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

数学、生物学和医学研究中的许多问题都以运动界面的存在为特征，这些界面可能具有复杂的形状并经历拓扑变化。本项目的目标是开发一种自适应变分水平集方法来数值模拟这类问题。一个主要的挑战是需要保持对流水平集函数的有符号距离性质并保证不可压缩流的质量守恒。在现有的水平集方法中，当不可恢复的损害已经造成时，这些约束通常在后处理步骤中强制执行。在所提出的有限元公式中，数值解使用Galerkin有限元方法变分公式中的拉格朗日乘子进行约束。这消除了后处理的需要和相关的数值误差。为了满足离散极大值原理和保证约束问题的非线性稳定性，采用了代数通量修正。结果是一个高分辨率的有限元方案，它保留了精确解的所有重要性质。采用一种新的网格自适应策略，将局部网格细化/粗化与节点的任意拉格朗日欧拉（ALE）位移相结合，进一步提高了精度。这项跨学科的研究将有助于科学家和医生更好地了解发生在人体中的流体流动。计算机模拟几乎对心血管系统的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。然而，以这种方式获得的信息的有用性取决于所采用的数值方法的准确性。开发一个生成漂亮的彩色图片的代码很容易，但很难保证结果在数量上是正确的，特别是对于本项目中考虑的一类自由边界问题。数值解表现出虚假的振荡，或者观察到自发的质量损失，这并不罕见。更糟糕的是，其他与物理现实的背离可能被忽视，并导致在适当的医疗方面做出错误的决定。拟议的方法旨在排除这种情况。修正后的水平集方法以数学理论为基础，具有许多独特的特征，可以高精度地捕捉不断演化的界面的变形和运动。这项研究为药物传递、肿瘤生长和其他生物过程的可靠模拟铺平了道路。

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.