Inverse Elasticity Problem Research Articles

Circumventing the solution of inverse problems in mechanics through deep learning: Application to elasticity imaging

The ability to make decisions based on quantities of interest that depend on variables inferred from measurement finds application in different fields of mechanics and physics. The evaluation of the inferred variables, and hence the quantities of interest, from the measurement typically requires the solution of an inverse problem. For example, in medical imaging the elastic heterogeneity of a tumor and its nonlinear elastic response can be used to distinguish benign tumors from their malignant counterparts. These images of linear and nonlinear elastic parameters of tissue are typically obtained by using a measured displacement field and solving a complex inverse elasticity problem. In this paper we consider circumventing the solution of the inverse problem by using measured displacements as input to a deep convolutional neural network (CNN) and training it to classify tumors on the basis of their elastic heterogeneity and nonlinearity. For a simple, 5-layer CNN trained with 8,000 samples for heterogeneity, and a 4-layer CNN trained with 4,000 samples for nonlinear elasticity we report classification accuracies in the range of 99.7%−99.9%. The training and testing data are both obtained from the forward solution of finite element models of samples. We also analyze the weights of the trained model to understand the process through which the network extracts features of elastic moduli from the input displacement images. Finally, we apply the nonlinear elasticity classifier, which is trained entirely using simulated data, to displacement images obtained from ten patients with breast lesions and note that it correctly classifies eight out of ten cases. This application illustrates how data from physics-based models can be used in improving the performance of a data-driven algorithm in data-sparse scenarios.

Inverse Problem of Elasticity for a Plate Weakened by Hole and Cracks

An isotropic elastic plate weakened by a hole and arbitrary system of cracks is considered. Using the principle of equal strength and minimization of stress intensity factors a theoretical analysis is made of the equistrong hole shape. A criterion and a method for solving the inverse problem of fracture prevention for the plate with a hole under the action of given system of external loads are proposed. A closed system of algebraic equations allowing minimization of the stress state and stress intensity factors depending on geometrical and mechanical characteristics of the plate is obtained. The obtained optimal hole shape provides an increase in the load-bearing capacity of the plate.

Method of automorphic functions for an inverse problem of antiplane elasticity

A nonlinear inverse problem of antiplane elasticity for a multiply connected domain is examined. It is required to determine the profile of $n$ uniformly stressed inclusions when the surrounding infinite body is subjected to antiplane uniform shear at infinity. A method of conformal mappings of circular multiply connected domains is employed. The conformal map is recovered by solving consequently two Riemann-Hilbert problems for piecewise analytic symmetric automorphic functions. For domains associated with the first class Schottky groups a series-form representation of a ($3n-4$) parametric family of conformal maps solving the problem is discovered. Numerical results for two and three uniformly stressed inclusions are reported and discussed.

In Vivo Tension Sensors Deliver Novel Insight into Mechanics of Zebrafish Gastrulation

Mechanical tension has recently been recognized as a key element to understand many biological processes such as collective cell migration during cancer invasion and embryogenesis. However, direct experimental access to determine tension in vivo in a non-destructive way remains a major challenge. Here, we present a novel experimental approach that allows direct measurement of stress inside in vivo tissue. By injecting fluorescent polyacrylamide (PAA) beads of known elasticity in the tissue, we can generate time resolved 3D images of the beads that are used to quantify the bead deformation. Solving the inverse elastic problem yields an approximation of the stress field inside the tissue. PAA beads are injected into zebrafish embryos to investigate the role of tissue stress for the collective cell migration during embryogenesis. At the same time positions of nuclei are detected and tracked over time using state of the art tracking techniques. Combining tissue flow, mechanical tension and cell density allows to directly differentiate between pure mechanical control of cell motion, for example via hydrodynamic or elastic effects, and active cell motion.

MFS fading regularization method for the identification of boundary conditions from partial elastic displacement field data

ABSTRACTA method is proposed to solve an inverse problem in two-dimensional linear isotropic elasticity. The inverse problem consists of the determination of both the entire displacement field and ...

