Mathematically Rigorous High-Fidelity Solvers

数值模拟是许多部门(例如航空航天和汽车)的标准做法，因为它们允许研究实验室中不可能的情景，并系统地探索工程设计空间。在工业解算器失效的地方(例如，空气动力学中的失速和冲击)，高阶方法承诺通过高效使用现代计算体系结构来提供预测模拟的变革性能力。然而，它们在实际问题上的应用受到稳定性问题的限制。这项提议概述了在这种情况下具有预测性的可证明稳定的高阶求解器的发展。由于这项研究的重点是声学和高强度聚焦超声(HIFU)在空气动力学和气泡破裂方面的应用，因此这项研究与价值数十亿美元的航空航天和医疗技术行业相关。一种非结构化的四维时空方法被追求，因为它产生的自由度(DOF)比备选方法少得多。更重要的是，当与时空自适应和激波跟踪相结合时，所产生的问题规模比备选方案小许多倍。自适应通过以最佳方式分布自由度来最小化自由度，以减小问题规模，因为其多尺度性质，非常适合于可压缩流动。三种方法被认为是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，以及3)r适应，其中自由度重新分布。在这方面，将致力于开发能够支持所有三种适应形式的有效算法，同时保持可证明的稳定性。声学和HIFU的空气动力学和气泡崩塌中的许多问题都与冲击有关。不幸的是，对于这类问题，高阶方法简化为围绕冲击的一阶方法，使得它们的效率较低。此外，在这种情况下，高阶求解器可能会因为形成近激波的振荡和非物理解而失败。在这个方案中，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。这是一种吸引人的方法，因为维持了高秩序，冲击被急剧地局限于局部。这项研究将产生变革性的解算器，能够以前所未有的规模和精度解决工业解算器无法预测的问题。他们将支持加拿大航空航天工业设计下一代飞机，并让人们深入了解可能导致这两种技术革命性应用的声学和HIFU技术。重要的是，这项研究将培养学生在数学、流体力学和软件工程方面具有非常理想的技能。这一技能不仅具有很高的可转移性，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。

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).