Abstract
We investigate the numerical reconstruction of smooth star-shaped voids (rigid inclusions and cavities) which are compactly contained in a three-dimensional isotropic linear elastic medium from a single set of Cauchy data (i.e. nondestructive boundary displacement and traction measurements) on the accessible outer boundary. This inverse geometric problem in three-dimensional elasticity is approximated using the method of fundamental solutions (MFS). The parameters describing the boundary of the unknown void, its centre, and the contraction and dilation factors employed for selecting the fictitious surfaces where the MFS sources are to be positioned, are taken as unknowns of the problem. In this way, the original inverse geometric problem is reduced to finding the minimum of a nonlinear least-squares functional that measures the difference between the given and computed data, penalized with respect to both the MFS constants and the derivative of the radial coordinates describing the position of the star-shaped void. The interior source points are anchored and move with the void during the iterative reconstruction procedure. The feasibility of this new method is illustrated in several numerical examples.
Introduction
In direct problems in solid mechanics, one has to determine the response of a system when the governing partial differential equations (equilibrium equations), the constitutive and kinematics equations, the initial and boundary conditions for the displacement and/or traction vectors and the geometry of the domain occupied by the solid are all known. However, if at least one of the above conditions is partially or entirely lacking, then one has a so-called inverse problem. Moreover, it is well-known that inverse problems are in general unstable, in the sense that small measurement errors in the input data may amplify significantly the errors in the solution, see e.g. [16] . Such inverse problems have been extensively studied, both theoretically and numerically, over the last three decades and an overview of these developments can be found in [10] .
Introduction
In the case of inverse geometric problems in solid mechanics, which represent an important subclass of inverse problems, the geometry of the domain occupied by the solid is not entirely known, however some additional information is available. More specifically, part of the boundary of the solution domain is not known but either the displacements or the tractions are known on this portion, whilst the remaining boundary is known and both displacement and traction measurements are available on it. The inverse geometric problems described above can be further subdivided into two subclasses, depending on the location of the unknown boundary, namely (i) identification of an unknown boundary or corrosion-type problems (the unknown boundary is a part of the outer boundary of the solution domain), see e.g. [27] [28] [29] , and (ii) identification of voids, i.e. cavities and rigid inclusions (the unknown boundary is an inner boundary), see e.g. [12, [21] [22] [23] .
Introduction
There are important studies that are devoted to the latter subclass of inverse geometric problems in elasticity.
Introduction
Alessandrini et al. [1, 2] proved that the volume (size) of a rigid inclusion in an elastic isotropic body can be estimated by an easily expressed quantity related to work, depending only on the boundary traction and displacement. Morassi and Rosset [33] provided upper and lower bounds on the size of unknown defects, such as cavities or rigid inclusions, in an elastic body, from boundary measurements of tractions and displacements.
Introduction
Later, they considered the inverse problem of determining a rigid inclusion inside an isotropic elastic body from a single set of Cauchy data on the outer boundary and proved its uniqueness and conditional stability [34] . The issue of uniqueness in determining cavities in a heterogeneous isotropic elastic medium in two dimensions was investigated by Ang et al. [4] , who used the unique continuation for the isotropic Lamé system and geometric considerations. Ben Ameur et al. [8] developed a rather general approach for identifiability and local Lipschitz stability of cavities in two and three spatial dimensions in linear elasticity and thermo-elasticity. Ikehata and Itou [19] considered the reconstruction problem of an unknown polygonal cavity in a homogeneous isotropic elastic body and provided an extraction formula of the convex hull of the cavity using the enclosure method.
