Mathematically Rigorous High-Fidelity Solvers

数学严谨的高保真解算器

• 批准号: RGPIN-2022-03211
• 负责人: DelReyFernandez, David
• 金额: $ 2.7万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=743672
• 关键词: Mathematically; Rigorous; Fidelity; Solvers

Numerical simulations are standard practice in many sectors (e.g., aerospace and automotive) because they allow for the study of scenarios not possible in the laboratory and the systemic exploration of engineering design spaces. Where industry solvers fail (e.g, stall and shocks in aerodynamics), high-order methods promise transformative capabilities to render predictive simulations through the efficient use of modern compute architectures. However, their application to practical problems have been limited by stability issues. This proposal outlines the development of provably stable high-order solvers that are predictive in such contexts. Because the focus is on applications in aerodynamic and bubble collapse in sonoporation and high-intensity focused ultrasound (HIFU), the research is relevant to the multibillion-dollar aerospace and medical technology industries. An unstructured four-dimensional spacetime approach is pursued as it results in significantly fewer degrees of freedom (DOFs) than alternatives. More importantly, when coupled with spacetime adaptation and shock-tracking, the resultant problem size is many times smaller than alternatives.  Adaptation, where DOFs are minimized by distributing them in an optimal fashion, to reduce problem size is ideally suited to compressible flows because of their multiscale nature. Three approaches are considered 1) p adaptation where element DOFs are increased or decreased, 2) h adaptation where elements are subdivided or agglomerated, and 3) r adaptation where the DOFs are redistributed. In this regard, developing efficient algorithms that can support all three forms of adaptation while retaining provable stability will be pursued. Many problems in aerodynamics and bubble collapse for sonoporation and HIFU are concerned with shocks. Unfortunately, for such problems, high-order methods reduce to first-order around shocks, rendering them less efficient. Moreover, in this context, high-order solvers can fail as a result of the oscillatory and nonphyiscal solutions that form near shocks. In this proposal, a shock-tracking approach will be developed where the mesh is deformed so that the element interfaces align with the shock. This is an appealing approach because high-order is maintained and the shock is localized sharply. This research will result in in transformative solvers that can solve problems at unprecedented scale and accuracy for which industrial solvers fail to be predictive. They will support Canada's aerospace industry in the design of next generation aircraft and lend insight into sonoporation and HIFU that could lead to revolutionary applications in both. Importantly, this research will train students with highly desirable skills in mathematics, fluid dynamics, and software engineering. This skill set is not only highly transferable but also is in high demand in the closely related fields of aerospace (Bombardier), software development (ANSYS), and ultrasound design (Philips).

数值模拟是许多部门的标准做法（例如，航空航天和汽车），因为它们允许在实验室中不可能的场景的研究和工程设计空间的系统探索。当行业解决方案失败时（例如，空气动力学中的失速和冲击），高阶方法通过有效使用现代计算架构来提供预测模拟的变革能力。然而，它们在实际问题中的应用受到稳定性问题的限制。该提案概述了可证明稳定的高阶求解器的发展，在这种情况下是预测。由于重点是在声孔和高强度聚焦超声（HIFU）中的空气动力学和气泡破裂应用，因此该研究与数十亿美元的航空航天和医疗技术行业有关。一个非结构化的四维时空的方法进行，因为它导致显着更少的自由度（DOFs）比替代品。更重要的是，当与时空适应和冲击跟踪相结合时，所产生的问题规模比替代方案小很多倍。 自适应，其中自由度最小化的分布在一个最佳的方式，以减少问题的大小是非常适合于可压缩流，因为它们的多尺度性质。三种方法被认为是1）p适应的元素自由度增加或减少，2）h适应的元素被细分或聚集，和3）r适应的自由度被重新分配。在这方面，将寻求开发能够支持所有三种形式的适应同时保持可证明的稳定性的有效算法。 声致穿孔和HIFU的空气动力学和气泡破裂中的许多问题与冲击有关。不幸的是，对于这样的问题，高阶方法减少到一阶围绕冲击，使他们效率较低。此外，在这种情况下，高阶求解器可能会失败的振荡和nonphyiscal解决方案，形成近冲击的结果。在这个建议中，将开发一个冲击跟踪的方法，网格变形，使元素界面对齐的冲击。这是一种很有吸引力的方法，因为高阶被保持并且冲击被急剧地局部化。这项研究将产生变革性的解决方案，可以解决工业解决方案无法预测的前所未有的规模和准确性问题。他们将支持加拿大航空航天工业设计下一代飞机，并深入了解声致穿孔和HIFU，这可能导致革命性的应用。重要的是，这项研究将培养学生在数学，流体动力学和软件工程方面具有非常理想的技能。这一技能不仅具有高度的可转移性，而且在航空航天（庞巴迪），软件开发（ANSYS）和超声设计（飞利浦）等密切相关的领域也有很高的需求。