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doi: 10.3934/era.2021027

Simultaneous recovery of surface heat flux and thickness of a solid structure by ultrasonic measurements

1. 

School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, China

2. 

Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China

3. 

State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China

4. 

Center of Nondestructive Examination, China Special Equipment Inspection and Research Institute, Beijing 100029, China

* Corresponding author: Hongyu Liu

Received  December 2020 Revised  February 2021 Published  March 2021

This paper is concerned with a practical inverse problem of simultaneously reconstructing the surface heat flux and the thickness of a solid structure from the associated ultrasonic measurements. In a thermoacoustic coupling model, the thermal boundary condition and the thickness of a solid structure are both unknown, while the measurements of the propagation time by ultrasonic sensors are given. We reformulate the inverse problem as a PDE-constrained optimization problem by constructing a proper objective functional. We then develop an alternating iteration scheme which combines the conjugate gradient method and the deepest decent method to solve the optimization problem. Rigorous convergence analysis is provided for the proposed numerical scheme. By using experimental real data from the lab, we conduct extensive numerical experiments to verify several promising features of the newly developed method.

Citation: Youjun Deng, Hongyu Liu, Xianchao Wang, Dong Wei, Liyan Zhu. Simultaneous recovery of surface heat flux and thickness of a solid structure by ultrasonic measurements. Electronic Research Archive, doi: 10.3934/era.2021027
References:
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O. M. Alifaov, Inverse Heat Transfer Problems, Spriger-Verlag, Berlin, 1994. Google Scholar

[2]

A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272.   Google Scholar

[3]

X. CaoH. Diao and H. Liu, Determining a piecewise conductive medium body by a single far-field measurement, CSIAM Transactions on Applied Mathematics, 1 (2020), 740-765.  doi: 10.13140/RG.2.2.19443.76327.  Google Scholar

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X. CaoY.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011.  Google Scholar

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Y. H. Dai and Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177-182.  doi: 10.1137/S1052623497318992.  Google Scholar

[7]

Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring, Arch. Ration. Mech. Anal., 231 (2019), 153-187.  doi: 10.1007/s00205-018-1276-7.  Google Scholar

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Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring within a magnetohydrodynamic model, Arch. Ration. Mech. Anal., 235 (2020), 691-721.  doi: 10.1007/s00205-019-01429-x.  Google Scholar

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Y. Deng and Z. Liu,, Iteration methods on sideways parabolic equations, Inverse Problems, 25 (2009), 095004, 14 pp. doi: 10.1088/0266-5611/25/9/095004.  Google Scholar

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Y. DengH. Liu and X. Liu, Recovery of an embedded obstacle and the surrounding medium for Maxwell's system, J. Differential Equations, 267 (2019), 2192-2209.  doi: 10.1016/j.jde.2019.03.009.  Google Scholar

[11]

Y. DengH. Liu and W.-Y. Tsui, Identifying variations of magnetic anomalies using geomagnetic monitoring, Discrete Contin. Dyn. Syst., 40 (2020), 6411-6440.  doi: 10.3934/dcds.2020285.  Google Scholar

[12]

Y. DengH. Liu and G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Differential Equations, 267 (2019), 2471-2502.  doi: 10.1016/j.jde.2019.03.019.  Google Scholar

[13]

W. HuY. Gu and C.-M. Fan, A meshless collocation scheme for inverse heat conduction problem in three-dimensional functionally graded materials, Eng. Anal. Bound. Elem., 114 (2020), 1-7.  doi: 10.1016/j.enganabound.2020.02.001.  Google Scholar

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M. A. Kant and P. R. von Rohr, Determination of surface heat flux distributions by using surface temperature measurements and applying inverse techniques, International Journal of Heat and Mass Transfer, 99 (2016), 1-9.  doi: 10.1016/j.ijheatmasstransfer.2016.03.082.  Google Scholar

[15]

V. A. Khoa, G. W. Bidney, M. V. Klibanov, Loc H. Nguyen, Lam H. Nguyen, A. J. Sullivan and V. N. Astratov, Convexification and experimental data for a 3D inverse scattering problem with the moving point source, Inverse Problems, 36 (2020), 085007, 34 pp.  Google Scholar