Introduction
With respect to the numerical identification of voids in elasticity, most of the studies available in the literature are devoted to the two-dimensional case. A regularized boundary integral formulation for the detection of flaws in planar structural membranes from the displacement measurements given at some boundary locations and the applied loading was proposed in [9] . Hsieh and Mura [18] developed a combined boundary element method (BEM)-nonlinear optimization algorithm for the detection of both the location and the shape of an unknown cavity in an elastic medium. Mellings and Aliabadi [30] presented a dual boundary element formulation for the identification of the location and size of internal flaws in two-dimensional structures. Kassab et al. [24] and Ulrich et al. [37] investigated the non-destructive detection of internal cavities in the inverse elastostatic problem using the BEM. The level set method and a regularization technique related to the perimeter of the unknown inclusion were employed by Ben Ameur et al. [7] for the numerical reconstruction of a void from a single Cauchy data. We finally mention that some three-dimensional elastodynamic inverse problems have been solved using the BEM in [6, 11] .
Introduction
In recent years the method of fundamental solutions (MFS), originally proposed by Kupradze and Aleksidze [26] and introduced as a numerical method by Mathon and Johnston [31] , has been used extensively for the numerical solution of inverse and related problems primarily due to its ease of implementation. An extensive survey of the applications of the MFS to inverse problems is provided in [20] . It appears that the MFS was used for the first time for the solution of inverse geometric problems in linear elasticity by Alves and Martins [3] , who adapted to the detection of rigid inclusions or cavities in an elastic body the method of Kirsch and Kress [25] .
Introduction
The method of [3] decomposes the inverse problem into a linear and ill-posed part in which a Cauchy problem is solved using the MFS and a nonlinear part in which the unknown boundary of the void is sought as a zero level set for a rigid inclusion (or computed iteratively, in an optimization scheme for a class of approximating shapes, for a cavity). In contrast to this, Karageorghis et al. [21] adopted a fully nonlinear MFS in which the nonlinear and ill-posed parts are dealt with simultaneously using a nonlinear regularized minimization. The reconstructions obtained using this latter method are more accurate than those obtained by decomposition methods, see e.g. [36] .
Introduction
The purpose of this paper is to extend to three-dimensional elasticity the two-dimensional analysis of [21] , the same way we have done for the harmonic scalar case in [22, 23] . In particular, we extend the work of [23] to threedimensional inverse geometric problems, see also [12] . The paper is organized as follows: Section 2 is devoted to the mathematical formulation of the inverse geometric problem investigated. The MFS discretization for this problem is described in Section 3, while the implementational details are given in Section 4. In Section 5, we investigate four examples. Finally, some concluding remarks and possible future work are provided in Section 6.
Introduction
2 The Cauchy-Navier equations of elasticity
The problem
We consider the boundary value problem in a bounded domain Ω ⊂ R 3 for the Cauchy-Navier system of elasticity for the displacement u in the form (see e.g. [17] )
The problem
where µ > 0 is the shear modulus and ν ∈ (0, 1/2) is the Poisson ratio, subject to the Dirichlet boundary
The problem
and the homogeneous boundary conditions
The problem
where α is 0 or 1. The inverse problem we are concerned with consists of determining not only the displacement u, but also the unknown inclusion Ω 1 so that u satisfies the Cauchy-Navier equations (1a), given the Dirichlet data f in (1b), the homogeneous boundary condition (1c) and the Neumann traction measurements
The problem
In the above, Ω = Ω 2 \Ω 1 , where Ω 1 ⊂ Ω 2 , is a bounded annular domain with boundary ∂Ω = ∂Ω 1 ∪ ∂Ω 2 .
The problem
The void Ω 1 may have many connected components, but Ω should be connected. Equation (1c), covers both homogeneous Dirichlet (α = 1, i.e. Ω 1 is a rigid inclusion) and Neumann (α = 0, i.e. Ω 1 is a cavity) boundary conditions on ∂Ω 1 . In (1c), t represents the traction defined by
The problem
In (2), the outward normal unit vector to the boundary at the point (x 1 , x 2 , x 3 ) is denoted by n(x 1 , x 2 , x 3 ) = (n x1 , n x2 , n x3 ), whilst σ is the stress tensor given, in terms of the strain tensor ε = ( ∇u + (∇u) T ) /2, by Hooke's law [17] , namely
The problem
where I is the 3 × 3 identity matrix.
The problem
If the Dirichlet and Neumann data (1b) and (1d) are not identically zero, then the uniqueness of the solution pair (u, Ω 1 ) of the inverse problem (1a)-(1d) holds, see [3] .
