1 Introduction and problem statement
The reconstruction of inclusions in materials by nondestructive testing becomes more and more important and opens a wide mathematical
field in inverse problems. The applications cover engineering, geoscientific and medical problems.
This is the reason, why several authors dealt with the inverse problem of linear elasticity in order to recover the Lamé parameters.
For the two dimensional case, we refer the reader to [ikehata1990inversion, nakamura1993identification, imanuvilov2011reconstruction, lin2017boundary]. Further on, in three dimensions, [nakamura1995inverse, nakamura2003global] and [eskin2002inverse]
gave the proof for uniqueness results for both Lamé coefficients under the assumption that is close to a positive constant.
[beretta2014lipschitz, beretta2014uniqueness] proved the uniqueness for partial data, where the Lamé parameters are piecewise
constant and some boundary determination results were shown in [nakamura1999layer, nakamura1995inverse, lin2017boundary].
In this paper, the key issue of the shape reconstruction of inclusions is the monotonicity property of the corresponding NeumanntoDirichlet operator
(see [Tamburrino06, tamburrino2002new]).
These monotonicity properties were also applied for electrical impedance tomography
(see, e.g., [harrach2013monotonicity]) and for linear elasticity in [Eberle_Monotonicity], which is
the basis for our current work. Our approach relies on the monotonicity
of the NeumanntoDirichlet operator with respect to the Lamé parameters and the techniques of localized potentials
[harrach2018helmholtz, harrach2018localizing, harrach2012simultaneous, gebauer2008localized, harrach2009uniqueness].
Thus, we start with the introduction of the problem and summarize the main results from
[Eberle_Monotonicity]. After that, we go over to the shape reconstruction itself, where we consider the standard and linearized monotonicity method.
In doing so, we present and compare numerical experiments from the aforementioned two methods.
We start with the introduction of the problem of interest in the following way.
We consider a bounded and connected open set ( or ), occupied by an isotropic material with linear stressstrain relation,
where and with
are the corresponding Dirichlet and Neumann boundaries. Then the displacement vector
satisfies the boundary value problem(1) 
where , are the Lamé parameters, is the symmetric gradient, is the normal vector pointing outside of , the boundary load and the
For given constants satisfying , , we define the set of admissible Lamé parameters by
The weak formulation of the problem (1) is given by
(2) 
where
We want to remark that in continuum mechanics, the function from the test space can be seen as a virtual displacement,
while the weak formulation (2) itself can be interpreted as the principle of virtual work.
The existence and uniqueness of a solution to the above variational formulation (2) follows from
the LaxMilgram theorem, see e.g., in [Ciarlet].
For the sake of completeness, we state two important inequalities (see e.g., [Friedrichs]) in the framework of elasticity,
which play an essential role in the proof of the aforementioned existence and uniqueness of the solution of the weak formulation.
Korn’s first inequality:
Let be a bounded and connected open set in . Then, there exists a constant such that
Korn’s second inequality:
Let be a bounded and connected open set in . Then there exists a constant such that
Next, we introduce the NeumanntoDirichlet operator by
It is well known that is a selfadjoint compact linear operator. The associated bilinear form is given by
where solves the elastic problem (1) and the corresponding problem with boundary load .
The operator is Frêchet differentiable, which can be proven by similar arguments as in the corresponding proof in [Lec07] for the impedance tomography
problem.
For directions , the derivative
is the selfadjoint compact linear operator associated to the bilinear form so that
Note that for , we obviously have that for
it follows that
Remark 1.1
The inverse problem we consider here is the following:
Find knowing the NeumanntoDirichlet operator .
2 Monotonicity methods
In this section, we introduce two monotonicity methods in order to reconstruct inclusions in elastic bodies. The first method is the standard (or nonlinearized) monotonicity
method and the second the linearized monotonicity method. For both methods, we analyze the monotonicity properties for the NeumanntoDirichlet operator and
formulate the corresponding monotonicity test which we apply for the realization of the numerical experiments.
First, we summarize and present the required results concerning the monotonicity properties.
The details and proofs can be found in [Eberle_Monotonicity]
. We start with the monotonicity estimate and the monotonicity property itself, which is the key issue for our study and will be analyzed later on in detail.
Lemma 2.1 (Lemma 3.1 from [Eberle_Monotonicity])
Let , be an applied boundary load, and let , . Then
(4)  
(5) 
Lemma 2.2
Let , be an applied boundary load, and let , . Then
(6)  
(7) 
Proof We start with a result shown in the poof of Lemma 3.1 in [Eberle_Monotonicity]:
Based on this, we are led to
Lemma 2.2 leads directly to
Corollary 2.1 (Corollary 3.2 from [Eberle_Monotonicity])
For
Based on these results, we can go over to the standard monotonicity method.
2.1 Standard monotonicity method
Our aim is to prove that the opposite direction of Corollary 2.1 holds true in order to formulate the socalled standard monotonicity test
(Corollary 2.2 and 2.3).
