Non-intrusive Polynomial Chaos Expansion Research Articles

Robust optimization of a novel ultraviolet (UV) photoreactor for water disinfection: A neural network approach

To optimize the ultraviolet (UV) water disinfection process, it is crucial to determine the ideal geometric dimensions of a corresponding model that enhance performance while minimizing the impact of uncertain photoreactor inputs. As water treatment directly affects people's lives, it is crucial to eliminate the risks associated with the non-ideal performance of disinfection photoreactors. Input uncertainties greatly affect photoreactor performance, making it essential to develop a robust optimization algorithm in advance to mitigate these effects and minimize the physical and financial resources required for constructing the photoreactors. In the suggested algorithm, a two-objective genetic algorithm is integrated with a non-intrusive polynomial chaos expansion (PCE) technique. Additionally, the Sobol sampling method is employed to select the necessary samples for understanding the system's behavior. An artificial neural network surrogate model is trained using sufficient data points derived from computational fluid dynamics (CFD) simulations. A novel type of UV photoreactors working based on exterior reflectors is chosen to optimize the process with three uncertain input parameters, including UV lamp power, UV transmittance of water, and diffusive fraction of the reflective surface. In addition, four geometrical design variables are considered to find the optimal configuration of the photoreactor. The standard deviation (SD) and the reciprocal of log reduction value (LRV) are set as the objective functions, calculated using PCE. The optimal design provides a LRV of 3.95 with SD of 0.2. The coefficient of variation (CoV) of the model significantly declines up to 7%, indicating the decreased sensitivity of the photoreactor to the input uncertainties. Additionally, it is discovered that the robust model exhibits minimal sensitivity to changes in reflectivity in various flow rates, and its output variability aligns with the SD obtained through robust optimization.

Uncertainty Quantification Analysis in the Blade Element Momentum Method

The blade element momentum (BEM) method is a widely used non-linear model for the efficient performance evaluation and design of wind turbines. The aim of this paper is to quantify the uncertainty related to BEM model inputs and sub-models, and investigate how these propagate through the model. Uncertainties related to viscous dissipation in the wake, aerofoil force coefficients, and tip-loss models are considered. Global sensitivity to these parameters is analysed using non-intrusive polynomial chaos expansion, which provides a structured method for uncertainty propagation and sensitivity quantification. Sobol indices are employed to rank the relative importance of each factor to the overall uncertainty in the system. By analysing the NREL 5 MW and DTU 10 MW reference wind turbines, we observe that the different rotors may exhibit different levels of sensitivity to input parameters. The effect of viscous mixing in the turbine wake is found to have a significant impact on predicted rotor performance. Uncertainty in tip-loss model coefficients is also found to generally be important, particularly when evaluating spanwise variations in rotor loads. Uncertainty quantification has the potential to improve understanding of BEM modelling, and provide guidance on the importance of sub-model improvements.

Robust topology optimization for multi-material structures considering material uncertainties

Multi-material structures have attracted more and more attention from scholars and engineers, because they usually have good composite performance. Uncertainty factors are ubiquitous in practice, especially for material uncertainties. Multi-material structures contain many materials with different properties, material uncertainties will affect the allocation of material usage, thereby affecting structural performance. Therefore, it is crucial to consider material uncertainties in the design of multi-material structures. This paper detailedly explores the impacts of elastic modulus uncertainty and Poisson's ratio uncertainty on the design of multi-material structures. The topology optimization is to minimize the compliance subjected to a mass constraint, where multiple materials can be freely allocated during the optimization. Floating projection topology optimization is adopted to search for the structural topologies, and robust sensitivity information is derived. For the quantification of material uncertainties, this paper introduces a non-intrusive polynomial chaos expansion (PCE) method to implicitly quantify them, the computational efficiency and accuracy in PCE is compared with the classical Monte Carlo method. Finally, topology optimization examples are provided to demonstrate the effects of elastic modulus uncertainty, Poisson's ratio uncertainty and hybrid uncertainty on the design of multi-material structures.