[16]

M. V. Klibanov, J. Li and W. Zhang, Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data, Inverse Problems, 35 (2019), 035005, 33 pp. doi: 10.1088/1361-6420/aafecd.  Google Scholar

[17]

J. LiH. Liu and S. Ma, Determining a random Schrödinger operator: Both potential and source are random, Comm. Math. Phys., 381 (2021), 527-556.  doi: 10.1007/s00220-020-03889-9.  Google Scholar

[18]

J. LiH. Liu and S. Ma, Determining a random Schrödinger equation with unknown source and potential, SIAM J. Math. Anal., 51 (2019), 3465-3491.  doi: 10.1137/18M1225276.  Google Scholar

[19]

R.-E. Plessix, A review of the adjoint-state method for computing the gradient of a functional with geophysical applications, Geophysical Journal International, 167 (2006), 495-503.  doi: 10.1111/j.1365-246X.2006.02978.x.  Google Scholar

[20]

R. A. PonramB. H. Prasad and S. S. Kumar, Thickness mapping of rocket motor casing using ultrasonic thickness gauge, Materials Today: Proceedings, 5 (2018), 11371-11375.  doi: 10.1016/j.matpr.2018.02.104.  Google Scholar

[21]

M. J. D. Powell, Restart procedures for the conjugate gradient method, Math. Programming, 12 (1977), 241-254.  doi: 10.1007/BF01593790.  Google Scholar

[22]

W. Sun and Y. -X. Yuan, Optimization Theory and Methods, Nonlinear Programming, Springer, New York, 2006.  Google Scholar

[23]

D. WeiY.-A. ShiB.-N. ShouY.-W. GuiY.-X. Du and G.-M. Xiao, Reconstruction of internal temperature distributions in heat materials by ultrasonic measurements, Applied Thermal Engineering, 112 (2017), 38-44.  doi: 10.1016/j.applthermaleng.2016.09.169.  Google Scholar

[24]

D. WeiX. YangY. ShiG. XiaoY. Du and Y. Gui, A method for reconstructing two-dimensional surface and internal temperature distributions in structures by ultrasonic measurements, Renewable Energy, 150 (2020), 1108-1117.  doi: 10.1016/j.renene.2019.10.081.  Google Scholar

[25]

Y.-X. Yuan, Analysis on the conjugate gradient method, Optimization Methods and Software, 2 (1993), 19-29.  doi: 10.1080/10556789308805532.  Google Scholar

show all references

References:
[1]

O. M. Alifaov, Inverse Heat Transfer Problems, Spriger-Verlag, Berlin, 1994. Google Scholar

[2]

A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272.   Google Scholar

[3]

X. CaoH. Diao and H. Liu, Determining a piecewise conductive medium body by a single far-field measurement, CSIAM Transactions on Applied Mathematics, 1 (2020), 740-765.  doi: 10.13140/RG.2.2.19443.76327.  Google Scholar

[4]

X. CaoY.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011.  Google Scholar

[5]

X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter, Commun. Math. Sci., 17 (2019), 1861-1876.  doi: 10.4310/CMS.2019.v17.n7.a5.  Google Scholar

[6]

Y. H. Dai and Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177-182.  doi: 10.1137/S1052623497318992.  Google Scholar

[7]

Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring, Arch. Ration. Mech. Anal., 231 (2019), 153-187.  doi: 10.1007/s00205-018-1276-7.  Google Scholar

[8]

Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring within a magnetohydrodynamic model, Arch. Ration. Mech. Anal., 235 (2020), 691-721.  doi: 10.1007/s00205-019-01429-x.  Google Scholar

[9]

Y. Deng and Z. Liu,, Iteration methods on sideways parabolic equations, Inverse Problems, 25 (2009), 095004, 14 pp. doi: 10.1088/0266-5611/25/9/095004.  Google Scholar

[10]

Y. DengH. Liu and X. Liu, Recovery of an embedded obstacle and the surrounding medium for Maxwell's system, J. Differential Equations, 267 (2019), 2192-2209.  doi: 10.1016/j.jde.2019.03.009.  Google Scholar

[11]

