材料原子系统的拟连续耦合建模、算法设计和分析

Many fundamental problems in material science and engineering can be modeled by partial differential equations that reflect the macroscopic or continuum property of materials. In spite of the tremendous successes, continuum models also have limitations, accuracy being one of them. When a material has defects or atomistic features, a continuum model may not be able to capture them. For example, in material fracture, the atomic bond has to be considered. Because of this, one might be tempted to switch to a fully atomistic model that is accurate and takes the full atomistic structure into account. However, this is not an efficient strategy, because atomistic systems involved with extremely fine scales are too large to handle even using most powerful computers available. This is where multi-scale modeling is particularly useful. By coupling continuum and atomistic models, it takes advantage of both the simplicity and efficiency of the continuum (or coarse-grained) models, as well as the accuracy of the atomistic models. ..The quasicontinuum (QC) method is a typical approach to formulating such computational models. It has attracted a great deal of interest in the past decade among applied and computational mathematicians around the globe. The idea of the QC model is that in the material region where no defects occur the theory of continuum material elasticity (or coarse grid approximation) may apply. The QC method is based on the finite element method and the so-called Cauchy-Born rule which computes the energy contribution of a representative atom (taken in an element) and piecewise-linearly extending it over the whole element. This Cauchy-Born rule gives a good approximation only to crystalline materials with simple lattice structure. However, most crystalline materials have a complex lattice structure (a union of a number of simple lattice sites). The idea of the QC method may be extended to complex lattice cases, accounting for relative shifts between the comprising simple lattice sites. Error analysis for complex lattice QC method was not available until a fairly recent attempt by the PI and his collaborators which relates the method to an appropriate discrete homogenization process. Many mathematical problems remain unsolved. Numerical analysis and atomistic/continuum interface consistency study of the complex lattice QC are still in their infancy. They are, therefore, the major research focus of this proposal. In the proposal we aim to establish theoretical and/or numerical analysis for QC multi-scale models, especially those for complex lattice crystalline materials. The outcomes of the analysis will suggest, motivate and help to design more accurate, more robust and more efficient computational models and implementation techniques for solving real material problems.

材料科学和工程的许多问题，没法用反映宏观连续性的偏微分方程模型来精确描述。而需要用到微观原子模型，例如，材料断裂或错位等。但由于原子数量巨大，难以直接求解。迫切需要建立一种宏观/微观描述相结合的可计算模型来较精确地获得材料特性。本项目重点研究有复杂晶格结构的材料的拟连续(Quasi-Continuum)模型。它是一个将原子模型/连续模型（即微观/宏观）耦合在一起的典型的多尺度计算模型或方法。在过去十年中，拟连续方法已经成为发展迅猛广泛应用的材料模拟方法，并且吸引了越来越多的应用和计算数学家的研究兴趣。本课题的研究目标是建立相容原子/连续模型界面耦合的，特别是具有复杂晶格结构的，拟连续多尺度有限元可计算耦合模型，对这些模型提供严格的理论和数值判据，并用于计算实际材料问题。我们的研究成果将会启发和帮助在材料模拟设计方面工作人员和学者建立和使用更准确可靠、更快速有效的可计算模型及算法实现技术。

材料科学和工程的许多问题，没法用反映宏观连续性的偏微分方程模型来精确描述。而需要用到微观原子模型，例如，材料断裂或错位等。但由于原子数量巨大，难以直接求解。迫切需要建立一种宏观/微观描述相结合的可计算模型来较精确地获得材料特性。本项目重点研究材料的拟连续(Quasi-Continuum)模型。它是一个将原子模型/连续模型（即微观/宏观）耦合在一起的典型的多尺度计算模型或方法。本项目分析了二维拟连续模型的后验误差，给出了后验误差估计，并基于此估计，设计了网格自适应算法，数值结果验证了理论分析中得到了与先验误差分析一致的收敛率，该部分工作为首创。除此之外，基于一维非线性混合型波动方程黎曼问题的分析，本项目还研究了一维拟连续动态模型的粗粒化近似的收敛性。考虑原子模型的可计算建模方法的同时，也致力于学科交叉研究，将建模的方法和分析技巧应用于其他方向和学科的相关问题，或将其他问题的分析方法，例如从连续多尺度问题中建立新方法新理论，应用到材料原子系统的研究当中去。我们研究了相关问题的多尺度有限元方法，特别是稳定性分析方法。这些相关问题包括对流扩散反应方程、麦克斯韦方程、纳维尔斯托克斯方程。

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