Quantification of airfoil geometry-induced aerodynamic uncertainties - comparison of approaches

Uncertainty quantification in aerodynamic simulations calls for efficient numerical methods since it is computationally expensive, especially for the uncertainties caused by random geometry variations which involve a large number of variables. This paper compares five methods, including quasi-Monte Carlo quadrature, polynomial chaos with coefficients determined by sparse quadrature and gradient-enhanced version of Kriging, radial basis functions and point collocation polynomial chaos, in their efficiency in estimating statistics of aerodynamic performance upon random perturbation to the airfoil geometry which is parameterized by 9 independent Gaussian variables. The results show that gradient-enhanced surrogate methods achieve better accuracy than direct integration methods with the same computational cost

Quantifizierung von Unsicherheiten für die Flachwassergleichung

The present thesis proposes two novel numerical integration techniques as an endeavour to break the "curse of dimension" to high-dimensional integrations, and investigates the efficiency of some numerical techniques quantifying uncertainty in the solution of shallow water equations (SWE) for flood modelling. The novel uncorrelated dimensions (UD) quadrature and compound UD quadrature have convergence rates independent of the dimension number of the integration if the integrand can be expressed by a multilinear functional of any integrable functions. A stochastic SWE model is set up by a probabilistic parameterisation of the SWE, whereon UD and quasi-Monte Carlo quadrature show advantage on the integrations for statistics. The model is also approximated by polynomial chaos expansions and Karhunen-Loeve expansions which are shown to be effective data compression techniques.Die vorliegende Arbeit stellt zwei neue numerische Integrationstechniken vor, die versuchen, den "Fluch der Dimension" zu brechen, und untersucht und vergleicht die Effizienz verschiedener numerischer Methoden zur Quantifizierung von Unsicherheiten bei der Lösung der Flachwassergleichung (FWG) für die Hochwasser-Modellierung. Die neuen Quadraturverfahren der "Unkorrelierten Dimensionen" (UD) und der "Zusammengesetzten Unkorrelierten Dimensionen" (Compound UD) weisen eine Konvergenzrate auf, die unabhängig von der Anzahl der Dimensionen der Integration ist, falls der Integrand als multilineares Funktional integrierbarer Funktionen ausgedrückt werden kann. Ein stochastisches Modell der FWG wird über eine probabilistische Parametrisierung der Flachwassergleichung aufgestellt, bei welchem UD und Quasi-Monte-Carlo-Quadratur Vorteile bei der Integration von Statistiken zeigen. Das Modell wird auch über die polynomiale Chaos und die Karhunen-Loeve Entwicklung approximiert, von denen gezeigt werden kann, dass sie nützlich für die Datenkompression sind

To be or not to be intrusive? The solution of parametric and stochastic equations - the "plain vanilla" Galerkin case

In parametric equations - stochastic equations are a special case - one may want to approximate the solution such that it is easy to evaluate its dependence of the parameters. Interpolation in the parameters is an obvious possibility, in this context often labeled as a collocation method. In the frequent situation where one has a "solver" for the equation for a given parameter value - this may be a software component or a program - it is evident that this can independently solve for the parameter values to be interpolated. Such uncoupled methods which allow the use of the original solver are classed as "non-intrusive". By extension, all other methods which produce some kind of coupled system are often - in our view prematurely - classed as "intrusive". We show for simple Galerkin formulations of the parametric problem - which generally produce coupled systems - how one may compute the approximation in a non-intusive way

Characterization and Control of Pore Structural Heterogeneity for Low-Thermal-Maturity Shale: A Case Study of the Shanxi Formation in the Northeast Zhoukou Depression, Southern North China Basin

