Information-Theoretic Meshfree Approximation Schemes in Solid Mechanics

项目摘要- N. Sukumar， uc - davis在本研究中，采用信息理论的变分原理构建了求解偏微分方程（PDEs）的无网格逼近格式。所提出的方法从根本上不同于先前的无网格方法研究，旨在为近似设计提供新的途径，并为偏微分方程提供新的解决方案策略。将形状函数视为离散概率分布，多项式再现条件作为约束条件。作为最小偏差统计推断的一种手段，PI使用最大熵原理来推导近似值，该近似值在搭配方案和伽辽金变分公式中被采用。利用熵最大化算法计算形状函数。提出了一种用于构造高阶近似格式的熵泛函。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在国家实验室获得宝贵研究经验的绝佳机会，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，并将开发一个用于可视化无网格形状函数的JAVA applet，这将是一个有效的学习工具，可以更好地理解无网格逼近方案。

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.