【作者】 杨涛；

【导师】 杨柳；

【作者基本信息】 兰州交通大学 ， 计算数学， 2020， 硕士

【摘要】 数学物理反问题中抛物方程是一个非常重要的研究内容,此类研究无论是在军事、医学、金融、物理、地质探测等领域,还是其他方面,都产生了深远的影响。本文主要在终端观测值给定的情形下,对混凝土导热系数的反问题进行了研究。而混凝土的绝热温升过程可用一个非线性热传导方程来描述。这类问题不但在工程和工业应用方面有非常多的应用,而且在自然科学的很多领域都有着重要的使用,更是在生活的诸多其他领域都有着广泛的应用。此类问题在自然科学和工程技术方面的重要的应用,例如:热传导、扩散、油藏模拟等。本篇文章中,我们研究的是一个二阶非线性热传导方程的数学模型,该问题的主要困难有二:所需反演的导热系数是二阶抛物方程的主项系数;右端的源项是一个非线性函数。首先,我们使用最优控制框架,将问题P转化为问题P1,并证明了控制泛函极小元的必要条件、局部唯一性和稳定性。而这些证明结果也为该问题的数值实验,奠定了理论依据。其次,使用有限体积法给出了非线性热传导方程的差分格式,对差分格式的稳定性进行了讨论,并利用其求得正问题的数值解。最后,对混凝土导热系数通过梯度型迭代法进行了数值模拟,并给出算例相应的数值结果,数值结果表明对未知导热系数重构的效果很好。本文主要包含以下四个部分:第一章主要对社会生活领域和工程建设中,涉及的反问题研究内容进行了简要叙述,并对目前所研究得到的结果做了一些概述。其次,对本文所涉及的背景知识和得到的一些研究内容做了一些介绍。最后,对本篇文章当中的每个章节所做的工作,给出了具体介绍。第二章主要从理论上重点分析了重构混凝土导热系数的反问题。首先,我们利用最优控制方法将问题进行了转化,并证明了优化问题当中极小元的必要条件,局部唯一性和稳定性,并且给出了相应的一些证明过程。第三章主要研究了基于对第二章中的理论分析的结果,从数值模拟的角度对该问题进行验证,并且重构了混凝土导热系数。首先,我们先使用有限体积法得到该方程的差分格式,并对方程的差分格式进行了稳定性分析,并给出了证明过程;其次,对正问题的数值解进行计算,从给出的算例中得到算例的数值解,最后,利用梯度型迭代方法,对问题进行数值计算,获得较满意的重构结果,并且也将算例中相应的数值结果展示了出来。第四章对本文所做的工作做了一个简要概述与总结,并指出接下来的研究工作中,我们可以进一步考虑的问题。在以后的工作中,我们希望能对问题的控制函数的收敛性以及数值实验中的系数改为变系数的数值实验。更多还原

【Abstract】 The parabolic equation in the inverse problem of mathematical physics is a very important research content.Such research has had a profound impact no matter in the fields of military,medicine,finance,physics,geological exploration,and other fields.In this paper,the inverse problem of the thermal conductivity of concrete is studied with the terminal observations given.The adiabatic temperature rise process of concrete can be described by a nonlinear heat conduction equation.Such problems not only have a lot of applications in engineering and industrial applications,but also have important uses in many areas of natural sciences,and they are also widely used in many other areas of life.In this paper,we are studying a mathematical model of a second-order nonlinear heat conduction equation.The main difficulties of this problem are two: The required thermal conductivity is the main coefficient of the second-order parabolic equation,and the source term at the right is A non-linear function.First,we use the optimal control framework to transform problem P into problem P1,and prove the necessary conditions,local uniqueness,and stability for controlling functional minima.These proof results also lay a theoretical basis for the numerical experiments of this problem.Secondly,the finite volume method is used to give the difference scheme of the nonlinear heat conduction equation,the stability of the difference scheme is discussed,and the numerical solution of the positive problem is obtained by using it.Finally,the thermal conductivity of concrete is numerically simulated by a gradient-type iterative method,and the corresponding numerical results are given.The numerical results show that the reconstruction of unknown thermal conductivity is effective.This article consists of the following four sections:The first chapter mainly briefly describes the research content of the counter-problems involved in the field of social life and engineering construction,and summarizes the results obtained by the current research.Secondly,it introduces the background knowledge involved in this article and some research contents obtained.Finally,a detailed introduction is given to the work done in each chapter in this article.The second chapter mainly analyzes the inverse problem of thermal conductivity of reconstructed concrete mainly in theory.First,we use the optimal control method to transform the problem,and prove the necessary conditions,local uniqueness and stability of the minimum element in the optimization problem,and give some corresponding proof processes.The third chapter mainly studies the results based on the theoretical analysis in Chapter Two,verifies the problem from the perspective of numerical simulation,and reconstructs the thermal conductivity of concrete.First,we use the finite volume method to obtain the difference scheme of the equation,and perform stability analysis on the difference scheme of the equation,and give the proof process.Second,the numerical solution of the positive problem is calculated.The numerical solution of the example is obtained in the example.Finally,the gradient iteration method is used to numerically calculate the problem to obtain satisfactory reconstruction results.The corresponding numerical results in the example are also shown.The fourth chapter mainly gives a brief overview and summary of the work done in this article,and points out the issues that we can consider further in the next research work.In future work,we hope that the convergence of the control function of the problem and the numerical experiments where the coefficients in the numerical experiments are changed to variable coefficients.更多还原