The method of fundamental solutions (MFS)
In the application of the MFS to (1), we seek an approximation to the solution of the three-dimensional CauchyNavier equations of elasticity as a linear combination of fundamental solutions in the form [35] u
The method of fundamental solutions (MFS)
where
The method of fundamental solutions (MFS)
is the fundamental solution matrix for the displacement vector in threedimensional isotropic linear elasticity given by
The method of fundamental solutions (MFS)
and the vectors a
The method of fundamental solutions (MFS)
contain the unknown MFS coefficients. Similarly, from (2), (4) and (5), the tractions are approximated by [5] 
The method of fundamental solutions (MFS)
where
The method of fundamental solutions (MFS)
is the fundamental solution matrix for the traction vector in three-dimensional isotropic linear elasticity, whose components are given by
The method of fundamental solutions (MFS)
The sources (ξ
The method of fundamental solutions (MFS)
. In particular, the sources (ξ
The method of fundamental solutions (MFS)
to ∂Ω 2 . In the MFS, taking the pseudo-boundary similar to the boundary yields, in general, improved results as has been demonstrated by Gorzelańczyk and Kołodziej [15] . In (4), the singularities (ξ Without loss of generality, we shall assume that the (known) fixed exterior boundary ∂Ω 2 is a sphere of radius R. As a result, the outer boundary collocation and source points are chosen as
The method of fundamental solutions (MFS)
respectively, wherẽ
The method of fundamental solutions (MFS)
and
The method of fundamental solutions (MFS)
and the (unknown) dilation parameter η ext ∈ (1, S), with S > 1 prescribed.
The method of fundamental solutions (MFS)
We further assume that the unknown boundary ∂Ω 1 is a smooth, star-like surface with respect to its centre which has unknown coordinates (X, Y, Z). This means that its equation in spherical coordinates can be written
The method of fundamental solutions (MFS)
where r is a smooth function with values in (0, R). The discretised form of (10) for ∂Ω 1 becomes
The method of fundamental solutions (MFS)
and we choose the inner boundary collocation and source points as
The method of fundamental solutions (MFS)
where the (unknown) contraction parameter η int ∈ (0, 1).
Implementational details
The coefficients
Implementational details
in (4), the radii r = (r n,m ) n=1,N ,m=1,M ∈ (0, R) in (11), the contraction and dilation coefficients η int ∈ (0, 1) and η ext ∈ (1, S)
Implementational details
in (13) and (9), and the coordinates of the centre C = (X, Y, Z) so that
Implementational details
by imposing the boundary conditions (1b), (1c) and (1d) in a regularized least-squares sense. This leads to the minimization of the functional
Implementational details
where λ 1 , λ 2 ≥ 0 are regularization parameters to be prescribed and η = [η int , η ext ].
Implementational details
Remarks.
Implementational details
(i) The Dirichlet data (1b) and the traction data (1d) come from practical measurements which are inherently contaminated with noisy errors, and that is why we have replaced f and g by f
Implementational details
] T , respectively, where, in computation, the noisy data are generated as
Implementational details
where p f and p g represent the percentage of noise added to the Dirichlet and Neumann boundary data on ∂Ω 2 , respectively, and ρ m,n is a pseudo-random noisy variable drawn from a uniform distribution in [−1, 1] using the MATLAB c ⃝ command -1+2*rand(1, M N ). In our numerical experiments it was observed that the effect of noise added to the Dirichlet boundary data was similar to that of perturbing the Neumann data. As a result in the numerical results section we only present results for noisy Neumann data, i.e.
Implementational details
p g ̸ = 0 and p f = 0. In Section 5 we shall re-denote p g by p.
