Biswanath Banerjee

Quantitative photoacoustic tomography from boundary pressure measurements: noniterative recovery of optical absorption coefficient from the reconstructed absorbed energy map

We describe a noniterative method for recovering optical absorption coefficient distribution from the absorbed energy map reconstructed using simulated and noisy boundary pressure measurements. The source reconstruction problem is first solved for the absorbed energy map corresponding to single- and multiple-source illuminations from the side of the imaging plane. It is shown that the absorbed energy map and the absorption coefficient distribution, recovered from the single-source illumination with a large variation in photon flux distribution, have signal-to-noise ratios comparable to those of the reconstructed parameters from a more uniform photon density distribution corresponding to multiple-source illuminations. The absorbed energy map is input as absorption coefficient times photon flux in the time-independent diffusion equation (DE) governing photon transport to recover the photon flux in a single step. The recovered photon flux is used to compute the optical absorption coefficient distribution from the absorbed energy map. In the absence of experimental data, we obtain the boundary measurements through Monte Carlo simulations, and we attempt to address the possible limitations of the DE model in the overall reconstruction procedure.

A pseudo-dynamical systems approach to a class of inverse problems in engineering

A pseudo-dynamical approach for a class of inverse problems involving static measurements is proposed and explored. Following linearization of the minimizing functional associated with the underlying optimization problem, the new strategy results in a system of linearized ordinary differential equations (ODEs) whose steady-state solutions yield the desired reconstruction. We consider some explicit and implicit schemes for integrating the ODEs and thus establish a deterministic reconstruction strategy without an explicit use of regularization. A stochastic reconstruction strategy is then developed making use of an ensemble Kalman filter wherein these ODEs serve as the measurement model. Finally, we assess the numerical efficacy of the developed tools against a few linear and nonlinear inverse problems of engineering interest.

A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements

We propose a pseudo-dynamic form of a sub-optimal Kalman filter for elastography of plane-strain models of soft tissues under strictly static deformations and partial measurements. Since the tissue material is nearly incompressible and is thus prone to volumetric locking via standard displacement-based finite element formulations, we use a Cosserat point approach for deriving the static equilibrium equations. A pseudo-dynamical form of the equilibrium equations, with added noise and appropriate augmentation by the discretized shear modulus as additional states, is then adopted as the process equation such that its steady-state solution approaches the static response of the plane-strain model. A fictitious noise of small intensity is also added to the measurement equation and, following linearization of the process equation, a Kalman filter is applied to reconstruct the shear modulus profile. We present several numerical experiments, some of which also bring forth the relative advantages of the proposed approach over a deterministic reconstruction based on a quasi-Newton search.

Efficient implementations of a pseudodynamical stochastic filtering strategy for static elastography

A computationally efficient pseudodynamical filtering setup is established for elasticity imaging (i.e., reconstruction of shear modulus distribution) in soft-tissue organs given statically recorded and partially measured displacement data. Unlike a regularized quasi-Newton method (QNM) that needs inversion of ill-conditioned matrices, the authors explore pseudodynamic extended and ensemble Kalman filters (PD-EKF and PD-EnKF) that use a parsimonious representation of states and bypass explicit regularization by recursion over pseudotime. Numerical experiments with QNM and the two filters suggest that the PD-EnKF is the most robust performer as it exhibits no sensitivity to process noise covariance and yields good reconstruction even with small ensemble sizes.

Anisotropic linear elastic parameter estimation using error in the constitutive equation functional

A modified error in the constitutive equation-based approach for identification of heterogeneous and linear anisotropic elastic parameters involving static measurements is proposed and explored. Following an alternating minimization procedure associated with the underlying optimization problem, the new strategy results in an explicit material parameter update formula for general anisotropic material. This immediately allows us to derive the necessary constraints on measured data and thus restrictions on physical experimentation to achieve the desired reconstruction. We consider a few common materials to derive such conditions. Then, we exploit the invariant relationships of the anisotropic constitutive tensor to propose an identification procedure for space-dependent material orientations. Finally, we assess the numerical efficacy of the developed tools against a few parameter identification problems of engineering interest.