Multifrequency Iterative Methods for the Inverse Medium Scattering Problems in Elasticity

This paper concerns the reconstruction of multiple elastic parameters (Lamé parameters and density) of an inhomogeneous medium embedded in an infinite homogeneous isotropic background in $\mathbb{R}^2$. The direct scattering problem is reduced to an equivalent system on a bounded domain by introducing an exact transparent boundary condition and the well-posedness of the corresponding variational problem is established. The Fréchet differentiability of the near-field scattering map is studied with respect to the elastic parameters. Based on the multifrequency measurement data and its phaseless term, two Landweber iterative algorithms are developed for the reconstruction of the multiple elastic parameters. Numerical examples, indicating that plane pressure incident wave is a better choice, are presented to show the validity and accuracy of our methods.

Extended Sampling Method for Inverse Elastic Scattering Problems Using One Incident Wave

We consider the inverse elastic scattering problems using the far field data due to one incident plane wave. A simple method is proposed to reconstruct the location and size of the obstacle using different components of the far field pattern. The method sets up linear ill-posed integral equations for sampling points in the domain of interrogation and uses the (approximate) solutions to compute indicators. Using the far field patterns of rigid disks as the kernels of the integral equations and moving the measured data to the right-hand side, the method has the ability to process limited aperture data. Numerical examples show that the method can effectively determine the location and approximate the support of the obstacle with little a priori information.

MFS fading regularization method for the identification of boundary conditions from partial elastic displacement field data

A method is proposed to solve an inverse problem in twodimensional linear isotropic elasticity. The inverse problem consists of the determination of both the entire displacement field and the boundary conditions inaccessible to the measurement from the partial knowledge of the displacement field. The algorithm is based on a fading regularization method (FRM) and is numerically implemented using the method of fundamental solutions (MFS). The inverse technique is first validated with synthetic data and is then applied to the interpretation of experimental measurements obtained by digital image correlation (DIC).

Interactive Two-Way Shape Design of Elastic Bodies

We present a novel system for interactive elastic shape design in both forward and inverse fashions. Using this system, the user can choose to edit the rest shape or the quasistatic shape of an elastic solid, and obtain the other shape that matches under the quasistatic equilibrium condition at the same time. The development of this system is based on the discovery that inverse quasistatic simulation can be immediately solved by Newton's method with a direct solver. To implement our simulator, we propose a Jacobian matrix evaluation scheme for the inverse elastic problem and we present step length and matrix evaluation techniques that improve the simulation performance. While our simulator is efficient, it is still not fast enough for the system to generate the result in real time. Our solution is a shape initialization method using the recent projective dynamics technique. Shape initialization not only works as a fast preview function during the user editing process, but also speeds up the convergence of quasistatic or inverse quasistatic simulation afterwards. The use of a heterogeneous algorithm structure allows the system to further reduce its preview cost, by utilizing the power of both the CPU and the GPU. Our experiment demonstrates that the whole system is fast, robust, and convenient for the designer to use in both forward and inverse elastic shape design. It can handle a variety of nonlinear elastic material models, and its runtime performance has space for more improvement.

Robust Reconstruction of Elasticity Using Ultrasound Imaging and Multi-Frequency Excitations.

Biomedical parameters of tissue can be important indicators for clinical diagnosis. One such parameter that reflects tissue stiffness is elasticity, the imaging of which is called elastography. In this paper, we use displacements from harmonic excitations to solve the inverse problem of elasticity based on a finite-element method (FEM) formulation. This leads to iterative solution of nonlinear and nonconvex problems. In this paper, we show the importance and selection of viable initializations in numerical simulation studies and propose techniques for the fusion of multiple initializations for ideal reconstructions of unknown tissue as well as combining information from excitations at multiple frequencies. Results show that our method leads up to 76% decrease in root-mean-squared error (RMSE) and 9.9 dB increase in contrast-to-noise ratio (CNR) in simulations with noise, when compared to conventional iterative FEM without multiple initializations and frequencies. As the wave patterns in individually selected frequencies may introduce artifacts, a joint inverse-problem solution of multi-frequency excitations is introduced as a robust solution, where CNR improvements of up to 11.9 dB are observed. We also present the methods on a tissue-mimicking gelatin phantom study using mechanical excitation and ultrafast plane-wave ultrasound imaging, where the RMSE was improved by up to 51%. An experiment of ablation via heating an ex-vivo bovine liver shows that reconstruction artifacts are reduced with our proposed method.

