Three-dimensional transient elastodynamic inversion using an error in constitutive relation functional

Abstract

This work is concerned with large-scale three-dimensional inversion under transient elastodynamic conditions by means of the modified error in constitutive relation (MECR), an energy-based, cost functional. In contrast to quasi-static or frequency-domain contexts, time-domain formulations have so far seen very limited investigation. A peculiarity of time-domain MECR formulations is that each evaluation involves the solution of two elastodynamic problems (one forward, one backward), which moreover are coupled (unlike the case of L 2 misfit functionals, where the forward state does not depend on the adjoint state). This coupling creates a major computational bottleneck, making MECR-based inversion difficult for spatially 2D or 3D configurations. To overcome this obstacle, we propose an approach whose main ingredients are (a) setting the entire computational procedure in a consistent timediscrete framework that incorporates the chosen time-stepping algorithm, and (b) using an iterative SOR-like method for the resulting stationarity equations. The resulting MECR-based inversion algorithm is formulated under quite general conditions, allowing for three-dimensional transient elastodynamics, straightforward use of available parallel solvers, a wide array of time-stepping algorithms commonly used for transient structural dynamics, and flexible boundary condition and measurement settings. The proposed MECR algorithm is then demonstrated on computational experiments involving 2D and 3D transient elastodynamics and up to over 500,000 unknown elastic moduli.

Analysis of the Error in Constitutive Equation Approach for Time-Harmonic Elasticity Imaging

“…Moreover, for time-harmonic conditions, the FEM-discretized coupled stationarity problem was shown in [4] to remain uniquely solvable at resonant prescribed frequencies. In addition, as shown in [8,12], ECE approaches can naturally accommodate configurations with partially or completely unknown BCs, a feature also used in [6] for parameter identification using elastostatic interior data (i.e. unknown BCs).…”

Robust energy-based model updating framework for random processes in dynamics: Application to shaking-table experiments

“…Besides, the CRE part of the residual, computed over the whole structure, allows to select the most erroneous areas in order to restrain the updating process to a few parameters; this provides another additional regularization (in the Tikhonov sense), particularly when the number of parameters to update becomes important. The mCRE model updating methodology has proved robustness and efficiency through a large number of applications involving defect detection [41][42][43][44], very noisy or even corrupted measurements [38,[45][46][47], tolerance to incomplete boundary conditions [40,48,49], full-field material identification [50][51][52], acoustics [53,54], wall-slab RC joint characterization [55], real-time model updating and data-assimilation [56] or coupling with model order reduction techniques like Proper Generalized Decomposition (PGD) [57]. Recently, a unified formulation of the mCRE has been proposed (in the time domain) for the full updating of constitutive relations and evolution laws in a nonlinear context [58].…”

Direct error in constitutive equation formulation for plane stress inverse elasticity problem

“…λ is a Lagrange multiplier used to enforce the equilibrium equation constraint. Because this formulation depends on minimizing ‖ σ − μ A ‖ 2 , it falls under the umbrella of ECE methods [27, 42, 28]. It is quadratic in the unknowns, however, which allows us to solve for the modulus field directly.…”

“…A popular objective function to minimize in such problems is based on the error in the constitutive equation, and so are called “ECE approaches” [25, 26, 27, 28, 29]. Such iterative methods tend to be relatively robust in dealing with noise, accommodate multiple measurements naturally, and can even handle missing measurements.…”