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

Empirical Assessment of Non-Intrusive Polynomial Chaos Expansions for High-Dimensional Stochastic CFD Problems

Empirical Assessment of Non-Intrusive Polynomial Chaos Expansions for High-Dimensional Stochastic CFD Problems

Systematic analysis of non-intrusive polynomial chaos expansion to determine rotary blood pump performance over the entire operating range

This study applies non-intrusive polynomial chaos expansion (NIPCE) surrogate modeling to analyze the performance of a rotary blood pump (RBP) across its operating range. We systematically investigate key parameters, including polynomial order, training data points, and data smoothness, while comparing them to test data. Using a polynomial order of 4 and a minimum of 20 training points, we successfully train a NIPCE model that accurately predicts pressure head and axial force within the specified operating point range ([0–5000] rpm and [0–7] l/min). We also assess the NIPCE model's ability to predict two-dimensional velocity data across the given range and find good overall agreement (mean absolute error = 0.1 m/s) with a test simulation under the same operating condition. Our approach extends current NIPCE modeling of RBPs by considering the entire operating range and providing validation guidelines. While acknowledging computational benefits, we emphasize the challenge of modeling discontinuous data and its relevance to clinically realistic operating points. We offer open access to our raw data and Python code, promoting reproducibility and accessibility within the scientific community. In conclusion, this study advances comprehensive NIPCE modeling of RBP performance and underlines how critically NIPCE parameters and rigorous validation affect results.

Robust Constrained Multi-Objective Guidance of Supersonic Transport Landing Using Evolutionary Algorithm and Polynomial Chaos

Landing of supersonic transport (SST) suffers from a large uncertainty due to its highly sensitive aerodynamic properties in the subsonic domain, as well as the wind gusts around runways. At the vehicle design stage, a landing trajectory optimization under wind uncertainty in a multi-objective solution space is desired to explore the possible trade-off in its key flight performance metrics. The proposed algorithm solves this robust constrained multi-objective optimal control problem by integrating non-intrusive polynomial chaos expansion into a constrained evolutionary algorithm. The computationally tractable optimization is made possible through the conversion of a probabilistic problem into an equivalent deterministic representation while maintaining a form of the multi-objective problem. The generated guidance trajectories achieve a significant reduction of the uncertainty in their terminal states with a marginal modification in the control history of the deterministic solutions, validating the importance of the consideration of robustness in trajectory optimization.

Analyzing the sensitivity of wave frequency responses of floating vessels to uncertain system variables utilizing Polynomial Chaos Expansion

Abstract From a mathematical viewpoint, the frequency domain analysis of vessel motion responses due to wave actions is based on integration of system dynamics idealized in terms of response amplitude operators (RAOs) for 6 DOF rigid body motions and an input wave spectrum in order to obtain the response spectrum. Various quantities of interest can be deduced from the response spectrum which are then used for deriving response-based operational limits for marine operations, also including extreme value and fatigue analysis. The variation of such quantities, owing to the uncertainties associated with the vessel system parameters, can be quantified by performing uncertainty propagation (UP) and consequent sensitivity analysis (SA). This study emphasizes and proposes a computational-efficient way of assessing the sensitivity of the system model output with respect to the uncertainties residing in the input parameters by operating on a surrogate model representation. In this respect, the global sensitivity analysis was effectively carried out by deploying an efficient non-intrusive polynomial chaos expansion (PCE) surrogate model built using a point collocation strategy. Successively, the Sobol' indices were obtained from the analytical decomposition of the polynomial coefficients. The indices, eventually, are employed to quantitatively measure the effects of input uncertainties on the output 6 DOF vessel Root Mean Square responses.

PyThia: A Python package for Uncertainty Quantification based on non-intrusive polynomial chaos expansions

PyThia: A Python package for Uncertainty Quantification based on non-intrusive polynomial chaos expansions

Robust optimization of a marine current turbine using a novel robustness criterion

