Mathematical Sciences: Nonlinear Waves, Nonlinear Materials and Chaotic Mixing

数学科学：非线性波、非线性材料和混沌混合

• 批准号: 9500568
• 负责人: John Grove
• 金额: $ 30万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-08-01 至 1998-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500568&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Waves; Materials

9500568 Grove This project seeks to develop a fundamental approach to the study of mixing layers, nonlinear materials, and fluid chaos. Nonlinear waves in hyperbolic conservation laws commonly display sensitive dependence to the problem formulation and to numerical solution algorithms. A completed theory of a hyperbolic wave should contain the following ingredients: (1) jump conditions describing the influence of the wave on the flow in which it is embedded, (2) the width or growth rate of the layer, (3) a complete theory of the internal structure, and (4) a systematic unfolding of all possible sensitive dependencies for the wave. The mathematical tools and theories used in the analysis are varied, and include: partial differential equations (hyperbolic conservation laws), random fields, perturbation theory (ordinary and renormalized), renormalization group methods, traveling wave analysis, bifurcation theory, and the geometric theory of ordinary differential equations. A representative mixing layer problem is acceleration driven layers, arising in instabilities of a fluid interface. Such mixing layers arise in many areas of basic science and technology, including supernovae, inertial confinement fusion, and injection jets in carburetors for internal combustion engines. Detailed mathematical modeling and analysis will seek to modify and refine the defining mathematical equations describing such flows and develop an improved understanding of the structure of these equations. This analysis will in turn be used to develop high resolution numerical methods for the solution of these equations. The nonlinear material wave patterns we study describe important metal forming processes such as punching and cutting (shear bands) and flow instabilities in forming plastic components by injection molding (viscoelastic materials). The analysis of waves in nonlinear materials uses a similar integrated approach: modeling, theor y, computations, and applications. Interactions with collaborators will allow transfer of this technology to appropriate applied physics and engineering communities.

9500568格罗夫 这个项目旨在发展一种基本的方法来研究混合 层、非线性材料和流体混沌。非线性双曲波 守恒定律通常表现出对问题的敏感依赖性 公式化和数值解算法。完整的理论 双曲波应包含以下成分：（1）跳跃 描述波对它所处的流动的影响的条件 嵌入式，（2）层的宽度或生长速率，（3）完整的理论， 内部结构，以及（4）系统地展开所有可能的敏感 依赖于浪潮。数学中使用的数学工具和理论 分析是多种多样的，包括：偏微分方程（双曲 守恒定律），随机场，扰动理论（普通和 重整化），重整化群方法，行波分析， 分歧理论和常微分几何理论 方程 代表性的混合层问题是加速驱动层， 流体界面的不稳定性。 这种混合层出现在许多地区 基础科学和技术，包括超新星，惯性约束 内燃机化油器中的融合和喷射喷嘴。 详细的数学建模和分析将寻求修改和完善 描述这种流动的定义数学方程，并开发了一个 更好地理解这些方程的结构。 该分析 将反过来用于开发高分辨率的数值方法， 解这些方程。我们研究的非线性物质波型 描述重要的金属成形工艺，如冲压和切割 （剪切带）和流动不稳定性， 注塑成型（粘弹性材料）。波的分析 非线性材料使用类似的集成方法：建模， 理论上， 计算和应用。与合作者的互动将允许 将这一技术转移到适当的应用物理学和工程学领域 社区.