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

Youjun Deng  School of Mathematics and Statistics, Central South University,
Changsha, Hunan, China<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Hongyu
Liu  Department of Mathematics, City University of Hong Kong, Kowloon, Hong
Kong, China<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Xianchao Wang
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong,
China<EMAIL_ADDRESS>,  Dong Wei  State Key Laboratory of Aerodynamics,
China Aerodynamics Research and Development
Center,Mianyang,Sichuan,621000,China
Center of Nondestructive Examination, China Special Equipment Inspection and
Research Institute, Beijing, 100029, China<EMAIL_ADDRESS>and  Liyan Zhu
School of Mathematics and Statistics, Central South University, Changsha,
Hunan, China

###### 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: thermoacoustic coupling, inverse problem, heat flux and thickness,
ultrasonic measurements, adjoint state method, convergence analysis

##  1\. Introduction

The heat conduction is a ubiquitous phenomenon which forms the basis for many
practical applications. Given the geometrical and material configurations of a
material structure as well as the heat source including the initial and
boundary temperature distributions, finding the temperature distribution as
well as its change on the material structure constitutes the so-called direct
or forward heat conduction problem. In many practical applications, one is
more interested in the so-called inverse heat conduction problem which
reverses the above forward problem through direct or indirect measurement
data; see e.g. [12, 17, 8, 9, 11, 1, 10, 16, 19, 13, 5, 7] and the references
cited therein for some related studies in the literature.

In this paper, motivated by practical applications (with experimental real
data from the lab), we are mainly concerned with the reconstruction of the
surface heat flux and the thickness of a solid structure by using the
associated ultrasonic measurements. The reconstruction of the surface heat
flux is one of the most typical inverse heat conduction problems, and is
widely encountered in aerospace, nuclear physics, metallurgy, and other
industrial fields; see [9, 11] and the references cited therein for more
related discussions. Ultrasonic thickness measurement is a commonly used non-
destructive testing method, and is widely used in petroleum, machinery, ship,
chemical industry and other fields [13, 5]. For most of existing results in
the literature, one either recovers the surface heat flux by assuming the
thickness of the material structure is a priori known, or recovers the
thickness of the material structure by assuming the surface heat flux is a
priori known. However, it is a more practical scenario that both the surface
heat flux and the thickness of the material structure are unknown and one
recovers both of them.

In this paper, based on the ultrasonic echo method and the inverse analysis
method of the heat conduction, we propose a novel scheme for simultaneously
recovering the surface heat flux and the thickness of the material structure
through the pulse-echo measurements by the ultrasonic probe. The study is
posed as an inverse problem associated with a thermoacoustic coupling model.
We recast the inverse problem as a PDE (partial differential
equation)-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. Rigours convergence analysis is provided for the
proposed numerical scheme. Finally, by using experimental real data from the
lab, we conduct extensive numerical experiments to verify effectiveness and
efficiency of the method.

The rest of this paper is organized as follows. In Section 2, we present the
mathematical formulation of the direct and inverse problems for our study and
also briefly discuss the corresponding physical setup. In Section 3, we give
the optimization formulation of the inverse problem and then derive the
alternating iteration scheme for solving the optimization problem. Sections 4
and 5 are, respectively, devoted to the theoretical convergence analysis and
numerical experiments.

##  2\. Mathematical and physical setups

The physical principle of the ultrasonic thickness measurement is to use the
propagation time of the ultrasonic waves in the medium to infer the thickness
of the underlying solid structure. The propagation time is mainly determined
by the thickness, material properties, and internal temperature field of the
solid structure; see Figure 1 for a schematic illustration of the physical
setup. The propagation time of the ultrasonic wave in the solid structure can
be expressed as (see [18]):

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

$\Lambda_{L}(t)=2\int_{0}^{L}\frac{1}{V(T(x,t))}\,\mathrm{d}x,\quad
t\in(0,+\infty),$ (2.1)

where $L\in\mathbb{R}_{+}$ denotes the unidirectional propagation distance of
the ultrasonic wave in the medium, i.e. the thickness of the material
structure being under detection. Here, $V$ is the propagation velocity of the
acoustic wave in the solid medium and is related to the material properties
and the structure temperature. Usually, it has an approximately linear
relationship with the temperature, i.e.,

$V(T)=aT+b,\quad a,\,b\in\mathbb{R},$ (2.2)

where $a$ and $b$ are determined by the properties of the material and
calibrated by experiments. $T(x,t)$ is the internal temperature of the
structure, which satisfies the following heat conduction system for $T\in
W_{2}^{2,1}\left(\left[0,L\right]\times(0,\tau)\right)$:

$\left\\{\begin{split}&\rho c\frac{\partial T}{\partial
t}=\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial
x}\right),\qquad\qquad\qquad\qquad\ \,(x,t)\in[0,L]\times[0,\tau],\\\
&-k\frac{\partial T}{\partial x}\Big{|}_{x=0}=q,\quad-k\frac{\partial
T}{\partial x}\Big{|}_{x=L}=0,\qquad\quad q(t)\in L^{2}\left[0,\tau\right],\\\
&T\mid_{t=0}=T_{0},\qquad\qquad\qquad\qquad\qquad\qquad\quad
x\in[0,L],\end{split}\right.$ (2.3)

where $k(x,t)$, $c(x,t)$ and $\rho(x)$ are the thermal conductivity, specific
heat capacity and density of the material, respectively, and $q(t)$ denotes
the heat flux density on the boundary.

