Computational approaches to solve inverse problems can provide generalized frameworks for treating and distinguishing between the various contributions to a system response, while providing physically meaningful solutions that can be applied to predict future behaviors. However, there are several common challenges when using any computational inverse mechanics technique for applications such as material characterization. These challenges are typically connected to the inherent ill-posedness of the inverse problems, which can lead to a nonexistent solution, non-unique solutions, and/or prohibitive computational expense.
Toward reducing the effects of inverse problem ill-posedness and improving the capability to accurately and efficiently estimate inverse problem solutions, a suite of computational tools was developed and evaluated. First, an approach to NDT design to maximize the capabilities to use computational inverse solution techniques for material characterization and damage identification in structural components, and more generally in solid continua, is presented. The approach combines a novel set of objective functions to maximize test sensitivity and simultaneously minimize test information redundancy to determine optimal NDT parameters. The NDT design approach is shown to provide measurement data that leads to consistent and significant improvement in the ability to accurately inversely characterize variations in the Young's modulus distributions for simulated test cases in comparison to alternate NDT designs. Next, an extension of the NDT design approach is presented, which includes a technique to address potential system uncertainty and add robustness to the resulting NDT design, again in the context of material characterization. The robust NDT design approach uses collocation techniques to approximate the modified objective functionals that not only maximize the test sensitivity and minimize the test information redundancy, but now also maximize the test robustness to system uncertainty. The capability of this probabilistic NDT design method to provide consistent improvement in the ability to accurately inversely characterize variations in the Young's modulus distributions for cases where systems have uncertain parameters, such as uncertain boundary condition features, is again shown with numerically simulated examples. Lastly, an approach is presented to more directly address the computational expense of solving an inverse problem, particularly for those problems with significant system uncertainties. The sparse grid method is used as the foundation of this solution approach to create a computationally efficient polynomial approximation (i.e., surrogate model) of the system response with respect to both deterministic and uncertain parameters to be used in the inverse problem solution process. More importantly, a novel generally applicable algorithm is integrated for adaptive generation of a data ensemble, which is then used to create a reduced-order model (ROM) to estimate the desired system response. In particular, the approach builds the ROM to accurately estimate the system response within the expected range of the deterministic and uncertain parameters, to then be used in place of the traditional full order modeling (i.e., standard finite element analysis) in constructing the surrogate model for the inverse solution procedure. This computationally efficient approach is shown through simulated examples involving both solid mechanics and heat transfer to provide accurate solution estimates to inverse problems for systems represented by stochastic partial differential equations with a fraction of the typical computational cost
