A New Finite Element Formulation of the Level Set Method for Free Boundary Problems

自由边界问题水平集法的新有限元公式

• 批准号: 1015002
• 负责人: Suncica Canic
• 金额: $ 15.7万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1015002&HistoricalAwards=false
• 关键词: New; Finite; Element; Formulation; Level

Many problems in mathematical biology and medical research are characterized by the presence of moving interfaces that may have a complex shape and undergo topological changes. The goal of this project is the development of an adaptive variational level set method for numerical simulation of such problems. A major challenge is the need to maintain the signed distance property of the convected level set function and guarantee mass conservation for incompressible flows. In existing level set methods, these constraints are commonly enforced at a postprocessing step when an irrecoverable damage has already been done. In the proposed finite element formulation, numerical solutions are constrained using Lagrange multipliers in the variational formulation for the Galerkin finite element method. This eliminates the need for postprocessing and the associated numerical errors. Algebraic flux correction is performed to satisfy the discrete maximum principle and secure nonlinear stability of the constrained problem. The result is a high-resolution finite element scheme that preserves all important properties of the exact solution. A further gain of accuracy is achieved with a new mesh adaptation strategy that combines local mesh refinement/coarsening with Arbitrary Lagrangian Eulerian (ALE) displacement of nodes.This interdisciplinary research will help scientists and medical doctors to gain a better understanding of fluid flows that take place in human body. Computer simulations are feasible for almost every part of the cardiovascular system, and multiple experiments can be performed without causing any hazard to the patient. However, the usefulness of information obtained in this way depends on the accuracy of the employed numerical methods. It is easy to develop a code that produces beautiful colorful pictures but it is difficult to guarantee that the results are quantitatively correct, especially for the class of free boundary problems considered in this project. It is not unusual that numerical solutions exhibit spurious oscillations, or a spontaneous loss of mass is observed. To make matters worse, other departures from physical reality may remain unnoticed and lead to wrong decisions regarding the appropriate medical treatment. The proposed methodology is designed to rule out such situations. The revised level set method is backed by mathematical theory and has a number of unique features which make it possible to capture the deformation and motion of evolving interfaces with high precision. This research paves the way to reliable simulation of drug delivery, tumor growth, and other biological processes.

数学生物学和医学研究中的许多问题的特征是存在可能具有复杂形状并经历拓扑变化的移动界面。该项目的目的是开发自适应变分级集方法，用于对此类问题进行数值模拟。一个主要的挑战是需要维持对流级设置功能的签名距离属性，并保证不可压缩流的质量保护。在现有的级别集方法中，这些约束通常是在发生不可恢复的损害后的后处理步骤执行的。在提出的有限元公式中，使用拉格兰奇乘数在Galerkin有限元方法的变分公式中限制数值解。这消除了对后处理和关联的数值错误的需求。执行代数通量校正以满足离散的最大原理并确保受约束问题的非线性稳定性。结果是一个高分辨率有限元方案，该方案保留了精确解决方案的所有重要特性。通过一种新的网格适应策略，将本地网格的精致/变厚与任意的拉格朗日欧拉（ALE）相结合，从而实现了进一步的准确性。这项跨学科研究将帮助科学家和医生对人体中的流体流有更好的了解。计算机模拟对于心血管系统的几乎每个部分都是可行的，并且可以进行多次实验而不会对患者造成任何危害。但是，以这种方式获得的信息的实用性取决于使用的数值方法的准确性。很容易开发产生美丽的色彩图片的代码，但是很难确保结果在定量上是正确的，尤其是对于本项目中考虑的一类自由边界问题。数值溶液表现出虚假振荡或观察到质量自发损失并不罕见。更糟糕的是，其他偏离物理现实的偏离可能仍未被注意，并导致关于适当医疗的错误决定。所提出的方法旨在排除这种情况。修订后的集合方法得到数学理论的支持，并且具有许多独特的特征，这些特征使得捕获具有高精度不断发展的接口的变形和运动。这项研究为对药物输送，肿瘤生长和其他生物学过程的可靠模拟提供了铺平的道路。