Therefore, we consider the case where contains inclusions in which the Lamé parameters and differ from otherwise
known background Lamé parameters.
For the precise formulation, we will now introduce the concept of the inner and the outer support of a measurable function in a similar way as in
[harrach2013monotonicity].
Definition 2.1
A relatively open set is called connected to if is connected and .
Definition 2.2
For a measurable function , we define

the support as the complement (in ) of the union of those relatively open , for which ,

the inner support as the union of those open sets , for which
, 
the outer support as the complement (in ) of the union of those relatively open that are connected to and for which .
For ease of notations, we assume that the background Lamé parameters are equal to . Further on, we consider . Our goal is to determine the inclusion
(8) 
from the knowledge of the NeumanntoDirichlet operator .
Hence, the inclusion in our nomenclature always contains the support of as well as all holes which cannot be connected to the boundary.
Let us consider the setting (8) as depicted in Figure 1. By proving the opposite direction of Corollary 2.1,
we show that can be reconstructed by monotonicity
tests, which simply compare (in the sense of quadratic forms) to the NeumanntoDirichlet
operators of test parameters . To be more precise, the support of
can be reconstructed under the assumption that has a connected complement, in which case we have
(c.f. [harrach2013monotonicity]).
Otherwise, what we can reconstruct is the support of together
with all holes that have no connection to the boundary , i.e, (c.f. [Factorization_Harrach_2013]).
Hence, in this paper we only take a look at inclusions with .
This leads to the formulation of the standard monotonicity test which is implemented in the next part. In the following, we define and as the
contrasts and as well as
as the characteristic function w.r.t. the inclusion
and the socalled test inclusion , respectively.Theorem 2.1
Let and . For every open set (e.g. ball or cube) and every , ,
implies
and
implies
Hence, the set
fulfills
Proof Let and . Let be an open set and , and
We start with Lemma 2.2 and get
This shows that with , , the condition , implies
Hence, we get with Corollary 2.1 that
It remains to show that
Let . Corollary 2.1 states that shrinking the open set only makes larger, so that we can assume without loss of generality that . We can apply Theorem 3.3 from [Eberle_Monotonicity] and obtain a sequence so that the solutions , of
(9) 
fulfill
(10)  
(11)  
(12)  
(13) 
From Lemma 2.1 and Equation (10)(13) it follows with as in system (9), and , where the index represents the reference medium and the index indicates the parameters of the inclusion, that
and hence
In addition, we state the theorem for the case , .
Theorem 2.2
Let and . For every open set (e.g. ball or cube) and every , ,
implies
and
implies
Hence, the set
fulfills
Proof The proof follows the lines of the proof of Theorem 2.1.
Based on this, we introduce the monotonicity tests.
Corollary 2.2
Standard monotonicity test: 1. version
Let ,
with and , where the inclusion is open and has a connected complement.
Further on, let , with , .
Then for every open set
(14) 
Proof Let and , . This means, that the conditions
, of Theorem 2.1 are fulfilled
which immediately leads to .
For the opposite direction we assume that there exits a , which fulfills . By applying the second part of Theorem 2.1 we obtain
which contradicts
Hence, implies that .
Further on, we formulate the corresponding corollary for the case and .
Corollary 2.3
Standard monotonicity test: 2. version
Let ,
with and , where the inclusion is open and has a connected complement.
Further on, let , with , .
Then for every open set
(15) 
Proof The proof follows the lines of the proof of Corollary 2.2 but we have to consider Theorem 2.2, where
,
implies
.
Further on, we use that
implies .
Next, we apply Theorem 2.1 to difference measurements
(16) 
which leads directly to the following lemma.
Lemma 2.3
Let . Under the same assumptions on and as in Theorem 2.1, we have for every open set (e.g. ball or cube) and every , ,
(17) 
implies
(18) 
and
implies
Proof We take a look at the difference .
The assumption , leads via Theorem 2.1 directly to the desired results.
Next, we go over to the consideration of noisy difference measurements
(19) 
with
(20) 
where and formulate a monotonicity test (c.f Corollary 2.2). We have to be aware of the fact that (18) will not hold in general for all . Thus, we have to modify the testing in the following way.
Corollary 2.4
Standard monotonicity test for noisy difference measurements
Let ,
with and , where the inclusion is open and has a connected complement.
Further on, let , with , .
Then for noisy data with
and every open set we mark as inside the inclusion only if
Proof We base our considerations on Remark 3.5 from [harrach2013monotonicity] which deals with the handling of noisy data. Monotonicity tests for noisy difference measurements can be stably implemented in the following sense. Let
(21) 
By replacing by its symmetric part, e.g. by , without loss of generality, we can assume that is selfadjoint. Hence, we have for , that
for all boundary loads . Thus,
holds in the quadratic sense. If , then