The present paper aims to establish a systematic robust optimization framework for the hydrodynamic performance of marine current turbines against uncertain conditions. To this end, we developed a novel robustness criterion based on the cumulative distribution function and employed an XFOIL-based blade element momentum method to predict turbine performance. The non-intrusive polynomial chaos expansion was also applied for uncertainty propagation and probabilistic computations. Initially, a set of sensitivity analysis studies was conducted on a model turbine to determine the most influential parameters. The results indicated that uncertainties have a significant impact on turbine performance, even under controlled experimental conditions. In the next stage, we applied deterministic and robust optimization approaches to optimize a real-world marine turbine design under practical conditions. We utilized a hybrid evolutionary optimization algorithm based on the Genetic Algorithm and Particle Swarm Optimization. Additionally, the novel robustness criterion was introduced to address the issues associated with conventional criteria. The robust optimization was performed independently using three criteria: the worst-case scenario, the mean value penalty, and the novel criterion. By investigating the probabilistic characteristics of the performance in the deterministic and robust optimum turbines, we demonstrated that the power generation capacity of the deterministic optimum is expectedly higher than that of the robust optimum in deterministic hypothetical conditions. However, during turbine operation in real conditions, which are accompanied by large uncertainties, the optimum design achieved using the new robustness criterion is more likely to produce greater power. Furthermore, the power variations of the turbine are more limited, indicating increased reliability and stability of the optimized design.

Active learning-based domain adaptive localized polynomial chaos expansion

The paper presents a novel methodology to build surrogate models of complicated functions by an active learning-based sequential decomposition of the input random space and construction of localized polynomial chaos expansions, referred to as domain adaptive localized polynomial chaos expansion (DAL-PCE). The approach utilizes sequential decomposition of the input random space into smaller sub-domains approximated by low-order polynomial expansions. This allows the approximation of functions with strong nonlinearities, discontinuities, and/or singularities that often appear in dynamical systems. Decomposition of the input random space and local approximations alleviates the Gibbs phenomenon for these types of problems and confines error to a very small vicinity near the non-linearity. The global behavior of the surrogate model is therefore significantly better than existing methods, as shown in numerical examples, including an engineering dynamical system exhibiting discontinuous response. The whole process is driven by an active learning routine that uses the recently proposed Θ criterion to assess local variance contributions (Novák et al., 2021). The proposed approach balances both exploitation of the surrogate model and exploration of the input random space and thus leads to efficient and accurate approximation of the original mathematical model. The numerical results show the superiority of the DAL-PCE in comparison to (i) a single global polynomial chaos expansion and (ii) the recently proposed stochastic spectral embedding (SSE) method (Marelli et al., 2021) developed as an accurate surrogate model and which is based on a similar domain decomposition process. This method represents a general framework upon which further extensions and refinements can be based and which can be combined with any technique for non-intrusive polynomial chaos expansion construction.

Research on Structural Failure of MEMS Safety and Arming Device Considering Uncertainty in Fabrication Errors

This paper presents an efficient methodology for assessing the structural failure of microelectromechanical systems (MEMS) used in safety and arming devices (SAD) while taking into account the effect of fabrication errors. The proposed methodology uses statistical characteristics that are assessed using non-intrusive polynomial chaos expansion (PCE) to consider uncertainty. The effectiveness of the methodology is demonstrated by considering the example of the latching feature in MEMS SAD, and comparing the results with experimental data obtained from a fabricated prototype. The proposed methodology accurately determines the failure probability and sensitive parameters of the sample, and provides insights into the impact of fabrication errors on the structural failure of the sample. It achieves the objective of reducing the burden of repetitive finite element analysis (FEA) in uncertainty analysis, and provides an efficient way to account for uncertainty in the study of MEMS devices.

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 on the Cooling Performance of a Transonic Turbine Vane With Upstream Endwall Misalignment

Abstract With increasing aerodynamic and thermal loads, film cooling has been a popular technology integrated into the design of the modern gas turbine vane endwall, especially for the first-stage vane endwall. A staggering amount of research has been completed to quantify the effect of operating conditions and cooling hole geometrical properties. However, most of these investigations did not address the influence of the manufacturing tolerances, assembly errors, and operation degradations on the endwall misalignment. In this paper, uncertainty quantification (UQ) analysis was performed to quantify the impacts of upstream endwall misalignment uncertainties on the endwall film cooling performance and vane surface phantom cooling performance. The upstream endwall misalignment, step geometry with various step heights, commonly exists between the combustor exit and the first-stage vane endwall. Based on the non-intrusive polynomial chaos expansion (NIPC) and the uniform probability distribution assumption, the deviation (step height) uncertainties of the upstream endwall misalignment were quantified. To predict the endwall secondary flow and film cooling effectiveness in the transonic linear vane passage, the commercial computational fluid dynamic solver ANSYS FLUENT was used to numerically solve the three-dimensional steady-state Reynolds-Averaged Navier–Stokes (RANS) equations. The robustness analysis of endwall film cooling performance and phantom cooling performance to the upstream endwall misalignment was conducted for three design upstream step heights (ΔH): a baseline configuration (ΔH = 0 mm), two misaligned configurations with forward step (ΔH = −5 mm) and backward step (ΔH = 5 mm), respectively. Results show that the actual cooling performance has a high probability of deviating from the nominal value for the baseline configuration. The critical regions that are most sensitive to the upstream step misalignment are also identified by variances. The UQ results also show that the design geometry with a forward step has a more robust film cooling performance on endwall and phantom cooling performance on the vane pressure side surface, which means a smaller variance and a better expectation than the no-step configuration. In contrast, the design geometry with a backward step induces the reductions in the expectation of the film cooling effectiveness and coolant coverage and the amplification of performance fluctuations. This work provides a certain guiding direction for the optimization design for the upstream step geometry.

