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# Journal Club Theme of January 2013: Error in Constitutive Equations Approach for Materials Characterization

In this journal club, I would like to discuss the Error in Constitutive Equations (ECE) approach as an emerging and exciting avenue to materials identification in the context of inverse problems. In the ECE approach discussed herein, we define a cost functional based on an energy norm that connects a set of kinematically admissible displacements and a set of dynamically admissible stresses. The set of kinematically admissible displacements is composed of fields that satisfy essential boundary conditions and possess sufficient regularity (i.e. smoothness). The set of dynamically admissible stresses is composed of fields that satisfy conservation of linear momentum and natural (i.e. traction) boundary conditions. The inverse problem is solved by finding material properties along with admissible displacement and stress fields such that the ECE functional (along with a measure of the data misfit) is minimized. Experimental data is introduced in the formulation as a quadratic penalty term added to the ECE functional.

**Figure 1.** Target and reconstructed shear "G" and bulk "B" moduli fields reconstructed using the ECE approach. [Banerjee et al. CMAME, 253, 60-72 (2013)]

ECE approaches possess significant advantages such as incorporating the relevant physics directly into the cost functional as well as offering natural metrics for a posteriori error estimation, among others. Furthermore, in many of our applications, we have observed faster convergence with ECE-type functionals than with conventional least squares objective functions.

Our group, in collaboration with the Mayo Clinic Ultrasound Research Laboratory, has been working with the ECE technique in the context of biomechanics problems. Specifically, we have focused on the identification of viscoelastic material properties in arteries, the heart wall, and breast tumors with the end goal of early disease diagnosis.

The ECE approach may be widely applied to problems involving complex nonlinear mechanics, coupled physics, and multiple temporal and spatial scales. There are many exciting (and still unexploited) opportunities in using ECE methods in the latter fronts. For an overview of the method and other references, please see

1) B. Banerje, T.F. Walsh, W. Aquino, and M. Bonnet (2013). Large Scale Parameter Estimation Problems in Frequency-Domain Elastodynamics Using an Error in Constitutive Equation Functional. Computer Methods in Applied Mechanics and Engineering, 253, 60-72.

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## Comments

## Potential impact of the ECE method

Wilkins, thanks for posting this interesting introduction to the Error in Constitutive Equations (ECE) approach. Could you provide more details about the potential impact of this method?

## Potential impact of the ECE method

Xuanhe:

The ECE method could be applied to a wide variety of problems. One of the salient features of the method is that by incorporating information (or lack of) about the constitutive equations into the cost functional, we can monitor errors not only in observable quantities (e.g displacements), but also in unobservable, hidden quantities in the material models. In addition, we have found in elasticity and viscoelasticity imaging problems that we can obtain sharp reconstructions in just a few optimization iterations.

Lastly, I would like to point out that this method allows for material inversions at resonant frequencies (even without damping). If a conventional least squares approach were used, the latter problem would require that the inverse problem be based on an eigenvalue problem, leading to new formulations, coding efforts, etc.

The above are, in my opinion, some of the relevant features.

## path dependent materials

Hi Wilkins , thanks for the interesting discussion. I just wonder if it is possible to extend the current approach to path-dependent materials (say material with plastic dissipation/damage)? In that case, do you need any modification for the ECE functional?

## path dependent materials

Hi WaiChing!

Yes, the method can be and has been extended to dissipative materials. The ECE functional definitely needs to be modified. The usual approach is to incorporate the parts of the constitutive relation that are unknown into the cost functional and the certain or known parts of the constitutive equations are added as constraints in the optimization problem. See for instance this paper that treats viscoplasticity and damage in the context of an ECE approach.

. H.-M. Nguyen, O. Allix, P. Feissel, A robust identification strategy for rate-dependent models in dynamics, Inverse Problems 24 (2008) 065006.

I hope this is helpful.