复相材料分子裂纹热传导模型的多尺度混合有限元

Multi-scale and Mixed Finite Element of Heat Transfer Model of Molecular Cracks of Composite Materials

随着材料科学和技术的发展,由于复相材料高度的敏感性和其它物理特性,它们在工程学、物理学和理论数学研究中变得越来越重要.因此,研究复相材料的物理特性就显得十分重要.迄今为止,对复相材料分子裂纹物性方面的研究非常少.针对复相材料的定态随机系数的椭圆偏微分方程问题,Jikov和Kozlov提出了一种均匀化方法,并证明了均匀化逼近解的存在性.然而,他们没有给出计算均匀化参数和解均匀化方程的数值方法.因此,本文中介绍了复相材料分子裂纹的均匀化热传导模型;然后,运用多尺度混合有限元方法来逼近复相材料分子裂纹的热传导模型;最后,给出了该热传导模型的多尺度混合有限元逼近解的存在唯一性和相应的误差估计.

With the rapid advance technology, composite materials of material science and are of more and more importance in engineering, physics and mathematics theory owing to their high intensity and physical performance. Therefore it is essential to accurately predict the physical performances of these composite materials. Up to now, the papers on the method for predicting the physical parameters of composite materials with random distribution are very few. For the elliptic PDF problems with stationary random coefficients, Jikov and Kozlov developed the homogenization method, and proved the existence of the homogenization solution. However, they did not give the numerical method for computing the homogenization parameters and solve the homogenization equation. In this paper, first of all, homogenization heat transfer model of molecular cracks of composite materials will be introduced. Secondly, the author would like to apply multi-scale and mixed finite method to approximation the heat transfer model. Finally, the existence, uniqueness and convergence of the multi-scale and mixed finite approximation solution are proved.