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Abstract
In this contribution, an inverse problem of determining a space-dependent vector source in a thermoelastic system of type-I, type-II and type-III is studied using information from a supplementary measurement at a fixed time. These thermoelastic systems consist of two equations that are coupled: a parabolic equation for the temperature and a vectorial hyperbolic equation for the displacement
. In this latter one, the source is unknown, but solely space dependent. A spacewise-dependent additional measurement at the final time ensures that the inverse problem corresponding with each type of thermoelasticity has a unique solution when a damping term
(with
componentwise strictly monotone increasing) is present in the hyperbolic equation. Despite the ill-posed nature of these inverse problems, a stable iterative algorithm is proposed to recover the unknown source in the case that
is also linear. This method is based on a sequence of well-posed direct problems, which are numerically solved at each iteration, step by step, using the finite element method. The instability of the inverse source problem is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results support the theoretically obtained results.
1 Introduction
Thermoelasticity is the change in the size and shape of a solid object as the temperature of that object fluctuates. A material that is elastic expands when heated and contracts when cooled. There are mathematical systems established that describe how the shape of an object changes when there are fluctuations in the temperature. More specifically, a thermoelastic system consists of two equations that are coupled: a parabolic equation for the temperature and a vectorial hyperbolic equation for the displacement. In [Citation1], three theories of thermoelasticity each with a corresponding system are developed: type-I, type-II and type-III thermoelasticities. The first type is based on Fourier’s law to describe the heat conduction in the body. This theory has the shortcoming that a thermal disturbance at one point of the body is instantly felt everywhere (infinite speed of propagation phenomena). This is physically not acceptable for materials with memory and is overcome by taking memory effects into account in the models for type-II and type-III thermoelasticities. The main difference between type-II and type-III is that in type-II thermoelasticity, the heat conduction is independent of the present values of the temperature gradient, see also [Citation2, Citation3]. A topic of great interest is the recovery of unknown (vector or heat) sources in thermoelastic systems from additional measurements.
In this contribution, an inverse problem of determining a space-dependent source in a thermoelastic system of type-III using information from a supplementary measurement at a given single instant of time is studied. The mathematical setting is the following. An isotropic and homogeneous thermoelastic body occupying an open and bounded domain with Lipschitz continuous boundary
is considered. Let
and
for a given final time
. The convolution product of a kernel
and a function
is denoted with the sign ‘
’
The following thermoelastic system of type-III describing the elastic and thermal behaviour in
is discussed:
1
1 with initial conditions:
2
2 where the unknown vector source
is of the form
3
3 with the vector fields
known and
unknown. The heat source
is known. Here,
and
denote, respectively, the displacement and the temperature difference from the reference value (in Kelvin) of the solid elastic material at the location
and time
. The Lamé parameters
and
, the coupling (absorbing) coefficient
and the thermal coefficient
are assumed to be positive constants because the medium is supposed to be isotropic homogeneous. The goal of this paper is to determine the spatial vector function
by an additional measurement (the condition of final overdetermination)
4
4 This means that the displacement is measured at the final time. The kernel function (also called relaxation function)
decays to zero as time goes to infinity. Moreover, it is assumed that
with
denoting the order of the derivative. These assumptions imply that
is strongly positive definite [Citation4, Corollary
]. This is equivalent with the existence of a positive constant
independent of
such that [Citation4, Lemma
], [Citation5]
These assumptions on the kernel
are natural, because usually
takes the form [Citation6]
with
and
, two positive constants. Note that a damping term
is added in the hyperbolic equation in comparison with the classic thermoelasticity system,
. This term is also considered in [Citation4, Chapter
], [Citation7, Citation8]. Moreover, this term is essential to establish the uniqueness of a solution to the inverse problem, see Theorem 2.1.
Recently, inverse source problems related to the classic thermoelastic system have been studied by several authors. Without taking memory effects into account, i.e. and
, Bellassoued and Yamamoto [Citation9] investigated an inverse heat source problem for type-I thermoelasticity. The main subject of their paper is the inverse problem of determining the heat source
instead of the vector source
. This is done by measuring
and
, where
is a subdomain of
such that
and
. They do not need data of
over the whole domain
. Using a Carleman estimate, a Hölder stability for the inverse source problem is proved, which implies the uniqueness of the inverse source problem. Wu and Liu [Citation10] studied an inverse source problem of determining
for type-II thermoelasticity, i.e.