In this paper, the inverse problem that we are concerned with is described as
follows:

######  Problem 1.

Given the measured propagation time of the ultrasonic wave $\Lambda_{m}(t)$
and the measured boundary temperature $T_{m}(L,t)$, determine the surface heat
flux $q(t)$ and the thickness $L$, i.e.,

$\left\\{\Lambda_{m}\left(t\right),T_{m}(L,t)\right\\}\rightarrow\\{q\left(t\right),L\\},\quad
t\in\left[0,\tau\right].$ (2.4)

We would like to point out that in the measured data, the temperature at the
end of the solid structure can be measured. However the thickness $L$ of the
structure is unknown in the practical application of our interest. It can be
directly verified that the inverse problem (2.4) is nonlinear.

##  3\. An alternating iteration scheme for the inverse problem

In this section, we first recast the inverse problem (2.4) as an optimization
problem by following the general framework of Tikhonov regularization for
inverse problems; see e.g. [2] and the references cited therein. Then we
present the newly proposed alternating iteration scheme. To that end, we
introduce the following objective functional with respect to the unknown heat
flux $q(t)$ and thickness $L$:

$J\left(q,L\right)=\frac{1}{2}\int_{0}^{\tau}\left(\Lambda_{L}\left(t\right)-\Lambda_{m}\left(t\right)\right)^{2}\mathrm{d}t+\frac{\alpha}{2}\int_{0}^{\tau}\left(T(L,t)-T_{m}(L,t)\right)^{2}\mathrm{d}t,$
(3.1)

where $\alpha\in\mathbb{R}_{+}$ signifies a regularization parameter. We then
recast the inverse problem (2.4) as the following PDE-constrained optimization
problem:

$\min_{q\in L^{2}(0,\tau),\,L\in\mathbb{R}_{+}}J\left(q,L\right)\ \
\mbox{subject to}\ T\in
W_{2}^{2,1}\left(\left[0,L\right]\times(0,\tau)\right)\
\mbox{satisfying}\quad(\ref{eq:master}).$ (3.2)

We next convert the constrained optimization problem into an unconstrained one
by using the Lagrange multiplier method. Using the heat conduction equation
(2.3), with the boundary conditions and the initial condition, the augmented
functional is given as follows:

$\displaystyle J\left(q,L\right)=$
$\displaystyle\frac{1}{2}\int_{0}^{\tau}\left(\Lambda_{L}\left(t\right)-\Lambda_{m}\left(t\right)\right)^{2}\mathrm{d}t+\frac{\alpha}{2}\int_{0}^{\tau}\left(T(L,t)-T_{m}(L,t)\right)^{2}\mathrm{d}t$
(3.3) $\displaystyle-\int_{0}^{L}\int_{0}^{\tau}\left\\{\rho c\frac{\partial
T(x,t)}{\partial t}-\frac{\partial}{\partial x}\left(k\frac{\partial
T(x,t)}{\partial x}\right)\right\\}\lambda_{1}(x,t)\,\mathrm{d}t\mathrm{d}x$
$\displaystyle+\int_{0}^{\tau}\left(k\frac{\partial T(x,t)}{\partial
x}+q(t)\right)\bigg{|}_{x=0}\lambda_{2}(t)\,\mathrm{d}t+\int_{0}^{\tau}\left(k\frac{\partial
T(x,t)}{\partial x}\right)\bigg{|}_{x=L}\lambda_{3}(t)\,\mathrm{d}t$
$\displaystyle-\int_{0}^{L}\bigg{(}T(x,t)-T_{0}(x,t)\bigg{)}\bigg{|}_{t=0}\
\lambda_{4}(x)\,\mathrm{d}x.$

where $\lambda_{1}(x,t)$, $\lambda_{2}(t)$, $\lambda_{3}(t)$ and
$\lambda_{4}(x)$ denote the Lagrange multipliers.

###  3.1. Gradients with respect to the parameters

To solve the optimization problem (3.3), the gradients of the objective
functional $J$ with respective to $q$ and $L$ are required. However, it is
difficult to solve the gradients directly. Thus, we refer to [15] and use the
adjoint state method to derive them. Noting that

$\displaystyle\int_{0}^{L}\int_{0}^{\tau}\rho c\frac{\partial T(x,t)}{\partial
t}\lambda_{1}(x,t)\,\mathrm{d}t\mathrm{d}x$
$\displaystyle\quad=-\int_{0}^{L}\int_{0}^{\tau}\rho
c\frac{\partial\lambda_{1}(x,t)}{\partial
t}T(x,t)\,\mathrm{d}t\mathrm{d}x+\int_{0}^{L}\bigg{(}\rho
c\lambda_{1}(x,t)T\left(x,t\right)\bigg{)}\bigg{|}_{0}^{\tau}\,\mathrm{d}x.$

Similarly, one can deduce that

$\displaystyle\int_{0}^{L}\int_{0}^{\tau}\frac{\partial}{\partial
x}\left(k\frac{\partial T(x,t)}{\partial
x}\right)\lambda_{1}(x,t)\,\mathrm{d}t\mathrm{d}x=\int_{0}^{L}\int_{0}^{\tau}\frac{\partial}{\partial
x}\left(k\frac{\partial\lambda_{1}(x,t)}{\partial
x}\right)T(x,t)\,\mathrm{d}t\mathrm{d}x$
$\displaystyle\qquad\qquad\qquad+\int_{0}^{\tau}\left(k\lambda_{1}(x,t)\frac{\partial
T(x,t)}{\partial
x}\right)\bigg{|}_{0}^{L}\,\mathrm{d}t\quad-\int_{0}^{\tau}\left(k\frac{\partial\lambda_{1}(x,t)}{\partial
x}T(x,t)\right)\bigg{|}_{0}^{L}\,\mathrm{d}t.$

