Mathematically Rigorous High-Fidelity Solvers

数值模拟是许多领域（例如，航空航天和汽车）的标准做法，因为它们允许对实验室中不可能的场景进行研究，并对工程设计空间进行系统探索。在行业求解器失败的地方（例如，空气动力学中的失速和冲击），高阶方法有望通过有效使用现代计算架构来实现预测模拟的变革性能力。然而，它们在实际问题中的应用受到稳定性问题的限制。本建议概述了可证明稳定的高阶解算器的发展，这些解算器在这种情况下具有预测性。由于该研究的重点是超声和高强度聚焦超声（HIFU）在空气动力学和气泡破裂方面的应用，因此该研究与数十亿美元的航空航天和医疗技术行业相关。采用一种非结构化的四维时空方法，因为它比其他方法的自由度（dof）要少得多。更重要的是，当与时空自适应和冲击跟踪相结合时，得到的问题规模比替代方案小很多倍。由于可压缩流的多尺度特性，通过以最优方式分配dof来最小化问题规模的自适应，非常适合于可压缩流。考虑了三种方法：1)p适应，其中元素自由度增加或减少；2)h适应，其中元素被细分或聚集；3)r适应，其中自由度被重新分配。在这方面，将寻求开发有效的算法，既能支持所有三种形式的适应，又能保持可证明的稳定性。超声和超声的空气动力学和气泡破溃的许多问题都与冲击有关。不幸的是，对于这样的问题，高阶方法在冲击周围降低到一阶，使它们效率较低。此外，在这种情况下，由于振荡和非物理解在冲击附近形成，高阶解可能会失败。在本提案中，将开发一种冲击跟踪方法，其中网格变形使单元界面与冲击对齐。这是一种很有吸引力的方法，因为它保持了高阶，而且冲击是明显局部化的。这项研究将产生变革性的解决方案，能够以前所未有的规模和精度解决工业解决方案无法预测的问题。他们将为加拿大航空航天工业设计下一代飞机提供支持，并为sonopation和HIFU提供见解，这可能会导致两者的革命性应用。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的高度理想技能。这种技能不仅具有很强的可转移性，而且在航空航天（庞巴迪）、软件开发（ANSYS）和超声波设计（飞利浦）等密切相关的领域也有很高的需求。

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).