and
. Based on a Carleman estimate, again a Hölder stability for the inverse source problem has been established from a displacement measurement
provided that
is known in a neighbourhood
of
. Note that there is no temperature measurement needed. In both contributions, no numerical scheme is provided to recover the unknown source. From now on, consider the most general situation that
and
, i.e. type-III thermoelasticity. As mentioned before, the goal of this contribution is to determine
from the additional final in time measurement (Equation4
4
4 ). The uniqueness of a solution to problem (Equation1
1
1 ) is proved using a variational approach instead of using a Carleman estimate. One of the key assumptions is that
is componentwise strictly monotone increasing. Unfortunately, this inverse problem is ill-posed since small errors present in any practical measurements give rise to large errors into the solutions. However, in the case that
is also linear, a stable iterative algorithm is proposed to recover the unknown source. This method is based on a sequence of well-posed direct problems, which are numerically solved at each iteration step using the finite element method. The instability of this inverse source problem is overcome by stopping the iterations using the discrepancy principle.[Citation11] This scheme is of the Landweber–Friedman type and is similar to that of Johansson and Lesnic for the heat conduction equation.[Citation12] This procedure is also used for the heat conduction equation with time-dependent coefficients in [Citation13].
The remainder of this paper is organized as follows. The uniqueness of a solution to the inverse problem under consideration is established in Section 2 under the assumption that is componentwise strictly monotone increasing (
can be linear or non-linear). In Section 3, a convergent and stable algorithm is proposed for the recovery of the unknown vector source in the case that
is also linear. Finally, some numerical experiments are developed in Section 4.
2 Uniqueness
Using Green’s formulas, the following coupled variational formulation for (Equation11
1 ) is obtained: find
such that
5
5
6
6 for all
and
. The spacewise-dependent measurement (Equation4
4
4 ) ensures that the inverse problem has a unique solution. This is stated in the following theorem.
Theorem 2.1
(Uniqueness). Let ,
and
componentwise. Then there exists at most one triplet
such that problem (Equation1
1
1 ) together with the condition (Equation4
4
4 ) hold.
Proof
Suppose that there are two solutions and
to (Equation1
1
1 )–(Equation4
4
4 ). Subtract, equation by equation, the variational formulation (Equation5
5
5 ) and (Equation6
6
6 ) corresponding with the solution
from the variational formulation for
. Set
,
and
. Then is
,
,
and
. Then, we obtain
7
7
8
8 for all
and
. First, we prove that
and
. Afterwards, we show that
. Therefore, in the first part of the proof, we should get rid of
. This can be done in a simple way. The main idea is
Indeed, putting
in (Equation7
7
7 ) and integrating in time over
gives
9
9 because
. Taking
in (Equation8
8
8 ) yields
10
10 due to
. Now, adding (Equation9
9
9 ) and (Equation10
10
10 ) implies
11
11 The strongly positive definiteness of
implies that
Thus, from (Equation11
11
11 ), it follows that
and
12
12 Due to the fact that
on
, we deduce that
Here, we can also see why the damping term is necessary. Without this term, we would only have that
, which gives no guarantee that
. Employing the fact that the vector field
is componentwise strictly monotone increasing, we get that
, i.e.
is constant in time. Therefore,
Putting the obtained information in (Equation7
7
7 ) gives
From this, we conclude that
in
.
Remark 2.2
From this theorem follows also immediately the uniqueness to the inverse problem corresponding with type-I thermoelasticity (). For type-II thermoelasticity (
), the proof of uniqueness of a solution is less straightforward. Then (Equation12
12
12 ) becomes
Therefore,
for all
and
. Hence, since the Laplace transform is one-to-one, it is clear that
in
. The uniqueness of a solution follows from
on
.
Remark 2.3
In fact, to prove the uniqueness of a solution in the case of type-III thermoelasticity, it is sufficient that the kernel is positive definite instead of strongly positive definite. But, the strongly positive definiteness of
is immediately considered because under this assumption, the unicity of a solution to the inverse problem is valid for all types of thermoelasticities, as mentioned in Remark 2.2.
Remark 2.4
The main trick of the proof is not applicable if the heat source should be unknown, i.e.
.
Remark 2.5
The uniqueness of the solution can also be obtained for more general coefficients. For instance, forthe following assumptions have to be satisfied
3 Reconstruction of the source term in a linear case
From now on, it is assumed that is linear. Then, the principle of linear superposition is applicable on problem (Equation1
1
1 )–(Equation4
4
4 ). The solution
is given by
, where
is a solution to
13
13 and
is solving
14
14 First, the well posedness of problem (Equation14
14
14 ) is discussed. Rivera and Qin [Citation14] proved the global existence and uniqueness of solutions to problem (Equation14
14
14 ) in one dimension when
and
. In the same situation, a more dimensional case for type-III thermoelasticity is studied in [Citation15]. The following lemma summarizes the available results. For a more general setting, see also [Citation16–Citation18].
