## Abstract

A modified error in the constitutive equation-based approach for identification of heterogeneous and linear anisotropic elastic parameters involving static measurements is proposed and explored. Following an alternating minimization procedure associated with the underlying optimization problem, the new strategy results in an explicit material parameter update formula for general anisotropic material. This immediately allows us to derive the necessary constraints on measured data and thus restrictions on physical experimentation to achieve the desired reconstruction. We consider a few common materials to derive such conditions. Then, we exploit the invariant relationships of the anisotropic constitutive tensor to propose an identification procedure for space-dependent material orientations. Finally, we assess the numerical efficacy of the developed tools against a few parameter identification problems of engineering interest.

## 1. Introduction

The material parameter estimation problem typically aims to determine unknown coefficient (parameter) functions from an imperfectly known system response. Numerical solutions of such problems have great practical significance in diverse engineering applications, namely seismic imaging, biomechanical imaging, material characterization and structural health monitoring. However, the available numerical algorithms are often challenged by the inherent non-uniqueness and ill-posedness of the inverse problem. As a result, a great number of estimation methodologies have already been developed and reported in the scientific literature. In the case of elastic parameter estimation, comprehensive reviews on different identification techniques can be found in [1–3]. While the identification techniques of isotropic elastic parameters are relatively well established [4–9], anisotropic constitutive parameter estimation from measured displacement, mode shapes or strain data is difficult owing to the increased number of material parameters and is comparatively less well addressed in the scientific literature. One of the major difficulties for the anisotropic parameter estimation problem is the generation of parameter-sensitive measurements for realistic reconstruction. Thus, the number of effective physical tests to be performed to identify full anisotropy is a still valid question to be answered. Additionally, if the material heterogeneity is considered, then multiple tests and strain heterogeneity in measurements represent a paradigm shift in the identification problem.

With the advancement of full-field measurement technology, several attempts have been made to identify homogeneous anisotropic material parameters. Bruno *et al.* [10] and Bruno & Poggialini [11] described the experimental procedures of characterization of isotropic and anisotropic plates by using full-field measurement under flexural loading conditions. Genovese *et al.* [12] suggested a novel hybrid procedure for the identification of in-plane material properties of woven reinforced fibreglass–epoxy laminate by considering full-field measurements from speckle interferometry. Lecompte *et al.* [13] have suggested a numerical–experimental characterization procedure of the orthotropic material properties of a composite plate performing a single biaxial tensile test on cruciform specimens. An open hole tensile test is also proposed in [14] to identify in-plane orthotropic stiffness of the material. Vibration signature-based orthotropic material parameter identification is proposed in [15,16]. A virtual field method (VFM) is also applied to identify homogeneous anisotropic material parameters [17–20]. Nigamaa & Subramanian [21] used the eigenfunction virtual fields method (EVFM) to determine orthotropic material properties from full-field measurement. Also, a parameter identification technique based on the VFM has been suggested by Rahmani *et al.* [22], who considered a regularization term induced from a micromechanical model of composite lamina. VFM is also used to identify thermomechanical material parameters [23].

Estimation of heterogeneous anisotropic elastic parameters is considerably less well addressed than the homogeneous case. This is mainly owing to the large number of inverse unknowns to be tackled in an ill-posed optimization framework. Generally, the displacement gap functional under the least-squares (*L*_{2}) framework with gradient-based optimizations are used. Liu *et al.* [24] identified full anisotropic constants in a heterogeneous medium from computed tomography data involving specific knowledge about the exterior and interior subdomain boundaries. They assumed that the displacement information was available at all domain boundaries (interior and exterior) and the force information was available at a part of the exterior boundary. Estimation of inhomogeneous and anisotropic material properties of saccular cerebral aneurysms by inverse reconstruction has been suggested by Kroon & Holzapfel [25]. More recently, Genovese *et al.* [26] developed a method for the inverse characterization of the heterogeneous elastic properties of soft tissues of gallbladder using full-field (digital image correlation) measurement. A direct identification technique is also proposed in [27]. Shore *et al.* [28] identified transverse anisotropic properties of cancellous bone under an *L*_{2} framework with a semi-norm-based regularization functional. However, the main disadvantages of the *L*_{2} functional are its sensitivity to the initial guess, and the large number of iterations that are required to obtain an acceptable solution when quasi-Newtonian methods are used. In recent work, Bal *et al.* [29] proposed a direct reconstruction strategy of a fully anisotropic elastic tensor from full-field displacement measurements exploiting the equilibrium elasticity equation and derived explicit update formulae. Detailed discussions have been provided on the stability, uniqueness and positivity constraints of the reconstructed material parameters.

