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

加拿大是世界上最大的水能和风能生产国之一。优化设计的船用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。在过去的一个世纪里，加拿大发生了200多起洪水，夺走了许多人的生命，造成了数十亿美元的损失。当河流泛滥时，对洪水平原的准确预测可能会防止生命损失，并有助于保护加拿大的家园和企业。这只是计算机模拟工具带来的两个巨大好处。然而，这些模拟工具的成功取决于所实施的算法。在这个项目中，我们将开发新的算法，以实现更高效、更准确的模拟工具。 最近，我们引入了一类新的求解小变形区域上流体流动的高精度算法。这一类有可能比许多其他的数值方法更准确、更高效。然而，许多流动模拟需要能够处理大变形区域上的流体流动的算法。这种大变形的例子包括：拓扑变化，区域可能分裂为几个子区域(例如，沸腾过程中的合并和气泡的破裂)以及绕旋转部件的流动(例如，在船舶或风力涡轮机的设计中)。拟议研究的总体目标是在我们的高阶精确算法的基础上，开发在大变形区域上模拟流体流动所需的新的方法学。 拟议的模拟研究通过为下一代模拟工具开发算法而使加拿大受益。这些工具将能够模拟工业、环境和社会中应用的各种流动。这些应用可能包括：通过设计最优的海洋和风力涡轮机叶片来增加可再生能源的生产；预测由于大范围降雨或融雪径流而导致河流泛滥的洪水平原；以及预测人造心脏瓣膜中的血液流动。此外，拟议的研究将培训具有开发和实施精确数值模拟方法所需的关键技能的高素质人员。这套技能对加拿大北部的软件开发至关重要，因为人们普遍认为，高阶数值方法很快就会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.