Y. DengH. Liu and W.-Y. Tsui, Identifying variations of magnetic anomalies using geomagnetic monitoring, Discrete Contin. Dyn. Syst., 40 (2020), 6411-6440.  doi: 10.3934/dcds.2020285.  Google Scholar

[12]

Y. DengH. Liu and G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Differential Equations, 267 (2019), 2471-2502.  doi: 10.1016/j.jde.2019.03.019.  Google Scholar

[13]

W. HuY. Gu and C.-M. Fan, A meshless collocation scheme for inverse heat conduction problem in three-dimensional functionally graded materials, Eng. Anal. Bound. Elem., 114 (2020), 1-7.  doi: 10.1016/j.enganabound.2020.02.001.  Google Scholar

[14]

M. A. Kant and P. R. von Rohr, Determination of surface heat flux distributions by using surface temperature measurements and applying inverse techniques, International Journal of Heat and Mass Transfer, 99 (2016), 1-9.  doi: 10.1016/j.ijheatmasstransfer.2016.03.082.  Google Scholar

[15]

V. A. Khoa, G. W. Bidney, M. V. Klibanov, Loc H. Nguyen, Lam H. Nguyen, A. J. Sullivan and V. N. Astratov, Convexification and experimental data for a 3D inverse scattering problem with the moving point source, Inverse Problems, 36 (2020), 085007, 34 pp.  Google Scholar

[16]

M. V. Klibanov, J. Li and W. Zhang, Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data, Inverse Problems, 35 (2019), 035005, 33 pp. doi: 10.1088/1361-6420/aafecd.  Google Scholar

[17]

J. LiH. Liu and S. Ma, Determining a random Schrödinger operator: Both potential and source are random, Comm. Math. Phys., 381 (2021), 527-556.  doi: 10.1007/s00220-020-03889-9.  Google Scholar

[18]

J. LiH. Liu and S. Ma, Determining a random Schrödinger equation with unknown source and potential, SIAM J. Math. Anal., 51 (2019), 3465-3491.  doi: 10.1137/18M1225276.  Google Scholar

[19]

R.-E. Plessix, A review of the adjoint-state method for computing the gradient of a functional with geophysical applications, Geophysical Journal International, 167 (2006), 495-503.  doi: 10.1111/j.1365-246X.2006.02978.x.  Google Scholar

[20]

R. A. PonramB. H. Prasad and S. S. Kumar, Thickness mapping of rocket motor casing using ultrasonic thickness gauge, Materials Today: Proceedings, 5 (2018), 11371-11375.  doi: 10.1016/j.matpr.2018.02.104.  Google Scholar

[21]

M. J. D. Powell, Restart procedures for the conjugate gradient method, Math. Programming, 12 (1977), 241-254.  doi: 10.1007/BF01593790.  Google Scholar

[22]

W. Sun and Y. -X. Yuan, Optimization Theory and Methods, Nonlinear Programming, Springer, New York, 2006.  Google Scholar

[23]

D. WeiY.-A. ShiB.-N. ShouY.-W. GuiY.-X. Du and G.-M. Xiao, Reconstruction of internal temperature distributions in heat materials by ultrasonic measurements, Applied Thermal Engineering, 112 (2017), 38-44.  doi: 10.1016/j.applthermaleng.2016.09.169.  Google Scholar

[24]

D. WeiX. YangY. ShiG. XiaoY. Du and Y. Gui, A method for reconstructing two-dimensional surface and internal temperature distributions in structures by ultrasonic measurements, Renewable Energy, 150 (2020), 1108-1117.  doi: 10.1016/j.renene.2019.10.081.  Google Scholar

[25]

Y.-X. Yuan, Analysis on the conjugate gradient method, Optimization Methods and Software, 2 (1993), 19-29.  doi: 10.1080/10556789308805532.  Google Scholar