In recent years, another technique for elastic parameter estimation was proposed based on the concept of error in the constitutive equation (ECE). The ECE concept was first introduced by Ladevèze *et al.* [30] for *a posteriori* error estimation in finite-element computations. Over the last two decades, the ECE-based technique has been explored to recover material parameters from the frequency domain response [31,32], the linear transient response [33], the linear static response [34–37] and the elastoplastic response [38,39]. In [8], a procedure based on a modified error in the constitutive equation (MECE) functional is proposed that identifies linear elastic isotropic material parameters with very few iterations when compared with the standard displacement discrepancy-based least-squares approach. This approach has also been explored recently to identify heterogeneous equivalent stiffnesses of composite plates from free vibration signatures [40].

Our present objective is to consider the MECE-based optimization approach to estimate spatially heterogeneous general anisotropic elastic parameters from a measured partial/full-field quasi-static response. However, a straightforward extension of the MECE-based technique, as proposed in [8], for the full anisotropic constitutive tensor is difficult owing to an increased number of material constants. In particular, material parameter update equations are nonlinear with the standard ECE functional for an anisotropic material (owing to the

The rest of the paper is organized as follows. After briefly introducing the material parameter estimation problem in §2, the MECE method is detailed. In §3, instead of using standard ECE-based minimization, we have used the trace norm of the constitutive discrepancy to propose an explicit update equation for a general anisotropic material. As an example, we show the use of this new explicit update procedure via a two-dimensional orthotropic material. Here, we exploit the invariant relationships of the anisotropic constitutive tensor to propose an identification procedure for space-dependent material orientations. In §4, we then consider the solvability of the developed update procedure to derive restrictions on the measured data for physically meaningful reconstructions of some common anisotropic materials. We present a short discussion on the possible choices of the penalty parameter in §5. Numerical experimentations are considered in §6 based on the proposed procedure. Finally, conclusions are drawn in §7.

## 2. Brief background

We are interested in the inverse problem associated with linear elasticity, which comprises estimating the spatial distribution of the constitutive tensor (*N*_{M} is the number of sets of available measurements and each *u*_{i} is governed by the forward elasticity equations as follows:
*a*
*b*
*c*
*d*

where *σ*_{i} denotes the stress tensor, ** ϵ**[

*u*_{i}] is the strain tensor,

*b*_{i}is the body force,

*n*_{i}denotes the outward unit normal. All the quantities defined above are for each separate loading case.

*Λ*) can also be constructed depending upon the measured quantities. However, the main objective of the inverse approach remains in determining (numerically) the unknown implicit relationship between the measured quantities and the material parameters. The approximate solution of the inverse problem is obtained through the following PDE constrained nonlinear optimization

## 3. Elastic parameter estimation: modified error in the constitutive equation approach

The ECE approach is based on a cost functional that measures the discrepancy between a given strain field and a given stress field, subject to different admissibility constraints, via a constitutive equation in terms of an energy norm [30,32]. For linear elastic materials, it can be given as
*κ*_{i} is a penalization parameter. Basically, the MECE functional is the sum of two errors: (i) the error in the constitutive equation and (ii) the error in the measurement. An important point to note here is that the constitutive discrepancy arises as different admissibilities are imposed on stresses and displacements. Also, the motive for using the penalty term in the above cost functional is to constrain the kinematically admissible displacement field such that it satisfies the given measurements approximately. Now, an optimization problem is cast to obtain the updated material parameter by minimizing the above discrepancy, given by
*κ*_{i}) acts as a regularizer (strictly speaking the inverse of the regularization parameter) that controls the magnitude of the data discrepancy to be imposed in the inverse estimation problem [8]. In contrast, in the *L*_{2}-based approach regularization is done by using a particular regularization functional based on identifiable parameters as shown in equation (2.1).