Finally, the equation (3.3) can be rewritten as

$\displaystyle J(q,L)=$
$\displaystyle\frac{1}{2}\int_{0}^{\tau}\left(\Lambda_{L}\left(t\right)-\Lambda_{m}\left(t\right)\right)^{2}\mathrm{d}t+\frac{\alpha}{2}\int_{0}^{\tau}\left(T(L,t)-T_{m}(L,t)\right)^{2}\mathrm{d}t$
$\displaystyle+\int_{0}^{L}\int_{0}^{\tau}\left\\{\rho
c\frac{\partial\lambda_{1}(x,t)}{\partial t}+\frac{\partial}{\partial
x}\left(k\frac{\partial\lambda_{1}(x,t)}{\partial
x}\right)\right\\}T(x,t)\,\mathrm{d}t\mathrm{d}x$
$\displaystyle+\int_{0}^{\tau}\left(k\lambda_{1}(x,t)\frac{\partial
T(x,t)}{\partial
x}\right)\bigg{|}_{0}^{L}\,\mathrm{d}t-\int_{0}^{\tau}\left(k\frac{\partial\lambda_{1}(x,t)}{\partial
x}T(x,t)\right)\bigg{|}_{0}^{L}\,\mathrm{d}t$
$\displaystyle-\int_{0}^{L}\bigg{(}\rho
c\lambda_{1}(x,t)T(x,t)\bigg{)}\bigg{|}_{0}^{\tau}\,\mathrm{d}x+\int_{0}^{\tau}\left(k\frac{\partial
T(x,t)}{\partial x}+q(t)\right)\bigg{|}_{x=0}\lambda_{2}(t)\,\mathrm{d}t$
$\displaystyle+\int_{0}^{\tau}\left(k\frac{\partial T(x,t)}{\partial
x}\right)\bigg{|}_{x=L}\lambda_{3}(t)\,\mathrm{d}t-\int_{0}^{L}\bigg{(}T(x,t)-T_{0}(x,t)\bigg{)}\bigg{|}_{t=0}\
\lambda_{4}(x)\,\mathrm{d}x.$

To obtain the adjoint state equation, we set

$\frac{\partial J}{\partial T}=0,$

and it yields

$\left\\{\begin{aligned} &\rho c\frac{\partial\lambda_{1}(x,t)}{\partial
t}+\frac{\partial}{\partial x}\left(k\frac{\partial\lambda_{1}(x,t)}{\partial
x}\right)=S(x,t),\\\
&-k\frac{\partial\lambda_{1}(x,t)}{\partial{x}}\bigg{|}_{x=0}=0,\quad-k\frac{\partial{\lambda_{1}(x,t)}}{\partial{x}}\bigg{|}_{x=L}=0,\\\
&\lambda_{1}(x,\tau)=0,\\\ &\lambda_{2}(t)=\lambda_{1}(0,t),\\\
&\lambda_{3}(t)=-\lambda_{1}(L,t),\\\ &\lambda_{4}(x)=-\rho
c\lambda_{1}(x,0).\end{aligned}\right.$

Here the source term is given by

$S(x,t)=2\left(\Lambda_{L}\left(t\right)-\Lambda_{m}\left(t\right)\right)\frac{a}{(V(x,t))^{2}}+\frac{\alpha}{L}\left(T(L,t)-T_{m}(L,t)\right).$

From (3.3), through a straightforward calculation, the gradients with respect
to the model parameters are given by

$\left\\{\begin{aligned} &\frac{\partial J}{\partial
q}(t)=\lambda_{2}(t)=\lambda_{1}(0,t),\\\ &\frac{\partial J}{\partial
L}=\int_{0}^{\tau}\left(\Lambda_{L}\left(t\right)-\Lambda_{m}\left(t\right)\right)\frac{\partial\Lambda_{L}\left(t\right)}{\partial
L}\,\mathrm{d}t=\int_{0}^{\tau}\frac{2\left(\Lambda_{L}\left(t\right)-\Lambda_{m}\left(t\right)\right)}{V(L,t)}\,\mathrm{d}t.\end{aligned}\right.$
(3.4)

In order to change the final condition to initial condition, a change of
variables can be employed :

$\mu(x,t)=\lambda_{1}(x,\tau-t).$

Consequently, the adjoint state equation is rewritten as

$\left\\{\begin{aligned} &\rho c\frac{\partial\mu(x,t)}{\partial
t}+\frac{\partial}{\partial x}\left(k\frac{\partial\mu(x,t)}{\partial
x}\right)=S(x,\tau-t),\\\
&-k\frac{\partial\mu(x,t)}{\partial{x}}\bigg{|}_{x=0}=0,\quad-k\frac{\partial{\mu(x,t)}}{\partial{x}}\bigg{|}_{x=L}=0,\\\
&\mu(x,0)=0.\end{aligned}\right.$

Therefore, according to (3.4), the gradients with respect to $q(t)$ and $L$
can be represented by

$\left\\{\begin{aligned} &\frac{\partial J}{\partial
q}(t)=\lambda_{1}(0,t)=\mu(0,\tau-t),\\\ &\frac{\partial J}{\partial
L}=\int_{0}^{\tau}\left(\Lambda_{L}\left(t\right)-\Lambda_{m}\left(t\right)\right)\frac{\partial\Lambda\left(t\right)}{\partial
L}\,\mathrm{d}t=\int_{0}^{\tau}\frac{2\left(\Lambda_{L}\left(t\right)-\Lambda_{m}\left(t\right)\right)}{V(L,t)}\,\mathrm{d}t.\end{aligned}\right.$

Next, we use the conjugate gradient method and the steepest descent method to
identify the heat flux $q(t)$ and the thickness $L$, respectively.

