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.

本研究的目的是开发一类新的方法来求解具有界面条件的偏微分方程（PDE）及其在工程和计算科学问题中的应用。由于准确地表示和演化接口与正确离散算子以开发具有预测能力的数值方法同样重要，因此我们的方法将是三管齐下的。 首先，研究了使用水平集方法的小型结构的表示和跟踪。水平集方法使用欧拉网格上定义的水平集函数来演化曲面或曲线。常用的方法会遭受质量损失。这项研究建议通过提供附加方程将梯度信息纳入计算中来解决上述问题。进化梯度信息的知识不允许在子网格尺度上实际模拟某些过程，但它将允许捕获和跟踪小于网格尺寸的结构。其次，梯度增强方法可以提高梯度（或高阶导数）发挥作用的微分量的计算精度。我们将研究所提出的梯度增强“哲学”到其他类偏微分方程的扩展，例如哈密尔顿-雅可比方程、具有界面跳跃条件的泊松方程和两相不可压缩纳维-斯托克斯方程。所提出研究的长期目标是发展梯度增强方案的完整理论，并提供一套用于求解具有界面的偏微分方程的工具和方法。第三，将研究涉及界面条件或不连续性的各种问题的许多应用。我们建议应用这些方法来研究各种情况下的不可压缩两相纳维-斯托克斯方程，包括复杂流体。我们将研究梯度增强框架对固液相互作用问题的适用性，以及作为具有巨大实际意义的非线性守恒定律原型的交通流冲击。