Figure 1.  A one-dimensional model based on ultrasonic detection
Figure 2.  The definition of figure
Figure 3.  Heat flux inversion results with the acoustic time accuracy fixed to be $ 10^{-10} $. (a): under a fixed initial heat flux and different initial thicknesses, (b): under a fixed initial thickness and different initial heat fluxes
Figure 4.  Heat flux inversion results with the acoustic time accuracy fixed to be $ 10^{-11} $. (a): under a fixed initial heat flux and different initial thicknesses, (b): under a fixed initial thickness and different initial heat fluxes
Table 1.  Convergence of the iteration method with different initial guesses and measurement errors
Acoustic time initial heat flux initial thickness reconstructed iterations
accuracy $ (s) $ $ q^0(\rm{{J}/{s}}) $ $ L^0(\rm{mm}) $ thickness $ L(\rm{mm}) $ $ \rm{n} $
$ 10^{-9} $ 0 3 50.0006 138
$ 10^{-10} $ $ 0 $ 3 50.0006 64
$ 10^{-11} $ $ 0 $ 3 50.0006 63
$ 10^{-9} $ $ 1\times10^3 $ 45 50.0032 208
$ 10^{-10} $ $ 1\times10^3 $ 45 50.0028 115
$ 10^{-11} $ $ 1\times10^3 $ 45 50.0025 108
Acoustic time initial heat flux initial thickness reconstructed iterations
accuracy $ (s) $ $ q^0(\rm{{J}/{s}}) $ $ L^0(\rm{mm}) $ thickness $ L(\rm{mm}) $ $ \rm{n} $
$ 10^{-9} $ 0 3 50.0006 138
$ 10^{-10} $ $ 0 $ 3 50.0006 64
$ 10^{-11} $ $ 0 $ 3 50.0006 63
$ 10^{-9} $ $ 1\times10^3 $ 45 50.0032 208
$ 10^{-10} $ $ 1\times10^3 $ 45 50.0028 115
$ 10^{-11} $ $ 1\times10^3 $ 45 50.0025 108
Table 2.  Convergence of the proposed iteration method with different initial guesses and measurement errors
acoustic time initial heat flux initial thickness reconstructed thickness iterations
accuracy $ q^0(\rm{{J}/{s}}) $ $ L^0(\rm{mm}) $ $ L(\rm{mm}) $ $ n $
$ 10^{-10} $ 0 3 50.0006 64
$ 10^{-10} $ $ 1\times10^3 $ 3 50.0016 58
$ 10^{-10} $ $ 1\times10^5 $ 3 50.0000 133
$ 10^{-10} $ 0 45 50.0025 126
$ 10^{-10} $ $ 1\times10^3 $ 45 50.0028 115
$ 10^{-10} $ 0 80 50.0028 77
$ 10^{-10} $ $ 1\times10^3 $ 80 50.0046 92
$ 10^{-11} $ 0 3 50.0006 63
$ 10^{-11} $ $ 1\times10^3 $ 3 50.0016 57
$ 10^{-11} $ $ 1\times10^5 $ 3 50.0006 147
$ 10^{-11} $ 0 45 50.0025 126
$ 10^{-11} $ $ 1\times10^3 $ 45 50.0025 108
$ 10^{-11} $ 0 80 50.0028 77
$ 10^{-11} $ $ 1\times10^3 $ 80 50.0029 80
acoustic time initial heat flux initial thickness reconstructed thickness iterations
accuracy $ q^0(\rm{{J}/{s}}) $ $ L^0(\rm{mm}) $ $ L(\rm{mm}) $ $ n $
$ 10^{-10} $ 0 3 50.0006 64
$ 10^{-10} $ $ 1\times10^3 $ 3 50.0016 58
$ 10^{-10} $ $ 1\times10^5 $ 3 50.0000 133
$ 10^{-10} $ 0 45 50.0025 126
$ 10^{-10} $ $ 1\times10^3 $ 45 50.0028 115
$ 10^{-10} $ 0 80 50.0028 77
$ 10^{-10} $ $ 1\times10^3 $ 80 50.0046 92
$ 10^{-11} $ 0 3 50.0006 63
$ 10^{-11} $ $ 1\times10^3 $ 3 50.0016 57
$ 10^{-11} $ $ 1\times10^5 $ 3 50.0006 147
$ 10^{-11} $ 0 45 50.0025 126
$ 10^{-11} $ $ 1\times10^3 $ 45 50.0025 108
$ 10^{-11} $ 0 80 50.0028 77
$ 10^{-11} $ $ 1\times10^3 $ 80 50.0029 80
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