Non-intrusive Polynomial Chaos Expansion Method Research Articles

Data assimilation method and application of shear stress transport turbulence model for complex separation of internal shock boundary layer flow

Data assimilation method and application of shear stress transport turbulence model for complex separation of internal shock boundary layer flow

Global sensitivity and uncertainty analysis of a coastal morphodynamic model using Polynomial Chaos Expansions

Predicting coastal erosion requires an accurate morphodynamic model. XBeach has been widely adopted to simulate storm-induced coastal erosion. Because of the large number of model input parameters and their uncertainties, sensitivity analysis is a crucial step toward reliable results. This requires running a large number of computationally expensive simulations. Here, we adopt a computationally cost-effective approach based on the Non-Intrusive Polynomial Chaos Expansion method to quantify the sensitivity of XBeach to its input parameters. The method is applied to the coastal erosion event caused by Hurricane Sandy along the barrier islands of New Jersey in the United States. The results show a spatial variation of the model sensitivity with increasing boundary conditions. Parameters’ interaction is found to decrease as the magnitude of boundary conditions increases, causing a reduction in the nonlinearity of the model behavior, and consequently leading to a reduction in the uncertainty of the model output.

Uncertainty quantification and sensitivity analysis of aerothermal performance for the turbine blade squealer tip

In this paper, an uncertainty quantification (UQ) method is proposed using the non-intrusive polynomial chaos expansion method and Smolyak sparse grids. Then coupled with three-dimensional Reynolds-Averaged Navier-Stokes (RANS) solutions, an uncertainty quantification procedure is carried out for aerodynamic and heat transfer performance of GE-E3 rotor blade squealer tip. A parameter sensitivity analysis using the Sobol Indice method is carried out to identify the key parameters for aerothermal performance of the squealer tip. Wherein, the inlet total pressure and the inlet flow angle are considered as flow condition uncertainty parameters and cavity depth is considered as geometrical uncertainty parameters. The uncertainty analysis results showed that under the influence of the uncertain geometry and operating conditions, the heat flux of squealer tip basically conforms to the normal distribution and the statistical mean value of it increased by 28.30% relative to the design value and the probability of it deviating from the design value by 10% is as high as 86.59%. The statistical average of the squealer tip film cooling effectiveness is reduced by 23.89% compared to the design value, and the probability of it deviating from the design value by 10% is as high as 85.08%. The result of sensitivity analysis reveals that the uncertainty of the aerodynamic characteristics of the squealer tip is almost entirely caused by the inlet flow angle which accounts for 94.01% of the variance of the leakage flow rate. And it is also the dominant variable for the uncertainty of the heat transfer performance considering that its variance indexes for tip heat flux QTip and film cooling effectiveness η are 67.71% and 54.42% respectively. Compared with the main effects, the influence of the interaction effects among the variables on the squealer tip aerothermal performance is almost negligible.

Topology optimization of continuum structures under geometric uncertainty using a new extended finite element method

In this article, robust topology optimization under geometric uncertainty is proposed. The design domain is discretized by an extended finite element method. A bi-directional evolutionary structural optimization carries out the optimization process. The performance of the proposed method is compared with the Monte Carlo, solid isotropic material with penalization, perturbation and non-intrusive polynomial chaos expansion methods. The novelty of the present method lies in the following three aspects: (1) this article is among the first to use the extended finite element method in studying the topology optimization under uncertainty; (2) by adopting the extended finite element method for boundary elements in the finite element framework, there is no need for any remeshing techniques; and (3) the numerical results show that the present method has a smoother boundary region and minimum value of the mean and standard deviation of compliance than the other methods, in particular mesh size.

Uncertainty quantification and sensitivity analysis of an ultraviolet water disinfection photo-reactor

In recent years, computational fluid dynamics (CFD) has been utilized as a powerful simulation tool to design and analyze the ultraviolet (UV) disinfection systems. A fact that is ignored in deterministic simulations is that most of industrial problems are associated with inevitable uncertainties in input parameters. In the present study, a non-deterministic CFD simulation of the UV disinfection process using non-intrusive polynomial chaos expansion method is performed to quantify the impact of uncertain input parameters on the performance of an annular photo-reactor. Five input variables with their probability distribution functions are determined as the uncertain inputs to evaluate their effects on the quantities of interest for different working conditions. Besides, the contribution of each uncertain variable is quantified by a sensitivity analysis using the Sobol’ indices method. The results show that the disinfection process of contaminated water with higher UV transmittances and lower flow rates is associated with more uncertainty.

