Numerical and Analytical Studies of Boundary Value Problems for PDE's. Direct and Inverse Problems

偏微分方程边值问题的数值和分析研究。

• 批准号: 9704575
• 负责人: Michael Vogelius
• 金额: $ 10.2万
• 依托单位: Rutgers University New Brunswick
• 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: 1997-07-15 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704575&HistoricalAwards=false
• 关键词: Numerical; Analytical; Studies; Boundary; Value

9704575 Vogelius The research funded by this grant will focus on analytical and numerical studies of partial differential equations. There will be an emphasis on direct as well as on inverse problems. As far as direct problems are concerned, the effective boundary layer behavior encountered in connection with PDE's with rapidly oscillating coefficients will be investigated. PDE's of this form are used to model the behavior of composite materials. Other ``homogenization" problems for composite materials, for example, certain problems associated with the relationship between microscopic and macroscopic failure (including debonding of closely spaced fibers and delamination) will be studied. Concerning inverse problems, those to be investigated include 1) inverse "coefficient" problems for linear elliptic equations with spatially varying coefficients and 2) inverse "source" problems for semilinear elliptic equations. The data used for identification in both cases consists of overdetermined (Dirichlet or Neumann) boundary data. The analytical component of this work concerns such questions as identifiability and continuous dependence. For the inverse "coefficient" problem, the investigation of the identification of cracks, inhomogeneities and corrosion damage will be continued and a study related to the ``imaging" of thin films will be initiated; for the inverse "source problem," connections with the well known Schiffer (or Pompeiu) conjecture will be investigated with particular emphasis on the relation between the ``smoothness" of the domain and the ability to identify. The numerical work will be devoted to the design of effective reconstruction methods, which to the largest extent possible rely on structural information about the solutions of the underlying PDE. The research funded by this grant will focus on analytical and numerical problems related to continuum mechanics. The research on direct problems has applications to several impor tant practical problems, for instance the assessment of the strength and potential failure of composite materials (including fracture, debonding of fibers and delamination). Special emphasis is put on the understanding of the relationship between microscopic and macroscopic behavior. The research on inverse problems has immediate applications to 1) medical impedance imaging, 2) nondestructive testing of mechanical parts (and sensors) as well as 3) the interpretation of magnetic diagnostics for Tokamak (fusion) devices. Part of this research is concerned with determining the sufficiency of the proposed boundary data for the various identifications (proving uniqueness and continuous dependence results). Another part of this research involves the design of effective algorithms, for instance for the detection and location of cracks and inhomogeneities in metal components as well as for the determination of the level of oxidation of thin films (gas sensors) using real experimental data. There will be an active involvement of post-doctoral researchers as well as graduate students and hopefully even some advanced undergraduate students in various aspects of the research.

9704575 Vogelius这项赠款资助的研究将集中于部分微分方程的分析和数值研究。将要强调直接和反问题。就直接问题而言，将研究与PDE遇到的有效边界层行为与迅速振荡系数有关。该形式的PDE用于建模复合材料的行为。例如，将研究与显微镜和宏观衰竭之间关系有关的某些问题（包括紧密间隔纤维和分层的脱键）的其他问题。具有空间变化系数的线性椭圆方程的“系数”问题和2）逆向“源”的问题，用于椭圆形方程。关注诸如识别性和持续依赖性的问题。对于逆向“源问题”，将研究与众所周知的Schiffer（或Pompeiu）猜想的联系，特别强调域的``平稳性''与识别能力之间的关系。数字工作将用于投入。有效的重建方法在最大程度上依赖于基础PDE的解决方案的结构信息，这将集中在与连续问题有关的分析和数值问题上。对于一些重要的实际问题，例如，对复合材料的强度和潜在失败的评估（包括骨折，纤维脱束和分层）逆问题立即应用于1）医疗阻抗成像，2）机械零件（和传感器）的非破坏性测试以及3）解释Tokamak（Fusion）设备的磁性诊断。这项研究的一部分与确定所提出的边界数据的充分性有关各种标识（证明独特性和持续依赖性结果）。这项研究的另一部分涉及有效算法的设计，例如，用于检测金属成分中裂纹和不均匀性的位置，以及使用实际实验数据确定薄膜（气体传感器）的氧化水平。博士后研究人员以及研究生将积极参与研究，甚至希望在研究的各个方面都有一些高级本科生。