Lemma 3.1
(i) | Assume that | ||||
(ii) | Assume that |
In the remainder of the paper, it is assumed that the assumptions of Lemma 3.1(ii) are valid. This is important because only in this situation are the boundary conditions satisfied since and
for
. Moreover, also the restriction
is well defined for
. This means, in particular, that the final displacement measurement
is well defined. Following Theorem 2.1 and Lemma 3.1(ii), the solution
to problem (Equation13
13
13 ) is unique if the additional final measurement is satisfied, i.e.
16
16 where
is the solution to problem (Equation14
14
14 ). Note that for given
, problem (Equation13
13
13 ) is a special case of problem (Equation14
14
14 ). In the following subsection, an algorithm for the recovery of the unknown source term is proposed.
Remark 3.2
Also, the well posedness of problem (Equation1414
14 ) can be obtained for more general coefficients. More specific, when the coefficients are space dependent, the following assumptions have to be satisfied
3.1 Algorithm for finding the source term
The procedure for the stable reconstruction of the solution and the source term
of problem (Equation1
1
1 )–(Equation4
4
4 ) is similar to the one presented in [Citation12, Citation13, Citation19]. It runs as follows:
(i) | First solve problem (Equation14 | ||||
(ii) | Choose an initial guess | ||||
(iii) | Assume that | ||||
(iv) | Define | ||||
(v) | The procedure continues by repeating steps (ii) and (iii) until a desired level of accuracy is achieved, see Sections 3.2 and 3.3. Suppose that the algorithm stopped after |
3.2 Convergence of the proposed algorithm in Section 3.1
Let be the unique solution to (Equation13
13
13 ) for given
, see Lemma 3.1. Define the corresponding operator
by
Finding a solution to the inverse problem is then equivalent to solving the following operator equation
17
17 The following theorem shows the convergence of the proposed algorithm in the previous subsection.
Theorem 3.3
Assume that the assumptions of Lemma 3.1(ii) are satisfied and suppose that the relaxation parameter satisfies
. Denote by
the unique solution to the original inverse problem (Equation1
1
1 )–(Equation4
4
4 ), where
is the solution to problem (Equation14
14
14 ) and
is solving (Equation13
13
13 )–(Equation16
16
16 ). Let
the
th approximation in the iterative algorithm of Section 3.1. Then
and
for every function
.
Proof
From the iterative algorithm and the linearity of the operator , it is possible to deduce that
Therefore,
This is a Landweber–Friedmann iteration scheme for solving the operator Equation (Equation17
17
17 ). The standard proof of convergence for Landweber’s iterations is given for
and
with
and
abstract Hilbert spaces in [Citation20, Theorem
]. A more general version of the proof of convergence for two not self-adjoint operators
is given in [Citation21, Theorem 3]. This implies thanks to the assumption
that the sequence
converges to
in
for arbitrary
. Inequality (Equation15
15
15 ) implies that
and
in
. Furthermore,
in
and
in
.
3.3 Stopping criterion
Reconsider the algorithm given in Section 3.1. There are errors present in each practical experiment. In this contribution, the displacement at the final time is measured to obtain a solution to problems (Equation11
1 )–(Equation3
3
3 ). Therefore, the case is considered when there is some error in the additional measurement (Equation4
4
4 ), i.e.
18
18 with
. This implies that also
is perturbated, see (Equation16
16
16 ). The perturbated field is designated analogously by
. The functions
and
are obtained using the algorithm with no noise on the initial data (Equation2
2
2 ). Note that this latter is a good assumption since the inverse problem is stable with respect to small perturbations in the initial data. The absolute
-error between this final measurement
and the
th approximation
at
is denoted by
19
19 Given the noise level
, the discrepancy principle [Citation11] can be used to obtain a stopping criterion for the algorithm [Citation20, Proposition 6.4]. This principle suggests to finish the iterations at the smallest index
for which
4 Numerical experiment
In the numerical experiments, it is assumed that the density, the thermal coefficient, the Lamé parameters and the coupling coefficient are normalized to be one, i.e. . Moreover, for ease of implementation, only the
D linear model of type-I thermoelasticity is considered. It is given by: find
such that
20
20 and the final overdetermination condition is satisfied
21
21 The solution to problem (Equation20
20
20 ) and (Equation21
21
21 ) is recovered by applying the algorithm proposed in Section 3.1. Let
in all the experiments. The forward mixed problems in this procedure are discretized in time according to the backward Euler method. The timestep for the equidistant time partitioning is chosen to be 0.001. At each timestep, the resulting elliptic mixed problems are solved numerically by the finite element method (FEM) using first-order (P1-FEM) Lagrange polynomials for the space discretization.