Aerodynamic Design Optimization of Transonic Natural-Laminar-Flow Airfoil at Low Reynolds Number

The position and size of laminar separation bubble on airfoil surfaces exert a profound impact on the efficiency of transonic natural-laminar-flow airfoil at low Reynolds number. Based on the particle swarm algorithm, an optimization methodology in the current work would be established with the aim of designing a high and robust performance transonic natural-laminar-flow airfoil at low Reynolds number. This methodology primarily includes two design processes: a traditional deterministic optimization at on-design point and a multi-objective of uncertainty-based optimization. First, a multigroup cooperative particle swarm optimization was used to obtain the optimal deterministic solution. The crowing distance multi-objective particle swarm optimization and the non-intrusive polynomial chaos expansion method were then adopted to determinate the Pareto-optimal front of uncertainty-based optimization. Additionally, the γ−Re¯θt transition model was employed to predict the laminar-turbulent transition. Regarding to the established optimization methodology, a propeller tip airfoil of solar energy unmanned aerial vehicle was finally designed. During optimization processes, the minimized pressure drag was particularly chosen as the optimization objective, while the friction drag increment served as a constraint condition. The deterministic results indicate that the optimized airfoil has a good ability to control the separation and reattachment positions, and the pressure drag can be greatly reduced when the laminar separation bubble is weakened. The multi-objective results show that the uncertainty-based optimized airfoil possesses a significant robust performance by considering the uncertainty of Mach number. The findings evidently demonstrate that the proposed optimization methodology and mathematical model are valuable tools to design a high-efficiency airfoil for the propeller tip.

NIPC-based uncertainty analysis of infrared radiation from rocket exhaust plumes caused by nozzle exit conditions

In this study, an uncertainty analysis for the infrared radiation characteristics of rocket exhaust plumes at representative trajectory points is performed. Owing to the instability of the rocket motor’s working characteristics, numerical rocket plume infrared radiation predictions possess significant uncertainties. In this study, four epistemic uncertain variables (freestream velocity, nozzle exit pressure, temperature, and velocity) are considered for uncertainty and sensitivity analyses. Based on the infrared signature analysis tool, the response surface of statistical samples is established through the point collocation nonintrusive polynomial chaos expansion method. Polynomial chaos expansion coefficients are solved using the quadrature method to calculate the statistical characteristics and uncertainty of random input variables. The tensor-product quadrature sparse grid method is utilized to reduce the number of samples for multiple input variables. Based on these models, the uncertainty quantification of infrared radiation for Atlas-IIA rocket plumes is analyzed, including the flows, radiation images, spectra, and radiance. The results show that the uncertainty mainly results from afterburning at low altitude, and the nozzle exit velocity has a significant influence on the radiation intensity of the plume. With an increase in altitude, the uncertainty of infrared radiation owing to the afterburning effect decreases, and the influence of the freestream velocity increases. In addition, the proportion of radiation intensity in the 4.3-μm band is higher than that in the 2.7-μm band, and the corresponding uncertainty band is gradually widened. The nozzle exit temperature is the dominant factor that affects the radiation characteristics of the plume at high altitudes. These results of uncertainty and sensitivity analyses are helpful for improving numerical models of the plume infrared signature.

A phase-field based robust topology optimization method for phononic crystals design considering uncertain diffuse regions

Uncertain manufacturing errors in fabricating phononic crystals (PnCs) may notably affect their band gap properties. One thus needs to consider uncertainties in the design and optimization of microstructures formed from the unit cell of a PnC. This study focuses on proposing a robust topology optimization method of PnC microstructures against random diffuse regions between material phases. The material distribution of the unit cell is performed using the phase-field method, which is able to simulate the motion and the uncertain width of the diffuse regions. A stochastic band-gap analysis is performed using the expansion optimal linear estimation method and the polynomial chaos expansion (PCE)-based framework. The objective function of the robust optimization formulation is taken as the weighted sum of the mean value and standard deviation of the specified band gap. With the sensitivities of the stochastic band gap obtained by the nonintrusive PCE method, the phase-field function is updated with the Allen–Cahn equation. Numerical examples show that the proposed method provides meaningful optimal designs of microstructures of the unit cell in achieving a robust band gap.

