Numerical Capturing of Multiscale Solutions in Partial Differential Equations

偏微分方程多尺度解的数值捕捉

• 批准号: 9704976
• 负责人: Thomas Hou
• 金额: $ 14.55万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704976&HistoricalAwards=false
• 关键词: Numerical; Capturing; Multiscale; Solutions; Partial

Numerical Capturing of Multiscale Solutions in PDEs Thomas Y. Hou Applied Math Caltech Abstract of Proposed Research We propose developing a new finite element method of computing multiscale and multicomponent problems in composite materials and porous media. In this method, the base functions are constructed in such a way that they contain the local microscopic properties of the differential operators. These oscillatory bases are independent from each other and can be constructed independently. In effect, we break a large scale computation into many smaller and independent pieces, which can be carried out perfectly in parallel. Thus, the size of the computation is drastically reduced. Due to this drastic memory saving, numerical computations of practical problems with many small scales, which were previously unobtainable using a traditional finite element method, now become possible using our method. An advantage of our approach is that it can handle general multiscale problems without the requirement of scale separation, a property which is very important for practical applications. Several numerical examples are included to demonstrate this effect. We will perform a careful numerical error analysis of our method. The effect of grid resonance, and geometric singularities of local inclusions, such as corners, cusps, will be analyzed. Effective numerical simulations of multiscale phenomena play an essential role in many engineering and industrial applications. A direct numerical solution of the multiple scale problems is difficult even with modern supercomputers. The major difficulty of direct solutions is the scale of computation. For oil reservoir simulations, it is common that millions of grid blocks are involved, with each block having a dimension of tens of meters, whereas the permeability measured from cores is at a scale of several centimeters. This gives more th an hundred thousands of degrees of freedom per spatial dimension in the computation. Therefore, a tremendous amount of computer memory and CPU time are required, and they can easily exceed the limit of today's computing resources. The goal of this proposed research is to provide a coarsened reservoir model which gives simulation results in close agreement with those of the original fine scale description but at considerable computational savings. Many existing up-scaling methods currently used in the oil industry are based on the homogenization theory with periodic structures and separable scales. There is no guarantee that they work for general random permeability without scale separation. In comparison, our approach is more systematic and general. We are very hopeful that this proposed research will provide an effective alternative to perform oil reservoir simulations.

偏微分方程中多尺度解的数值捕获 Thomas Y. Hou 应用数学加州理工学院拟议研究摘要 我们建议开发一种新的有限元方法来计算复合材料和多孔介质中的多尺度和多组分问题。 在该方法中，基函数的构造方式使得它们包含微分算子的局部微观性质。这些振荡基座彼此独立，可以独立构造。实际上，我们将大规模计算分解为许多较小且独立的部分，这些部分可以完美地并行执行。因此，计算量大大减少。由于这种巨大的内存节省，许多小尺度的实际问题的数值计算以前使用传统的有限元方法无法实现，现在使用我们的方法成为可能。 我们的方法的一个优点是它可以处理一般的多尺度问题，而不需要尺度分离，这一特性对于实际应用非常重要。包括几个数值示例来证明这种效果。我们将对我们的方法进行仔细的数值误差分析。将分析电网共振的影响以及局部夹杂物（例如角、尖点）的几何奇点。 多尺度现象的有效数值模拟在许多工程和工业应用中发挥着重要作用。即使使用现代超级计算机，直接数值求解多尺度问题也很困难。直接解决方案的主要困难是计算规模。 对于油藏模拟，通常涉及数百万个网格块，每个网格块的尺寸为数十米，而岩心测量的渗透率则为几厘米尺度。这在计算中为每个空间维度提供了超过十万个自由度。 因此，需要大量的计算机内存和CPU时间，并且它们很容易超出当今计算资源的限制。这项研究的目标是提供一个粗化的油藏模型，该模型给出的模拟结果与原始精细尺度描述的结果非常一致，但节省了大量的计算量。目前石油工业中使用的许多现有的放大方法都是基于具有周期性结构和可分离尺度的均质化理论。无法保证它们在没有尺度分离的情况下适用于一般随机渗透率。相比之下，我们的方法更加系统化和通用化。 我们非常希望这项研究将为油藏模拟提供有效的替代方案。