### (a) An alternating solution approach: coupled problem

To solve the MECE-based optimization problem, generally an alternating minimization approach is followed after setting up the appropriate Lagrangian corresponding to the cost functional [33,37]. This results in the formation of a coupled system with the primary and Lagrange variables. Now, the Lagrangian functional in conjunction with the MECE cost functional equation (3.2) can be written as
*u*_{i} and *w*_{i} are defined as
** w_{i}** plays the role of the Lagrange multiplier in equation (3.4). Now, the minimization problem of equation (3.3) can be solved by converting it to a saddle point problem with the Lagrangian (i.e.

**and**

*a***:**

*b*

*u*_{i},

*w*_{i},

*σ*_{i}and equating them to zero. At first, considering the partial derivative

*σ*_{i}for the

*i*th dataset can be defined as

*δ*

*σ*_{i}, we have

*u*_{i}and

*w*_{i}and substituting the expression of

*σ*_{i}from equation (3.10) and equating them to zeros, we get

*a*and

*b*

Equations (3.11*a,b*) are coupled in *u*_{i} and *w*_{i}. The admissible mechanical fields are obtained by solving these coupled equations. The coupled problem is discretized, using the standard finite-element method (see [8] for details), and the discrete coupled system of equations is obtained as
*K*] is the global stiffness matrix and [*D*] is a Boolean matrix that extracts the measured displacement for data collected sparsely in space. {*P*_{i}} represents the applied traction vector and {*R*_{i}} denotes a sparse vector containing the nodal measured displacements for the *i*th loading case, respectively. The uncoupled form of equation (3.12) can be expressed as

### (b) Material parameter update

In this section, material parameters are updated using the second optimality criterion of equation (3.9). At this stage, as generally followed in classical optimization, using the ECE functional for the material parameter update will lead to the nonlinear parameter update equation for the anisotropic material parameter. This is owing to the presence of the *U* with respect to the material parameter *C*_{pqrs}, we have the following scalar equation:
*U* with respect to all independent material parameters (*C*_{pqrs}), we finally obtain a system of simultaneous linear equations in terms of material constants as the unknown.

The rank of the coefficient matrix, composed of the kinematically admissible strain ** ϵ**[

*u*_{i}] components as constructed via equation (3.16), determines the uniqueness of the material update. For example, in the case of a general three-dimensional linear elastic anisotropic material, equation (3.16) will lead to 21 such linear equations. But the number of independent quantities (i.e. stresses and strains) is six, leading to a singular coefficient matrix for

*i*=1. Nevertheless, the rank deficiency can be removed by considering multiple load cases (thus the multiple strain field). However, the choice of such strains (thus measurements as evident from equation (3.14)) cannot be arbitrary. In particular, the strain components (for each loading case) must form the basis (of dimension equal to the number of scalar unknowns in

We show an example of the above parameter update equation (equation (3.16)) for on-axis and off-axis orthotropic two-dimensional geometry in subsequent sections. It may be noted here that update equations are valid locally (e.g. either domain-wise or element-wise or point-wise); the following results apply for homogeneous as well as heterogeneous parameter estimation. However, our present derivations are based on a domain-wise constant material parameter assumption, as followed in the numerical experimentations (element-wise constant).