An Adaptive Least Angle Regression Method for Uncertainty Quantification in FDTD Computation

The nonintrusive polynomial chaos expansion method is used to quantify the uncertainty of a stochastic system. It potentially reduces the number of numerical simulations in modeling process, thus improving efficiency while ensuring accuracy. However, the number of polynomial bases grows substantially with the increase of random parameters, which may render the technique ineffective due to the excessive computational resources. To address such problems, methods based on the sparse strategy such as the least angle regression (LARS) method with hyperbolic index sets can be used. This paper presents the first work to improve the accuracy of the original LARS method for uncertainty quantification. We propose an adaptive LARS method in order to quantify the uncertainty of the results from the numerical simulations with higher accuracy than the original LARS method. The proposed method outperforms the original LARS method in terms of accuracy and stability. The L2 regularization scheme further reduces the number of input samples while maintaining the accuracy of the LARS method.

Uncertainty Quantification of NOx Emission Due to Operating Conditions and Chemical Kinetic Parameters in a Premixed Burner

Many sources of uncertainty exist when emissions are modeled for a gas turbine combustion system. They originate from uncertain inputs, boundary conditions, calibration, or lack of sufficient fidelity in a model. In this paper, a nonintrusive polynomial chaos expansion (NIPCE) method is coupled with a chemical reactor network (CRN) model using Python to quantify uncertainties of NOx emission in a premixed burner. The first objective of uncertainty quantification (UQ) in this study is development of a global sensitivity analysis method based on the NIPCE method to capture aleatory uncertainty on NOx emission due to variation of operating conditions. The second objective is uncertainty analysis (UA) of NOx emission due to uncertain Arrhenius parameters in a chemical kinetic mechanism to study epistemic uncertainty in emission modeling. A two-reactor CRN consisting of a perfectly stirred reactor (PSR) and a plug flow reactor (PFR) is constructed in this study using Cantera to model NOx emission in a benchmark premixed burner under gas turbine operating conditions. The results of uncertainty and sensitivity analysis (SA) using NIPCE based on point collocation method (PCM) are then compared with the results of advanced Monte Carlo simulation (MCS). A set of surrogate models is also developed based on the NIPCE approach and compared with the forward model in Cantera to predict NOx emissions. The results show the capability of NIPCE approach for UQ using a limited number of evaluations to develop a UQ-enabled emission prediction tool for gas turbine combustion systems.

Historical carpentry corner log joints—Numerical analysis within stochastic framework

The paper presents the results of numerical analysis performed on historical, traditional carpentry corner log joints of two basic topologies: the short-corner dovetail connection and the saddle notch connection. These types of carpentry joints are commonly used in currently preserved objects of wooden architecture. All connections have been modelled in pinewood, which has been defined in the Finite Element software MSC.Marc/Mentat as an orthotropic material. The numerical calculations have been carried out for two types of connections with two different boundary conditions and load types. The contact phenomenon between the individual elements of the connections has been taken into account. The main purpose of the research is to select the most damage-resistant type of connection and to determine the stress distributions on the contact surfaces, which demines the damage areas. However, a lot of uncertainties appear in the studied models, e.g. due to the natural variability of the material properties of wood and the uncertainty of friction coefficient. Therefore the uncertainty quantification and global sensitivity analysis has been performed in order to include these uncertainties and study their effect on variation of the mechanical response of the connections. A regression-based non-intrusive polynomial chaos expansion method has been employed to complete the task.The state-of-the-art knowledge about the damage-prone zones in the considered connections is immensely important since many wooden buildings, mostly historical, require maintenance, renovation and the reinforcement of existing, especially historical elements. On the contrary, there are not many results of related research published yet.

Reliability-based trajectory optimization using nonintrusive polynomial chaos for Mars entry mission

This paper presents the reliability-based sequential optimization (RBSO) method to settle the trajectory optimization problem with parametric uncertainties in entry dynamics for Mars entry mission. First, the deterministic entry trajectory optimization model is reviewed, and then the reliability-based optimization model is formulated. In addition, the modified sequential optimization method, in which the nonintrusive polynomial chaos expansion (PCE) method and the most probable point (MPP) searching method are employed, is proposed to solve the reliability-based optimization problem efficiently. The nonintrusive PCE method contributes to the transformation between the stochastic optimization (SO) and the deterministic optimization (DO) and to the approximation of trajectory solution efficiently. The MPP method, which is used for assessing the reliability of constraints satisfaction only up to the necessary level, is employed to further improve the computational efficiency. The cycle including SO, reliability assessment and constraints update is repeated in the RBSO until the reliability requirements of constraints satisfaction are satisfied. Finally, the RBSO is compared with the traditional DO and the traditional sequential optimization based on Monte Carlo (MC) simulation in a specific Mars entry mission to demonstrate the effectiveness and the efficiency of the proposed method.

