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A direct linear inversion for discontinuous elastic parameters recovery from internal displacement information only

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The aim of this paper is to present and analyze a new direct method for solving the linear elasticity inverse problem. Given measurements of some displacement fields inside a medium, we show that a stable reconstruction of elastic parameters is possible, even for discontinuous parameters and without boundary information. We provide a general approach based on the weak definition of the stiffness-to-force operator which conduces to see the problem as a linear system. We prove that in the case of shear modulus reconstruction, we have an \(L^2\) stability with only one measurement under minimal smoothness assumptions. This stability result is obtained through the proof that the linear operator to invert has closed range. We then describe a direct discretization which provides stable reconstructions of both isotropic and anisotropic stiffness tensors.

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A Notations and tools

1.1 A.1 Tensor notations

Definition 5

We denote \({\mathbb {R}}^{d\times d}\) the space of real matrices and \({\mathbb {R}}^{d\times d}_{\text {sym}}\) the space of real symmetric matrices. Notice that \({\mathbb {R}}^{d\times d}_{\text {sym}}\sim {\mathbb {R}}^{d(d+1)/2}\). We denote \(T^4= {\mathbb {R}}^{d^4}\) the space of order 4 real tensors. We have:

  1. (i)

    \(A:B=\sum _{ij}A_{ij}B_{ij}\in {\mathbb {R}}\) for \(A,B\in {\mathbb {R}}^{d\times d}\);

  2. (ii)

    \((A\otimes B)_{ijkl}=A_{ij}B_{kl}\in T^4\) for \(A,B\in {\mathbb {R}}^{d\times d}\);

  3. (iii)

    \((\mathbf {A}:B)_{ij}=\sum _{kl}\mathbf {A}_{ijkl}B_{kl}\in {\mathbb {R}}^{d\times d}\) for \(\mathbf {A}\in T^4\) and \(B\in {\mathbb {R}}^{d\times d}\);

  4. (iv)

    \((B:\mathbf {A})_{ij}=\sum _{kl}B_{kl} \mathbf {A}_{klij}\in {\mathbb {R}}^{d\times d}\) for \(\mathbf {A}\in T^4\) and \(B\in {\mathbb {R}}^{d\times d}\);

  5. (v)

    \((\mathbf {A}:\mathbf {B})_{ijkl}=\sum _{mn}\mathbf {A}_{ijmn}\mathbf {B}_{mnkl}\in T^4\) for \(\mathbf {A},\mathbf {B}\in T^4\);

  6. (vi)

    \(\mathbf {A}|\mathbf {B}=\sum _{ijkl}\mathbf {A}_{ijkl}\mathbf {B}_{ijkl}\in {\mathbb {R}}\) for \(\mathbf {A},\mathbf {B}\in T^4\).

We define \(T^4_{\text {sym}}\) to be the space of all tensors \(\mathbf {T}\) such that for any symmetric matrix \(S\in {\mathbb {R}}^{d\times d}_{\text {sym}}\), the matrix \(\mathbf {T}:S\) is also symmetric and for any antisymmetric matrix A, we have \(\mathbf {T}:A=0\). Remark that in dimension 2, \(T^4_{\text {sym}}\sim {\mathbb {R}}^6\) and in dimension 3, \(T^4_{\text {sym}}\sim {\mathbb {R}}^{21}\).

1.2 A.2 Sobolev spaces

Definition 6

For any Lipschitz domain \({\varOmega }\subset {\mathbb {R}}^d\), we define

$$\begin{aligned} W^{1,p}({\varOmega }):= \left\{ u \in L^p({\varOmega }):\ |\nabla u|\in L^2({\varOmega }) \right\} . \end{aligned}$$

We also define the following space:

$$\begin{aligned} H^1_0({\varOmega },{\mathbb {R}}^d):=\left\{ \mathbf {u} \in L^2({\varOmega },{\mathbb {R}}^{d}):\ |\nabla \mathbf {u}|\in L^2({\varOmega }), \mathbf {u}|_{\partial {\varOmega }} = \mathbf {0}\right\} , \end{aligned}$$

equipped with the norm:

$$\begin{aligned} \left\| {\mathbf {u}}\right\| _{H^1_0({\varOmega })} := \left\| {\nabla ^s\mathbf {u}}\right\| _{L^2({\varOmega })}, \end{aligned}$$

where \(\nabla ^s\mathbf {u}=(\nabla \mathbf {u}+\nabla \mathbf {u}^T)/2\).

Remark 14

The fact that this definition for the norm is correct is a direct consequence of Korn’s inequality and Poincaré’s inequality.

Proposition 6

Properties of \(W^{1,p}({\varOmega })\):  If \(p>d\), the following results hold.

  1. (i)

    \(W^{1,p}({\varOmega }) \hookrightarrow L^{\infty }({\varOmega })\);

  2. (ii)

    \(u,v \in W^{1,p}({\varOmega }) \Rightarrow uv \in W^{1,p}({\varOmega })\);

  3. (iii)

    \(u\in W^{1,p}({\varOmega })\), \(\varphi \in H^1_0({\varOmega }) \Rightarrow u\varphi \in H^1_0({\varOmega })\);

  4. (iv)

    \(u\in W^{1,p}({\varOmega })\), \(f\in H^{-1}({\varOmega })\) implies that \(uf\in H^{-1}({\varOmega })\) and \(\Vert uf\Vert _{H^{-1}}\le C \Vert u \Vert _{W^{1,p}} \Vert f \Vert _{H^{-1}}\) for some constant C independent of u and f.

