The imaging of constitutive parameters is of interest in many science and engineering fields. Indeed, non-invasive and nondestructive techniques are used to characterize key properties of a system given its response due to an external excitation. Then, assuming a priori a given model of the system, the measured response and an inverse approach are used to identify material properties. This work was undertaken in the context of identification of spatially-varying elastic and viscoelastic parameters of solids using vibroacoustics based techniques.

Two optimization approaches, nongradient and gradient-based optimization, were investigated in this work. Initially, nongradient-based algorithms were preferred over gradient-based algorithms because of there ability to find global minima irrespective of initial guesses. For instance, Gaussian radial basis functions were used to construct a finite-dimensional representation of the elastic moduli. Then, an inverse approach was used to approximate the spatiallyvarying elastic moduli through the system response induced by the radiation force of ultrasound. The inverse problem was cast as an optimization problem in which a least-square error functional that quantified the misfit between the experimental and finite element representation system response is minimized by searching over a space of admissible vectors that best describe the spatial distribution of the elastic moduli. Subsequently, gradient-based optimization was preferred over nongradient-based optimization as the number of design variables increased due to the increment in computational cost.

Two inverse approaches, L2-adjoint and concept of error in constitutive equation, were investigated in the context of gradient-based optimization. First, the L2-adjoint inverse approach was used to characterize spatially-varying viscoelastic properties because of its advantage to efficiently calculate the gradient of the error functional with respect to the design variables by solving the corresponding adjoint equations. The inverse problem was cast as an optimization problem in which a least-square error functional that quantified the misfit between the experimental and the finite element representation system response is minimized by searching over a space of admissible functions that best describe the spatially-varying viscoelastic properties.

Given that the least-square error functional is non-convex, an inverse approach based on the concept of error in constitutive equation was investigated. The convexity property of the error in constitutive equation functionals, shown extensively for elliptic boundary value problems, reduce the sensitivity of the inverse solution to parameter initialization. The inverse problem was cast as an optimization problem in which an error in constitutive equation functional that quantified the misfit between the kinematically and dynamically admissible stress fields is minimized by searching over a space of admissible functions that best describe the spatially-varying viscoelastic properties. Contrary to the L2-adjoint inverse approach, the gradient equation is easily derived by taking the direct derivative of the error in constitutive equation functional with respect to the design variables.

The feasibility of the proposed inverse approaches is demonstrated through a series of numerical and physical experiments. Results show that the proposed inverse approaches have the potential to characterize spatially-varying elastic and viscoelastic properties of solids in realistic settings. Furthermore, it will be shown that the inverse approach based on the concept of error in constitutive equation outperformed the L2-adjoint inverse approach.