Validation exercises for computational models of materials under impact must contend with sparse experimental data as well as with uncertainties due to microstructural stochasticity and variabilities in thermomechanical properties of the material. This paper develops statistical methods for determining confidence levels for verification and validation of computational models subject to aleatoric and epistemic uncertainties and sparse stochastic experimental datasets. To demonstrate the method, the classical problem of Taylor impact of a copper bar is simulated. Ensembles of simulations are performed to cover the range of variabilities in the material properties of copper, specifically the nominal yield strength A, the hardening constant B, and the hardening exponent n in a Johnson–Cook material model. To quantify uncertainties in the simulation models, we construct probability density functions (PDFs) of the ratios of the quantities of interest, viz., the final bar diameter D f to the original diameter D 0 and the final length L f to the original length L 0 . The uncertainties in the experimental data are quantified by constructing target output distributions for these QoIs ( D f / D 0 and L f / L 0 ) from the sparse experimental results reported in literature. The simulation output and the experimental output distributions are compared to compute two metrics, viz., the median of the model prediction error and the model confidence at user-specified error level. It is shown that the median is lower and the model confidence is higher for L f / L 0 compared to D f / D 0 , implying that the simulation models predict the final length of the bar more accurately than the diameter. The calculated confidence levels are shown to be consistent with expectations from the physics of the impact problem and the assumptions in the computational model. Thus, this paper develops and demonstrates physically meaningful metrics for validating simulation models using limited stochastic experimental datasets. The tools and techniques developed in this work can be used for validating a wide range of computational models operating under input uncertainties and sparse experimental datasets.

To represent input variability accurately, an input distribution model for random variables should be constructed using many data. However, for certain input variables, engineers may have only their intervals, which represent input uncertainty. In practical engineering applications, both random and interval variables could exist at the same time. To consider both input variability and uncertainty, inverse reliability analysis should be carried out considering both random and interval variables—mixed variables—and their mathematical correlation in a performance measure. In this paper, an iterative most probable point (MPP) search method has been developed for the mixed-variable problem. The update procedures for MPP search are developed considering the features of mixed variables in the inverse reliability analysis. MPP search for random and interval variables proceed simultaneously to consider the mathematical correlation. An interpolation method is introduced to find a better candidate MPP without additional function evaluations. Mixed-variable design optimization (MVDO) has been formulated to obtain cost-effective and reliable design in the presence of mixed variables. In addition, the design sensitivity of a probabilistic constraint has been developed for an effective and efficient MVDO procedure. Using numerical examples, it is found that the developed MPP search method finds an accurate MPP more efficiently than the generic optimization method does. In addition, it is verified that the developed method enables the MVDO process with a small number of function evaluations.

For reliability-based design optimization (RBDO), generating an input statistical model with confidence level has been recently proposed to offset inaccurate estimation of the input statistical model with Gaussian distributions. For this, the confidence intervals for the mean and standard deviation are calculated using Gaussian distributions of the input random variables. However, if the input random variables are non-Gaussian, use of Gaussian distributions of the input variables will provide inaccurate confidence intervals, and thus yield an undesirable confidence level of the reliability-based optimum design meeting the target reliability β t . In this paper, an RBDO method using a bootstrap method, which accurately calculates the confidence intervals for the input parameters for non-Gaussian distributions, is proposed to obtain a desirable confidence level of the output performance for non-Gaussian distributions. The proposed method is examined by testing a numerical example and M1A1 Abrams tank roadarm problem.