复合材料非局部扩散问题的多尺度计算及其收敛性分析

国内外虽然对均质材料非局部扩散问题的数值算法取得了很大进展，但复合材料中该问题的多尺度算法研究甚少。本项目将以复合材料的非局部扩散问题为研究对象，创新发展一套数学上可靠的微-细-宏观耦合的高精度和高效率的多尺度计算方法，部分地解决相关的数学、力学理论问题。首先，针对复合材料非局部扩散问题建立和发展相应的微-细-宏观耦合多尺度模型及含有多尺度信息的积分微分方程适定性分析的数学理论，并建立复合材料非局部扩散问题的多尺度算法及其收敛性分析。其次，部分解决复合材料非局部扩散问题多尺度算法设计中的难点问题，为复合材料非局部模型的多尺度算法提供严格的数值分析理论基础，并对复合材料小周期参数与近场范围尺寸参数的尺度效应建立严格数学分析理论。最后，编程实现复合材料非局部扩散问题的多尺度算法，为复合材料非局部微-细-宏观扩散行为的精细分析提供多尺度计算平台和材料优化设计与制备提供理论依据。

Although great progress has been made in numerical methods for non-local diffusion problems of homogeneous materials at home and abroad, there is few research on multi-scale algorithms for the problems in composite materials. This project will take the non-local diffusion problems of composite materials as the research object, innovatively developing a mathematically reliable multi-scale calculation method with high accuracy and efficiency of micro-fine-macro coupling, and partially solving the related mathematical and mechanical theoretical problems. .Firstly, the mathematical theory of well-posed analysis of micro-fine-macro coupled multi-scale model and integro-differential equation with multi-scale information is established and developed for the non-local diffusion problem of composite materials, and the multi-scale algorithm and its convergence analysis are established.Secondly, it partially solves the difficult problems in the design of multi-scale algorithm for the non-local diffusion problem of composite materials, provides a rigorous theoretical basis for the multi-scale algorithm for the non-local model of composite materials, and establishes a rigorous mathematical analysis theory for the scale effects of small periodic parameters and near-field size parameters of composite materials.Finally, a multi-scale algorithm for non-local diffusion of composite materials is programmed, and a multi-scale computing platform for fine analysis of non-local micro-micro-macro diffusion behavior of composite materials is established, which provides a theoretical basis for material optimization design and preparation.