一类三维逆时热传导问题的数值求解

数学物理学报 ›› 2022, Vol. 42 ›› Issue (1): 187-200.

一类三维逆时热传导问题的数值求解

孟庆春^1,²,张磊^1,*()

¹ 黑龙江大学数学科学学院哈尔滨 150080
² 利沃夫国立理工大学应用数学与基础科学学院乌克兰利沃夫 79013

收稿日期:2021-01-11 出版日期:2022-02-26 发布日期:2022-02-23
通讯作者: 张磊 E-mail:zhanglei@hlju.edu.cn
基金资助:
国家自然科学基金(11871198);国家自然科学基金(11801116);黑龙江省高校基本科研业务费-青年创新团队(RCYJTD201804);中央高校基本科研业务费(3072020CFT2401)

Numerical Solution of the Three-Dimensional Inverse Heat Conduction Problems

Qingchun Meng^1,²,Lei Zhang^1,*()

¹ Department of Mathematical Sciences, Heilongjiang University, Harbin 150080
² Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, Lviv, Ukraine 79013

Received:2021-01-11 Online:2022-02-26 Published:2022-02-23
Contact: Lei Zhang E-mail:zhanglei@hlju.edu.cn
Supported by:
the NSFC(11871198);the NSFC(11801116);the Fundamental Research Funds for the Universities of Heilongjiang Province-Youth Innovation Team(RCYJTD201804);the Fundamental Research Funds for the Central Universities(3072020CFT2401)

摘要/Abstract

摘要：

该文考虑一类三维逆时热传导问题的数值解法.基于有限差分时间离散，并结合伽辽金（Galerkin）方法对空间进行有限元离散，导出刚度矩阵及载荷向量，对热传导问题进行数值求解.针对反问题，利用分离变量法建立T时刻温度场与初始温度场之间的对应关系，给出了反演公式，并在一定先验假设条件下证明了反问题的局部稳定性.为克服反问题求解的不适定性，使用吉洪诺夫（Tikhonov）正则化和终值数据扰动正则化方法反演了初始温度场，通过数值实验验证了算法的有效性.

关键词: 三维逆时热传导问题, 有限元, 不适定性, 正则化方法

Abstract:

In this paper, we consider the numerical solution of a three-dimensional inverse heat conduction problem. Based on the finite difference and the finite element method, the stiffness matrix and load vector are derived to solve the heat conduction problem. We use the variable separation method to establish the corresponding relationship between the temperature field at time T and the initial temperature field for the inverse problem. The inversion formulation is obtained. The local stability for the inverse problems is proved under certain priori assumptions. To overcome the ill-posedness for solving the inverse problem, we used the Tikhonov regularization and perturbation regularization method to reconstruct the initial temperature field. We verified the effectiveness of the algorithm through several numerical experiments.

Key words: 3-D inverse heat conduction problem, Finite element, Ill-posedness, Regularization method

中图分类号:

O241.8

孟庆春,张磊. 一类三维逆时热传导问题的数值求解[J]. 数学物理学报, 2022, 42(1): 187-200.

Qingchun Meng,Lei Zhang. Numerical Solution of the Three-Dimensional Inverse Heat Conduction Problems[J]. Acta mathematica scientia,Series A, 2022, 42(1): 187-200.

参考文献

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