# American Institute of Mathematical Sciences

doi: 10.3934/era.2021027
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## 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 Early access 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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##### References:
A one-dimensional model based on ultrasonic detection
The definition of figure
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
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
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
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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