Computational Inverse Solution Strategies for Characterization of Localized Variations of Material Properties in Solids and Structures

Computational inverse characterization approaches that combine computational physical modeling and nonlinear optimization minimizing the difference between measurements from experimental testing and the responses from the computational model are uniquely well-suited for quantitative characterization of structures and systems for a variety of engineering applications. Potential applications that are suited for computational inverse characterization range from damage identification of civil structures to elastography of biological tissue. However, certain challenges, primarily relating to accuracy, efficiency, and stability, come along with any computational inverse characterization approach. As such, proper application-specific formulation of the inverse problem, including parameterization of the field to be inversely determined and selection/implementation of the optimization approach are critical to ensuring an accurate solution can be estimated with minimal (i.e. practically applicable) computational expense.\ud The present work investigates strategies to optimally utilize the available measurement data in combination with a priori information about the nature of the unknown properties to maximize the efficiency and accuracy of the solution procedure for applications in inverse characterization of localized material property variations. First, a strategy using multi-objective optimization for inverse characterization of material loss (i.e., cracks or erosion) in structural components is presented. For this first component, the assumption is made that sufficient a priori information is available to restrict the parameterization of the unknown field to a known number and shape of material loss regions (i.e., the inverse problem is only required to identify size and location of these regions). Since this type of parameterization would typically be relatively compact (i.e., low number of parameters), the inverse problem is well suited for non-gradient-based optimization approaches, which can provide accuracy through global search capabilities. The multi-objective inverse solution approach shown divides the available measurement data into multiple competing objectives for the optimization process (rather than the typical single objective for all measurement data) and uses a stochastic multi-objective optimization technique to identify a Pareto front of potential solutions, and then select one "best" inverse solution estimate. Through simulated test problems of damage characterization, the multi-objective optimization approach is shown to provide increased solution estimate diversity during the search process, which results in a substantial improvement in the capabilities to traverse the optimization search space to minimize the measurement error and produce accurate damage size and location estimates in comparison with analogous single objective optimization approaches. An extension of this multi-objective approach is then presented that addresses problems for which the quantity of localized changes in properties is unknown. Thus, a self-evolving parameterization algorithm is presented that utilizes the substantial diversity in the Pareto front of potential solutions provided by the multi-objective optimization approach to build up the parameterization iteratively with an ad hoc clustering algorithm, and thereby determine the quantity, size, and location of localized changes in properties with minimal computational expense. Similarly as before, through simulated test problems based on characterization of damage within plates, the solution strategy with self-evolving parameterization is shown to provide an accurate and efficient process for the solution of inverse characterization of localized property changes.\ud \ud For the second half of the present work, a substantial change in the inverse problem assumptions is made, in that the nature (i.e., shape) of the property variation is no longer assumed to be known as precisely a priori Thus, a more general (e.g., mesh-based) parameterization of the unknown field is needed, which would typically come at a cost of significantly increased computational expense and/or loss of solution uniqueness. To balance the generalization of the approach and still utilize some amount of the knowledge that the solution is localized in nature, while maintaining efficiency, a hybrid compact-generalized parameterization approach is presented. The initial incarnation of this hybrid approach combines a machine learning data reconstruction strategy known as gappy proper orthogonal decomposition (POD) with a least-squares direct inversion approach to estimate material stiffness distribution in solids (i.e., to solve elastography problems). The direct inversion approach uses a generalized mesh-based parameterization of the unknown field, but full-field response measurements (i.e., measurements everywhere in the solid) are required, which are not available for most practical inverse characterization problems. Therefore, the gappy POD technique first identifies the pattern of potential response fields of the solid through a collection of a priori forward numerical analyses of the solid response with a specified compact parameterization and a corresponding collection of arbitrarily generated parameter sets. Once the pattern is identified, the gappy POD technique is able to use the available partial-field measurement data to estimate the full-field response of the solid to be used by the direct inversion. Thus, the computational cost of the inverse characterization is negligible once the gappy POD process has been completed. Through simulated test problems relating to characterization of inclusions in solids, the direct inversion approach with gappy POD is shown to provide highly efficient and relatively accurate inverse characterization results for the prediction of Young's modulus distributions from partial-field measurement data. This direct inversion approach is further validated through an example problem regarding characterization of the layered stiffness properties of an engineered vessel from ultrasound measurements. Lastly, an extension of this hybrid approach is presented that uses the characterization results provided by the previous direct inversion approach as the initial estimate for a gradient-based optimization process to further refine/improve the inverse solution estimate. In addition, the adjoint method is used to calculate the gradient for the optimization process with minimal computational expense to maintain the overall computational efficiency of the inverse solution process. Again, through simulated test problems based on the characterization of localized, but arbitrarily shaped, inclusions within solids, the three-step (gappy POD - direct inversion - gradient-based optimization) inverse characterization approach is shown to efficiently provide accurate and relatively unique inverse characterization estimates for various types of inclusions regardless of inclusion geometry and quantity