#### (i) Two-dimensional on-axis (specially) orthotropic material

The constitutive relationship of a two-dimensional specially orthotropic linear elastic material (material axes lie along the global coordinate axes) can be given by
*Q*_{ij}s unknown) as follows:
*A*^{i} is the coefficient matrix for the *i*th loading case in terms of strains. The integration of equation (318) is performed numerically. The explicit form of *A*^{i} for the present case can be written as

#### (ii) Two-dimensional off-axis (rotated) orthotropic material

In the case of a two-dimensional off-axis (material axes not lying along the global coordinate axes) orthotropic linear material model, the constitutive relation can be represented as follows:
*Q*_{ij}s in equation (3.17)) and a material orientation angle *θ*. However, this representation will lead to a nonlinear constitutive matrix in terms of five unknowns. To avoid any such nonlinearity, we have exploited the invariance relationships as given in [42]. At first, we represent *U*_{j}s) and an orientation angle *θ* as follows:
*U*_{j}s are related with four on-axis material constants as follows:
*m*_{i}s. Then, we follow the same procedure as per equation (3.16) to obtain the linear update equations with six *m*_{i}s as unknowns as follows:
*A*^{i} are given in appendix A. After finding out the unknown variables (*m*_{i}s), we can find out the values of the four invariants by using the following equation:
*U*_{2} and *U*_{3} are related to distinct physical interpretations of the orthotropic material. For instance, as shown in [43], *U*_{2}>0 represents the direction of orthotropy, *U*_{3}>0 represents the low shear modulus orthotropy (i.e. 4*Q*_{66}<(*Q*_{11}+*Q*_{22})−2*Q*_{12}) and *U*_{3}<0 represents the high shear modulus orthotropy (i.e. 4*Q*_{66}>(*Q*_{11}+*Q*_{22})−2*Q*_{12}) [43,44]. The choice of proper signs of *U*_{2} and *U*_{3} has to be made *a priori* for a meaningful reconstruction. The material orientation angle is then found by either of the following equations:
*U*_{j}s) are evaluated, we can obtain independent material constants from equation (3.22). Note, here, that the above sequential procedure for obtaining the material orientation angle and components of {*Q*} from

*Step*1. Find out the solution of the coupled problem using equations (3.12) and calculate the stresses and strains.*Step*2. Find the values of variables (*m*_{1}to*m*_{6}) using equations (3.24).*Step*3. Compute the updated values of the invariants (*U*_{1}to*U*_{4}) from*m*_{i}s (equations (3.25)).*Step*4. Obtain a reconstruction of the material orientation angle by using equation (3.26).*Step*5. Calculate the rotation-independent orthotropic material constants by using equation (3.22).

## 4. Solvability of the material update equations

The main difficulty of estimating the anisotropic material parameter is to find a suitable parameter-sensitive measurement that gives a realistic reconstruction. This difficulty finally imposes restrictions on the boundary conditions (e.g. loads, support fixity) and the number of physical tests to be performed for inverse characterization. For instance, it is well established experimentally [13,14,19] that identification of the two-dimensional orthotropic material parameters requires at least two independent measurements if the orientation of the material axis is known *a priori* and the rotated orthotropic material (i.e. full two-dimensional anisotropic) requires at least three loading cases [11,20,45]. However, it is very difficult to throw light on such restrictions owing to the implicit relationship between the measured response and the estimated coefficients. The objective of the present section is to find such restrictions, within the MECE framework, by analysing the proposed update equation.

At first, we briefly remark on some important observations of the MECE-based identification procedure for the quasi-static case. In the first step of the MECE approach, we generate statically admissible stress and kinematically admissible displacement (so strain) fields. It is to be noted here that the kinematically admissible displacement (or strain) field approximately satisfies (via the penalty parameter) the measured displacement, but the stress field does not satisfy the measurements. Also, in contrast to the elastodynamic case [8,40], the basic problem equation (3.12) is actually uncoupled for the static case. This is substantiated via equations (3.13) and (3.14), wherein the first equation served as a forward elasticity solution (with displacement ** v**:=

**+**

*u***) and the second equation is a measurement penalized forward elasticity solution. The energy equivalent of equation (3.14) can also be written as**

*w**Ω*

^{m}⊆

*Ω*

_{0}. In other words, the kinematically admissible displacement field

**(defined over**

*u**Ω*

_{0}) represents the complete data (filtered via penalization) set similar to the data completion approach as presented for the inverse Cauchy problem [46]. We exploit this observation to propose restrictions on the measured data. In particular, kinematically admissible strains (

