Inverse Elasticity Problem Research Articles

Exact solutions of the compressible inverse elasticity problem

In elastic modulus imaging or quantitative elastography, tissue stiffness is inferred from an (ultrasonically) measured displacement field. Doing so requires the solution of an elastic inverse problem. We present several exact solutions of the compressible inverse elasticity problem: Given the (possibly transient) dispalcement field measured everywhere in an isotropic, compressible, linear elastic solid, and given density ρ, determine the Lamé parameters λ(x) and μ(x). The cases we treat are: (a) for μ(x) known a priori, find λ; (b) for λ(x) known a priori, find μ; (c) neither λ(x) nor μ is known a priori, but Poisson’s ratio ν is known; (d) neither λ(x) nor μ nor ν is known a priori. We present several example applications including 2D and 3D problems, quasistatic and time dependent displacement fields. Many of the results here are drawn from P.E. Barbone and A.A. Oberai [‘‘Elastic Modulus Imaging: Some exact solutions of the compressible elastography inverse problem,’’ Phys. Med. Biol. (in press)].

Performance analysis of steady-state harmonic elastography

Shear modulus estimation can be confounded by the ill-posed nature of the inverse elasticity problem. In this paper, we report the results of experiments conducted on simulated and gelatin phantoms to investigate the effect of various parameters (i.e., regularization, spatial filtering and the subzone generation process) associated with shear modulus reconstruction on the statistical accuracy (mean squared error), and image quality (i.e., contrast and spatial resolution) of the recovered mechanical properties. The results indicate several interesting observations. Firstly, the intrinsic spatial resolution of magnetic resonance elastography (MRE) is dependent on both regularization and spatial filtering. Secondly, the elastographic contrast-to-noise ratio (CNRe) increases with increasing regularization and spatial filtering, but it was not affected by the zoning parameters (i.e., the subzones and the extent of the overlap). Thirdly, the statistical accuracy (MSE) of the recovered property improved with increasing regularization, and spatial filtering weight, but the size of the subdomains and their overlap had no significant effect.

Cavity identification in linear elasticity and thermoelasticity

AbstractThe aim of this paper is an analysis of geometric inverse problems in linear elasticity and thermoelasticity related to the identification of cavities in two and three spatial dimensions. The overdetermined boundary data used for the reconstruction are the displacement and temperature on a part of the boundary.We derive identifiability results and directional stability estimates, the latter using the concept of shape derivatives, whose form is known in elasticity and newly derived for thermoelasticity. For numerical reconstructions we use a least‐squares formulation and a geometric gradient descent based on the associated shape derivatives. The directional stability estimates guarantee the stability of the gradient descent approach, so that an iterative regularization is obtained. This iterative scheme is then regularized by a level set approach allowing the reconstruction of multiply connected shapes. Copyright © 2007 John Wiley & Sons, Ltd.

Numerical computation of an inverse contact problem in elasticity

An inverse contact problem is a classical ill-posed problem in the sense of Hadamard. In this paper we develop a numerical computational method for solving an inverse contact problem in elasticity. Based on the Tikhonov regularization technique, we transform the original problem into an optimization problem so that a stable numerical approximation to the solution can be obtained by a standard finite element method. A convergence analysis on the proposed numerical algorithm is also given by a conditional stability estimate. © VSP 2006.

Numerical computation of an inverse contact problem in elasticity

An inverse contact problem is a classical ill-posed problem in the sense of Hadamard. In this paper we develop a numerical computational method for solving an inverse contact problem in elasticity. Based on the Tikhonov regularization technique, we transform the original problem into an optimization problem so that a stable numerical approximation to the solution can be obtained by a standard finite element method. A convergence analysis on the proposed numerical algorithm is also given by a conditional stability estimate.

A Combined FEM/Genetic Algorithm for Vascular Soft tissue Elasticity Estimation

Tissue elasticity reconstruction is a parameter estimation effort combining imaging, elastography, and computational modeling to build maps of soft tissue mechanical properties. One application is in the characterization of atherosclerotic plaques in diseased arteries, wherein the distribution of elastic properties is required for stress analysis and plaque stability assessment. In this paper, a computational scheme is proposed for elasticity reconstruction in soft tissues, combining finite element modeling (FEM) for mechanical analysis of soft tissues and a genetic algorithm (GA) for parameter estimation. With a model reduction of the discrete elasticity values into lumped material regions, namely the plaque constituents, a robust, adaptive strategy can be used to solve inverse elasticity problems involving complex and inhomogeneous solution spaces. An advantage of utilizing a GA is its insistence on global convergence. The algorithm is easily implemented and adaptable to more complex material models and geometries. It is meant to provide either accurate initial guesses of low-resolution elasticity values in a multi-resolution scheme or as a replacement for failing traditional elasticity estimation efforts.