###  3.2. Update $q$ with the conjugate gradient method

To numerically reconstruct the heat flux $q$, we shall discretize the heat
flux with respect to the time $t$. Suppose that $[0,\tau]$ is discretized as
follows

$0=t_{0}\leq t_{1}\leq\cdots\leq t_{i}\leq t_{i+1}\leq\cdots\leq t_{N}=\tau.$

The reconstruction schemes of the heat flux based on the conjugate gradient
(CG) method is described as follows

$q_{i}^{n+1}=q_{i}^{n}+\beta^{n}p_{i}^{n},$ (3.5)

where the subscript $i$ indicates the discretization of the heat flux in time,
and the superscripts $n$ and $n+1$ denote the iteration steps. $p_{i}^{n}$
signifies the conjugate direction and it is generated by the rule

$p_{i}^{n}=\begin{cases}&-g_{i}^{n},\qquad\qquad\quad\ n=1,\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ &-g_{i}^{n}+\alpha^{n}p_{i}^{n-1},\quad n\geq 2,\\\
\end{cases}$

where $\alpha^{n}$ is the CG update parameter given by

$\alpha^{n}=\frac{\displaystyle\sum_{i=0}^{N}(g_{i}^{n})^{T}(g_{i}^{n}-g_{i}^{n-1})}{\displaystyle\sum_{i=0}^{N}\left\|g_{i}^{n-1}\right\|^{2}},$

with $\left\|\cdot\right\|$ denoting the Euclidean norm, and

$g_{i}^{n}=\frac{\partial J}{\partial q}\Big{|}_{t=t_{i}}^{n}.$

In addition, the step size $\beta^{n}$ is obtained by the exact line search
and can be described as

$\beta^{n}=\frac{\displaystyle\sum_{i=0}^{N}\left(\Lambda^{n}_{L}\left(t_{i}\right)-\Lambda_{m}\left(t_{i}\right)\right)\Delta
t_{in}\,}{\displaystyle\sum_{i=0}^{N}{\left[\Delta t_{in}\right]^{2}}\,},$

where $\Lambda^{n}_{L}\left(t\right)$ is the solution of the forward problem
and $\Delta t_{n}$ is the change in transmission time and can be expressed as:

$\Delta t_{n}(t)=\int_{0}^{L}{\frac{1}{V_{g^{n}}(x,t)}}\mathrm{d}x.$

Here $V_{g^{n}}=aT_{g^{n}}+b$ and $T_{g^{n}}(x,t)$ is the solution of the
following sensitivity equation

$\left\\{\begin{aligned} &\rho c\frac{\partial T}{\partial
t}=\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right),\\\
&-k\frac{\partial T}{\partial x}\bigg{|}_{x=0}=g^{n},\quad-k\frac{\partial
T}{\partial x}\bigg{|}_{x=L}=0,\\\ &T|_{t=0}=T_{0}.\end{aligned}\right.$

###  3.3. Update $L$ with the steepest descent method

The reconstruction of the thickness based on the steepest descent method is
described as follows:

$L^{n+1}=L^{n}+\lambda^{n}d^{n},$ (3.6)

where the superscripts $n$ and $n+1$ denote the iteration steps, and $d^{n}$
denotes the negative gradient direction respect to $L$,

$d^{n}=-\frac{\partial J}{\partial
L}\bigg{|}_{L=L^{n}}=\int_{0}^{\tau}\frac{2\left(\Lambda_{m}(t)-\Lambda_{L}(t)\right)}{V(L^{n},t)}\,\mathrm{d}t.$

And the step size $\lambda^{n}$ is determined by an inexact line search
technique called Wolfe-Powell search method. Assuming that $f(L)=J(q,L)$ is
differentiable, the Wolfe-Powell search method is used to find $\lambda^{n}$
along $d^{n}$ such that

$\triangledown f(L^{n}+\lambda^{n}d^{n})^{T}d^{n}\geq\sigma\triangledown
f(L^{n})^{T}d^{n},$ $f(L^{n}+\lambda^{n}d^{n})\leq f(L^{n})+\rho\triangledown
f(L^{n})^{T}d^{n},\quad\rho\in(0,1/2),\sigma\in(\rho,1).$

Assuming that $\varphi(\lambda^{n})=f(L^{n}+\lambda^{n}d^{n})$, the strategy
for computing the step length $\lambda^{n}$ can be described as follows:

Step1: Let $\lambda^{0}=0,\lambda^{max}>0$, and choose
$\lambda^{1}\in[\lambda^{0},\lambda^{max}],\rho\in(0,1/2),\sigma\in(\rho,1)$.
Evaluate $\varphi(\lambda^{0})$ and $\varphi^{{}^{\prime}}(\lambda^{0})$. Let
$a_{0}=\lambda^{0},b_{0}=\lambda^{max},n=0$.
Step2: Evaluate $\varphi(\lambda^{n})$. If
$\varphi(\lambda^{n})\leq\varphi(\lambda^{0})+\rho\lambda^{n}\varphi^{{}^{\prime}}(\lambda^{0}),$
go to Step3. Else, go to Step4, set $a_{n+1}=a_{n},b_{n+1}=\lambda^{n}$.
Step3: Evaluate $\varphi^{{}^{\prime}}(\lambda^{n})$. If
$\varphi^{{}^{\prime}}(\lambda^{n})\geq\sigma\varphi^{{}^{\prime}}(\lambda^{0})$,
stop. Else, set $a_{n+1}=\lambda^{n},b_{n+1}=b_{n}$, go to Step4.
Step4: Let $\lambda^{n+1}=\frac{\displaystyle a_{n+1}+b_{n+1}}{\displaystyle
2}$, set $n=n+1$, go to Step2.

