一类三维逆时热传导问题的数值求解

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.

图/表 8

表 1

图 1

表 2

图 2

表 3

图 3

表 4

图 4

参考文献 19

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T	J=10			J=20
T	$\left\\|e_{1}\right\\|_{L^{2}}$	$\left\\|e_{1}\right\\|_{L ^{\infty}}$	$\left\\|e_{2}\right\\|$	$\left\\|e_{1}\right\\|_{L^{2}}$	$\left\\|e_{1}\right\\|_{L ^{\infty}}$	$\left\\|e_{2}\right\\|$
2	0.0972	0.0087	0.0111	0.0961	0.0030	0.0039
4	0.1537	0.0137	0.0222	0.1514	0.0048	0.0077
6	0.1824	0.0163	0.0335	0.1790	0.0057	0.0116
8	0.1923	0.0172	0.0449	0.1881	0.0059	0.0155

T	J=30			J=40
T	$\left\\|e_{1}\right\\|_{L^{2}}$	$\left\\|e_{1}\right\\|_{L ^{\infty}}$	$\left\\|e_{2}\right\\|$	$\left\\|e_{1}\right\\|_{L^{2}}$	$\left\\|e_{1}\right\\|_{L ^{\infty}}$	$\left\\|e_{2}\right\\|$
2	0.0961	0.0030	0.0039	0.1435	0.0016	0.0020
4	0.1514	0.0048	0.0077	0.2260	0.0025	0.0041
6	0.1790	0.0057	0.0116	0.2669	0.0030	0.0061
8	0.1881	0.0059	0.0155	0.2803	0.0031	0.0082

δ	α = 0.0001			α = 0.0005
δ	$\left\\|e_{3}\right\\|_{L^{2}}$	$ \left\\|e_{3}\right\\|_{L ^{\infty}}$	$\left\\|e_{4}\right\\|$	$\left\\|e_{3}\right\\|_{L^{2}}$	$ \left\\|e_{3}\right\\|_{L ^{\infty}}$	$\left\\|e_{4}\right\\|$
0.001	0.3704	0.0119	0.0117	0.3165	0.0100	0.0100
0.005	0.4378	0.0134	0.0138	0.3815	0.0126	0.0121
0.01	0.5333	0.0198	0.0169	0.4628	0.0156	0.0146
0.05	1.3499	0.0734	0.0427	1.1337	0.0432	0.0359

δ	α = 0.001			α = 0.005
δ	$\left\\|e_{3}\right\\|_{L^{2}}$	$ \left\\|e_{3}\right\\|_{L ^{\infty}}$	$\left\\|e_{4}\right\\|$	$\left\\|e_{3}\right\\|_{L^{2}}$	$ \left\\|e_{3}\right\\|_{L ^{\infty}}$	$\left\\|e_{4}\right\\|$
0.001	0.2490	0.0080	0.0079	0.2778	0.0088	0.0088
0.005	0.3122	0.0100	0.0099	0.2147	0.0068	0.0068
0.01	0.3950	0.0122	0.0125	0.1381	0.0048	0.0044
0.05	1.0394	0.0332	0.0329	0.5074	0.0152	0.0160

α	T=4			T=6
α	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$
0.0001	0.2490	0.0080	0.0079	0.2778	0.0088	0.0088
0.0005	0.3122	0.0100	0.0099	0.2147	0.0068	0.0068
0.001	0.3950	0.0122	0.0125	0.1381	0.0048	0.0044
0.005	1.0394	0.0332	0.0329	0.0377	0.0011	0.0013

α	T=8			T=10
α	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$
0.0001	0.4306	0.0141	0.0154	0.5406	0.0201	0.0193
0.0005	0.3987	0.0130	0.0142	0.4987	0.0159	0.0178
0.001	0.3616	0.0117	0.0129	0.4513	0.0142	0.0161
0.005	0.0686	0.0023	0.0024	0.0789	0.0024	0.0028

α	T=20			T=30
α	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$
0.0001	1.0701	0.0332	0.0382	1.5610	0.0492	0.0557
0.0005	0.9424	0.0297	0.0336	1.1354	0.0359	0.0405
0.001	0.7844	0.0247	0.0280	0.6202	0.0196	0.0221
0.005	0.4205	0.0133	0.0150	2.9273	0.0926	0.1044

α	T=40			T=50
α	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$
0.0001	1.8839	0.0595	0.0672	1.6395	0.0519	0.0585
0.0005	0.5125	0.0162	0.0183	2.3469	0.0742	0.0837
0.001	1.0346	0.0327	0.0369	6.0408	0.1910	0.2154
0.005	9.2018	0.2910	0.3281	17.8107	0.5632	0.6350

α	T=4			T=6
α	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$
0.0001	0.6014	0.0454	0.0214	0.9492	0.0748	0.0338
0.0005	0.2497	0.0135	0.0089	0.3730	0.0185	0.0133
0.001	0.2042	0.0101	0.0073	0.3015	0.0144	0.0108
0.005	0.0210	0.0015	0.0007	0.0585	0.0031	0.0021

α	T=8			T=10
α	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$
0.0001	1.0726	0.0650	0.0382	1.1132	0.0726	0.0397
0.0005	0.4502	0.0205	0.0161	0.5333	0.0230	0.0190
0.001	0.3932	0.0199	0.0140	0.4762	0.0163	0.0170
0.005	0.0865	0.0044	0.0031	0.0953	0.0037	0.0034

α	T=20			T=30
α	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$
0.0001	1.1529	0.0486	0.0411	1.5826	0.0521	0.0564
0.0005	0.9618	0.0333	0.0343	1.1500	0.0356	0.0401
0.001	0.7997	0.0240	0.0285	0.6346	0.0203	0.0226
0.005	0.4066	0.0131	0.0145	2.9149	0.0922	0.1039

α	T=40			T=50
α	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$	$\left\\|e_{5}\right\\|_{L^{2}}$	$\left\\|e_{5}\right\\|_{L ^{\infty}}$	$\left\\|e_{6}\right\\|$
0.0001	1.8994	0.0617	0.0677	1.6546	0.0525	0.0590
0.0005	0.5265	0.0166	0.0188	2.3341	0.0738	0.0832
0.001	1.0211	0.0325	0.0364	6.0298	0.1907	0.2150
0.005	9.1925	0.2907	0.3278	17.8057	0.5631	0.6349

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