Enhancing the performance of model-based elastography by incorporating additional a priori information in the modulus image reconstruction process

Model-based elastography is fraught with problems owing to the ill-posed nature of the inverse elasticity problem. To overcome this limitation, we have recently developed a novel inversion scheme that incorporates a priori information concerning the mechanical properties of the underlying tissue structures, and the variance incurred during displacement estimation in the modulus image reconstruction process. The information was procured by employing standard strain imaging methodology, and introduced in the reconstruction process through the generalized Tikhonov approach. In this paper, we report the results of experiments conducted on gelatin phantoms to evaluate the performance of modulus elastograms computed with the generalized Tikhonov (GTK) estimation criterion relative to those computed by employing the un-weighted least-squares estimation criterion, the weighted least-squares estimation criterion and the standard Tikhonov method (i.e., the generalized Tikhonov method with no modulus prior). The results indicate that modulus elastograms computed with the generalized Tikhonov approach had superior elastographic contrast discrimination and contrast recovery. In addition, image reconstruction was more resilient to structural decorrelation noise when additional constraints were imposed on the reconstruction process through the GTK method.

Comparative evaluation of strain-based and model-based modulus elastography

Elastography based on strain imaging currently endures mechanical artefacts and limited contrast transfer efficiency. Solving the inverse elasticity problem (IEP) should obviate these difficulties; however, this approach to elastography is often fraught with problems because of the ill-posed nature of the IEP. The aim of the present study was to determine how the quality of modulus elastograms computed by solving the IEP compared with those produced using standard strain imaging methodology. Strain-based modulus elastograms ( i.e., modulus elastograms computed by simply inverting strain elastograms based on the assumption of stress uniformity) and model-based modulus elastograms ( i.e., modulus elastograms computed by solving the IEP) were computed from a common cohort of simulated and gelatin-based phantoms that contained inclusions of varying size and modulus contrast. The ensuing elastograms were evaluated by employing the contrast-to-noise ratio (CNR e) and the contrast transfer efficiency (CTE e) performance metrics. The results demonstrated that, at a fixed spatial resolution, the CNR e of strain-based modulus elastograms was statistically equivalent to those computed by solving the IEP. At low modulus contrast, the CTE e of both elastographic imaging approaches was comparable; however, at high modulus, the CTE e of model-based modulus elastograms was superior. (E-mail: marvin.m.doyley@dartmouth.edu)

Level set methods for geometric inverse problems in linear elasticity

In this paper, we investigate the regularization and numerical solution of geometric inverse problems related to linear elasticity with minimal assumptions on the geometry of the solution. In particular, we consider the probably severely ill-posed reconstruction problem of a two-dimensional inclusion from a single boundary measurement. In order to avoid parametrizations, which would introduce a priori assumptions on the geometric structure of the solution, we employ the level set method for the numerical solution of the reconstruction problem. With this approach, we construct an evolution of shapes with a normal velocity chosen depending on the shape derivative of the corresponding least-squares functional in order to guarantee its descent. Moreover, we analyse penalization by perimeter as a regularization method, based on recent results on the convergence of Neumann problems and a generalization of Golab's theorem. The behaviour of the level set method and of the regularization procedure in the presence of noise are tested in several numerical examples. It turns out that reconstructions of good quality can be obtained only for simple shapes or for unreasonably low noise levels. However, it seems reasonable that the quality of reconstructions improves by using more than a single boundary measurement, which is an interesting topic for future research.

Inverse Equivalent Elastic and Viscoelastic Loading Problems—Analytical and Computational Simulations of Experiments

Inverse Equivalent Elastic and Viscoelastic Loading Problems—Analytical and Computational Simulations of Experiments

Inverse Equivalent Elastic and Viscoelastic Loading Problems—Analytical and Computational Simulations of Experiments

Analytical protocols are presented for the determination of experimentally realizable relatively “simple” loading functions, consisting of concentrated loads that mimic actual quasistatic and/or dynamic more complicated loadings, to produce similar nearly equal equivalent deflection patterns. These computational simulations in essence represent a series of virtual experiments. Two protocols are formulated for the determination of equivalent loads. The first consists of “lumped loads” and in the second, Galerkin's method is used to determine the system of equivalent concentrated loads. The accuracy of the protocol is examined and shown that the simulation error can be made negligible with a manageable finite number of equivalent concentrated loads.