###  3.4. Optimize algorithm iteration format

In this paper, we iterate the heat flux $q$ and the thickness $L$
alternatively, and the proposed algorithm is listed as follows:

Step1: Choose an initial point $q_{i}^{0}$, $L^{0}$, $\varepsilon\in(0,1)$.
Step2: Fixed $L^{n}$. Update $q_{i}$ using the formular (3.5).
Step3: Fixed $q_{i}^{n+1}$ update $L$ using the formular (3.6).
Step4: Evaluate $J(q_{i}^{n+1},L^{n+1})$. If
$J(q_{i}^{n+1},L^{n+1})<\varepsilon$, stop. Else, set $n=n+1$, go to Step2.

Algorithm 1 Alternating iteration algorithm

1:$q^{0}(N),L^{0},crl,n_{max},\varepsilon$.

2:$q(N)$, $L$, $T(nl)$.

3:$q^{n}\leftarrow q^{0}$, $L^{n}\leftarrow L^{0}$, $J\leftarrow J^{0}$.

4:while $abs(J)>crl.AND.n<n_{max}$ do

5: call gradient

6: $g^{n}\leftarrow g1$,

7: call cgm

8: $p^{n}\leftarrow p1$,

9: call bet

10: $\beta^{n}\leftarrow\beta 1$,

11: for $i=1,N$ do

12: $q^{n}(i)\leftarrow q^{n}(i)-bet*p^{n}(i)$

13: $g2(i)\leftarrow g1(i)$

14: $p2(i)\leftarrow p1(i)$

15: end for

16: $//$ update $q$

17: if $abs(L-j1)>\varepsilon$ then

18: $j1\leftarrow L$

19: call wolfe(j)

20: $//$ update $L$

21: else

22: $aa\leftarrow aa/10$

23: $//$ update regularization parameter

24: end if

25: compute objective function $J$.

26:end while

##  4\. Convergence Analysis

In this section, we shall analyze the convergence of the reconstruction scheme
proposed in the previous section. Let $\left(q_{i}^{*},L^{*}\right)$ be the
optimal solution to the optimization problem (3.3), i.e.,

$J\left(q_{i}^{*},L^{*}\right)\leq
J\left(q_{i}^{n},L^{n}\right),\quad\forall\,q_{i}^{n}\in\mathbb{R}^{N},\
L^{n}\in\mathbb{R}.$ (4.1)

It is clear that the necessary condition of (4.1) is:

$\triangledown
J\left(q_{i}^{*},L^{*}\right)=\left(g_{i}^{*},-d^{*}\right)^{T}=0,$

and hence it is sufficient for us to prove

$\lim_{n\rightarrow\infty}\inf\|\left(g_{i}^{n},d^{n}\right)\|=0.$ (4.2)

Next, we prove that the optimization algorithm that consists of (3.5) and
(3.6) satisfies the convergence condition (4.2). Before we discuss the
convergence, we introduce some notations and important lemmas.

######  Definition 4.1.

$Polka-Ribi\acute{e}re-Polyak(PRP)$ method is a nonlinear conjugate gradient
method, and it has the following form:

$\displaystyle q^{n+1}=q^{n}+\beta^{n}p^{n},$ (4.3) $\displaystyle
p^{n}=\left\\{\begin{aligned} &-g^{n},&\quad n=1,\\\
&-g^{n}+\alpha^{n}p^{n-1},&\quad n\geq 2,\end{aligned}\right.$

where

$\alpha^{n}_{PRP}=\frac{g^{nT}(g^{n}-g^{n-1})}{\left\|g^{n-1}\right\|^{2}}.$
(4.4)

######  Definition 4.2.

Exact line search: at each iteration, the step size $\beta^{n}$ is selected so
that

$f(q^{n}+\beta^{n}p^{n})=\underset{\beta}{\min}f(q^{n}+\beta p^{n}),$

where the objective functional is $f(q)=J(q,L),q(t)\in
L^{2}\left(\left[0,\tau\right]\right)$.

######  Remark 4.1.

Iteration algorithm (3.5) is a PRP conjugate method with the exact line
search.

Next, we prove the convergence of the PRP conjugate method with an exact line
search as well as its convergence condition. To that end, we first derive
several auxiliary lemmas.