Surrogate models for the blade element momentum aerodynamic model using non-intrusive polynomial chaos expansions

Abstract. In typical industrial practice based on IEC standards, wind turbine simulations are computed in the time domain for each mean wind speed bin using a few unsteady wind seeds. Software such as FAST, BLADED, or HAWC2 can be used to capture the unsteadiness and uncertainties of the wind in the simulations. The statistics of these aeroelastic simulation outputs are extracted and used to calculate fatigue and extreme loads on the wind turbine components. The minimum requirement of having six seeds does not guarantee an accurate estimation of the overall statistics. One solution might be running more seeds; however, this will increase the computation cost. Moreover, to move beyond blade element momentum (BEM)-based tools toward vortex/potential flow formulations, a reduction in the computational cost associated with the unsteady flow and uncertainty handling is required. This study illustrates the unsteady wind aerodynamic statistics' stationary character based on the standard turbulence models. This character is shown based on the output of National Renewable Energy Lab (NREL) 5MW reference machine BEM simulations. Afterwards, we propose a non-intrusive polynomial chaos expansion (PCE) to build a surrogate model of the loads' statistics, the rotor thrust, and torque, at each time step, to estimate the extreme statistics more accurately and efficiently.

Stochastic boundary condition effects on supersonic leading edge blowing

The present manuscript studies the robustness of leading edge blowing for active control of the upstream bow shock and overall aerothermal characteristics of a supersonic airfoil subject to variability in the injection conditions. The injection boundary conditions were considered stochastic and their randomness was propagated to the flow topology and aerothermal quantities of interest by means of non-intrusive Polynomial Chaos Expansions. The evaluations needed to build these expansions were carried out via steady two-dimensional Reynolds-Averaged Navier Stokes simulations. Two different injection port sizes were considered, with the smallest promoting a Coanda effect that gives rise to non-symmetric flow topologies. The uncertain boundary conditions lead to significant variations in the flow topology, particularly at the injection exit and the bow shock position, with variations of up to 75% of the nominal value. This study identifies the key operating conditions to ensure the optimal application of leading-edge injection for active control of supersonic airfoils.

Probabilistic CFD analysis on the flow field and performance of the FDA centrifugal blood pump

The present study is set out to systematically investigate the combined impact of operational, geometrical, and model uncertainties on the hemodynamics and performance characteristics in the U.S. Food and Drug Administration (FDA) benchmark centrifugal blood pump. Non-intrusive Polynomial Chaos Expansion (NIPCE) has been utilized to propagate the uncertainty of 12 random input variables in the flow field and the performance characteristics of the blood pump at three working conditions. The global sensitivity of the Quantities of Interest (QoI) to the uncertain input parameters was measured through the Sobol’ indices. The Multiple Reference Frames (MRF) approach and the SST k−ω turbulence model were employed to conduct the 3D CFD computations of the pump. In addition, we quantified the device-related hemolysis using the semi-empirical power-law model. The stochastic CFD results of the pump velocity field and hydraulic performance parameters showed a satisfactory agreement with measurements. Statistical analysis indicated that the effect of operational and geometrical uncertainties on the velocity field of the blood pump chiefly emerges at the outlet diffuser region, more specifically along the center-line of the fluid jet formed inside the diffuser. This study has also clarified that the uncertainties in the flow field and the hydraulic performance of the blood pump are mainly due to the variations of impeller diameter, rotational speed, radial clearance, and flow rate. By contrast, the hemolysis index is profoundly affected by the model parameters. Additionally, higher robustness against uncertainties was observed for hydraulic efficiency compared to the pump head. Eventually, it was shown that the maximum value of the distribution function of the relative hemolysis index lies within the bounds of the measurements.

