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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, the inverse problem theory, calculation and application in the quantitative analysis of the mechanical properties of nanomaterials are discussed.These problems have important applications in the field of nanomaterials science and the development of functional devices, etc. The measurement, optimization design Research and development and application have great guiding significance.According to different engineering measurement methods, the quantitative analysis methods of mechanical properties of nanomaterials can be generally divided into two categories, static method and dynamic method.This paper focuses on two methods, the first study of the Euler-Bernoull equation In this paper, we obtain the convergence in terms of probability for the inversion of stochastic source terms, and for the inverse spectrum Problem, this paper transforms it into the optimization problem, and gives the numerical example verification.Finally, we put forward the new application of these inverse problems and the new research directions in mathematics.