Implementational details
(ii) For ∂Ω 2 a sphere, the outward normal vector n is defined as follows:
Implementational details
In the case of the boundary ∂Ω 1 , we know that the position vector of a boundary point is given by
Implementational details
and that the normal to the parametrised surface is given by
Implementational details
where the subscripts ϑ and ϕ denote the partial derivatives with respect to ϑ and ϕ, respectively. Now, and thus
Implementational details
[ −r ϕ sin ϕ + r ϑ sin ϑ cos ϑ cos ϕ − r sin 2 ϑ cos ϕ, r ϕ cos ϕ + r ϑ sin ϑ cos ϑ sin ϕ − r sin 2 ϑ sin ϕ, − sin ϑ(r ϑ sin ϑ + r cos ϑ) ] , and
Implementational details
[ −r ϕ sin ϕ + r ϑ sin ϑ cos ϑ cos ϕ − r sin 2 ϑ cos ϕ, r ϕ cos ϕ + r ϑ sin ϑ cos ϑ sin ϕ − r sin 2 ϑ sin ϕ, − sin ϑ(r ϑ sin ϑ + r cos ϑ) ] on ∂Ω 1 .
Implementational details
As a result, in the expressions for the tractions (6) the normal derivatives are given by (16) and (20) for x ∈ ∂Ω 2 and x ∈ ∂Ω 1 , respectively. In (20), we use the finite-difference approximations
Implementational details
with the convention that r n,M +1 = r n,1 , r n,0 = r n,M , and (iv) Since the inverse problem is ill-posed, in (14) , the regularization terms
Implementational details
are added in order to achieve the stability of the numerical MFS solution u N M and the smooth boundary ∂Ω 1 . We do not include regularization terms λ 3 |η| 2 and λ 4 |C| 2 since both η and C only have a small number of components and the numerical solution is expected to be stable in both η and C.
Non-linear minimization
The minimization of functional (14) is carried out using the MATLAB c ⃝ [32] optimization toolbox routine lsqnonlin which solves nonlinear least squares problems. This routine by default uses the so-called trust-regionreflective algorithm based on the interior-reflective Newton method [13, 14] , and terminates when (i) the change in the solution vector is less than a specified tolerance, or (ii) the change in the residual is less than a specified tolerance, or (iii) the specified number of iterations or number of function evaluations is exceeded. The routine lsqnonlin does not require the user to provide the gradient and, in addition, it offers the option of imposing lower and upper bounds on the elements of the vector of unknowns (a 1 , a 2 , b 1 , b 2 , c 1 , c 2 , r, η, C) through the vectors lb and up. We can thus easily impose the constraints 0 < r n,m < 1, n = 1, N , m = 1, M , 0 < η int < 1, 1 < η ext < S and −R < X < R, −R < Y < R, −R < Z < R . In our numerical experiments we choose S = 6. Moreover, we choose the initial guess vector of unknowns (a 
Numerical examples
In all numerical examples considered in this section we took µ = 1 and ν = 0.3. In all figures presented the reconstructed values of r, i.e. the numerically reconstructed object Ω 1 , are presented as red colour dots.
Example 1 (Rigid inclusion)
We consider an example in a hollow sphere domain Ω = Ω 2 \Ω 1 , where
Example 1 (Rigid inclusion)
We consider the following exact solutions for the displacements
Example 1 (Rigid inclusion)
where A and B are constants chosen such that u 1 = u 2 = u 3 = 0, on the inner boundary. We choose A = 1 and B = −0.125 so that the internal sphere has radius r int = 0.5.
Example 1 (Rigid inclusion)
In Figures 1-3 In Figure 4 we present the variation of E r for M = N = 6, M = N = 8, M = N = 8, M = N = 10, M = N = 10, M = N = 12, respectively, for various numbers of iterations.
Example 2 (Cavity)
We consider again the domain Ω = Ω 2 \Ω 1 where Ω 2 and Ω 1 are spheres defined by (23) and (24), respectively.
Example 2 (Cavity)
The exact solution has the form (25) but now, in order to have t 1 = t 2 = t 3 = 0, on the inner boundary we need to choose the constants A and B differently. In particular, we choose A = 1 and B = 0.0625(1 + ν)/(1 − 2ν) so that the internal sphere has radius r int = 0.5.
