Capturing subgrid structures with level set methods

该项目解决了使用水平集方法寻找，表示和跟踪小结构的需要所产生的重要问题。一个特别的重点是开发的新方法的流体动力学应用。 水平集方法使用定义在欧拉网格上的水平集函数对曲面进行编码，并通过演化函数来演化曲面。 通常使用的实现遭受质量损失，并且小结构可以在计算过程中消失。 为了解决这些问题，局部网格细化和拉格朗日功能已经相当成功，但在该方法的基本的简单性和透明性的牺牲。本研究介绍了一种新的解决方案的困难：将梯度信息的过程。 目前的方法不携带，也不更新这一信息。梯度信息的知识不足以允许实际模拟亚网格尺度过程，但它可以捕获和跟踪亚网格大小的对象。它还有望提高计算梯度起作用的量（例如应力）的准确性。梯度数据必须及时更新，保持函数值和导数之间的一致性，同时利用导数携带的额外信息。 这是利用基本方程的精确解的特征性质来完成的。 所提出的方法的优点是，它捕捉小的结构，同时保留了一个纯粹的欧拉方法在一个规则的网格简单。 这种新方法利用梯度信息，其计算量与忽略梯度的现有技术具有相同的数量级。识别和精确跟踪小或薄的结构，以及分隔具有不同属性的区域的边界，在模拟许多物理和生物过程以及许多其他计算应用中是基本的。薄液体和固体膜、晶片和纤维的演变;气泡流;液滴形成;胶体;等。 该项目的研究将有助于更好地模拟此类过程。水平集方法提供了一种非常有用的表面跟踪技术： 关键思想是将表面建模为某些属性/功能改变符号的轨迹， 并移动表面平流的功能-而不是表面本身。 这有很多好处;例如，它允许与其他相关计算的简单接口，其中表面起作用-其中通常优选在规则网格上具有数据，其中表面难以直接表示（例如：用于表示图像的像素）。然而，这种方法的一个标准困难是，当分辨率低于某个级别时，部分界面可能会丢失。 在这项研究中，作者研究了一种改善这一困难的新方法，除了水平集函数之外，还通过引入计算梯度信息。与现有补救措施不同，这种方法不会破坏水平集方法的基本简单性。 在许多实际应用中，梯度信息是可用的，但目前没有得到充分利用。 示例1：计算机图形中的数据结构存储表面法线，这些法线在对象的模拟中没有完全使用。 为数据配备梯度应提高进一步处理步骤的质量，例如在用于逼真渲染的可视化技术中。示例2：2-D和3-D MRI或CT-SCAN图像的动态范围很高，但当前技术不利用这种梯度信息。将其纳入计算应该会增加有效分辨率，从而改善婴儿肿瘤的检测和小异常的识别。

This project tackles important problems arising from the need to find, represent and track small structures using level set methods. A particular focus are fluid dynamics applications of the new approaches developed. Level set methods encode surfaces using level set functions defined on Eulerian grids, and evolve them by evolving the function. Commonly used implementations suffer from mass loss, andsmall structures can vanish over the course of a computation. To remedy theseproblems, local mesh refinements and Lagrangian features have been reasonably successful, but at the expense of the method's basic simplicity and transparency.This research introduces a new solution to the difficulty: incorporate gradient information into the process. Current approaches do not carry, nor update this information. Instead (when/if needed) it is approximated from the grid function.Knowledge of gradient information is not enough to allow actual simulation of subgrid scale processes, but it enables the capture and tracking of subgrid size objects. It is also expected to improve accuracy in calculating quantities (e.g.stresses) where gradients play a role. The gradient data must be updated in time, maintaining coherence between function values and derivatives, while exploiting the extra information carried by derivatives. This is done using characteristicproperties of the exact solutions to the underlying equation(s). The advantageof the proposed approach is that it captures small structures, while preserving the simplicity of a purely Eulerian approach on a regular grid. This new method uses gradient information with a computational effort which is of the same order of magnitude as that of the current techniques that ignore gradients.Identifying and accurately tracking small or thin structures, and the boundaries separating regions with different properties, is fundamental in simulating many physical and biological processes, and in many other computational applications.Examples arise in: medical imaging; image processing; evolution of thin liquid and solid films, wafers, and fibers; bubbly flows; droplet formation; colloids;etc. The research in this project should contribute to a better simulation ofsuch processes. A very useful technology for surface tracking is provided by thelevel set method: the key idea is to model the surface as the locus where someproperty/function changes sign, and to move the surface advecting the function--- rather than the surface itself. This has many advantages; e.g. it allows an easy interface with other associated calculations where the surface plays a role--- in which it is usually preferable to have the data on a regular grid, where the surface is hard to represent directly (e.g.: the pixels used to represent an image). However, one standard difficulty with this approach is that parts of the interface may be lost when below some level of resolution. In this research the authors investigate a new approach to ameliorating this difficulty, by carrying in the calculation gradient information, in addition to the level set function.Unlike prior remedies, this approach does not tamper with the basic simplicity of the level set method. In many practical applications gradient information is available, but currently not fully used. Example 1: Data structures in computer graphics store surface normals, which are not fully used in simulations of the object. Equipping the data with gradients should improve the quality of further processing steps, such as in visualization techniques for realistic rendering.Example 2: The dynamic range of 2-D and 3-D MRI or CT-SCAN images is high, but current technology does not make use this gradient information. Incorporating it into the calculations should increase the effective resolution, thus improving the detection of tumors in infants and the identification of small anomalies.