A computationally efficient approach for inverse material characterization combining Gappy POD with direct inversion

Highlights

• Gappy POD was used to reconstruct full-field response from partial-field measurements.

• A physics-based direct inversion procedure using full-field response was presented.

• Once the POD modes are obtained, the inversion is equivalent to a single FEA.

• Practical applicability of the presented approach was experimentally validated.

Abstract

An approach for computationally efficient inverse material characterization from partial-field response measurements that combines the Gappy proper orthogonal decomposition (POD) machine learning technique with a physics-based direct inversion strategy is presented and evaluated. Gappy POD is used to derive a data reconstruction tool from a set of potential system response fields that are generated from available a priori information regarding the potential distribution of the unknown material properties. Then, the Gappy POD technique is applied to reconstruct the full spatial distribution of the system response from whatever portion of the response field has been measured with the chosen system testing method. Lastly, a direct inversion strategy is presented that is derived from the equations governing the system response (i.e., physics of the system), which utilizes the full-field response reconstructed by Gappy POD to produce an estimate of the spatial distribution of the unknown material properties. The direct inversion technique is a particularly computationally efficient inversion technique, requiring a cost equivalent to a single numerical analysis. Therefore, the majority of the computational expense of the presented approach is the one-time potential response generation for the Gappy POD technique, which leads to an approach that is substantially computationally efficient overall. Two numerically simulated examples are shown in which the elastic modulus distribution was characterized based on partial-field displacement response measurements, both static and dynamic. The inversion procedure was shown to have the capability to efficiently provide accurate estimates to material property distributions from partial-field response measurements. The direct inversion with Gappy POD response estimation was also shown to be substantially tolerant to noise in comparison to the direct inversion given measured full-field response. Lastly, a physical example regarding elastography of an arterial construct from ultrasound imaging response measurements is shown to validate the practical applicability of the direct inversion approach with Gappy POD response reconstruction.

Introduction

Inverse problems relating to the characterization of various material properties of a variety of solids/structures and systems are of paramount importance in a wide range of science and engineering fields. For instance, structures, from industrial to biological, could be evaluated to determine their current state of health based upon their material properties, whether mechanical, thermal, or electrical. Corresponding to this substantial interest in material property characterization, a wide variety of inverse problem solution strategies have been developed relating to a variety of applications, such as structural health monitoring and nondestructive evaluation  [1], [2] and biomechanical imaging  [3], [4], [5], [6], [7], among other applications  [8], [9].

Since it is often not possible to find analytical solutions for inverse problems in practice, due to problem ill-posedness and/or complexities in geometry and boundary conditions, among other challenges, computational inverse characterization solution approximation approaches have become common. Overall, computational inverse characterization approaches, which are typically based around some type of computational representation of the mechanics of the system of interest (e.g., finite element analysis), have been shown through several studies  [9], [5], [3] to provide generalized frameworks for treating and distinguishing between various contributions to a system response, while providing physically meaningful solutions that can be applied to predict future behaviors. However, there is as wide a variety of computational inverse problem approaches as there are applications, with each having different strengths and weaknesses, and their effectiveness is significantly dependent upon the specifics of the application of interest. Moreover, the different computational approaches often differ significantly in the tradeoff between computational efficiency and solution accuracy.

One general way that computational inverse problem solution approaches can be divided is into those that use iterative optimization and those that are non-iterative/direct. Typically, the optimization-based approaches attempt to determine the unknown properties that minimize the difference between the measured system response and the response predicted by the computational representation of the mechanics of the system  [2], [10], [11]. The optimization approaches can be divided further into those that use gradient-based optimization (e.g., Newton’s method, conjugate gradient, etc.) and those that use non-gradient-based optimization (e.g., random search, genetic algorithm, etc.) to minimize the response error. Gradient-based optimization methods typically require substantially less iterations to converge to a solution approximation in comparison to non-gradient-based methods (i.e., are less computationally expensive), but often become trapped in local minima (i.e., an inaccurate solution), while non-gradient-based (since they commonly include some stochastic component) often have closer to global search capabilities. Alternatively, non-iterative methods attempt to somehow directly relate the measured response to the unknown parameters  [12], [13]. As such, once set up, non-iterative methods are naturally almost negligible in computing cost in comparison to the optimization-based approaches.

Non-iterative methods include machine learning approaches that use experimentally and/or numerically generated datasets of potential inverse problems solution parameters and the corresponding system responses to train a “surrogate” mapping (e.g., artificial neural network) to approximate the relationship between the system response (as inputs to the mapping) and inverse unknowns (as outputs of the mapping)  [14], [15]. Then, provided with a measured system response, the surrogate mapping can estimate inverse solution parameters in fractions of a second. Although relatively simple to implement, these machine learning approaches completely exclude any knowledge of the mechanics of the system beyond what can be naturally discerned from the dataset, and can have significant problems creating the surrogate mapping at all, since the input–output relationship is often not one-to-one, leading to substantial accuracy concerns in many cases. Alternatively, there are several different approaches (referred to as “direct inversion” methods herein) that instead manipulate the equations governing the mechanics of the system (i.e., the forward boundary value problem) to create a solution process similar to that of solving the forward problem itself (e.g., similar in process to a finite element analysis to predict the deformation response of a solid given geometry, material properties, and boundary conditions). Therefore, the direct inversion methods provide a solution estimate based upon the measured response and the mechanics assumed to govern the response of the system considered at a cost on the order of a single numerical analysis of the forward problem.

