Cansmart 2009

International Workshop SMART MATERIALS AND STRUCTURES 22 - 23 October 2009, Montreal, Quebec, Canada

2009 Cansmart Workshop

A TWO-STEP IDENTIFICATION PROCEDURE FOR THE

IDENTIFICATION OF DISTRIBUTED MATERIAL CONSTITUTIVE PARAMETERS FROM SURFACE DISPLACEMENT

MEASUREMENTS

L. Li, F. Ghrib, W. Polies

Department of Civil and Environmental Engineering, University of Windsor, On, Canada

fghrib@uwindsor.ca

ABSTRACT

The present work describes an inverse identification technique based on the finite element model updating for computing the spatial distribution of Young’s modulus using full-field displacement data obtained from digital image correlation. The problem is formulated as a Tikhonov-regularized of a minimization problem of a data discrepancy functional using the equilibrium equations as constraints. The gradient of the objective functional is evaluated using the adjoint method; a spatial filtering is introduced to the gradient image during the iterative process to improve the numerical stability in the presence of noise in displacement data. Numerical example is presented to demonstrate the performance of the presented methodology.

Keywords: Finite element model update, Adjoint method, Inverse problem, Elasticity imaging, Field measurement.

INTRODUCTION

Surface field measurement techniques, such as the digital image correlation, are

becoming very popular in experimental mechanics. These techniques are potentially useful for the identification of heterogeneous spatial distribution of material properties. This has become a new and promising branch of research and development attracting widespread academic and industrial interests.

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A recent issue of the journal of “Experimental Mechanics” had been dedicated to this

specific area of research [1]. In a review paper published in this issue, Avril et al gave an extensive review of recently developed techniques that identify material parameters from full-field measurements [2]. The term full-field measurement, currently, mostly is equivalent to measurement of displacements and/or strain from digital image correlation [1]. It is also interesting to note that this area of research has long been an area undergoing extensive research in bio-medicine and biomechanics, usually under the name elasticity imaging or elastography [3-5].

The identification of distribution of material parameters is much more difficult than the

problem of parameter identification of homogeneous material parameters. From parameterization of a continuous distribution would give rise to a huge number of unknown parameters. As a general approach, the problem is usually formulated as a nonlinear optimization problem minimizing the data discrepancy functional between the measured and predicted displacement fields, and control equations are included as constraints. This approach results in partial-differential-equations (PDE) - constrained optimization problem [6]. In the case of elasticity, Navier’s equations are used as PDE constraint. From mathematical point of view, this formulation is an elliptic distributed control problem, where the unknown variables can be viewed as controls and the measured variables (displacements or strain) are viewed as states [7].

In the following, we present an optimization-based finite element model updating

technique for the reconstruction of the Young’s modulus from full-field displacement measurements. Many problems in applied mechanics, such as inclusions problem and damage accumulation, are directly related to the identification of distributed Young’s modulus. The solution process of the optimisation makes use of the adjoint method to evaluate the sensitivity parameters for constructing efficient optimization algorithms; a spatial filtering is used to modify the updating gradient vector, and stabilize the updating process with the presence of noise in data.

The adjoint method had been used in conjunction of an elasticity imaging technique

developed by Oberai et al [3]. The presented method is formulated in the context of finite element with no reference to the continuous settings. The objective function is regularized using the Tikhonov method. The Tikhonov regularized finite element model update technique is also used by Doyley et al [5], where the sensitivities are evaluated using a direct differentiation approach. Numerical examples show that the presence of noises in the data, the reconstruction deteriorates severely; the development of inverse technique robust to noise in data is still an open question.

THE RECONSTRUCTION METHODS

Problem set-up

A general approach in solving inverse problem of structural parameter identification is to

minimize an objective function defined as a data discrepancy to measure the fit-to-data between computed and measured mechanical response (i.e. either strain or displacements).

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While the direct problem is solved using finite element method, this is also termed as finite element model updating.

The commonly used objective function is defined using the sum of squared differences

between the measured data and the corresponding simulated values of the displacement field on the surface of the structure. A Tikhonov regularization term is added to overcome the ill-posedness and stabilize the solution. In the case of static equilibrium, the constrained optimization problem is as follows:

2

21

1min ( ) ( )2

. . ( )

Nm

i ii

J u u

s t K

α=

⎧= − +⎪

⎨⎪ =⎩

∑θu,θ θ

θ u f2θ

(1)

where is the total number of measurements, is the vector of unknown parameters, is the measured data and the corresponding simulated value using a trial distribution of parameters .

N θ mu)(θu

θ2

θ is the Euclidian norm ofθ . The constraint in equation (1) is the expression of equilibrium formulated within the finite element. The parameterα is the Tikhonov regularization parameter; the detailed analysis of the Tikhonov regularization and the selection of the regularization parameter can be found in [9]. The adjoint sensitivity analysis

The numerical solution of problem (1) requires the evaluation of the sensitivity of the

observable variables (displacement) with regards to the control parameters . If the number of control parameters is relatively small, for example less than ten, direct differentiation can be used to evaluate the sensitivity [8]. If the cost of direct problem simulation is small, powerful direct optimization techniques such as the genetic algorithms (GA) can be used without sensitivity analysis; the GA is especially powerful due to its convergence property to find the global optimal solution.

