Computational Approaches to Efficiently Maximize Solution Capabilities and Quantify Uncertainty for Inverse Problems in Mechanics

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

Adaptive Reduced-Basis Generation for Reduced-Order Modeling for the Solution of Stochastic Nondestructive Evaluation Problems

A novel algorithm for creating a computationally efficient approximation of a system response that is defined by a boundary value problem is presented. More specifically, the approach presented is focused on substantially reducing the computational expense required to approximate the solution of a stochastic partial differential equation, particularly for the purpose of estimating the solution to an associated nondestructive evaluation problem with significant system uncertainty. In order to achieve this computational efficiency, the approach combines reduced-basis reduced-order modeling with a sparse grid collocation surrogate modeling technique to estimate the response of the system of interest with respect to any designated unknown parameters, provided the distributions are known. The reduced-order modeling component includes a novel algorithm for adaptive generation of a data ensemble based on a nested grid technique, to then create the reduced-order basis. The capabilities and potential applicability of the approach presented are displayed through two simulated case studies regarding inverse characterization of material properties for two different physical systems involving some amount of significant uncertainty. The first case study considered characterization of an unknown localized reduction in stiffness of a structure from simulated frequency response function based nondestructive testing. Then, the second case study considered characterization of an unknown temperature-dependent thermal conductivity of a solid from simulated thermal testing. Overall, the surrogate modeling approach was shown through both simulated examples to provide accurate solution estimates to inverse problems for systems represented by stochastic partial differential equations with a fraction of the typical computational cost

IMECE2010-38693 Multi-objective crashworthiness optimization of Composite Hat-shape Energy Absorber using GMDH-type Neural Networks and Genetic Algorithms

Abstract Reducing the weight of car body and increasing the crashworthiness capability of car body are two important objectives of car design. In this paper, a multi-objective optimization for optimal composite hat-shape energy absorption system is presented At the first, the behaviors of the hat shape under impact, as simplified model of side member of a vehicle body, are studied by the finite element method using commercial software ABAQUS. Two meta-models based on the evolved group method of data handling (GMDH) type neural networks are then achieved for modeling of both the absorbed energy (E) and the Tsai-Hill Failure Criterion (TS) with respect to geometrical design variables using those training and testing data obtained models. The obtained polynomial neural metamodels are finally used in a multi-objective optimum design procedure using NSGA-II with a new diversity preserving mechanism for Pareto based optimization of hat-shape. Two conflicting objectives such as maximizing the energy absorption capability (E), minimizing the Tsai-Hill Failure Criterion are considered in this work