In each experiment, the exact solution for is compared with the numerical solution
obtained when the algorithm finishes after a finite number of
iterations. The index
is the smallest index
for which
or is the maximum number of iterations when this number is reached. In both experiments, the maximum number of iterations equals 10,000. The finite element library DOLFIN from the FEniCS project is used for the implementation.[Citation22]
4.1 Experiment 1
The exact solution to problem (Equation20
20
20 ) and (Equation21
21
21 ) is prescribed as follows
22
22 Some simple calculations with the use of this exact solution give the exact data for the numerical experiment
For the space discretization, a fixed uniform mesh consisting of 50 intervals is used. In this experiment, an uncorrelated noise is added to the additional condition (Equation21
21
21 ) in order to simulate the errors present in real measurements. The noise is generated randomly with given magnitude
and 5% and the resulting final measurement is denoted by
, see also Section 3.3. This gives for (Equation18
18
18 ) that
23
23 According to the discrepancy principle, the algorithm is finished at the smallest index
such that (Equation19
19
19 ) is satisfied, see Table . The obtained results, see Figure , are in accordance with the numerical experiments performed for the heat conduction equation in [Citation12]. As
or
increases, the attainability of the stopping criterion (Equation19
19
19 ) becomes faster. In this experiment, the numerical solution
, with
the stopping index, approximates the unknown source better for larger values of
. Even for a large amount of noise (5%), an accurate approximation for the source is obtained. Note that the algorithm is divergent for
.
Table 1. The stopping iteration number for Experiment 1 given by (Equation19
19
19 ), with
given by (Equation23
23
23 ).
Figure 1. The exact solution (Equation2424
24 ) and the numerical solution for the source (Equation22
22
22 ) for
and
for different values of
.
![Figure 1. The exact solution (Equation2424 p(x)=0for0⩽x<131for13⩽x<230for23<x⩽124 ) and the numerical solution for the source (Equation2222 u(x,t)=(1+t)2x(x-1),θ(x,t)=(1+t)x(1-x),p(x)=x(x-1).22 ) for e~=1% and e~=5% for different values of κ.](/cms/asset/46d8962f-d7fe-4738-b4af-598f066c2ef2/gipe_a_959008_f0001_oc.gif)
Figure 2. The exact solution (Equation2424
24 ) and the numerical solution for the source (Equation24
24
24 ) for
and
.
![Figure 2. The exact solution (Equation2424 p(x)=0for0⩽x<131for13⩽x<230for23<x⩽124 ) and the numerical solution for the source (Equation2424 p(x)=0for0⩽x<131for13⩽x<230for23<x⩽124 ) for e~=1% and κ=10.](/cms/asset/d5cd386e-e311-4407-8b2e-b9073971d42b/gipe_a_959008_f0002_oc.gif)
4.2 Experiment 2
In this experiment, a discontinuous source given by24
24 is reconstructed. Since the direct problems in the algorithm for given
do not have an analytical solution, the data (Equation21
21
21 ) is obtained by solving the direct problem using the FEM for
. Now, a fixed uniform mesh consisting of
intervals is used for the space discretization. Only a randomly generated noise with magnitude
is considered on the resulting final measurement. In this experiment,
. The stopping index is equal to
and
for
and
, respectively. There are more than 10,000 iterations needed for
. The numerical solution for the source in the case that
is given in Figure . The approximation is in accordance with the experiment performed in [Citation12] with the same unknown source for only the heat conduction equation. Note that there is a large number of iterations needed to obtain this solution. In the light of this, other stopping criteria were also considered, but they need the same amount of iterations to obtain the same accuracy.
5 Conclusion
In this contribution, the determination of a space-dependent vector source in a thermoelastic system of type-I, type-II and type-III is studied using information from a supplementary measurement at the final time. The uniqueness of a solution is proved when is componentwise strictly monotone increasing. A convergent and stable algorithm is developed to approximate the unknown source when
is also linear. This is demonstrated by numerical experiments.
Acknowledgements
The authors are supported by the IAP P7/02-project of the Belgian Science Policy.
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