######  Lemma 4.1.

[14] Let $\theta_{n}$ be the angle between the searching direction $p^{n}$ and
the negative gradient direction $-g^{n}$. Then

$\cos\theta_{n}=\frac{-g^{nT}p^{n}}{\|g^{n}\|\|p^{n}\|}.$

When the line search is the exact line search, the angle $\theta_{n}$ is
represented by Figure 2. If $\alpha$ is given by (4.4), we have

$\tan\theta_{n+1}\leq\sec\theta_{n}\frac{\|g^{n+1}-g^{n}\|}{\left\|g^{n}\right\|}.$
(4.5)

Figure 2. The definition of figure

###### Proof.

Figure 2 gives the equation

$\left\|p^{n}\right\|=\sec\theta_{n}\left\|g^{n}\right\|.$ (4.6)

Further, if $n$ is replaced by $n+1$ in Figure 2 , we find the identity

$\alpha^{n+1}\left\|p^{n}\right\|=\tan\theta_{n+1}\|g^{n+1}\|.$ (4.7)

By (4.4), one has

$\alpha^{n+1}=\frac{(g^{n+1})^{T}(g^{n+1}-g^{n})}{\left\|g^{n}\right\|^{2}},$

and hence by the Cauchy-Schwarz inequality,

$\alpha^{n+1}\leqslant\frac{\left\|g^{n+1}\right\|\left\|g^{n+1}-g^{n}\right\|}{\left\|g^{n}\right\|^{2}}.$
(4.8)

Next, by the elimination of $\left\|g^{n}\right\|$ from (4.6) and(4.7), one
can show the following equality,

$\alpha^{n+1}=\frac{\tan\theta_{n+1}\left\|g^{n+1}\right\|}{\sec\theta_{n}\left\|g^{n}\right\|},$
(4.9)

which in combination with (4.8) and (4.9) readily yields that

$\tan\theta_{n+1}\leqslant\sec\theta_{n}\frac{\left\|g^{n+1}-g^{n}\right\|}{\left\|g^{n}\right\|}.$

The proof is complete. ∎

######  Lemma 4.2.

[3] Consider the iterative method of the form $q^{n+1}=q^{n}+\beta^{n}p^{n}$,
where $p^{n}$ satisfies the descent condtion $p^{nT}g^{n}\leq 0$, the step
size $\beta^{n}$ is obtained by the exact line search, the objective
functional $f(q)$ is bounded below, and its gradient $\triangledown f(q)$
satisfies the Lipschitz condition, then

$\sum_{n\geqslant
1}\frac{(g^{nT}p^{n})^{2}}{\left\|p^{n}\right\|^{2}}<\infty,$ (4.10)

and according to the definition of $\theta_{n}$, (4.10) is equivalent to

$\sum_{n\geqslant 1}\left\|g^{n}\right\|^{2}\cos\theta_{n}<\infty.$ (4.11)

######  Lemma 4.3.

[4] If the step size $\beta^{n}$ is obtained by the exact line search and the
objective function $f(q)$ is uniformly convex, then

$f(q^{n})-f(q^{n}+s^{n})\geqslant c\left\|s^{n}\right\|^{2}$

holds, where $s^{n}=q^{n+1}-q^{n}$, $c>0$ is a constant, and further there is
$\left\|s^{n}\right\|\rightarrow 0$ as $n\rightarrow\infty$.

######  Lemma 4.4.

Assume that the objective function $f(q)$ is uniformly convex and bounded from
below, and its gradient $\triangledown f(q)$ satisfies the Lipschitz
condition. Consider the algorithm (4.3)–(4.4), if the step size $\beta^{n}$ is
obtained by the exact line search, then

$\lim_{n\rightarrow\infty}\inf\left\|g^{n}\right\|=0.$

###### Proof.

The proof follows a similar spirit to that of Theorem 1 in [14]. By an
absurdity argument, we assume that the theorem does not hold. Then there is a
constant $\gamma$, such that for any $n\geqslant 1$,

$\left\|g^{n}\right\|\geqslant\gamma.$ (4.12)

By Lemma 4.3, there is $\left\|s^{n}\right\|\rightarrow 0,n\rightarrow\infty$,
which combines with the fact that $\triangledown f(q)$ is Lipschitz continuous
implies that there exists a positive integer $m$, such that

$\left\|g^{n+1}-g^{n}\right\|\geqslant\frac{1}{2}\gamma,$ (4.13)

which holds for any $n\geqslant m$. Noticing that for any
$\theta_{n}\in[0,\pi/2)$, there is

$\sec\theta_{n}\leqslant 1+\tan\theta_{n},$

which together with (4.5), (4.12) and (4.13) further implies that

$\displaystyle\tan\theta_{n+1}$
$\displaystyle\leqslant\frac{1}{2}+\frac{1}{4}+\cdots+(\frac{1}{2})^{n-1-m}(1+\tan\theta_{m})$
$\displaystyle=1+\tan{\theta_{m}},\quad\forall n\geqslant m.$

Therefore, the angle $\theta_{n}$ between the searching direction $p_{n}$ and
the negative gradient direction $-g^{n}$ is always smaller than an angle
bounded above by ${\pi}/{2}$. Therefore, by (4.11), we have
$\left\|g^{n}\right\|\rightarrow 0$, which is a contradiction to (4.12). Thus
the theorem must hold true.

The proof is complete. ∎

Next, we establish the convergence of the steepest descent algorithm with the
Wolfe-Powell step search method.

