Shape Reconstruction in Linear Elasticity: Standard and Linearized Monotonicity Method

In this paper, we deal with the inverse problem of the shape reconstruction of inclusions in elastic bodies. The main idea of this reconstruction is based on the monotonicity property of the Neumann-to-Dirichlet operator presented in a former article of the authors. Thus, we introduce the so-called standard as well as linearized monotonicity tests in order to detect and reconstruct inclusions. In addition, we compare these methods with each other and present several numerical test examples.

• 3 publications
• 8 publications
05/05/2021

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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 Neumann-to-Dirichlet 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 Neumann-to-Dirichlet 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 stress-strain relation, where and with

are the corresponding Dirichlet and Neumann boundaries. Then the displacement vector

satisfies the boundary value problem

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩−μΔu−(λ+μ)∇(∇⋅u)=0inΩ,(λ(∇⋅u)I+2μ^∇u)n=gonΓN,u=0onΓD, (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

 A={(λ,μ):(λ,μ)∈L∞(Ω)×C0,1(Ω),α1≤λ≤α2,β1≤μ≤β2}.

The weak formulation of the problem (1) is given by

 ∫Ω2μ^∇u:^∇v+λ∇⋅u∇⋅vdx=∫ΓNg⋅vdsforallv∈V, (2)

where

 V:={v∈H1(Ω)d:v|ΓD=0}.

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 Lax-Milgram 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

 c∥v∥H1(Ω)d≤∥^∇v∥L2(Ω)d×dforallv∈V.

Korn’s second inequality:
Let be a bounded and connected open set in . Then there exists a constant such that

 c∥v∥H1(Ω)d≤∥^∇v∥L2(Ω)d×d+∥v∥L2(Ω)dforallv∈V.

Next, we introduce the Neumann-to-Dirichlet operator by

 Λ(λ,μ):L2(ΓN)d→L2(ΓN)d:g↦u|ΓN.

It is well known that is a self-adjoint compact linear operator. The associated bilinear form is given by

 ⟨g,Λ(λ,μ)h⟩=∫Ω2μ^∇u(g)(λ,μ):^∇u(h)(λ,μ)+λ∇⋅u(g)(λ,μ)∇⋅u(h)(λ,μ)dx,

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

 Λ′(λ,μ)(^λ,^μ):L2(ΓN)d→L2(ΓN)d

is the self-adjoint compact linear operator associated to the bilinear form so that

 ⟨ Λ′(λ,μ)(^λ,^μ)g,h⟩ =−∫Ω2^μ^∇u(g)(λ,μ):^∇u(h)(λ,μ)+^λ∇⋅u(g)(λ,μ)∇⋅u(h)(λ,μ)dx.

Note that for , we obviously have that for

 ^λ0≤^λ1and^μ0≤^μ1

it follows that

 Λ′(λ,μ)(^λ0,^μ0)≥Λ′(λ,μ)(^λ1,^μ1).
Remark 1.1

The inverse problem we consider here is the following:

Find knowing the Neumann-to-Dirichlet 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 non-linearized) monotonicity method and the second the linearized monotonicity method. For both methods, we analyze the monotonicity properties for the Neumann-to-Dirichlet 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

 ∫Ω2(μ0−μ1)^∇u1:^∇u1+(λ0−λ1)∇⋅u1∇⋅u1dx (4) ≥⟨g,Λ(λ1,μ1)g⟩−⟨g,Λ(λ0,μ0)g⟩ ≥∫Ω2(μ0−μ1)^∇u0:^∇u0+(λ0−λ1)∇⋅u0∇⋅u0dx. (5)
Lemma 2.2

Let , be an applied boundary load, and let , . Then

 ⟨g,Λ(λ1,μ1)g⟩−⟨g,Λ(λ0,μ0)g⟩ (6) ≥∫Ω2(μ1−μ21μ0)^∇u1:^∇u1dx+∫Ω(λ1−λ21λ0)∇⋅u1∇⋅u1dx =∫Ω2μ1μ0(μ0−μ1)^∇u1:^∇u1dx+∫Ωλ1λ0(λ0−λ1)∇⋅u1∇⋅u1dx. (7)

