A modified error in constitutive equation approach for frequency-domain viscoelasticity imaging using interior data
Introduction
Inverse characterization of viscoelastic properties is of high relevance in many areas of science, engineering, and medicine. In particular, in the medical field, it is well-known that viscoelastic properties are correlated to tissue pathology [1]. This observation has spurred the development of a group of techniques, commonly referred to as elastography or elasticity imaging techniques, whose goal is to identify the elastic or viscoelastic properties of tissue non-invasively (see [2], [3], [4], [5], [6] for reviews). Other applications abound and include the nondestructive evaluation of structural systems such as buildings, bridges, and aircraft components.
In this work, we are concerned with the quantification of viscoelastic material properties using noisy interior data, in situations where traction and/or displacement boundary conditions are unknown or uncertain. This problem is of high relevance in the field of elasticity imaging where displacement or velocity fields are obtained using ultrasound or MRI, and there is a high degree of uncertainty regarding the magnitude and nature of the excitation sources. Different techniques have been developed that address this problem, including algebraic direct inversion [7], [1], [8] and the adjoint weighted equations (AWE) methods [9], [10]. These techniques have the advantage of being non-iterative but need the evaluation of derivatives of the data, making them very sensitive to noise. Moreover, optimization approaches [11], [12], [13], [14], [15], which have the advantage of handling sparse and imperfect data, have received limited or no attention for problems with interior data that lack knowledge of the boundary conditions due to the complication of the forward problem then being ill-posed.
Our goal in this work is to develop a methodology for reconstructing viscoelastic properties from imperfect interior data and underspecified boundary conditions. The proposed methodology is based on the minimization of a modified error in constitutive equation (MECE) functional [16], [13], [17]. MECE functionals combine the error in constitutive equation (ECE) [18], which measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses, and a quadratic error term that incorporates the measurement data. MECE-based approaches have found applications in model updating with vibrational data [16], [19], time-domain formulations [20], [13], and large scale identification problems in both elastodynamics [17] and coupled acoustic-structure systems [21]. MECE-based identification methods investigated thus far assume situations where well-posed boundary conditions are available. By contrast, one of the main features of the present approach is the fact that the proposed MECE formulation leads to optimality systems that are invertible, subject to a solvability condition that is easily met in practice, even in cases where (i) boundary conditions are (totally or partially) underspecified, and (ii) interior data is available only in a portion of the solid under investigation. This result is moreover achieved by exploiting classical first-order optimality conditions only. Some of the limitations inherent to other optimization-based formulations are therefore avoided. In addition, we adapt previous ideas regarding the solution strategy for the minimization problem to the realms of viscoelasticity, where positivity requirements are enforced using inequality constraints.
The rest of this article is organized as follows. Section 2 describes the steady-state viscoelasticity problem and the inverse problem of interest. The MECE-based minimization strategy is derived in Section 3, which also addresses the adjustment of its main parameters. Section 4 is devoted to a set of numerical experiments designed to demonstrate the capabilities of the method. Concluding remarks are finally given in Section 5.
Section snippets
Problem setting
Governing equations of motion. Let a solid viscoelastic body occupy a bounded and connected domain with boundary . The time-harmonic motion of this body is governed by the balance equations where is the displacement field, represents the specified angular frequency, denotes the known mass density, is a given body force density, represents the stress tensor, and are the given surface force density (traction) and its support,
MECE functional
Following the approach presented in [17], [21], the inverse problem addressed in this work is formulated as an optimization problem in which the unknown constitutive tensor is estimated by minimizing an objective function that additively combines two error terms: (1) an error in constitutive equation (ECE) functional [18] that has been adapted for viscoelastic materials and that measures the discrepancy in the constitutive equation that connects kinematically admissible strains and
Numerical experiments
This section is devoted to a series of numerical experiments, inspired by the field of biomedical imaging, that are intended to showcase the capabilities of the methodology of Section 3. They consist in imaging a viscoelastic inclusion (with unknown location, geometry, and characteristics) embedded in a viscoelastic background medium (all materials being assumed to be isotropic) by reconstructing the complex bulk and shear moduli fields and , treated as completely unknown. In particular,
Conclusions
The intended contributions of this work, devoted to the imaging of linearly viscoelastic heterogeneous moduli using interior data, include (i) the development of an algorithm exploiting a generalized error in constitutive equation functional which handles in a very natural way situations where boundary conditions are not known and (ii) a demonstration of its feasibility on 2D and 3D synthetic imaging problems involving up to tens of thousands of unknown viscoelastic moduli and using incomplete
Acknowledgment
Wilkins Aquino and Manuel Diaz were partially supported by NIH Grant # R01CA174723.
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