优化设计的船舶和风力涡轮机叶片可以进一步增加这些可再生能源的产量。在过去的世纪里，加拿大发生了200多次洪水，夺走了许多人的生命，造成了数十亿美元的损失。准确预测河流泛滥时的洪泛平原可能会防止生命损失，并有助于保护加拿大的家庭和企业。这只是两个从计算机模拟工具中受益匪浅的应用程序。然而，这些模拟工具的成功取决于实现的算法。在这个项目中，我们将开发新的算法，更有效，更准确的仿真工具。 最近，我们介绍了一类新的高阶精度算法的小变形域上的流体流动。这一类有可能比许多其他数值方法更准确，更有效。然而，许多流动模拟需要能够科普具有大变形的域上的流体流动的算法。这种大变形的例子包括：拓扑变化，其中域可以分裂成几个子域（例如，沸腾过程中气泡的聚结和破裂）和围绕旋转部件的流动（例如，在船舶或风力涡轮机的设计中）。拟议研究的首要目标是在我们的高阶精确算法的基础上开发模拟大变形域上的流体流动所需的方法。 拟议的研究有利于加拿大开发下一代仿真工具的算法。这些工具将能够模拟工业、环境和社会中各种各样的流动。这些应用可能包括：通过设计最佳的海洋和风力涡轮机叶片来增加可再生能源的生产;预测由于大量降雨或融雪径流造成河流泛滥时的洪泛平原;预测人工心脏瓣膜中的血流。此外，拟议的研究将培养具有开发和实施精确数值方法所需关键技能的高素质人才。这种技能组合是必不可少的软件开发在加拿大，因为人们普遍认为，高阶数值方法将很快成为模拟软件的标准。

Canada is one of the largest producers of hydro and wind energy in the world. Optimally designed marine and wind turbine blades can further increase the production of these sources of renewable energy. More than two hundred floods have occurred in Canada over the past century taking many lives and causing billions of dollars in damage. An accurate prediction of flood plains when rivers overflow may prevent the loss of lives and help protect Canadian homes and businesses. These are just two applications that benefit tremendously from computer simulation tools. The success of these simulation tools, however, depends on the implemented algorithms. In this project we will develop new algorithms for more efficient and more accurate simulation tools. Recently we introduced a new class of higher-order accurate algorithms for fluid flows on domains with small deformations. This class has the potential of being more accurate and more efficient than many other numerical methods. Many flow simulations, however, require algorithms that can cope with fluid flows on domains with large deformations. Example of such large deformations include: topological changes in which the domain may split into several sub-domains (e.g. coalescence and break-up of bubbles in boiling processes) and flow around rotating components (e.g. in the design of marine or wind turbines). The overarching objective of the proposed research is to develop the methodology needed for simulating fluid flow on domains with large deformations, building upon our higher-order accurate algorithms. The proposed research benefits Canada by developing algorithms for next generation simulation tools. These tools will enable simulation of a wide variety of flows for applications in industry, environment and society. These applications may include: increasing the production of renewable energy through the design of optimal marine and wind turbine blades; predicting flood plains in the event of rivers overflowing due to extensive rainfall or snow-melt run-off; and predicting blood flow in artificial heart valves. Furthermore, the proposed research will train highly qualified personnel with the critical skills needed to develop and implement accurate numerical methods. This skill-set is essential for software development in Canada as it is widely believed that higher-order numerical methods will soon become the standard in simulation software.