Uncertainty analysis of NOx and CO emissions in industrial ethylene cracking furnace using high-precision sparse polynomial chaos expansion

ABSTRACT Traditional designing of ethylene-cracking furnaces by computational fluid dynamics (CFD) does not consider the uncertainties in actual engineering. This paper introduces one uncertainty analysis framework based on non-intrusive polynomial chaos expansion (NIPCE) and multi-order Sobol indices to perform uncertainty quantification (UQ) and sensitivity analysis for NOx and CO emissions in the combustion process. In this framework, several operating parameters of the bottom burners that easily cause uncontrollable fluctuations in the flame and emission characteristics are selected as uncertainty variables and characterized as a probability distribution. Computationally expensive CFD simulation of cracking furnace is used to generate a small number of samples, which are carried out to develop high-precision sparse PCE models by the degree-adaptive scheme and least angle regression (LAR) algorithm. And multi-order Sobol sensitivity analysis based on PCE models is performed efficiently to research the influence of uncertain parameters on pollution generation. Under 3% of burner’s uncertainty operating parameters, the results find that the excess air coefficient of 1.20 can obviously reduce the generation of NOx and CO emissions and control the fluctuation caused by uncertainties. Moreover, sensitivity analysis determines the critical variables that affect pollutant emissions to help the actual process avoid the unknown effects of uncertainties.

Stochastic finite element analysis of composite cycloidal propeller blade during crash-stop ship maneuver

This article quantifies uncertainty in the dynamic responses of marine cycloidal propeller (MCP) blade by Stochastic Finite Element Method (SFEM) during crash-stop ship maneuver. Stopping a ship in an emergency is known as crash-stop astern maneuver. A MCP contains a horizontal circular metallic disc on which six aerofoil blades are hung vertically. All blades are made up of carbon fiber-reinforced composite laminates. Uncertainty in ship speed and composite material properties are propagated to find out the statistical response of natural frequencies, displacement, and stresses on the blade. SFEM is a combination of Stochastic Response Surface Method (SRSM) and Finite Element Method (FEM). SRSM is a collocation-based non-intrusive polynomial chaos expansion to transform any random distribution into a stochastic response surface equation (surrogate model). During ship maneuvering motion, the whole unit rotates about a vertical axis at the centre. For each rotation of the disc, all blades undergo periodic oscillations. The SRSM for uncertainty propagation is computationally more efficient than the standard Monte Carlo Simulation (MCS) technique without compromising the accuracy of the results. It is observed that uncertainty in vibration responses of MCP blade is significant and worth study.

Stochastic Investigation on the Robustness of Laminar-Flow Wings for Flight Tests

Natural laminar-flow (NLF) wings and hybrid laminar-flow control (HLFC) wings are efficient drag reduction technologies. It is vital to develop robust laminar-flow wings to research the stochastic characteristics of both NLF and HLFC wings and reveal the generation mechanism for differences in their statistical responses. Wing-glove flight experiments and deterministic numerical simulations using the method with a suction-velocity model are performed, which are in good agreement. Uncertainty and correlation analyses based on nonintrusive polynomial chaos expansion are then performed to evaluate the influence of operational and geometric uncertainties. The simulations indicate that operational condition uncertainties have a greater impact on their statistical responses. Pressure coefficients near the leading edge and end region of the favorable pressure gradient are the most sensitive to disturbances in the angle of attack (AOA) and Mach number, respectively. Whereas HLFC wings significantly delay the laminar–turbulent transition, NLF wings may have better robustness against disturbances if the development of Tollmien–Schlichting (TS) waves is not completely suppressed in the suction region. This is because disturbances affect statistical responses of HLFC wings by changing both the local pressure gradients and suction coefficient distributions. Positive correlations between the AOA, local pressure gradients, and suction coefficients amplify the influence of AOA disturbances. Meanwhile, Mach-number disturbances have negative and positive correlations with local pressure gradients and suction coefficients, respectively. The opposite correlation weakens the influence of Mach-number disturbances.