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

Abstract

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.

Appendices

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:

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

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

• (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 $B\in {\mathbb {R}}^{d\times d}$;

• (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 $B\in {\mathbb {R}}^{d\times d}$;

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

• (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.

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

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

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

• (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

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

Proof

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

Proof

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

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

(37)

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.

$\square$