Identification of the Impulsive Load on an Elastic Rectangular Plate

The paper presents a stable solution to an ill-posed problem of identification of the time-dependent impulsive load acting on a plate (an inverse elastic problem). The time dependence of the deformation at one point of the plate is known from a numerical analysis, where errors occurring in a real experiment are modeled. The results obtained are compared with data for a rectangular plate under impact loading

Regularized boundary element solution for an inverse boundary value problem in linear elasticity

AbstractIn this paper the Tikhonov regularization method is analysed for the ill‐conditioned system of linear equations which arises from the discretization of an ill‐posed boundary value problem in linear elasticity. The choice of the regularization parameter is achieved according to the L‐curve method. Then the regularization method is numerically implemented using the boundary element method (BEM). Copyright © 2002 John Wiley & Sons, Ltd.

On uniqueness for an inverse problem in inhomogeneous elasticity

We consider the scattering of time‐harmonic elastic waves in an isotropic medium which is characterized by constant Lamé coefficients and an inhomogeneous mass density. We prove that a knowledge of the far‐field pattern for all incident plane waves and a single frequency provides sufficient information to determine the mass density uniquely. To this end we construct a periodic Faddeev‐type fundamental solution for the Navier equation and derive the existence of complex geometrical optics solutions to the Navier equation.

Optimization algorithms for identification inverse problems with the boundary element method

In this paper the most suitable algorithms for unconstrained optimization now available applied to an identification inverse problem in elasticity using the boundary element method (BEM) are compared. Advantage is taken of the analytical derivative of the whole integral equation of the BEM with respect to the variation of the geometry, direct differentiation, which can be used to obtain the gradient of the cost function to be optimized.

The effective properties of a perforated elastic plate Numerical optimization by genetic algorithm

We consider a perforated elastic plate in which identical traction-free holes form square or hexagonal periodic arrays. The optimization problem of finding the hole shape in this structure that maximizes either of its effective moduli is posed as a full scale inverse problem of elasticity. In this context, only limiting cases of a single hole and cellular solids with thin cell walls have been studied thus far. Here, we cover the gap between these cases numerically solving the problem by the genetic algorithm approach. A new time-saving scheme of the fitness evaluation provides reliable data even near the percolation limit. The presented numerical results comprehensively describe the optimal behavior of perforated plates in terms of equivalent homogeneous medium. Similar, though less sophisticated approach was used by the author Vigdergauz, (Int. J. Solids Struct. 38 (2001) 6851) to optimize an isolated inclusion in a plate.

A study on an estimation technique for the transverse impact of plates

This paper describes a numerical algorithm to determine the response of the structure due to a known force. The inverse dynamics problem in structural dynamics is to estimate the unknown forcing function using dynamic response. Theory of plates is employed to solve inverse problems in elasticity with specific applications to position and force sensing. In this paper, the contact force during the transverse impact of a plate is given and the response is solved. A Kalman filter technique was used to numerically estimate an applied force on the plate with the numerical response by an inverse method. In order to examine the accuracy of the proposed method, the plate is subjected to four type forces, which are triangular, sinusoidal, rectangular and random force. The estimated results have a good agreement with the exact values in all cases tested. Copyright © 2001 John Wiley & Sons, Ltd.

The Local Conditional Stability Estimate for an Inverse Contact Problem in the Elasticity

In this paper, we consider an inverse problem in elasticity of determining stress on a contact domain from measurements outside the contact domain. The local conditional stability estimate for determining the stress vector is obtained. This conditional stability can give the convergence rate for Tikhonov's regularized solutions.

An optimal control approach to an inverse nonlinear elastic shell problem applied to car windscreen design

A numerical procedure to optimise the sag bending process for car windscreens is proposed. In particular a simplified elastic model with geometric nonlinearities is considered. The optimisation approach is based on the numerical resolution of an optimal control problem in nonlinear elasticity. The objective of the optimisation is to match a desired shape by controlling the temperature distribution and minimising a quadratic functional. Also, we define and analyse the finite element approximation of the optimality system and a gradient method for the solution of the resulting discrete variational inequality. Finally, numerical experiments for the simulation of the simplified model are discussed.

A variational inequality formulation of an inverse elasticity problem

We present some systematic approaches to the mathematical formulation and numerical resolution of an optimal control problem in linear elasticity. The objective of the optimization is to match a desired displacement by controlling the Young's modulus so as to minimize a quadratic functional. Theoretical results are presented in the general framework of linear elastic theory which lead to a variational inequality. Also, we define and analyze a finite element approximation of the optimality system and a gradient method for the solution of the discrete variational inequality. Finally, numerical experiments for the simulation of a simplified model for the “sag bending process” in the manufacturing of automobile windscreens are discussed.