Capturing subgrid structures with level set methods

使用水平集方法捕获子网格结构

• 批准号: 0813648
• 负责人: Rodolfo Rosales
• 金额: $ 49.2万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-15 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0813648&HistoricalAwards=false
• 关键词: Capturing; subgrid; structures; level; set

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.

该项目解决了由于需要使用水平集方法来寻找、表示和跟踪小型结构而产生的重要问题。一个特别的焦点是开发的新方法的流体动力学应用。水平集方法使用欧拉网格上定义的水平集函数对曲面进行编码，并通过进化该函数来进化曲面。通常使用的实现遭受质量损失，并且小的结构可能在计算过程中消失。为了解决这些问题，局部网格加密和拉格朗日特征取得了相当大的成功，但代价是该方法的基本简单性和透明度。本研究引入了一种新的解决方案：在过程中加入梯度信息。目前的方法不携带、也不更新这一信息。相反，(当需要时/如果需要)它是从网格函数近似而来的。梯度信息的知识不足以允许实际模拟亚网格尺度的过程，但它能够捕获和跟踪子网格大小的对象。它还有望提高在梯度起作用的情况下计算量(例如应力)的精度。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(S)。所提出的方法的优点是它捕捉到了小结构，同时保持了规则网格上纯欧拉方法的简单性。这种新方法使用梯度信息，其计算工作量与当前忽略梯度的技术具有相同的数量级。识别和准确跟踪细小或薄的结构以及分隔不同性质区域的边界，是模拟许多物理和生物过程的基础，也是许多其他计算应用的基础。例如：医学成像；图像处理；薄液体和固体薄膜、晶片和纤维的演化；泡状流；液滴形成；胶体等。该项目的研究应该有助于更好地模拟这些过程。水平集方法提供了一种非常有用的曲面跟踪技术：其关键思想是将曲面建模为某些属性/函数改变符号的轨迹，并将曲面平流到函数-而不是曲面本身。这有很多优点；例如，它允许与曲面起作用的其他相关计算的容易接口-其中，通常最好将数据放在规则网格上，其中曲面很难直接表示(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，可能会丢失部分界面。在这项研究中，作者探索了一种新的方法来改善这一困难，除了水平集函数之外，还引入了计算梯度信息，该方法不像以前的补救方法那样，不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，但目前还没有得到充分的利用。示例1：计算机图形中的数据结构存储曲面法线，这些法线在对象的模拟中并未完全使用。为数据配备梯度应该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。例如2：2和3-D MRI或CT扫描图像的动态范围很高，但目前的技术没有利用这种梯度信息。将其纳入计算应提高有效分辨率，从而提高对婴儿肿瘤的检测和对微小异常的识别。