######  Lemma 4.5.

[6] Let the objective function $f(L)=J(q,L),L\in\mathbb{R}$ be differentiable
and bounded from below, and $g(L)=\triangledown f(L)$ satisfy the Lipschitz
condition. Suppose that the steepest descent method generates a sequence
$L^{n},n\geq 1$, using the recurrence

$L^{n+1}=L^{n}+\lambda^{n}d^{n},$

where the direction $d^{n}$ denotes the negative gradient direction. If the
step size $\lambda$ satisfies

$\left\\{\begin{aligned} &(g^{n+1})^{T}d^{n}\geq\sigma(g^{n})^{T}d^{n},\\\
&f(L^{n+1})\leq
f(L^{n})-\rho\lambda^{n}(d^{n})^{2},\quad\rho\in(0,1/2),\sigma\in(\rho,1),\end{aligned}\right.$

where $g^{n+1}=g(L^{n+1})=g(L^{n}+\lambda^{n}d^{n})$, then one has

$f(L^{n})-f(L^{n+1})\geq\frac{\rho(1-\sigma)}{M}\left\|g^{n}\right\|^{2}\cos^{2}\theta_{n}.$

######  Lemma 4.6.

If the function $f(L)$ is continuously differentiable and satisfys the
conditions of Lemma 4.5, then sequence $L^{n}$ generated by the steepest
descent method satisfys:

$\lim_{n\rightarrow\infty}\left\|g^{n}\right\|^{2}=0.$

###### Proof.

By Lemma 4.5,

$f(L^{n})-f(L^{n+1})\geq\frac{\rho(1-\sigma)}{M}\left\|g^{n}\right\|^{2}\cos^{2}\theta_{n}.$

Thus

$\displaystyle
f(L^{0})-f(L^{n+1})=[f(L^{0}-f(L^{1})]+[f(L^{1})+f(L^{2})]+\cdots+[f(L^{n})-f(L^{n+1})]$
(4.14)
$\displaystyle=\sum_{k=0}^{n}[f(L^{k})-f(L^{k+1})]\geq\sum_{k=0}^{n}\frac{\rho(1-\sigma)}{M}\left\|g^{k}\right\|^{2}\cos^{2}(d^{k},-g^{k}).$

Notice that $\cos^{2}(d_{k},-g_{k})=1,k=1,2,\cdots,{n}$. We therefore have
from (4.14) that

$\sum_{k=0}^{n}\frac{\rho(1-\sigma)}{M}\left\|g_{k}\right\|^{2}\leq
f(L^{0})-f(L^{n+1}).$ (4.15)

Since $f(L)$ is bounded from below, one sees that
$f(L^{0})-f(L^{n+1})<\infty$, which together with (4.15) readily implies that

$\lim_{n\rightarrow\infty}\left\|g^{n}\right\|^{2}=0.$

The proof is complete. ∎

Next we prove that the optimization algorithm consisting of (3.5) and (3.6)
satisfies the convergence condition (4.2).

######  Theorem 4.1.

Consider the iterative algorithm consisting of (3.5) and (3.6):

$\left\\{\begin{aligned} &q_{i}^{n+1}=q_{i}^{n}+\beta^{n}p^{n}_{i},\\\
&L^{n+1}=L^{n}+\lambda^{n}d^{n},\end{aligned}\right.$

Assume that the objective functional $J(q,L)$ satisfies the following
conditions:
(a): $J(q,L)$ is continuously differentiable with respect to $q(t)$ and $L$;
(b): $J(q,L)$ is uniformly convex;
(c): $J(q,L)$ is bounded from below;
(d): Its gradient $\triangledown J(q,L)$ is Lipschitz continuous.
Then the optimization algorithm consisting of (3.5) and (3.6) satisfies the
convergence condition (4.2), i.e.,

$\liminf_{n\rightarrow\infty}\|\left(g_{i}^{n},d^{n}\right)\|_{1}=0,$

where $\|\cdot\|_{1}$ denotes the $l^{1}$-norm, namely
$\|\left(g_{i}^{n},d^{n}\right)\|_{1}=\|g_{i}^{n}\|_{1}+|d^{n}|.$

###### Proof.

We first consider the iterative algorithm (3.5). Since the step size
$\lambda^{n}$ is searched by the Powell-Wolfe method, it satisfies the
following conditions:

$\displaystyle\varphi(\lambda^{n})\leq\varphi(0)+\rho\lambda^{n}\varphi^{{}^{\prime}}(0),$
(4.16)
$\displaystyle\varphi^{{}^{\prime}}(\lambda^{n})\geq\sigma\varphi^{{}^{\prime}}(0),\quad\rho\in(0,1/2),\sigma\in(\rho,1),$

where $\varphi(\lambda^{n})=J(q^{n},L^{n}+\lambda^{n}d^{n})$. By using
conditions (a) (c) (d) and (4.16), we see that Lemma 4.6 holds. Hence by Lemma
4.6, we have

$\lim_{n\rightarrow\infty}\left|d^{n}\right|=0.$ (4.17)

We proceed to consider the iterative algorithm (3.6). By virtue of the
conditions (a) (b) (c) (d), we see that the iteration algorithm (3.5) is a PRP
conjugate method with the exact line search. By Lemma 4.4,

$\lim_{n\rightarrow\infty}\inf\left\|g_{i}^{n}\right\|=0,$ (4.18)

According to (4.17) and (4.18),

$\lim_{n\rightarrow\infty}\inf\left(\|g_{i}^{n}\|+|d^{n}|\right)=0.$

Thus,

$\lim_{n\rightarrow\infty}\inf\|\left(g_{i}^{n},d^{n}\right)\|_{1}=\lim_{n\rightarrow\infty}\inf\left(\|g_{i}^{n}\|+|d^{n}|\right)=0.$

The proof is complete. ∎

##  5\. Numerical Examples

In this section, we present several numerical examples to verify the
effectiveness and robustness of the proposed scheme in simultaneously
reconstructing the surface heat flux and the thickness of a solid structure
under different acoustic time accuracies, initial fluxes, and initial
thicknesses. It is emphasized that all the data in our numerical examples are
collected by lab experiments following the setup described in Figure 1.

