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.