Example 2 (Cavity)
In 
Example 3
We next consider the case R = 1, α = 1 and the rigid inclusion Ω 1 has an acorn shape [23, 36] described parametrically by
Example 3
and Ω 2 is the unit sphere. The Dirichlet data on ∂Ω 2 is taken as
Example 3
Since in this case no analytical solution is available, the Neumann data (1d) is numerically simulated by solving the direct Dirichlet well-posed problem given by equations (1a), (1b), and (1c) with α = 1, when ∂Ω 1 is given by (26) , using the MFS with M = N = 20, M = N = 16. In order to avoid committing an inverse crime, the inverse solver is applied using M = N = 8, M = N = 10.
Example 3
In Figures 9 and 10 we present the results obtained with X = Y = Z = 0.1, no noise and p = 5% noise, respectively, no regularization with M = N = 8, M = N = 10, for various numbers of iterations, as well as the correct shape (26) to be reconstructed. From Figure 9 it can be seen that for exact data, accurate retrievals of the acorn shape (26) are obtained as the number of iterations increase, but this conclusion is not maintained for noisy data, see Figure 10 . This is because, in the absence of regularization, the obtained numerical solution will amplify the small noise with which the input data is contaminated, thus becoming unstable. In order to deal with this instability, regularization should be included.
Example 3
In Figures 11 and 12 we present the results obtained with noise p = 5% after 100 iterations, and regularization with λ 1 (with λ 2 = 0) and λ 2 (with λ 1 = 0), respectively. From Figure 11 it can be seen that regularization with λ 1 is not very effective and better results are obtained for regularization with λ 2 , in particular when λ 2 is about 10 −1 , see Figure 12 .
Example 3
Similar results have been obtained for reconstructing an acorn shape cavity, i.e. α = 0, and are therefore not presented.
Example 4
We finally consider the case R = 1, α = 1 and the rigid inclusion Ω 1 is a pinched ball [23, 36] described parametrically by
Example 4
and Ω 2 is the unit sphere. The Dirichlet data on ∂Ω 2 is taken as in (27) .
Example 4
Since in this case no analytical solution is available, the Neumann data (1d) is numerically simulated by solving the direct Dirichlet well-posed problem given by equations (1a), (1b), and (1c) with α = 1, when ∂Ω 1 is given by (28) , using the MFS with M = N = M = N = 20. In order to avoid committing an inverse crime, the inverse solver is applied using M = N = 8, M = N = 10.
Example 4
In Figures 13 and 14 In Figures 15 and 16 we present the results obtained with noise p = 5% after 100 iterations, and regularization with λ 1 (with λ 2 = 0) and λ 2 (with λ 1 = 0), respectively. Finally, in Figure 17 we present the corresponding Lcurves with λ 1 and λ 2 regularizations for noise p = 5% after 100 iterations. The L-curve corner for regularization in λ 1 is not consistent with the best results which appear just above λ 1 = 10 −1 in Figure 15 whereas the L-curve corner for regularization in λ 2 is consistent with the the best results which appear around λ 2 = 10 −1 in Figure   16 .
Conclusions
The main features of this work can be summarized as follows:
Conclusions
• This is the first time a three-dimensional inverse geometric problem in elasticity has been considered using the MFS.
Conclusions
• We get the full benefits of the MFS since we are dealing with a nonlinear problem in three dimensions and in complex geometries. Here, the meshlessness and the boundary nature of the method and their concomitant ease of implementation become important.
Conclusions
• The numerical results indicate that the numerical method is accurate (for no noise) and stable with respect to noise added in the input data.
Conclusions
• Accurate results are obtained for relatively few degrees of freedom.
Conclusions
• The dynamic approach of the MFS has been used. It is well-suited for such nonlinear problems since, in addition to the parameters determining the shape of the sought void, we simultaneously determine the location of the unknown pseudo-boundaries as well as the coordinates of the centre of the object.
Conclusions
• The MFS implementations to three-dimensional isotropic linear thermo-elasticity, as well as to two-and three-dimensional anisotropic linear elasticity, of the corresponding inverse void problems are deferred to a future work. 

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