SBIR Phase I: Bifurcation analysis of nonlinear PDEs - a powerful Software Tool for Computational Design - Educational Applications

SBIR 第一阶段：非线性偏微分方程的分岔分析 - 强大的计算设计软件工具 - 教育应用

• 批准号: 0712091
• 负责人: Edward Kansa
• 金额: $ 10万
• 依托单位: Convergent Solutions
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2007-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0712091&HistoricalAwards=false
• 关键词: SBIR; Phase; Bifurcation; analysis; nonlinear

This Small Business Innovation Research (SBIR) Phase I research project will develop advanced software tools for efficient and accurate numerical bifurcation analysis of nonlinear elliptic partial differential equations (PDEs) applicable to such fields as computational design for educational applications. Nonlinear elliptic PDEs are the basis for many scientific and engineering problems, such as chemical reactions, pattern formation and propagation of action potentials in biology, blood coagulation cascades in biophysics, etc. In these problems it is crucial to understand the qualitative dependence of the solution on the problem parameters. The principal approach of numerical bifurcation analysis is based on continuation of solutions to well-defined operator equations. Such computational results give to student/researcher a deeper understanding of the solution behavior, stability, multiplicity, and bifurcations, and often provide direct links to underlying mathematical theories. Interactive interface and automatic PDE discretization let the student/teacher concentrate on the problem solving, increasing the learning process efficiency. The intellectual merit of the proposed activity is in the integration of efficient and accurate discretization methods of PDEs by radial-based functions (RBFs) into existing numerical bifurcation analysis software. The outcome of this project will show the feasibility of the proposed concept that can be applied to a variety of problems. As an example, this software tool will be validated on an important application - the analysis of a blood coagulation cascade.The numerical bifurcation analysis tools for educational applications are currently not available on the market. Although recent progress made in mesh less numerical methods creates a basis for incorporating these into numerical bifurcation software, high accuracy mesh less methods would allow researchers to elegantly identify the spatial solutions of the PDE systems that are used for mathematical description ofnonlinear processes. These methods could also help to identify new regimes otherwise missing, to bring insights into evolution of nonlinear systems dynamics and provide enhanced scientific and technological understanding. Additionally, if this tool could be integrated into MATLAB platform environment - a widely used mathematics tool, which would enable these software tools to be user-friendly, portable to all operating systems and allow a standard handling of data files, graphical output, etc, and would allow these tools to be attractive for education at various science and engineering departments of universities. This project will also have a broad impact by enabling open source model for software distribution.

这个小企业创新研究（SBIR）第一阶段的研究项目将开发先进的软件工具，用于有效和准确的非线性椭圆偏微分方程（PDE）的数值分叉分析，适用于教育应用的计算设计等领域。非线性椭圆偏微分方程是许多科学和工程问题的基础，如化学反应，图案的形成和传播的动作电位在生物学，血液凝固级联在生物物理学等，在这些问题中，它是至关重要的是要了解定性依赖的解决方案的问题参数。数值分歧分析的主要方法是基于定义良好的算子方程的解的延拓。这样的计算结果给学生/研究人员的解决方案的行为，稳定性，多重性和分叉的更深层次的理解，并经常提供直接的联系，基本的数学理论。交互式界面和自动PDE离散化让学生/教师专注于解决问题，提高学习过程的效率。 拟议的活动的智力优点是在集成的高效和准确的离散化方法的偏微分方程的径向基函数（RBFs）到现有的数值分岔分析软件。该项目的结果将表明所提出的概念的可行性，可以应用于各种问题。例如，该软件工具将在一个重要应用--血液凝固级联分析上进行验证。目前市场上还没有针对教育应用的数值分叉分析工具。虽然最近取得的进展，在无网格的数值方法创建了一个基础，将这些数值分叉软件，高精度的无网格方法将允许研究人员优雅地确定用于数学描述ofnonlinear过程的PDE系统的空间解决方案。这些方法还可以帮助确定新的制度，否则失踪，使洞察非线性系统动力学的演变，并提供增强的科学和技术的理解。此外，如果这个工具可以集成到MATLAB平台环境-一个广泛使用的数学工具，这将使这些软件工具是用户友好的，可移植到所有操作系统，并允许数据文件的标准处理，图形输出等，并将允许这些工具在大学的各种科学和工程部门的教育有吸引力。该项目还将通过启用软件分发的开源模型产生广泛的影响。