**[**

*ϵ*

*u*_{i}]) should be such that the coefficient matrix, composed of strain components of multiple load cases, must be invertible locally for a material update. This finally leads to the relation between each admissible strain field

**[**

*ϵ*

*u*_{i}] and thus in between each measurement

**[**

*ϵ*

*u*_{i}]) must be linearly independent (i.e. complete), as also shown in [29]. Finally, this should, in principle, bring out the restrictions on physical experimentations. In the following sections, we have analysed the invertibility conditions (thus rank) of the coefficient matrix given by update equation (3.16) for different anisotropic materials.

### (a) Two-dimensional orthotropic material

Let us consider the coefficient matrix *A*^{i} of updating of parameters element-wise for an orthotropic material and re-write the coefficient part of equation (3.18) as given below. Here, *Ω*^{e} represents the domain of an element typical of finite-element method discretization,

#### (i) Single measurement, i.e. *N*_{M}=1

For notational simplicity, let us assume that **Δ** becomes
**Δ**, we have
*θ*_{12}≠0. It is clear that with one set of measurements one cannot update four constants for an orthotropic two-dimensional material.

#### (ii) Two sets of measurements, i.e. *N*_{M}=2

Now, in addition to the previous case, we add one more measurement and assume that **Δ** as follows:

#### (iii) Three sets of measurements, i.e. *N*_{M}=3

We now consider three sets of measurements and assume that *Q*_{66}. However, *Q*_{11}, *Q*_{22} and *Q*_{12} can be updated.

### (b) Three-dimensional transverse isotropic material

As our next illustration, we consider the three-dimensional transverse isotropic material model represented by five material parameters. Using equation (3.16) for all five parameters, the coefficient matrix for updating the material parameters can be written in the same way as in equation (4.2).

#### (i) Single measurement, i.e. *N*_{M}=1

Let us assume that **Δ** for this material model becomes

#### (ii) Two sets of measurements, i.e. *N*_{M}=2

Here, we consider one additional set of measurement data. Similar to the previous case, we assume that **Δ**)

We want to point out here that ensuring the above restrictions on kinematically admissible strains (thus on measured data) in a point-wise (or domain-wise) manner, as discussed in [29,41], whereas loading is entirely controlled via boundary tractions, is a question that still needs further investigations. This also raises the question of the systematic choice of physical experimentations for realistic material parameter reconstruction. However, we have left these issues for further investigations and intend to explore elsewhere.

## 5. Selection of the penalty parameter

It can be noted that the success of the MECE approach is crucially dependent on the choice of penalty parameter. In this context, we follow the penalty continuation scheme as proposed in [8]. First, we non-dimensionalize the penalty parameter *κ*_{i} used in equations (3.14) as
*U*_{i} is the strain energy of the *i*th loading situation for the initial guess of material parameter *α*_{0}) has been taken before starting the inverse problem, and then subsequently, at each iteration, it is multiplied by a multiplication factor 10^{β} (*β*>0) to obtain the penalty parameter at each subsequent step. For the present illustrations, we prefer to choose the initial and incremental parameters in the penalty parameter selection scheme via a trial and error method. Then, we restrict the iterative recursions via a stopping rule governed by the discrepancy principle (owing to Morozov). At each iteration, we use the following data misfit-based termination criteria to stop the iterative recursions:
*f*_{m} is a relatively misfitting functional and *u*_{i} and *δ* is the noise intensity and *c* is a regularization parameter. In all the numerical experiments, we have taken *c*=1. However, a more rational approach [47] can be followed for selecting such penalty parameters. We have left this important topic for future investigation and it will be reported elsewhere.

