Large scale parameter estimation problems in frequency-domain elastodynamics using an error in constitutive equation functional

Abstract

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.

Inverse material identification in coupled acoustic-structure interaction using a modified error in constitutive equation functional

“…Like in earlier applications of MECE to parameter identification, e.g. [5, 3], this subdivision makes it natural to adopt an iterative alternating strategy of block Gauss-Seidel type whereby each solution iteration for (21) consists in (i) solving the set of linear equations (21a)–(21d) for the mechanical fields and Lagrange multipliers, with kept fixed, and (ii) updating via equation (21e). Here, this method will be seen to entail solving a 4 × 4 complex symmetric block linear system followed (when considering isotropic materials) by simple and explicit update formulas for the elastic moduli, similar to those used in [3].…”

“…More recent extensions of the MECE approach to time-domain formulations [2,16] were shown to be very robust in the presence of high levels of noise, while also providing an inherent error estimate through the ECE. The MECE method was extended to large scale identification problems in [3], where the authors also showed that this approach displayed fast convergence and accuracy as compared to conventional least-squares minimization approaches.…”

Efficient formulations of the material identification problem using full-field measurements

“…In this case it is considered an error included directly in the displacements vector by using the procedure applied in [5,43]. The results obtained in this work and in the papers cited are not directly comparable since more than one displacement field is used, thus a different MIP is solved.…”

“…The geometry considered in this case also corresponds to a problem usually used in the literature [5]. We start by describing the problem, and after that, we present the results obtained when different kinds of errors are introduced in the data.…”

“…This is a widely used formulation defined in the continuum [5,22,23] or in the discretized domain as in the formulation FU [11,16,18]. Let us consider the error defined in the continuous domain as:…”