Mathematical Sciences: Calculation of Young Measure Valued Solutions Arising in the Calculus of Variations

数学科学：变分法中产生的杨测值解的计算

• 批准号: 9203406
• 负责人: Noel Walkington
• 金额: $ 7.2万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9203406&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Calculation; Young; Measure

Algorithms will be constructed to calculate generalized solutions of problems arising in the calculus of variations that involve non-convex bulk energy functionals and which have no classical minima. The generalized solutions, consisting of a parameterized Young measure and a weak limit (of a minimizing sequence) do exist and will be calculated numerically by these algorithms. The proposed approach offers considerable economy over direct simulation of the oscillatory minimizing sequences since the weak limit and Young measure are slowly varying quantities in general, which may be well approximated using coarse meshes. In many physical problems, the observed macroscopic properties are averages of fine-scale quantities visible only through optical or possibly electron microscopes. For example, many common metals consist of layers of several distinctly different component metals. The problem of determining the properties of such a materials by direct averaging over the fine layers is beyond the scope of current computers. The Young measure characterizes the averaged properties very succinctly. In this project, the theory of Young measures will be utilized to develop computer codes capable of determining both the microscopic and macroscopic properties of such materials.

将构建算法来计算变分计算中出现的问题的广义解，这些问题涉及非凸体能量泛函并且没有经典最小值。 由参数化杨氏测度和（最小化序列的）弱极限组成的广义解确实存在，并将通过这些算法进行数值计算。 与直接模拟振荡最小化序列相比，所提出的方法提供了相当大的经济性，因为弱极限和杨氏测度通常是缓慢变化的量，可以使用粗网格很好地近似。 在许多物理问题中，观察到的宏观特性是仅通过光学显微镜或可能的电子显微镜可见的精细尺度量的平均值。 例如，许多常见金属由几种截然不同的金属成分层组成。 通过直接平均精细层来确定这种材料的特性的问题超出了当前计算机的范围。 杨氏测度非常简洁地描述了平均特性。 在该项目中，杨氏测度理论将用于开发能够确定此类材料的微观和宏观特性的计算机代码。