Interface tracking methods and gradient-augmented algorithms: theory and applications

界面跟踪方法和梯度增强算法：理论与应用

• 批准号: 402612-2011
• 负责人: Nave, JeanChristophe
• 金额: $ 2.33万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=573336
• 关键词: Interface; tracking; methods; gradient; augmented

The objective of the proposed research is the development of a new class of methods for the solution of Partial Differential Equations (PDEs) with interface conditions and their application to problems in engineering and computational science. Since it is as important to represent and evolve the interface accurately as it is to correctly discretize operators in order to develop numerical methods with predictive capabilities, our approach will be three-pronged. First, the representation and tracking of small structures using level set methods is investigated. Level set methods evolve a surface or a curve using a level set function defined on an Eulerian grid. Commonly used approaches suffer from a loss of mass. This investigation proposes to remedy the above-mentioned problems by incorporating gradient information into the calculation, by providing additional equations. The knowledge of evolved gradient information does not allow an actual simulation of some process on a sub-grid scale, but it will allow to capture and to track structures smaller than the grid size. Second, the gradient-augmented approach may improve the accuracy in calculating differential quantities where gradients (or higher derivatives) play a role. We will investigate the extension of the proposed gradient-augmented "philosophy" to other class of PDEs such as Hamilton-Jacobi equations, Poisson equation with interface jump conditions and the two-phase incompressible Navier-Stokes Equations. The long-term aim of the proposed investigations is to develop a full theory of gradient-augmented schemes, and to provide a set of tools and methods for the solution of PDEs with interface. Third, many application to various problems involving interface conditions or discontinuities will be investigated. We propose applying the methods to study incompressible two-phase Navier-Stokes equations in various situations, including complex fluids. We will investigate applicability of the gradient-augmented framework to solid-fluid interaction problems, and shocks in traffic flows as prototype non-linear conservation laws with great practical interest.

提出的研究的目的是开发一类新的方法，用于解决偏微分方程（PDE）的接口条件和它们的应用问题，在工程和计算科学。由于它是一样重要的代表和发展的接口准确，因为它是正确的离散运营商，以开发具有预测能力的数值方法，我们的方法将是三管齐下。 首先，研究了基于水平集方法的小结构表示与跟踪。水平集方法使用定义在欧拉网格上的水平集函数来演化曲面或曲线。通常使用的方法遭受质量损失。本研究提出通过将梯度信息纳入计算中，通过提供额外的方程来补救上述问题。演化梯度信息的知识不允许在亚网格尺度上实际模拟某些过程，但它将允许捕获和跟踪比网格尺寸更小的结构。其次，梯度增强方法可以提高计算梯度（或更高导数）起作用的微分量的准确性。我们将研究所提出的梯度增广的“哲学”的扩展到其他类的偏微分方程，如Hamilton-Jacobi方程，泊松方程与界面跳跃条件和两相不可压Navier-Stokes方程。本研究的长期目标是发展一套完整的梯度增广格式理论，并提供一套求解含界面偏微分方程的工具和方法。第三，许多应用程序涉及界面条件或不连续性的各种问题将进行调查。我们建议应用的方法来研究不可压缩的两相Navier-Stokes方程在各种情况下，包括复杂的流体。我们将研究梯度增强框架的适用性，以固体-流体相互作用的问题，并在交通流中的冲击作为原型非线性守恒律具有很大的实际意义。