Information-Theoretic Meshfree Approximation Schemes in Solid Mechanics

项目摘要-N.Sukumar，UC-Davis在本方案中，采用信息论变分原理来构造求解偏微分方程组(PDE)的无网格近似格式。该方法从根本上不同于以往无网格方法的研究，旨在为偏微分方程组的逼近设计和新的求解策略提供新的途径。将形函数视为离散概率分布，以多项式再现条件为约束条件。作为一种最小偏差统计推断的方法，PI使用最大熵原理来推导近似式，该近似式在配置格式和Galerkin变分公式中被采用。形状函数的计算采用了最大熵算法。文中还提出了构造高阶近似格式的一种熵泛函。与劳伦斯利弗莫尔国家实验室的Michael Puso的外部合作将为研究生提供一个极好的机会，让他们在国家实验室获得宝贵的研究经验，并为变形计算中无网格方法的理论和应用奠定坚实的基础。计划开设一门关于无网格方法的新课程，并将开发一个可视化无网格形函数的Java Applet，这将是一个有效的学习工具，有助于更好地理解无网格近似格式。

PROJECT SUMMARY - N. Sukumar, UC-DavisIn this proposal, information-theoretic variational principles are adopted to construct meshfree approximation schemes for the solution of partial differential equations (PDEs). The proposed approach is fundamentally distinct from prior research in meshfree methods, and purports to provide new pathways in the design of approximants as well as in providing novel solution-strategies for PDEs. The shape functions are viewed as a discrete probability distribution, and the polynomial reproducing conditions are the constraints. As a means for least-biased statistical inference, PI uses the maximum entropy principle to derive the approximant, which is adopted within collocationschemes and Galerkin variational formulations. Entropy maximization algorithms are used to compute the shape functions. An entropy functional for the construction of higher-order approximation schemes is also proposed. The external collaboration with Michael Puso at Lawrence Livermore National Laboratory will provide an excellent opportunity for a graduate student to gain valuable research experienceat a national laboratory and also to develop a strong foundation in the theory and application of meshfree methods in deformation computations. A new course offering on meshfree methods is planned, and a JAVA applet for visualizing meshfree shape functions will be developed, which will be useful as an effective learning tool to better understand meshfree approximation schemes.