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

Deng Youjun; Liu Hongyu; Wang Xianchao; Wei Dong; Zhu Liyan

doi:10.3934/era.2021027

Article Contents

2021, Volume 29, Issue 5: 3081-3096. Doi: 10.3934/era.2021027

This issue Previous Article Instability and bifurcation of a cooperative system with periodic coefficients Next Article Three types of weak pullback attractors for lattice pseudo-parabolic equations driven by locally Lipschitz noise

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
^* Corresponding author: Hongyu Liu

Received: December 2020

Revised: February 2021

Early access: March 2021

Published: March 2021

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35K05, 49N45, 65N21.

Citation:

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Figure 1. A one-dimensional model based on ultrasonic detection

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Figure 2. The definition of figure

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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

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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

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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

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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

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