Information-Theoretic Meshfree Approximation Schemes in Solid Mechanics

项目概要- N. Sukumar，UC-Davis在这个建议中，采用信息论变分原理来构造求解偏微分方程（PDE）的无网格逼近格式。所提出的方法是从根本上不同于以往的研究无网格方法，并声称提供新的途径，在设计的逼近，以及在提供新的解决方案的策略偏微分方程。形状函数被视为离散的概率分布，多项式再生条件是约束。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，该近似值在配置方案和Galerkin变分公式中被采用。熵最大化算法用于计算形状函数。本文还提出了一种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极好的机会，使其能够在国家实验室获得宝贵的研究经验，并在变形计算中的无网格方法的理论和应用方面奠定坚实的基础。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小程序，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.