Elastic Parameter Identification of Plate Structures Using Modal Response: An ECE Based Approach

AbstractThis article describes an inverse method for the identification of heterogeneous anisotropic elastic parameters of plate structure from an incomplete modal response. Many efforts in this area have shown that derivatives of the displacement mode shapes or curvature mode shapes are much more sensitive to the heterogeneous parameter distribution of the structure when compared with the mode shapes themselves. Thus, the curvature mode shapes are used widely to quantify the material profile. Following this observation, a strategy based on the error in constitutive equation (ECE) function, which naturally incorporates this fact seamlessly into the identification procedure, is proposed. The identification problem is posed as an optimization problem in that the cost function measures the discrepancy in the constitutive equation, which connects kinematically admissible curvature/strains and dynamically admissible couples/stresses. The resulting system becomes an extended system with primary and Lagrangian v...

Convergence analysis of the Newton algorithm and a pseudo-time marching scheme for diffuse correlation tomography

We propose a self-regularized pseudo-time marching scheme to solve the ill-posed, nonlinear inverse problem associated with diffuse propagation of coherent light in a tissuelike object. In particular, in the context of diffuse correlation tomography (DCT), we consider the recovery of mechanical property distributions from partial and noisy boundary measurements of light intensity autocorrelation. We prove the existence of a minimizer for the Newton algorithm after establishing the existence of weak solutions for the forward equation of light amplitude autocorrelation and its Fréchet derivative and adjoint. The asymptotic stability of the solution of the ordinary differential equation obtained through the introduction of the pseudo-time is also analyzed. We show that the asymptotic solution obtained through the pseudo-time marching converges to that optimal solution provided the Hessian of the forward equation is positive definite in the neighborhood of optimal solution. The superior noise tolerance and regularization-insensitive nature of pseudo-dynamic strategy are proved through numerical simulations in the context of both DCT and diffuse optical tomography.

An axisymmetric model for Taylor impact test and estimation of metal plasticity

The impact of a flat-ended cylindrical rod onto a rigid stationary anvil, often known as the Taylor impact test, is studied. An axisymmetric model is developed to capture the deformation behaviour of the rod after impact. The most distinctive feature of the proposed model is that it takes into account the spatial and temporal variation of both longitudinal and radial deformation and consequently the strains and strain rates. The final deformed shapes and time histories of different field variables, as obtained from the model, are found to be in good agreement with corresponding experimental and numerical results reported in the literature. The proposed model is then used to formulate an inverse framework to estimate the Johnson–Cook constitutive parameters. In the inverse formulation, the objective function is constructed using the final deformed length and diameter at the impact end of the retrieved rod. Finally, the potential of the proposed model in estimating material parameters is illustrated through some examples.

Thermo-elastic material parameters identification using modified error in constitutive equation approach

Abstract This paper presents an identification procedure for anisotropic thermo-elastic heterogeneous material profile based on modified error in constitutive equation (MECE) approach. The inverse problem is posed as an optimization problem where the objective functional evaluates the difference in constitutive relation that associates kinematically admissible strain field to the statically admissible stress field. An additional term due to corruption in measurement data is included in the cost functional as a penalty form. While following standard MECE-based identification procedure, we have proposed a trace norm of the constitutive discrepancy functional that arises due to two dissimilar fields for material parameter update. In the process, we obtain explicit parameter update formula for general anisotropic thermo-elastic material. However, unlike elastic case, parameter update equations are nonlinear due to thermo-elastic constitutive relation. Finally, the potential of the proposed procedure in estimating anisotropic material parameters is illustrated through some large-scale parameter estimation problems.

Noniterative inversion strategy for photoacoustic tomography: Recovery of absorbed energy map from boundary pressure measurements

The inverse problem in photoacoustic tomography (PAT) seeks to obtain the absorbed energy map from the boundary pressure measurements for which computationally intensive iterative algorithms exist. The computational challenge is heightened when the reconstruction is done using boundary data split into its frequency spectrum to improve source localization and conditioning of the inverse problem. The key idea of this work is to modify the update equation wherein the Jacobian and the perturbation in data are summed over all wave numbers, k, and inverted only once to recover the absorbed energy map. This leads to a considerable reduction in the overall computation time. The results obtained using simulated data, demonstrates the efficiency of the proposed scheme without compromising the accuracy of reconstruction.