Proof We start with a result shown in the poof of Lemma 3.1 in [Eberle_Monotonicity]:

 ⟨g,Λ(λ1,μ1)g⟩−⟨g,Λ(λ0,μ0)g⟩ =∫Ω2μ1^∇(u1−u0):^∇(u1−u0)+λ1∇⋅(u1−u0)∇⋅(u1−u0) +2(μ0−μ1)^∇u0:^∇u0+(λ0−λ1)∇⋅u0∇⋅u0dx.

Based on this, we are led to

 ⟨g,Λ(λ1,μ1)g⟩−⟨g,Λ(λ0,μ0)g⟩ =∫Ω2μ1^∇u1:^∇u1−4μ1^∇u1:^∇u0+2μ1^∇u0:^∇u0+λ1∇⋅u1∇⋅u1 −2λ1∇⋅u1∇⋅u0+λ1∇⋅u0∇⋅u0+2μ0^∇u0:^∇u0+λ0∇⋅u0∇⋅u0 −2μ1^∇u0:^∇u0−λ1∇⋅u0∇¸⋅u0dx =∫Ω2μ1∣∣^∇u1∣∣2−4μ1^∇u1:^∇u0+λ1|∇⋅u1|2−2λ1∇⋅u1∇⋅u0 +2μ0∣∣^∇u0∣∣2+λ0|∇⋅u0|2dx =∫Ω2μ0∣∣^∇u0−μ1μ0^∇u1∣∣2+λ0∣∣∣∇⋅u0−λ1λ0∇⋅u1∣∣∣2 +(2μ1−2μ21μ0)∣∣^∇u1∣∣2+(λ1−λ21λ0)|∇⋅u1|2dx ≥∫Ω2(μ1−μ21μ0)∣∣^∇u1∣∣2+(λ1−λ21λ0)|∇⋅u1|2dx.

Corollary 2.1 (Corollary 3.2 from [Eberle_Monotonicity])

For

 λ0≤λ1andμ0≤μ1impliesΛ(λ0,μ0)≥Λ(λ1,μ1).

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 so-called 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

 D:=out∂Ωsupp(φ) (8)

from the knowledge of the Neumann-to-Dirichlet 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 Neumann-to-Dirichlet 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 so-called test inclusion , respectively.

Theorem 2.1

Let and . For every open set (e.g. ball or cube) and every , ,

 αχB≤λ−λ0,βχB≤μ−μ0

implies

 Λ(λ0+αχB,μ0+βχB)≥Λ(λ,μ)

and

 B⊈D

implies

 Λ(λ0+αχB,μ0+βχB)≱Λ(λ,μ).

Hence, the set

 R:=⋃α,β≥0,α+β>0{B⊆Ω:Λ(λ0+αχB,μ0+βχB)≥Λ(λ,μ)},

fulfills

 innsupp((λ−λ0,μ−μ0)T)⊆R⊆D.

Proof Let and . Let be an open set and , and

 αχB≤λ−λ0,βχB≤μ−μ0.

 ⟨g,Λ(λ0+αχB,μ0+βχB)g⟩−⟨g,Λ(λ,μ)g⟩ ≥∫Ω2μ0+βχBμ(μ−μ0−βχB)^∇u(λ0+αχB,μ0+βχB):^∇u(λ0+αχB,μ0+βχB) +λ0+αχBλ(λ−λ0−αχB)∇⋅u(λ0+αχB,μ0+βχB)∇⋅u(λ0+αχB,μ0+βχB)dx.

This shows that with , , the condition , implies

 ⟨g,Λ(λ0+αχB,μ0+βχB)g⟩−⟨g,Λ(λ,μ)g⟩≥0.