## 6. Results and discussion

In this section, we have assessed the numerical performance of the proposed inverse procedure with full-field noisy measured displacement data. Here, for each example problem, parameter reconstruction is performed element-wise. Also, the positivity constraints of the elasticity parameters are enforced numerically. In particular, we update the material parameters (iteration-wise) only when the positive definiteness of the elasticity tensor remains unaffected. To mimic the experimental data, we have synthetically generated measurements by solving a forward problem in a finer mesh. The data were then polluted by adding artificial random noise. That is, the *j*th degrees of freedom of the displacement vector for the *i*th loading case *u*_{ij} is obtained as
*r*_{ij} denotes the *j*th component of the independent random variable ** r** for the

*i*th experiment. The random variable is uniformly distributed in [−1,1].

### (a) Specially orthotropic material parameter reconstruction

The irregularity of the position and orientation of fibres has influenced the mechanical properties of unidirectional composites. Thus, the detection of positions as well as fibre orientations has great importance in the mechanical characterization of unidirectional composite materials. In this numerical example, we have taken up the problem of inverse detection of the position of fibres in the cross section of a unidirectional composite lamina with known orientation (*θ*=0).

A representative area element (RAE) of a carbon–epoxy composite lamina consisting of an isotropic matrix reinforced with randomly distributed infinite orthotropic cylindrical (circular in two dimensions) fibres is taken for illustration. The assumed mechanical properties of the soft background (matrix) and hard inclusions (carbon fibres) are provided in the table 2. The diameter of the cylindrical fibre (carbon-AS4) is 7 μ*m*. The length of a single side of the square domain (RAE) considered for this problem is 56 μ*m*. The reconstruction is performed in a 100×100 bilinear quadrilateral finite-element mesh leading to 40 000 material parameters as unknowns. To assess the proposed methodology, we artificially choose two loading cases (as given in table 1) and the corresponding boundary conditions (as shown in figure 1). For this numerical example, all the applied uniform normal displacements and shear displacements are of the order of 1% and 0.5% of the size of the square domain. Practically, the choice of the spatial position of the selected RAE is random. So, to obtain proper reconstructions, we have to consider the actual varying boundary tractions on that RAE. This has been explained by Rahmani *et al.* [22]. We have taken *α*_{0}=10^{4} and *β*=0.125 for the noiseless condition, whereas *α*_{0}=0.15 and *β*=0.015 are taken for the noisy case. In figure 2, the reference and the reconstruction plots of the spatial distribution of on-axis orthotropic composite lamina properties *Q*_{11}, *Q*_{12}, *Q*_{22} and *Q*_{66} are shown, respectively. It can be noted from these figures that the proposed algorithm has been successful in distinctly capturing the position of the inclusions. The demarcation of the inclusions (fibres) can be clearly noted in these figures for all the reconstructions. However, the reconstruction of the *Q*_{12} parameter is more erroneous than the other reconstructions. This is mainly owing to the higher contrast ratio (table 2) of parameter *Q*_{11} compared with other parameters. In particular, *Q*_{12} is responsible for the axial stress owing to strain in the perpendicular direction while *Q*_{11} (and *Q*_{22}) is responsible for the axial stress owing to strain in the same direction. Thus, a higher contrast ratio between these parameters does not allow *Q*_{12} to be reconstructed with the same accuracy as axial moduli (*Q*_{11} and *Q*_{22}), even when strains are slightly perturbed. The proposed MECE algorithm took very few, only 23 and 38, iterations for successful reconstructions of the material parameters for the noiseless and 1% noisy case, respectively.

### (b) Rotated orthotropic material parameter reconstruction

Here, we have considered the spatial reconstruction of the four on-axis (specially orthotropic case) material parameters and the material fibre orientation angle (*θ*=30°) of the previous example. For the reconstruction of the five parameters, we have followed the proposed procedure described in §3. We consider here the four loading cases (figure 3) to obtain sensitive measurements. In this case, all four given uniform displacements are of the order of 1%. The displacements are applied to the specimen statically. The domain size, the diameter of the fibres and all the other requisite properties for this problem are the same as in the previous example problem. *α*_{0}=10^{−1} and *β*=0.25 are taken for the noiseless condition, whereas *α*_{0}=10^{−3} and *β*=0.04 are considered for the noisy case. The reconstruction is performed in a 100×100 bilinear quadrilateral finite-element mesh leading to 60 000 material parameters as unknowns. Figure 4 shows the reference and reconstruction plots of *Q*_{11}, *Q*_{12}, *Q*_{22}, *Q*_{66} and the fibre orientation angle (*θ*), respectively, for the off-axis orthotropic composite lamina. It can be noted from these figures that the proposed algorithm has been successful in distinctly capturing the position of the inclusion and the sharp boundary as well, at no noise and 1% noise. The proposed algorithm reaches acceptable reconstructions successfully with only 21 and 63 iterations for the noiseless and the 1% noisy case, respectively.

