纳米材料力学性质定量分析中的反问题 献给林群教授80华诞  

Inverse problems in quantifying mechanical properties in nanomaterials

作　　者：包刚[1] 刘华彦[2] 李明[1] 徐翔[1] 

机构地区：[1]浙江大学数学系,杭州310027 [2]太原理工大学数学系,太原030021

出　　处：《中国科学：数学》2015年第7期831-842,共12页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:91130004,11421110002,11426235,11471284和11401423);浙江大学科研启动经费资助项目

摘　　要：本文对纳米材料力学性质定量分析中出现的反问题理论、计算和应用进行了探讨.这类问题在纳米材料科学以及功能器件开发等方面中有着重要的应用,对纳米尺度下的测量、优化设计、研发及应用有着重大的指导意义.根据工程测量方法的不同,纳米材料力学性质的定量分析方法一般可以分为两类,静态法和动态法.本文针对两种方法,率先研究Euler-Bernoull方程的反演随机源项、反演系数和反谱问题,得到了对于一般非均匀纳米材料性质测定的方法,其中对于反演随机源项,本文得到在依概率意义下的收敛性;对于反谱问题,本文将其转化为优化问题求解,并给出数值算例验证.最后提出这些反问题新的应用和数学上新的研究方向.In this paper, we study three inverse problems on quantifying mechanical properties of nanomaterials, including an inverse source problem, an inverse coefficient problem and an inverse spectral problem. These problems are of significant importance to nanosciences, especially on developing functional devices, measurements, optimal designs and their applications. According to engineering applications, we mainly have two methods on quantifying mechanical properties of nanomaterials： static and dynamical methods. In this paper, we first investigate an inverse random source problem, inverse coefficient problem and inverse spectral problem based upon these two methods. Specifically, first convergence result is obtained with overwhelming probability for the inverse random source and coefficient problem. For the inverse spectral problem, we propose an iterative method to solve the approximated optimization problem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method. Several new applications of these inverse problems and future work are mentioned at the end of this paper.

关 键 词：反随机源问题 反系数问题 反谱问题 Euler-Bernoull方程 

分 类 号：TB383.1[一般工业技术—材料科学与工程]