Lemma 1

(\(\nabla \) has a closed range in \(\{\mu _0\}^\perp \)) Let \({\varOmega }\) be a Lipschitz domain of \({\mathbb {R}}^d\) and \(\mu _0\in L^\infty ({\varOmega })\) be such that \(\mu _0\ge m\ge 0\). Then, there exists a constant \(c>0\) such that

$$\begin{aligned} \forall \mu \in \{\mu _0\}^\perp ,\quad \left\| {\mu }\right\| _{L^2({\varOmega })}\le c\left\| {\nabla \mu }\right\| _{H^{-1}({\varOmega })}. \end{aligned}$$


Suppose that this is false. Take a sequence \((\mu _n)\) such that \(\left\| {\mu _n}\right\| _{L^2({\varOmega })}=1\) and \(\left\| {\nabla \mu _n}\right\| _{H^{-1}({\varOmega })}\rightarrow 0\). Up to an extraction \(\mu _n\overset{L^2({\varOmega })}{\rightharpoonup }\mu \) and \(\int _{\varOmega }\mu _n\mu _0\rightarrow \int _{\varOmega }\mu \mu _0 = 0\). Moreover, \(\nabla \mu = 0\) and so \(\mu \) is constant. Then \(\mu =0\). As the embedding \(L^2({\varOmega })\hookrightarrow H^{-1}({\varOmega })\) is compact, we get that \(\left\| {\mu _n}\right\| _{H^{-1}({\varOmega })}\rightarrow 0\). Saying now that \(\left\| {\mu _n}\right\| _{L^2({\varOmega })}^2=\left\| {\mu _n}\right\| _{H^{-1}({\varOmega })}^2+\left\| {\nabla \mu _n}\right\| _{H^{-1}({\varOmega })}^2\) we arrive at a contradiction. \(\square \)

B Proof of Theorem 3


Denote \({\varSigma }\) the closure of the discontinuity surface of S. The open set \({\varOmega }\backslash {\varSigma }\) can be decomposed as a countable union of connected open sets:

$$\begin{aligned} {\varOmega }\backslash {\varSigma }=\bigcup _{i\in I}{\varOmega }_i. \end{aligned}$$

One may apply Theorem 2 on each subset and say that there exists some \(\nu _i\in {{\mathcal {C}}}^0({\varOmega }_i)\) such that any solution of the problem is written as

$$\begin{aligned} \mu =\sum _{i\in I}\alpha _ie^{\nu _i}\mathbf{1}_{{\varOmega }_i}\quad \text { in }{\varOmega }\backslash {\varSigma }, \end{aligned}$$

where \(\alpha _i\)’s are some real numbers.

We show now that these numbers are linked by the jump condition over \({\varSigma }\). Consider two subdomain \({\varOmega }_i\) and \({\varOmega }_j\) in contact in the sense that their common boundary \({\varSigma }_{ij}:=\partial {\varOmega }_i\cap \partial {\varOmega }_j\) is of positive surface measure: \({{\mathcal {H}}}^{d-1}(\partial {\varOmega }_i\cap \partial {\varOmega }_j)>0\). As \({\varSigma }\) is rectifiable, there exists \(x_0\in {\varSigma }_{ij}\) and \(B:=B(x_0,{\varepsilon })\) such that \({\varOmega }_i^B:={\varOmega }_i\cap B\) and \({\varOmega }_j^B:={\varOmega }_j\cap B\) are Lipschitz domains. As \(\mu \) and S are \(W^{1,p}\) in \({\varOmega }_i^B\) and \({\varOmega }_j^B\), so is the product \(\mu S\) and it admits two-sided traces \(\mu _i S_i\) and \(\mu _j S_j\) defined as functions of \(L^p({\varSigma }_{ij}\cap B)\). From the variational formulation, the jump condition at \({\varSigma }_{ij}\cap B\) reads as

$$\begin{aligned} \mu _i S_i\nu = \mu _j S_j\nu ,\quad \text { almost everywhere on }{\varSigma }_{ij}\cap B. \end{aligned}$$

This jump condition gives a vectorial equation linking \(\alpha _i\) and \(\alpha _j\) which is

$$\begin{aligned} \alpha _i e^{\nu _i}S_i\nu = \alpha _j e^{\nu _j}S_j\nu . \end{aligned}$$

As \(\nu _i\), \(\nu _j\) are bounded in B and \(|\det S_i|\), \(|\det S_j|\ge c>0\), there exists \(c'>0\) such that \(|e^{\nu _i}S_i\nu |\ge c'\) and \(|e^{\nu _j}S_j\nu |\ge c'\). A first consequence is that if one \(\alpha _i=0\) then they are all zero and \(\mu =0\).

Now consider another solution \(\mu '=\sum _{i\in I}\beta _ie^{\nu _i}\mathbf{1}_{{\varOmega }_i}\) and assume that \(\frac{\mu '}{\mu }\) is not constant. There exist \({\varOmega }_i,\ {\varOmega }_j\) in contact such that \({\beta _i}/{\alpha _i}\ne {\beta _j}/{\alpha _j}\). Using (37) for both couples \((\alpha _i,\alpha _j)\) and \((\beta _i,\beta _j)\), it follows that there exists \(\gamma \ne 0\) such that \(\alpha _j=\gamma \alpha _i\) and \(\beta _j=\gamma \beta _i\), which leads to \({\beta _i}/{\alpha _i}= {\beta _j}/{\alpha _j}\). Since this is absurd, \(\mu '/\mu \) is constant.

\(\square \)

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Ammari, H., Bretin, E., Millien, P. et al. A direct linear inversion for discontinuous elastic parameters recovery from internal displacement information only. Numer. Math. 147, 189–226 (2021).

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