Uncertainty Propagation and Robust Design in CFD Using Sensitivity Derivatives

This study investigates and demonstrates a methodology for uncertainty propagation and robust design in Computational Fluid Dynamics (CFD). Efficient calculation of both first- and second-order sensitivity derivatives is requisite in the proposed methodology. In this study, first- and second-order sensitivity derivatives of code output with respect to code input are obtained through an efficient incremental iterative approach. An approximate statistical moment method for uncertainty propagation is first demonstrated on a quasi one-dimensional (1-D) Euler CFD code. This method is then extended to a two-dimensional (2-D) subsonic inviscid model airfoil problem. In each application, given statistically independent, random, normally distributed input variables, a first- and second-order statistical moment matching procedure is performed to approximate the uncertainty in the CFD output. In each model problem, a Sensitivity Derivative Enhanced Monte Carlo (SDEMC) method is also demonstrated. With this methodology, incorporation of the first-order sensitivity derivatives into the data reduction phase of a conventional Monte Carlo (MC) simulation allows for improved accuracy in determining the first moment of the CFD output. The statistical moment method and the SDEMC method are also incorporated into an investigation of output function variance. The methods that exploit the availability of sensitivity derivatives are found to be valid and computationally efficient when considering small deviations from input mean values. In both the 1-D and 2-D problems, uncertainties in the CFD input variables are incorporated into robust optimization procedures. For each optimization, statistical moments involving first-order sensitivity derivatives appear in the objective function and system constraints. The constraints are cast in probabilistic terms; that is, the probability that a constraint is satisfied is greater than or equal to some desired target probability. Gradient-based robust optimization of this stochastic problem is accomplished through use of both first and second-order sensitivity derivatives. For each robust optimization, the effect of increasing both input standard deviations and target probability of constraint satisfaction are demonstrated. This method provides a means for incorporating uncertainty when considering small deviations from input mean values

T1 mapping in cardiac MRI

Quantitative myocardial and blood T1 have recently achieved clinical utility in numerous pathologies, as they provide non-invasive tissue characterization with the potential to replace invasive biopsy. Native T1 time (no contrast agent), changes with myocardial extracellular water (edema, focal or diffuse fibrosis), fat, iron, and amyloid protein content. After contrast, the extracellular volume fraction (ECV) estimates the size of the extracellular space and identifies interstitial disease. Spatially resolved quantification of these biomarkers (so-called T1 mapping and ECV mapping) are steadily becoming diagnostic and prognostically useful tests for several heart muscle diseases, influencing clinical decision-making with a pending second consensus statement due mid-2017. This review outlines the physics involved in estimating T1 times and summarizes the disease-specific clinical and research impacts of T1 and ECV to date. We conclude by highlighting some of the remaining challenges such as their community-wide delivery, quality control, and standardization for clinical practice

Can the silicon content in hair be an indicator of atherosclerosis risk?

Silicon is the second most prevalent element in the Earth’s crust and the third most abundant trace element in the human body. Silicon compounds counteract the crystallization of minerals in the urinary tract, impede the deposition of lipid plaques in the walls of blood vessels and through their presence prevent the absorption of calcium compounds, which reduce the elasticity of blood vessels and may cause diseases of the circulatory system. The article presents the results of studies on the level of silicon and magnesium in the hair of patients with atherosclerosis (N=137, age 60-94) and the control group of healthy people (N=242, age 20-80). The measured silicon content by ICP-OES in healthy people decreases with age, especially after 40 years of age, and ranges from 43.3±7.8 to 22.4±8.4 µg g-1. The average level of silicon in patients with atherosclerosis is much lower and ranges from 14.0±6.7 to 7.9±4.9 µg g-1, depending on the age range. However, a wide spread of Si values is observed in every age group, even in the group of healthy military students living for several years in the same conditions and using the same diet. Among the patients, there is a group with Si levels below 10 µg g-1, a value that does not appear in healthy people, even those aged 70-80. Due to the presence of a concentration range of 10-20 µg g-1 among all tested samples, the Si content in the hair cannot be unequivocally considered a certain atherosclerosis marker, although particularly low Si content below 10 µg g-1 should be a clear signal heralding a disease. In such patients, there is also a significantly reduced level of magnesium (5-15 µg g-1) compared with the norms adopted in Poland (25-35 µg g-1)

Employing Sensitivity Derivatives for Robust Optimization under Uncertainty in CFD

A robust optimization is demonstrated on a two-dimensional inviscid airfoil problem in subsonic flow. Given uncertainties in statistically independent, random, normally distributed flow parameters (input variables), an approximate first-order statistical moment method is employed to represent the Computational Fluid Dynamics (CFD) code outputs as expected values with variances. These output quantities are used to form the objective function and constraints. The constraints are cast in probabilistic terms; that is, the probability that a constraint is satisfied is greater than or equal to some desired target probability. Gradient-based robust optimization of this stochastic problem is accomplished through use of both first and second-order sensitivity derivatives. For each robust optimization, the effect of increasing both input standard deviations and target probability of constraint satisfaction are demonstrated. This method provides a means for incorporating uncertainty when considering small deviations from input mean values