A comparative study of two constitutive models within an inverse approach to determine the spatial stiffness distribution in soft materials

A comparative study is presented to solve the inverse problem in elasticity for the shear modulus (stiffness) distribution utilizing two constitutive equations: (1) linear elasticity assuming small strain theory, and (2) finite elasticity with a hyperelastic neo-Hookean material model. Assuming that a material undergoes large deformations and material nonlinearity is assumed negligible, the inverse solution using (2) is anticipated to yield better results than (1). Given the fact that solving a linear elastic model is significantly faster than a nonlinear model and more robust numerically, we posed the following question: How accurately could we map the shear modulus distribution with a linear elastic model using small strain theory for a specimen undergoing large deformations? To this end, experimental displacement data of a silicone composite sample containing two stiff inclusions of different sizes under uniaxial displacement controlled extension were acquired using a digital image correlation system. The silicone based composite was modeled both as a linear elastic solid under infinitesimal strains and as a neo-Hookean hyperelastic solid that takes into account geometrically nonlinear finite deformations. We observed that the mapped shear modulus contrast, determined by solving an inverse problem, between inclusion and background was higher for the linear elastic model as compared to that of the hyperelastic one. A similar trend was observed for simulated experiments, where synthetically computed displacement data were produced and the inverse problem solved using both, the linear elastic model and the neo-Hookean material model. In addition, it was observed that the inverse problem solution was inclusion size-sensitive. Consequently, an 1-D model was introduced to broaden our understanding of this issue. This 1-D analysis revealed that by using a linear elastic approach, the overestimation of the shear modulus contrast between inclusion and background increases with the increase of external loads and target shear modulus contrast. Finally, this investigation provides valuable information on the validity of the assumption for utilizing linear elasticity in solving inverse problems for the spatial distribution of shear modulus associated with soft solids undergoing large deformations. Thus, this work could be of importance to characterize mechanical property variations of polymer based materials such as rubbers or in elasticity imaging of tissues for pathology.

Slit Maps in the Study of Equal-Strength Cavities in $n$-Connected Elastic Planar Domains

The inverse problem of plane elasticity on $n$ equal-strength cavities in a plane subjected to constant loading at infinity and in the cavities' boundaries is analyzed. By reducing the governing boundary value problem to the Riemann--Hilbert problem on a symmetric Riemann surface of genus $n-1$, a family of conformal mappings from a parametric slit domain onto the $n$-connected elastic domain is constructed. The conformal mappings are presented in terms of hyperelliptic integrals, and the zeros of the first derivative of the mappings are analyzed. It is shown that for any $n\ge 1$ there always exists a set of the loading parameters for which these zeros generate inadmissible poles of the solution.

A Mathematical Framework for Deep Learning in Elastic Source Imaging

An inverse elastic source problem with sparse measurements is our concern. A generic mathematical framework is proposed which extends a low-dimensional manifold regularization in the conventional source reconstruction algorithms thereby enhancing their performance with sparse data-sets. It is rigorously established that the proposed framework is equivalent to the so-called deep convolutional framelet expansion in machine learning literature for inverse problems. Apposite numerical examples are furnished to substantiate the efficacy of the proposed framework.

Improving the sensitivity to map nonlinear parameters for hyperelastic problems

The mechanical properties of tissues are important indicators of tissue “health”. Diseased tissues due to cancer and other causes tend to stiffen with increase in strain, exhibiting a nonlinear stress–strain behavior. The hyperelastic mechanical behavior of soft tissues can be characterized by an exponential model with two material parameters, namely the shear modulus μ and a nonlinearity parameter γ. A variety of methods and techniques have been developed to solve inverse problems in nonlinear elasticity to determine these properties given the mechanical response of the tissues. Reconstruction of the nonlinear parameter from noisy measurements of displacement response is a difficult problem, and obtaining a well-recovered solution is challenging. This article is directed towards the improvement in the reconstruction of the nonlinear parameter, γ, by introducing a new parameter, which is a combination of γ and the first invariant of the Cauchy–Green deformation tensor. Comparative study is carried out between reconstructions of γ directly from previously existing formulations and the reconstruction of γ from the new parameter, for 2D problems. The new method is compared with previous methods using numerical experiments in terms of shape of the stiff regions, the contrast in γ and robustness under different loading conditions. We find that the new method is a considerable improvement to previous methods and could be a valuable tool in biomedical applications.