Direct inversion approaches have seen considerable application in problems relating to characterization of the distribution of elastic properties from mechanical testing. Furthermore, these approaches include those that solve for the material properties over the entire (or almost the entire) domain at once (i.e., global methods), as well as those that break the domain into subregions and determine the properties for each subdomain in sequence (i.e., local). Global methods typically use some variation of either the finite difference method (FDM) or the finite element method (FEM) to create a system of equations based on the forward boundary value problem to solve the node or element-based elastic modulus distribution everywhere in the domain. FDM approaches for global direct inversion include the seminal work of Raghavan and Yagle  [16] for simultaneously characterizing stiffness and hydrostatic pressure distribution of tissue from strain measurements, as well as other subsequent similar approaches for characterization of relative elastic modulus  [17] and relative shear modulus  [18] distributions. FEM approaches have shown some benefits over FDM approaches for global direct inversion in that they can be cast in a way so as to only require the gradient of displacement (i.e., a single derivative) rather than the divergence of strain (i.e., two derivatives of displacement). In this way, FEM approaches can be more tolerant to noise and other measurement errors than FDM approaches. Examples of FEM approaches for global direct inversion include the work by Zhu et al.  [12] for elasticity reconstruction from displacement measurements and the work by Park and Maniatty  [13] for shear modulus reconstruction from measured steady-state dynamic displacement fields. The local methods are often similar to the global methods, with several approaches similarly using principles from FDM and FEM (although not all), with the fundamental difference being just how the system is discretized/solved. Oliphant et al. [19], [20] presented a local approach that used polynomial fit of the measured dynamic displacement field and the strong form of the governing equation of motion to characterize the elasticity distribution in tissues, while Romano et al.  [21], [22] used a weak form FEM-type approach for element-by-element characterization of the ratio between both Lamé constants and density from dynamic displacement measurements. In a further modification, Albocher et al. have proposed an adjoint-weighted equation approach based on Galerkin discretization, which results in a stable and convergent direct inversion method for estimating material property distributions  [23]. One particularly unifying aspect of the direct inversion approaches is that the entire (or nearly entire) spatial distribution (i.e., full-field) of the system response (e.g., displacement) must be measured/available to successfully characterize the unknown properties. Due to this requirement, the use of direct inversion (particularly of mechanical properties) has been mainly limited to biomechanical imaging applications (e.g., characterization of elastic properties of tissues from medical imaging data), where full-field or nearly full-field deformation information is regularly available. An additional common challenge of direct inversion approaches is their noise sensitivity. Most direct inversion approaches are not capable of producing a usable solution estimate with any significant level of measurement noise, which has resulted in several investigations into strategies for signal denoising prior to applying direct inversion  [13], [20].

This work presents an approach to utilize measurement data from only a portion of the system domain (i.e., partial-field data) for direct inversion of material properties. In particular, the approach is presented in the context of characterizing the spatial distribution of the elastic modulus of solids (i.e., elastography) provided with displacement response measurements over some portion of the solid domain. The core component of this approach is the use of the Gappy proper orthogonal decomposition (POD) machine learning strategy to build a data reconstruction tool that can predict the full-field response of a system from the available partial-field measurements. This data reconstruction tool is built from a set of potential solution fields that are generated based upon information known a priori about the nature of the potential inverse problem solutions (e.g., arbitrarily generated approximate potential elastic modulus distributions). Once the full-field response reconstruction is complete, the full-field response is applied in a finite element-type direct inversion strategy to estimate the unknown material property distribution everywhere in the domain at a computational expense approximately equivalent to a single finite element analysis of the system. In addition, 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.

The following section presents the details of the Gappy POD approach to reconstruct full-field data from partial-field measurements. Then, Section  3 shows the direct inversion algorithm to calculate the spatial distribution of elastic modulus provided with the full-field displacement response and the boundary conditions corresponding to the test method used to produce the measurements (i.e., the constraints on the solid and the excitation used to generate the displacement measurements). The complete direct inversion with Gappy POD algorithm is summarized in Section  4. Section  5 presents two simulated case studies relating to characterization of localized elastic modulus distributions in solids to evaluate the capabilities of the inverse characterization procedure, which are followed by an example utilizing (actual) experimentally obtained displacement measurements to evaluate the stiffness distribution of an arterial construct to validate the “real-life” applicability of the approach.