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Quantitative stochastic homogenization: periodic representative volume element approximationsin non-linear elasticity

定量随机均质化：非线性弹性中的周期性代表性体积元近似

基本信息

批准号：
405009441
负责人：
Professor Dr. Stefan Neukamm
金额：
--
依托单位：
Institut für Wissenschaftliches Rechnen
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2018
资助国家：
德国
起止时间：
2017-12-31 至 2020-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/405009441?language=en
关键词：
Quantitative stochastic homogenization periodic representative

项目摘要

Polycrystalline materials, reinforced rubber, foams, and biological tissues are examples for the huge class of random heterogeneous materials (RM). Those materials feature microstructural uncertainties: e.g. the distribution, geometry, and constitutive parameters of the individual phases of a composite might be known on a statistical level only. Typically RM display an effective behavior on large length-scales, which can be designed and optimized by changing the composition and geometry of the microstructure. Understanding the subtle interplay between microscopic and macroscopic scales and the question of how effective properties emerge in such systems is an exciting and fundamental problem --- both, for theory, as well as, for the development of new technologies. Science approaches this question in various ways, e.g.~experimentally, via modeling and simulation, and by mathematical analysis. In many cases, if the RM is statistically homogeneous, it displays on large length-scales an (almost) deterministic physical behavior. This allows for a tremendous reduction of complexity with regard to modeling and simulation: Instead of resolving microscopic and uncertain properties, one can consider a macroscopic and deterministic model describing a homogeneous material. This is at the bases of many methods in computational micromechanics, e.g. RVE-based methods that approximate an unknown macroscopic constitutive relation with help of a representative volume element (RVE) of the random microstructure. Although RVEs are widely used, only little is known about the convergence properties, and many questions are controversially discussed (e.g. size of the RVE, boundary conditions). The analytic study of RVEs in the random case (e.g. convergence rates, a priori error estimates) belongs to the field of quantitative stochastic homogenization (QSH) and is mainly based on methods from partial differential equations and Calculus of Variations. In the last ten years QSH developed into a very activearea in applied analysis. In a recent work (Gloria, Neukamm and Otto in Inventiones Mathematicae 2015), we obtained the first convergence results for a periodic RVE in the random case with optimal scaling in the size of the RVE and the number of Monte Carlo iterations. The result and method (in parts) are restricted to linear, scalar elliptic equations. In the present project, we develop a QSH theory for non-linearly elastic composites with random microstructures, which is a class of high relevance in mechanics. In particular, we study a priori estimates for periodic RVEs and investigate analytically the impact of statistical correlations on the convergence rates. The transition from linear QSH theory to the non-convex case is interesting and challenging: The latter already features a genuinely different classical homogenization theory that involves new objects, new phenomena, and difficulties that are absent in the linear case.

多晶材料、增强橡胶、泡沫和生物组织是一大类随机异质材料 (RM) 的示例。这些材料具有微观结构的不确定性：例如复合材料各相的分布、几何形状和本构参数可能仅在统计水平上已知。通常 RM 在大长度尺度上表现出有效的行为，可以通过改变微观结构的成分和几何形状来设计和优化。理解微观和宏观尺度之间的微妙相互作用以及如何在此类系统中出现有效属性的问题是一个令人兴奋的基本问题——对于理论和新技术的开发都是如此。科学以多种方式解决这个问题，例如通过实验、建模和模拟以及数学分析。在许多情况下，如果 RM 在统计上是同质的，它就会在大长度尺度上显示出（几乎）确定性的物理行为。这使得建模和仿真的复杂性大大降低：我们可以考虑描述均质材料的宏观和确定性模型，而不是解决微观和不确定的属性。这是计算微观力学中许多方法的基础，例如基于 RVE 的方法，借助随机微观结构的代表性体积元素 (RVE) 来近似未知的宏观本构关系。尽管 RVE 被广泛使用，但人们对收敛特性知之甚少，并且许多问题的讨论也存在争议（例如 RVE 的大小、边界条件）。随机情况下 RVE 的分析研究（例如收敛速度、先验误差估计）属于定量随机均质化 (QSH) 领域，主要基于偏微分方程和变分法的方法。在过去十年中，QSH 发展成为应用分析中非常活跃的领域。在最近的一项工作中（Gloria、Neukamm 和 Otto in Inventiones Mathematicae 2015），我们在随机情况下获得了周期性 RVE 的第一个收敛结果，并且在 RVE 大小和蒙特卡洛迭代次数方面具有最佳缩放比例。结果和方法（部分）仅限于线性标量椭圆方程。在本项目中，我们开发了一种用于具有随机微观结构的非线性弹性复合材料的QSH理论，这是一类与力学高度相关的理论。特别是，我们研究了周期性 RVE 的先验估计，并分析研究了统计相关性对收敛率的影响。从线性 QSH 理论到非凸情况的转变是有趣且具有挑战性的：后者已经具有真正不同的经典均质化理论，其中涉及线性情况中不存在的新对象、新现象和困难。