θ

However, the reconstruction of distribution of material parameters is a functional

identification problem; the discretization of the field often results in a large number of parameters to be evaluated. For problems with large number of unknowns, the adjoint-method based sensitivity evaluation is more efficient than any other sensitivity-evaluation techniques [8]. The constraint equations are the discretized governing equation using finite element; more generally they are written in the following form:

0fuθu(θθ,R =−= )())( K (2)

where is the state variables (displacements in our problems), which is an implicit function of the unknown material variables θ . We use Lagrangian multiplier to change the constrained optimization to an unconstrained problem. The augmented output function that enforces the governing equations via Lagrange multipliers is:

u

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(3) )()()( uθ,Rλuθ,uθ, TJL −=

Differentiating the Lagrangian with respect to iθ , we have the gradient of the Lagrangian

i

T

i

T

ii ddJJ

ddL

θθθθ ∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

+∂∂

=Rλu

uRλ

uuθ, )( (4)

In this gradient, the term idd θu is very difficult term to evaluate (sensitivity). However,

if we impose that the term TJ∂ ∂⎛ −⎜ ∂ ∂⎝ ⎠Rλ

u u⎞⎟be equals to zero, one can calculate the Lagrange

multiplier, , then the sensitivity term, λ idd θu , is no longer needed.

0=⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

uRλ

uTJ (5)

Equation-(5) is the adjoint equation and it is expressed as:

))( K(θλUU Tm =− (6)

where U is the displacement vector computed from the finite element equation-(2), and Um is the measured displacement vector. Therefore we have:

i

T

ii

Jd

dLθθθ ∂∂

+∂∂

=Rλuθ, )( (7)

The two derivatives in (7), i

Jθ∂∂ , and

iθ∂∂R , can be easily computed. In computing

iθ∂∂R ,

iii θθθ ∂∂

−∂

∂=

∂∂ fuK(θR ) (8)

The global stiffness in (8) matrix is an assembly of the elements’ stiffness matrix; and for linear elastic materials, the element stiffness matrix is proportional to the material modulus and geometric coefficient in the element, so that its derivative to θ can be easily defined for a given stiffness matrix expression.

The iterative optimization procedure

The iterative optimization algorithm for finding a gradient consists of two consecutive steps: 1) solve (6) for Tλ , then substitute Tλ into (7) to calculate the gradient of Lagrangian. With the gradient calculated from adjoint method, classical gradient-based optimization techniques can be used to find the unknown stiffness parameters. In the present paper, the gradient based pseudo-Newton optimization algorithm implemented in MATLAB

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optimization toolbox is used in the simulation [10]. The flowchart of the iterative process is given in Fig. 1.

FEM simulation of responses and calculation of gradient at Ek(x)

Update the distribution of Ek(x) using a Pseudo-Newton method

toleranceEE kk <−+

2

1 )()( xxNo, k=k+1

Initial guess of distribution of Young’s modulus E0(x)

Measurement of response

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Output the distribution E(x)

Yes

Fig. 1: Flowchart of the iterative procedure.

The steps given above are general and can be applied to other finite element schemes

providing the governing relation 0fuθu(θθ,R =−= )())( K holds.

Filtering the gradient

Doyley et al [5] proposed the idea of filtering the updated Young’s modulus at each iteration during an iterative reconstruction process. The filtering procedure proposed by the authors is of a type of edge-enhancing filtering; this type of filtering has been proven to

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generate better images with abrupt changes of Young’s modulus. The idea is adopted here; but the filtering is performed over the calculated gradient image within each iteration, rather than the overall updated Young’s modulus. Simulations show that this type of filtering stabilizes the solution in the presence of noise.

The filtering adopted for the gradient is a sliding average filtering, which slide a 3-by-3

moving average across the 2-dimensional image of gradients, producing a low-pass filtered version of the original gradient image. This two-dimensional filtering can be implemented using two-dimensional convolution. At the k-th iteration, the gradient to update the Young’s modulus at element A is given as the filtered value

∑=

=6

1)(

61)(

ii

kk ADAD (9)

where the superscript k indicates the iteration number; represent all the elements in the 6-by-6 block centered at the element A (including the element A itself), as shown in Fig. 2.

iA

Within the image processing, this is also an edge-enhancing filtering. Edge-enhancing

filtering are useful at reproducing blocky images, and stabilize the image-deblurring process. Numerical experiments show that this filtering is good for reconstructing piecewise-distribution of Young’s modulus in the presence of noise. Its relationship to image-deblurring and mathematical analysis is still an open question and need to be explored.

A

Fig. 2: The 3-by-3 block used in the sliding moving average of the gradient for element-A.

NUMERICAL EXPERIMENT In order to verify the proposed technique, a simple problem is illustrated. The simple

plate shown in Fig. 3 is considered. It is a composite specimen made of two different materials. The objective is to reconstruct the distribution of the Young’s modulus using the proposed algorithm. Using the finite element method, the specimen is discretized into 10 × 20 = 200 plane stress Q1 elements of equal size. Hence, there are in total 200 unknown parameters representing the distribution of Young’s modulus of the specimen. Prescribed tensile displacement is applied to the right side while the left side is constrained.