Bifurcation Analysis with Aerodynamic-Structure Uncertainties by the Nonintrusive PCE Method

An aeroelastic model for airfoil with a third-order stiffness in both pitch and plunge degree of freedom (DOF) and the modified Leishman–Beddoes (LB) model were built and validated. The nonintrusive polynomial chaos expansion (PCE) based on tensor product is applied to quantify the uncertainty of aerodynamic and structure parameters on the aerodynamic force and aeroelastic behavior. The uncertain limit cycle oscillation (LCO) and bifurcation are simulated in the time domain with the stochastic PCE method. Bifurcation diagrams with uncertainties were quantified. The Monte Carlo simulation (MCS) is also applied for comparison. From the current work, it can be concluded that the nonintrusive polynomial chaos expansion can give an acceptable accuracy and have a much higher calculation efficiency than MCS. For aerodynamic model, uncertainties of aerodynamic parameters affect the aerodynamic force significantly at the stage from separation to stall at upstroke and at the stage from stall to reattach at return. For aeroelastic model, both uncertainties of aerodynamic parameters and structure parameters impact bifurcation position. Structure uncertainty of parameters is more sensitive for bifurcation. When the nonlinear stall flutter and bifurcation are concerned, more attention should be paid to the separation process of aerodynamics and parameters about pitch DOF in structure.

Non-intrusive uncertainty quantification in structural-acoustic systems using polynomial chaos expansion method

A framework of non-intrusive polynomial chaos expansion method (PC) was proposed to investigate the statistic characteristics of the response of structural-acoustic system containing random uncertainty. The PC method does not need to reformulate model equations, and the statistics of the response can be evaluated directly. The results show that compared to the direct Monte Carlo method (MCM) based on the original numerical model, the PC method is effective and more efficient.

A Cholesky-Based SGM-MLFMM for Stochastic Full-Wave Problems Described by Correlated Random Variables

In this letter, the multilevel fast multipole method (MLFMM) is combined with the polynomial chaos expansion (PCE)-based stochastic Galerkin method (SGM) to stochastically model scatterers with geometrical variations that need to be described by a set of correlated random variables (RVs). It is demonstrated how Cholesky decomposition is the appropriate choice for the RVs transformation, leading to an efficient SGM-MLFMM algorithm. The novel method is applied to the uncertainty quantification of the currents induced on a rough surface, being a classic example of a scatterer described by means of correlated RVs, and the results clearly demonstrate its superiority compared to the nonintrusive PCE methods and to the standard Monte Carlo method.

Uncertainty Quantification by the Nonintrusive Polynomial Chaos Expansion with an Adjustment Strategy

This paper proposes the criteria to adjust the polynomial order and the number of sample points employed in the nonintrusive polynomial chaos expansion method for effective uncertainty quantification. These criteria are calculated based on the total variations in approximated responses between different polynomial orders and different numbers of sample points to avoid Runge’s phenomena. The polynomial chaos expansion method adjusted by the proposed criteria is tested in analytic functions and then applied to the sonic boom propagation analysis considering the uncertainty in atmospheric temperature, humidity, and wind velocity. It is confirmed that the polynomial chaos expansion method with the proposed adjustment strategy can estimate the statistics of the sonic boom signature with sufficient accuracy and only 0.35% of the computational cost compared with the Monte–Carlo method.

A fast and more universal algorithm for an uncertain acoustic filed in shallow-water

Non-intrusive polynomial chaos expansion (NPCE) method is a fast algorithm with the best performances for an uncertain acoustic filed currently, in which the selection of collocation points is an important factor for the computational accuracy, and some special processing methods such as piecewise probabilistic collocation method, should be adopted when the outputs of acoustic field vary severely with uncertain ocean environmental parameters. A new fast algorithm for uncertain acoustic filed in shallow-water is proposed based on Kriging model. The theoretical description of the new algorithm is given, and numerical simulations are conducted to verify the performances of the proposed algorithm. The physical interpretations are given in detail. The results demonstrate that the proposed algorithm is more accurate than the NPCE method under the same conditions, and any special processing method does not need to be adopted when the outputs of acoustic field vary severely with the uncertain ocean environmental parameters. The weakness of NPCE method can be overcome by the proposed algorithm, which is that the computational cost increases with the stochastic polynomial expanding to a higher order for enhancing the computational accuracy. The selection of sample point of the proposed algorithm is simpler and easier than that of NPCE method, and the calculation errors can be given directly. Thus, the proposed algorithm is more universal than NPCE method.