Solution of the nonlinear elasticity imaging inverse problem: The incompressible case

Abstract

We have recently developed and tested an efficient algorithm for solving the nonlinear inverse elasticity problem for a compressible hyperelastic material. The data for this problem are the quasi-static deformation fields within the solid measured at two distinct overall strain levels. The main ingredients of our algorithm are a gradient based quasi-Newton minimization strategy, the use of adjoint equations and a novel strategy for continuation in the material parameters. In this paper we present several extensions to this algorithm. First, we extend it to incompressible media thereby extending its applicability to tissues which are nearly incompressible under slow deformation. We achieve this by solving the forward problem using a residual-based, stabilized, mixed finite element formulation which circumvents the Ladyzenskaya–Babuska–Brezzi condition. Second, we demonstrate how the recovery of the spatial distribution of the nonlinear parameter can be improved either by preconditioning the system of equations for the material parameters, or by splitting the problem into two distinct steps. Finally, we present a new strain energy density function with an exponential stress-strain behavior that yields a deviatoric stress tensor, thereby simplifying the interpretation of pressure when compared with other exponential functions. We test the overall approach by solving for the spatial distribution of material parameters from noisy, synthetic deformation fields.

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

“…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:…”

A generalized computationally efficient inverse characterization approach combining direct inversion solution initialization with gradient-based optimization

“…As the associated eigenvalues from the solution of Eqn. (10) relate to the value of each mode for the representation of the given dataset, typically some heuristic is used based on the relative sum of the associated eigenvalues to determine the modes to retain for further use [21,22]. One additional note is that the Gappy POD approach can also act as somewhat of a noise filter during the reconstruction process, thereby providing an added benefit of reducing the effects of measurement noise on the subsequent direct inversion solution procedure.…”

“…Note that the only difference in the formulation presented herein to convert to static rather than steady-state dynamic is that the momentum terms are set to zero. This second example scenario was intended to relate to potential applications of tissue characterization (e.g., tumor characterization) from (quasi-) static mechanical testing [10,13].…”

“…[8,9] and those that use gradient-based optimization (e.g., Newton's method, conjugate gradient, etc.) [4,10]. Non-gradient-based methods often have more significant global search capabilities in comparison to the gradient-based optimization approaches, which can become trapped in local minima (i.e., an inaccurate solution).…”

Computational Time Reversal for NDT Applications Using Experimental Data

“…If the objective function surface is smooth enough, a gradienttype minimization scheme may be employed. The adjoint method [8,10,11] can then be used to calculate the gradient of the objective function efficiently. A different class of solution methods is based on probabilistic analysis, as in [12].…”