The Shanxi Formation layers in the northeast of the Zhoukou Depression, Southern North China Basin, mainly consist of dark mudstone interbed with tight stone and widely developed coal seam, which is a promising target for unconventional oil and gas exploration. A series of geochemical and geological methods were used to analyze the characterization and controls of the pores structural heterogeneity in low-thermal-maturity shale. These methods include the Rock-Eval analysis, total organic carbon (TOC) analysis, scanning electron microscope observation with an energy-dispersive spectrometer (SEM-EDS), X-ray diffraction, and low-pressure N2 adsorption. Based on these measurements, the pore diameter, specific surface area (SSA), and fractal dimension (D) were calculated, and then, the pore structure heterogeneity was analyzed. The result shows the pores of Shanxi Formation shale are mainly interparticle pores with low porosity and low permeability, and the pore structure is highly complex. The average fractal dimension of the micropore and the macropore are both 2.77, but that of the mesopore is 2.65, indicating a less-complex mesopore structure than the micropore and macropore. The S2, S1, and TOC exhibit no clear correlation with SSA and fractal dimension of pores, which proved the little impact of organic matter on the heterogeneity of pore structure in the low-maturity shale of the research area. The illite has a strong effect on the pore structural heterogeneity of Shanxi Formation shale. The samples with high content of illite show higher SSA, better physical properties, and low fractal dimension, reflecting low pore structural heterogeneity. However, the quartz and clay minerals show a slight correlation with SSA and no obvious relationship with the fractal dimension, indicating a little effect of them on the pore structure heterogeneity. The pore structural heterogeneity decreases along with the increase in porosity, while the permeability influenced by a variety of reasons under the compaction shows a poor relationship with SSA and fractal dimension. On the whole, the pore structural heterogeneity decreases for low-thermal-maturity shale with high content of illite and high porosity, which should be considered to be the better unconventional oil and gas reservoir in the research area

Efficient Quantification of Aerodynamic Uncertainies Using Gradient-Employing Surrogate Methods

Uncertainty quantification (UQ) in aerodynamic simulations is hindered by the high computational cost of CFD models. With gradient information obtained efficiently by using an adjoint solver, gradient-employing surrogate methods are promising in speeding up the UQ process. To investigate the efficiency of UQ methods we apply gradient-enhanced radial basis functions, gradient-enhanced point-collocation polynomial chaos, gradient-enhaced Kriging and quasi-Monte Carlo (QMC) quadrature to a test case where the geometry of an RAE2822 airfoil is perturbed by a Gaussian random field parameterized by 10 independant variables. The four methods are compared in their efficiency in estimating some statistics and the probability distribution of the uncertain lift and drag coefficients. The results show that with the same computational effort the gradient-employing surrogate methods achieve better accuracy than the QMC doe

Efficient Quantification of Aerodynamic Uncertainty due to Random Geometry Perturbations

The effort of quantifying the aerodynamic uncertainties caused by uncertainties in the airfoil geometry is hindered by the large number of the variables and the high computational cost of the CFD model. To identify e�fficient methods addressing this challenge, four promising methods, gradient-enhanced Kriging (GEK), gradient-assisted polynomial chaos (GAPC), maximum entropy method and quasi-Monte Carlo quadrature are applied to a test case where the geometry of an RAE2822 airfoil is perturbed by a Gaussian random �field parameterized by nine independent variables. The four methods are compared in their efficiency of estimating some statistics and probability distribution of the uncertain lift and drag coe�cients. The results show that the two surrogate method, GEK and GAPC, both utilizing gradient information obtained by an adjoint CFD solver, are more effi�cient in this situation. Their advantage is expected to increase as the number of variables increases

Uncertainty Quantification Using Surrogate Models

Uncertainty quantification (UQ) in aerodynamic simulations is retarded by the high computational cost of CFD models. With gradient information obtained at a reduced cost by using an adjoint solver, gradient-employing surrogate methods are promising in speeding up the UQ process. To investigate efficiency of UQ methods we apply gradient-enhanced radial basis functions, gradient-enhanced point-collocation polynomial chaos, gradient-enhanced Kriging and quasi-Monte Carlo quadrature to a test case where the geometry of a RAE2822 airfoil is perturbed by a Gaussian random field parameterized by 10 independent variables. The four methods are compared in their efficiency in estimating some statistics and probability distribution of the uncertain lift and drag coefficients. The results show that with the same computational effort the gradient-employing surrogate methods achieve better accuracy in the UQ than the QMC does