The specimens with a thickness of $L=50\,\rm{mm}$ are heated at one boundary
and the surface heat flux is $q(t)=10^{5}\,\rm{J/s}$. The ultrasonic wave
probes are stalled on the other boundary of the specimens with the detection
frequency set to be $\omega=1\,\mathrm{Hz}$, and the total detection time set
to be $\tau=500\,\rm{s}$. Moreover, the thermal conductivity of the specimens
is $k=50\,\rm{W/(m\cdot^{\circ}\\!C)}$, specific heat is
$c=400\,\rm{J/(kg\cdot^{\circ}\\!C)}$ and density of the material is
$\rho=7800\,\rm{kg/m^{3}}$. The initial tempareture is chosen as
$T_{0}=\rm{26\,^{\circ}C}$. The relationship between the velocity and
temperature is given as follows:

$V(T)=-0.4521T+3259.9.$

In the following numerical examples, the stopping criterion for the iterations
is set to be $J(q,L)<5\times 10^{-18}$. The Fortran software is used for
implementing of Algorithm 1.

The reconstruction results of thickness under the acoustic time accuracy of
$10^{-9}$, $10^{-10}$ and $10^{-11}$ are respectively shown in the Table 1.

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
$10^{-9}$ | $1\times 10^{3}$ | 45 | 50.0032 | 208
$10^{-10}$ | $1\times 10^{3}$ | 45 | 50.0028 | 115
$10^{-11}$ | $1\times 10^{3}$ | 45 | 50.0025 | 108
Table 1. Convergence of the iteration method with different initial guesses
and measurement errors.

It can be found that the thickness can be reconstructed effectively under
different acoustics time accuracies. The error and the number of iterations
show that under the same initial value, the accuracy of acoustic time will
affect the convergence speed of the algorithm. If the accuracy of acoustic
time reaches $10^{-10}$ or $10^{-11}$, one can achieve much accurate
reconstruction results. Thus, in the following numerical examples, we adopt
the measurement data with an acoustic time accuracy of $10^{-10}$ or
$10^{-11}$ to study the effect of the initial values on the inversion
procedure.

Figure 3 presents the reconstruction results of the heat flux under different
initial thicknesses and different initial heat flux conditions. The acoustic
accuracy is fixed to be $10^{-10}$.

(a)

(b)

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 shows the reconstruction results of the heat flux under different
initial thicknesses and different initial heat flux conditions. The acoustic
accuracy is fixed to be $10^{-11}$.

(a)

(b)

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.

By observing the surface heat flux reconstruction results in Figures 3 and 4,
it can be found that when the acoustic time accuracy is $10^{-10}$ and
$10^{-11}$, under different initial conditions, the inversion value of the
heat flux converges to the real value. Moreover, when the acoustic time
accuracy reaches $10^{-11}$, the inversion value of heat flux very close to
the real value, which achieves a much accurate reconstruction.

Table 2 lists the inversion results and the iteration times of the thickness
under different initial thicknesses and different initial heat fluxes initial
conditions. The acoustic time accuracy is $10^{-10}$ or $10^{-11}$.

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
$10^{-10}$ | $1\times 10^{3}$ | 3 | 50.0016 | 58
$10^{-10}$ | $1\times 10^{5}$ | 3 | 50.0000 | 133
$10^{-10}$ | 0 | 45 | 50.0025 | 126
$10^{-10}$ | $1\times 10^{3}$ | 45 | 50.0028 | 115
$10^{-10}$ | 0 | 80 | 50.0028 | 77
$10^{-10}$ | $1\times 10^{3}$ | 80 | 50.0046 | 92
$10^{-11}$ | 0 | 3 | 50.0006 | 63
$10^{-11}$ | $1\times 10^{3}$ | 3 | 50.0016 | 57
$10^{-11}$ | $1\times 10^{5}$ | 3 | 50.0006 | 147
$10^{-11}$ | 0 | 45 | 50.0025 | 126
$10^{-11}$ | $1\times 10^{3}$ | 45 | 50.0025 | 108
$10^{-11}$ | 0 | 80 | 50.0028 | 77
$10^{-11}$ | $1\times 10^{3}$ | 80 | 50.0029 | 80
Table 2. Convergence of the proposed iteration method with different initial
guesses and measurement errors.

The results show that the iterative algorithm converges very fast and robust
with different initial conditions.

##  6\. Conclusion

Based on the ultrasonic echo method and the inverse problem analysis method of
the heat conduction, combined with the optimization model, a method of
simultaneously reconstructing the thickness and the surface heat flux of a
solid structure is established in this paper. This approach is non-destructive
and non-contact and it can be used to recover the surface heat flux and the
wall thickness at the same time. It possesses a high engineering value. We
provide a rigorous convergence analysis of the proposed numerical scheme. By
using experimental lab data, we conducted extensive numerical experiments to
verify the effectiveness and efficiency of the newly developed method. It is
found that as long as the acoustic time accuracy reaches $10^{-10}$ or
$10^{-11}$, the proposed iteration method converges very fast and robust with
respect to different initial guesses.

## Acknowledgement

The work of H Liu was supported by a startup fund from City University of Hong
Kong and the Hong Kong RGC General Research Fund (projects 12301420, 12302919,
12301218).

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