This paper presents the formulation and implementation of an Error in Constitutive Equations (ECE) method suitable for large-scale inverse identification of linear elastic material properties in the context of steady-state elastodynamics. In ECE-based methods, the inverse problem is postulated as an optimization problem in which the cost functional measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses. Furthermore, in a more recent modality of this methodology introduced by Feissel and Allix (2007), referred to as the Modified ECE (MECE), the measured data is incorporated into the formulation as a quadratic penalty term. We show that a simple and efficient continuation scheme for the penalty term, suggested by the theory of quadratic penalty methods, can significantly accelerate the convergence of the MECE algorithm. Furthermore, a (block) successive over-relaxation (SOR) technique is introduced, enabling the use of existing parallel finite element codes with minimal modification to solve the coupled system of equations that arises from the optimality conditions in MECE methods. Our numerical results demonstrate that the proposed methodology can successfully reconstruct the spatial distribution of elastic material parameters from partial and noisy measurements in as few as ten iterations in a 2D example and fifty in a 3D example. We show (through numerical experiments) that the proposed continuation scheme can improve the rate of convergence of MECE methods by at least an order of magnitude versus the alternative of using a fixed penalty parameter. Furthermore, the proposed block SOR strategy coupled with existing parallel solvers produces a computationally efficient MECE method that can be used for large scale materials identification problems, as demonstrated on a 3D example involving about 400,000 unknown moduli. Finally, our numerical results suggest that the proposed MECE approach can be significantly faster than the conventional approach of L2 minimization using quasi-Newton methods.
This work presents an approach to inversely determine material properties for solids immersed in fluids through the use of steady-state dynamic response. The methodology uses measured acoustic pressure amplitudes in the fluid surrounding a structure being vibrated with a harmonic force to determine the parameters for elastic and viscoelastic material models. Steady-state dynamic finite element analysis is used to compute the frequency response function of homogeneous and heterogeneous solids. The frequency response is then used to inversely estimate material parameters. In order to solve the inverse problem, an optimization method is presented which combines the global search capabilities of the random search method with the reduced computational time of a surrogate model approach. Through numerical and laboratory experiments, this work shows that acoustic emissions hold sufficient information for quantifying both elastic and viscoelastic material behaviors. Furthermore, the examples show that the surrogate model accelerated random search algorithm is an efficient and accurate method for solving these types of inverse problems. This research has direct and important implications in nondestructive evaluation of material properties in general. However, the examples shown in this paper are based on applications related to noninvasive characterization of biological materials through methodologies currently applied to medical imaging.
In recent years, several new techniques based on the radiation force of ultrasound have been developed. Vibro-acoustography is a speckle-free ultrasound based imaging modality that can visualize normal and abnormal soft tissue through mapping the acoustic response of the object to a harmonic radiation force induced by ultrasound. In vibro-acoustography, the ultrasound energy is converted from high ultrasound frequencies to a low acoustic frequency (acoustic emission) that is often two orders of magnitude smaller than the ultrasound frequency. The acoustic emission is normally detected by a hydrophone. In medical imaging, vibroacoustography has been tested on breast, prostate, arteries, liver, and thyroid. These studies have shown that vibro-acoustic data can be used for quantitative evaluation of elastic properties. This paper presents an overview of vibro-acoustography and its applications in the areas of biomedicine.
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