Inverse Solution for Bolt Preload Using Surface Deformation

In this paper, an inverse biharmonic axisymmetric elasticity problem is solved by invoking measured out-of-plane surface deformation values at discrete locations around a preloaded bolt head, in order to calculate the underhead contact stress and joint clamp load that would have caused that out-of-plane surface deformation. Solution of this type of inverse problem promises to improve the automation process of bolted joint system assembly, especially in critical and safety-related applications. For example, a real-time optically measured joint surface deformation can be utilized for automating process control of bolted joint assembly in a reliable fashion. This would be a significant reliability improvement as compared to the commonly used method in mass production using torque-only control method in which there is wide scatter in the torque–tension correlation due to the normal scatter in frictional variables. Finite element analysis (FEA) method is used to validate the inverse problem solution provided in this paper.

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

We present a new computational formulation for inverse problems in elasticity with full field data. The formulation is a variant of an error in the constitutive equation formulation, but allows direct solution for the modulus field and accommodates discontinuous strain fields. The development of the formulation is motivated by the relatively poor performance of current direct formulations, reported so far in literature, in dealing with discontinuities in the strain and material property distribution. The formulation relies on minimizing the error in the constitutive equation with a momentum equation constraint. Numerical results on model problems show that the formulation is capable of handling discontinuous and noisy strain fields, and also converging with mesh refinement for continuous and discontinuous material property distributions. The application to reconstruct the elastic modulus distribution in solid breast tumors is shown.

Numerical Analysis of Inverse Elasticity Problemwith Signorini's Condition

AbstractAn optimal control problem is considered to find a stable surface traction, which minimizes the discrepancy between a given displacement field and its estimation. Firstly, the inverse elastic problem is constructed by variational inequalities, and a stable approximation of surface traction is obtained with Tikhonov regularization. Then a finite element discretization of the inverse elastic problem is analyzed. Moreover, the error estimation of the numerical solutions is deduced. Finally, a numerical algorithm is detailed and three examples in two-dimensional case illustrate the efficiency of the algorithm.

Factorization method in the geometric inverse problem of static elasticity

The factorization method, which has previously been used to solve inverse scattering problems, is generalized to geometric inverse problems of static elasticity. We prove that finitely many defects (cavities, cracks, and inclusions) in an isotropic linearly elastic body can be determined uniquely if the operator that takes the forces applied to the body outer boundary to the outer boundary displacements due to these forces is known.

Non-iterative regularized MFS solution of inverse boundary value problems in linear elasticity: A numerical study

The numerical reconstruction of the missing Dirichlet and Neumann data on an inaccessible part of the boundary in the case of two- and three-dimensional linear isotropic elastic materials from the knowledge of over-prescribed noisy measurements taken on the remaining accessible boundary part is investigated. This inverse problem is solved using the method of fundamental solutions (MFS), whilst its stabilization is achieved through several singular value decomposition (SVD)-based regularization methods, such as the Tikhonov regularization method [48], the damped SVD and the truncated SVD [18]. The regularization parameter is selected according to the discrepancy principle [40], generalized cross-validation criterion [14] and Hansen’s L-curve method [20].

A fast multiple-scale polynomial solution for the inverse Cauchy problem of elasticity in an arbitrary plane domain

The polynomial expansion method together with the collocation technique is a cheap yet simple method to solve the Navier equations of elasticity, which is easily arranged to satisfy the governing equations and boundary conditions pointwise. In this paper we propose a novel numerical algorithm for the solution of an overspecified/underspecified Cauchy problem of linear elasticity in an arbitrary plane domain, by using the multiple-scale Pascal polynomial expansion method (MSPEM), of which the scales are determined a priori by the collocation points, according to the idea of equilibrated matrix. In the numerical tests of a direct problem as well as the overspecified/underspecified Cauchy problems, the MSPEM is very accurate and stable against large relative noise up to 20% for the unknown displacements recovery problem, and up to 100% absolute noise for the recovery of unknown loading force. The present method is convergent very fast for most cases within 100 iteration steps.