### (c) Transversely isotropic elastic parameter reconstruction

Finally, we have considered a transversely isotropic material reconstruction problem to investigate the ability of the proposed algorithm to detect the region of osteopenic cancellous bone in a healthy cancellous bone background from full-field synthetic displacement data. Here, in this problem, the directions of anisotropy are assumed to be unknown *a priori*. The elastic properties of these two regions are taken from the work of Shore *et al.* [28] and are shown in table 3. Uniaxial compressive strains (1%) are applied to the specimen in each (*X*, *Y* , *Z*) principal material direction with the addition of an *X*–*Y* shear strain (1%). The type of support conditions for each loading are taken from Shore *et al.* [28]. Considering a homogeneous initial guess, we finally estimate heterogeneous transversely isotropic material parameter distributions over a 50×50×50 mesh of three-dimensional brick elements. The reconstructed parameter profiles are shown in figure 5 and in figure 6 with reference material profiles. Figures 5 and 6 clearly show the effectiveness of the proposed algorithm towards the reconstruction of all the material parameters *C*_{11},*C*_{12},*C*_{13},*C*_{33} and *C*_{44}. To assess the computational performance of the proposed reconstruction scheme with regard to standard reconstruction methods, we solve this problem via an *L*_{2}-based quasi-Newtonian approach [48], keeping the computational data the same. The relative misfit plot (figure 7) shows a comparison study between the *L*_{2}-based quasi-Newtonian approach and the proposed MECE-based approach. It clearly indicates superior performance of the MECE-based algorithm in terms of the required forward solves for reaching a specified tolerance of the relative misfit functional equation (5.2). The limiting value of the relative misfit functional is taken as 10^{−8} to stop the MECE algorithm. The MECE approach took nine solves of the basic/coupled problem (nbp) (i.e. 9×2 forward solves), whereas the quasi-Newtonian method took 3608 functions and gradient (nfg, [48]) evaluation (i.e. 3608×2 forward solves).

## 7. Closure

An inverse reconstruction technique based on the MECE functional, applicable for heterogeneous linear elastic anisotropic materials, is explored numerically. The material parameters are updated by considering an *L*_{2}-norm of constitutive discrepancy. When compared with the standard nonlinear ECE-based parameter update procedure, the proposed update equations are linear for anisotropic material models. The explicit update formulae are derived for the off-axis and on-axis orthotropic and as well as for three-dimensional transverse isotropic material models. In the case of an off-axis orthotropic material, invariance relationships are exploited to reconstruct space-dependent heterogeneous material orientations. Explicit update formula revealed the required constraints to be imposed on the measured data. Some common material models are investigated to find out the required number of independent loading cases. Numerical experimentations demonstrated the applicability and potential of the proposed procedure for large-scale parameter estimation problems with linear anisotropy. Also, a fair comparison with the standard displacement discrepancy-based quasi-Newtonian approach revealed a faster convergence speed of the proposed algorithm. Moreover, the present strategy could be exploited with stochastic filters, such as an ensemble filter or a Monte Carlo filter, to address possible uncertainties involved in mathematical modelling and experimental measurements. This may constitute the content of future work.

## Competing interests

We declare we have no competing interests.

## Funding

No funding has been received for this article.

## Appendix A. Coefficient matrix for two-dimensional rotated orthotropy

The components of the symmetric matrix ** A** and the right-hand side

*f*_{p}in equation (3.24) are as follows:

- Received March 25, 2016.
- Accepted July 1, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.