Hence, we get with Corollary 2.1 that

 Λ(λ0+αχB,μ0+βχB)≥Λ(λ,μ).

It remains to show that

 B⊈DimpliesΛ(λ0+αχB,μ0+βχB)≱Λ(λ,μ).

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

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩−(μ0+βχB)Δu(gn)−((λ0+αχB)+(μ0+βχB))∇(∇⋅u(gn))=0inΩ,((λ0+αχB)(∇⋅u(gn))I+2(μ0+βχB)^∇u(gn))n=gnonΓN,u(gn)=0onΓD, (9)

fulfill

 limn→∞∫B^∇u(gn):^∇u(gn)dx=∞, (10) limn→∞∫D^∇u(gn):^∇u(gn)dx=0, (11) limn→∞∫B∇⋅u(gn)∇⋅u(gn)dx=∞, (12) limn→∞∫D∇⋅u(gn)∇⋅u(gn)dx=0. (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

 ⟨gn,Λ(λ0+αχB,μ0+βχB)−Λ(λ,μ)gn⟩ =⟨gn,Λ(λ0+αχB,μ0+βχB)gn⟩−⟨gn,Λ(λ,μ)gn⟩ ≤∫Ω2(μ0+(μ1−μ0)χD)^∇u(gn):^∇u(gn) +(λ0+(λ1−λ0)χD)∇⋅u(gn)∇⋅u(gn)dx −∫Ω2(μ0+βχB)^∇u(gn):^∇u(gn)+(λ0+αχB)∇⋅u(gn)∇⋅u(gn)dx =∫Ω2(μ1−μ0)χD^∇u(gn):^∇u(gn)+(λ1−λ0)χD∇⋅u(gn)∇⋅u(gn) −2βχB^∇u(gn):^∇u(gn)−αχB∇⋅u(gn)∇⋅u(gn)dx
 =−∫B2β^∇u(gn):^∇u(gn)+α∇⋅u(gn)∇⋅u(gn)dx +∫D2(μ−μ0)^∇u(gn):^∇u(gn)+(λ−λ0)∇⋅u(gn)∇⋅u(gn)dx →−∞,

and hence

 Λ(λ0+αχB,μ0+βχB)≱Λ(λ,μ).

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 , ,

 αχB≤λ0−λ,βχB≤μ0−μ

implies

 Λ(λ0−αχB,μ0−βχB)≤Λ(λ,μ)

and

 B⊈D

implies

 Λ(λ0−αχB,μ0−βχB)≰Λ(λ,μ).

Hence, the set

 R:=⋃α,β≥0,α+β>0{B⊆Ω:Λ(λ0−αχB,μ0−βχB)≤Λ(λ,μ)},

fulfills

 innsupp((λ0−λ,μ0−μ)T)⊆R⊆D.

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

 B⊆DifandonlyifΛ(λ0+αχB,μ0+βχB)≥Λ(λ,μ). (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

 B⊆DifandonlyifΛ(λ0−αχB,μ0−βχB)≤Λ(λ,μ). (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

 ΛD=Λ(λ0,μ0)−Λ(λ,μ) (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 , ,

 αχB≤λ−λ0,βχB≤μ−μ0 (17)

implies

 ΛD−ΛB≥0 (18)

and

 B⊈D

implies

 ΛD−ΛB≱0.

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

 Λδ≈ΛD (19)

with

 ∥Λδ−ΛD∥≤δ, (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

 ∥Λδ−ΛD∥≤δ,

and every open set we mark as inside the inclusion only if

 Λδ−ΛB+δI≥0forδ>0sufficientlysmall.

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

 ∥Λδ−ΛD∥≤δ. (21)

By replacing by its symmetric part, e.g. by , without loss of generality, we can assume that is self-adjoint. Hence, we have for , that

 ⟨g,(Λδ−ΛB)g⟩≥−δ∥g∥L2(ΓN)d

for all boundary loads . Thus,

 Λδ−ΛB≥−δI

holds in the quadratic sense. If , then