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The measured displacement field on the surface is simulated using computational analysis results for the given reference parameters. The Tikhonov regularization parameter is set equal to . This value is not meant to be the optimal one, i.e. the identification could be improved if a better value of the parameter is found.

410−

Prescribed displacement E = 250 GPa ν = 0.3 Ω2

E = 200 GPa ν = 0.3 Ω1

30 mm

30 mm × 2 = 60 mm

Fig. 3: The plane stress bi-material specimen.

A simple additive Gaussian noise has been used to study the stability of the solution when

noise corrupts the data. Numerical simulations show that the identification without added noise is satisfactory; however, with added noise, the performance deteriorates severely. At 2% noise level, the identified image of Young’s modulus distribution is shown in Fig. 4. The border between the two different materials is clearly visible, but its exact location is blurred within two to three elements, which corresponds to 10~15% of the total length of the longer side of the specimen. At 5% noise level, the border is hardly recognizable. One should bear in mind that noises are introduced not only from measuring process, but also from the modelling process; both sources are physically inevitable.

GPa

Fig. 4: The identified image of Young’s modulus distribution

of the bi-material specimen.

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A numerical index, the weighted average error (WAE), is introduced here to measure the

performance of the algorithm:

2

1

1 ∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

N

itri

tri

idi

EEE

NWAE (10)

where and represent the identified and the true values of Young’s modulus at element i, respectively. N represents the total number of elements in the finite element mesh. Another measure of performance is the maximum deviation of identified values from corresponding true values:

idiE tr

iE

%100max ×⎟⎟⎠

⎞⎜⎜⎝

⎛ −tri

tri

idi

i EEE (11)

Although the solution is not smooth, as shown Fig. 4, one can recognize that there are

generally two different regions of the Young’s modulus. To improve the solution process, one can augment the previously presented algorithm by a parametric optimisation identifying the two zones separately. To proceed with a parametric modelling and identification of the modulus distribution. In this way the distribution of Young’s modulus is parameterized by a fewer parameters. For this particular case, , the modulus of the left part of the specimen,

, the modulus of the right part of the specimen, and a parameter , defining the distance from the border between the supposed two materials to the left side of the specimen. Mathematically the elasticity modulus is defined as a function,

1E

2E 1l

1 2 1( ) ( , , )E x E E E l= . That is to say, this function is assumed to describe completely the distribution . Then a direct optimization process without sensitivity analysis can be used to find the three parameters that provide the best fit between the simulated and measured displacement response. The direct optimization is performed using MATLAB optimization toolbox [10]. The performances of the parametric method and the proposed finite element model updating procedure are listed in Table-1.

1 2 1( , , )E E E l )(xE

Table-1: Performance of the identification

Method Noise level WAE Maximum deviation from true value

0 % 0.01 0.1 % 2 % 0.03 0.3 % Parametric

identification 5 % 0.09 1.1 % 0 % 0.12 5 % 2 % 0.28 9 % Adjoint-based FE

model update 5 % 1.01 27 %

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CONCLUSION

A finite element model updating procedure for the inverse reconstruction of the elasticity

modulus distribution with surface displacements measurements is proposed. The problem is formulated as a minimization of a data discrepancy functional augmented with Tikhonov regularization term. The optimisation procedure uses gradient based algorithms. The gradient of the objective functional with respect to the unknown parameters can be solved numerically following an adjoint formulation. Tikhonov regularization and a spatial filtering of the gradients in each iterate are adopted to stabilize the problem in the presence of noise in data. Once the gradient information is obtained, we solve the optimization problem using a gradient-based method. Once the different regions are identified, a second step consists at using a parametric optimisation formulation to smoothen the results.

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Lemosse, D., Pagano, S., Pagnacco, E., Pierron, F., “Overview of Identification Methods of Mechanical Parameters Based on Full-field Measurements”. Experimental Mechanics, 48, 2008, pp. 381–402.

3. Oberai, A.A., Gokhale, N.H., Feijoo, G.R., “Solution of inverse problems in elasticity imaging using the adjoint method”. Inverse Problems, 19, 2003, pp. 297-313.

4. Sumi, C., Suzuki, A., Nakayama, K., “Estimation of shear modulus distribution in soft tissue from strain distribution”. IEEE Trans. Biomed. Eng., 42, 1995, pp. 193–202.

5. Doyley, M.M., Meaney, P.M., Bamber, J.C., “Evaluation of an iterative reconstruction method for quantitative elastography”. Physics in Medicine and Biology, 45, 2000, pp. 1521-1540.

6. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S., Optimization with PDE Constraints. Springer, 2009.

7. Lions, J.L., Optimal control of systems governed by partial differential equations. Academic, NY, 1971.

8. Kleiber, M., Antunez, H., Hien, T.D., Kowalczyk, P., Parameter sensitivity in nonlinear mechanics, theory and finite element computations. John wiley & Sons, 1997.

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