Kirubel Teferra

Refined Stratified Sampling for efficient Monte Carlo based uncertainty quantification

Abstract A general adaptive approach rooted in stratified sampling (SS) is proposed for sample-based uncertainty quantification (UQ). To motivate its use in this context the space-filling, orthogonality, and projective properties of SS are compared with simple random sampling and Latin hypercube sampling (LHS). SS is demonstrated to provide attractive properties for certain classes of problems. The proposed approach, Refined Stratified Sampling (RSS), capitalizes on these properties through an adaptive process that adds samples sequentially by dividing the existing subspaces of a stratified design. RSS is proven to reduce variance compared to traditional stratified sample extension methods while providing comparable or enhanced variance reduction when compared to sample size extension methods for LHS – which do not afford the same degree of flexibility to facilitate a truly adaptive UQ process. An initial investigation of optimal stratification is presented and motivates the potential for major advances in variance reduction through optimally designed RSS. Potential paths for extension of the method to high dimension are discussed. Two examples are provided. The first involves UQ for a low dimensional function where convergence is evaluated analytically. The second presents a study to asses the response variability of a floating structure to an underwater shock.

Mapping model validation metrics to subject matter expert scores for model adequacy assessment

This paper develops a novel approach to incorporate the contributions of both quantitative validation metrics and qualitative subject matter expert (SME) evaluation criteria in model validation assessment. The relationship between validation metrics (input) and SME scores (output) is formulated as a classification problem, and a probabilistic neural network (PNN) is constructed to execute this mapping. Establishing PNN classifiers for a wide variety of combinations of validation metrics allows for a quantitative comparison of validation metric performance in representing SME judgment. An advantage to this approach is that it semi-automates the model validation process and subsequently is capable of incorporating the contributions of large data sets of disparate response quantities of interest in model validation assessment. The effectiveness of this approach is demonstrated on a complex real-world problem involving the shock qualification testing of a floating shock platform. A data set of experimental and simulated pairs of time history comparisons along with associated SME scores and computed validation metrics is obtained and utilized to construct the PNN classifiers through K-fold cross validation. A wide range of validation metrics for time history comparisons is considered including feature-specific metrics (phase and magnitude error), a frequency metric (shock response spectra), a time-frequency metric (wavelet decomposition), and a global metric (index of agreement). The PNN classifiers constructed using a Parzen kernel for the class conditional probability density function whose smoothing parameter is optimized using a genetic algorithm performs well in representing SME judgment.

Direct parameter estimation for generalised balanced power diagrams

ABSTRACT The statistical characterisation and synthetic reproduction of a polycrystalline materials microstructure is assisted by mathematically representing its morphology by a tessellation model. The generalised balanced power diagram (GBPD) is a tessellation model that was shown in previous studies to accurately reproduce the microstructure morphology of various materials by closely matching micrographs obtained through electron microscopy. These studies employed costly optimisation procedures to determine the best-fit model parameters, limiting the scalability of the model. In this work, it is shown that setting the tessellation cell parameters to values such that the shape moments of the corresponding grains are matched results in a quality of fit that is commensurate with optimisation procedures. This fitting approach decouples the interaction among grains when fitting the tessellation parameters and, most notably, provides analytical, closed-form expressions for all the model parameters. The performance of this parameter fitting approach is demonstrated on multiple micrographs of various materials, and it compares similarly to the performance of optimisation procedures reported in recently published literature. As the fitted parameter values are obtained through trivial computations, this approach enables extensive scalability of the GBPD model such that it can be used to represent extremely large characterisation data sets.

Uncertainty quantification for constitutive model calibration of brain tissue

The results of a study comparing model calibration techniques for Ogdens constitutive model that describes the hyperelastic behavior of brain tissue are presented. One and two-term Ogden models are fit to two different sets of stress-strain experimental data for brain tissue using both least squares optimization and Bayesian estimation. For the Bayesian estimation, the joint posterior distribution of the constitutive parameters is calculated by employing Hamiltonian Monte Carlo (HMC) sampling, a type of Markov Chain Monte Carlo method. The HMC method is enriched in this work to intrinsically enforce the Drucker stability criterion by formulating a nonlinear parameter constraint function, which ensures the constitutive model produces physically meaningful results. Through application of the nested sampling technique, 95% confidence bounds on the constitutive model parameters are identified, and these bounds are then propagated through the constitutive model to produce the resultant bounds on the stress-strain response. The behavior of the model calibration procedures and the effect of the characteristics of the experimental data are extensively evaluated. It is demonstrated that increasing model complexity (i.e., adding an additional term in the Ogden model) improves the accuracy of the best-fit set of parameters while also increasing the uncertainty via the widening of the confidence bounds of the calibrated parameters. Despite some similarity between the two data sets, the resulting distributions are noticeably different, highlighting the sensitivity of the calibration procedures to the characteristics of the data. For example, the amount of uncertainty reported on the experimental data plays an essential role in how data points are weighted during the calibration, and this significantly affects how the parameters are calibrated when combining experimental data sets from disparate sources.

Effect of human head morphological variability on the mechanical response of blast overpressure loading

A methodology is introduced to investigate the effect of intersubject head morphological variability on the mechanical response of the brain when subjected to blast overpressure loading. Nonrigid image registration techniques are leveraged to warp a manually segmented template model to an arbitrary number of subjects following a procedure to coarsely segment the subjects in batch. Finite element meshes are autogenerated, and blast analysis is conducted. The template model is initially constructed to enable the full automated implementation and application of the proposed methodology. The application of the proposed approach for an anterior-oriented blast has been demonstrated, and the results reveal that the pressure response in the brain does exhibit some dependence on head morphological variability. While the magnitude of the peak pressure response can vary by more than 30%, its location within the brain is unaffected by head morphological variability. A linear least squares analysis was conducted to demonstrate that the peak magnitude of pressure is uncorrelated with head volume while it is correlated with aspect ratio relating to the amount of exposed surface area to the blast. These features of the pressure response are likely due to the peak pressure occurring during the early stages of stress wave transmission and reflection. As a result, the pressure response due to blast overpressure loading is predominantly loading dependent while morphological variability has a secondary effect.