Point-collocation Non-intrusive Polynomial Chaos Research Articles

Opposing jets for heat flux reduction and uncertainty analysis on a V-shaped blunt leading edge

With the development of three-dimensional inward-turning inlets, the V-shaped blunt leading edge model has attracted increasing attention. In the current study, an approach of using opposing jets for heat flux reduction on the V-shaped blunt leading edge is put forward. The flow structures of the V-shaped model and the model with opposing jets are further analyzed. The mechanism of heat flux generation and heat flux reduction are pointed out. By utilizing the point-collocation non-intrusive polynomial chaos (NIPC) method, the uncertainty quantification and sensitivity analysis are carried out for the parameters of the freestream and opposing jet. When calculating the uncertainty intervals, it is recommended to use the Monto Carlo method to generate massive samples for the surrogate model instead of directly using the general confidence interval. Results show that the uncertainty for heat flux is much more severe than pressure. The most contributions of the input parameters come from the freestream Mach number Ma∞ and the freestream temperature Tinf. The heat flux reduction approach proposed in this work proves to be effective for heat flux reduction and has potential application for engineering practices.

Quantification of parametric uncertainty in γ-Reθ model for typical flat plate and airfoil transitional flows

Owing to the lack of physical knowledge of boundary layer transition, the γ-Reθ transition model introduces closure parameters, which increase the uncertainty of transition prediction. The objective of this work is to quantify the uncertainties of closure parameters in the quantities of interests and identify the key parameters. The six closure parameters in the uncertainty intervals are used as input variables, and the uncertainties of the output results are propagated by a stochastic expansion based on the point-collocation nonintrusive polynomial chaos method. The relative contribution of each parameter to uncertainty is evaluated by the Sobol index. The computational cases include natural and bypass transitional flows on zero-pressure-gradient flat plates, and subsonic and transonic flows around airfoils. For most cases, ce2, ca2, and ca1 dominate the uncertainty, and the influence of σθt is also significant when the history effects of flow are evident. The contribution of parameters in airfoils is more complex than that in flat plates. The transonic airfoil case shows that flow separation dramatically changes the distribution of Sobol indices, which poses a challenge to the accurate prediction of transition. Generally, ce2 and ca2 are the key parameters of the γ-Reθ model.

Bayesian Inference of Cavitation Model Coefficients and Uncertainty Quantification of a Venturi Flow Simulation

In the present work, uncertainty quantification of a venturi tube simulation with the cavitating flow is conducted based on Bayesian inference and point-collocation nonintrusive polynomial chaos (PC-NIPC). A Zwart–Gerber–Belamri (ZGB) cavitation model and RNG k-ε turbulence model are adopted to simulate the cavitating flow in the venturi tube using ANSYS Fluent, and the simulation results, with void fractions and velocity profiles, are validated with experimental data. A grid convergence index (GCI) based on the SLS-GCI method is investigated for the cavitation area, and the uncertainty error (UG) is estimated as 1.12 × 10−5. First, for uncertainty quantification of the venturi flow simulation, the ZGB cavitation model coefficients are calibrated with an experimental void fraction as observation data, and posterior distributions of the four model coefficients are obtained using MCMC. Second, based on the calibrated model coefficients, the forward problem with two random inputs, an inlet velocity, and wall roughness, is conducted using PC-NIPC for the surrogate model. The quantities of interest are set to the cavitation area and the profile of the velocity and void fraction. It is confirmed that the wall roughness with a Sobol index of 0.72 has a more significant effect on the uncertainty of the cavitating flow simulation than the inlet velocity of 0.52.

Uncertainty and sensitivity analysis of heat transfer in hypersonic three-dimensional shock waves/turbulent boundary layer interaction flows

Hypersonic shock waves/turbulent boundary layer interaction flow can cause severe localized thermal loads, while the fluctuations in freestream conditions and inevitable mechanical vibrations can strongly influence the flow structures and heat transfer. This study aims to quantify the uncertainties in heat transfer and pressure of fin-plate configurations and find their main contributors. Freestream velocity, freestream static pressure, freestream static temperature, wall temperature, and fin angles are chosen as uncertainty sources. The point-collocation non-intrusive polynomial chaos method is employed to propagate the uncertainties. Subsequently, Sobol indices are utilized to denote the contribution of each input parameter. The uncertainty and sensitivity results are plotted along representative lines and on the entire bottom surface to obtain the key parameters of uncertainties. As a consequence, the uncertainty in heat transfer can exceed 30%, which is higher than that in pressure. Freestream velocity, freestream static pressure, and fin angles are the key parameters of heat transfer. Freestream velocity is the biggest contributor to the uncertainty in heat flux near the heat flux peak by influencing flow reattachment. Fin angles dominate the developments of primary vortexes and are the source of the uncertainties near the leading edge of fins. The Sobol index of freestream static pressure is relatively high at the separation line, which reveals that the boundary of flow separation is sensitive to static pressure.

Uncertainty Analysis of Inflow Parameters for Mid-Lift-to-Drag Vehicle Transition

The boundary-layer transition and aerothermal environment for mid-lift-to-drag-ratio entry vehicles are highly complex and influenced by several factors. The uncertainty and sensitivity of transition and aeroheating predictions on such vehicles with respect to the inflow parameters are investigated in this paper. Five inflow parameters (freestream Mach number, freestream pressure, freestream temperature, wall temperature, and angle of attack) are chosen as the uncertainty variables. The transition predictions are obtained using Reynolds-averaged Navier–Stokes simulations with the -- transition model. The point-collocation nonintrusive polynomial chaos method is employed to describe and propagate the uncertainty of the output quantities of interests. Moreover, the Sobol index is adopted to quantify the contribution of each variable to the total uncertainty. Our results reveal that small changes (less than 3.4%) in the inflow parameters could yield roughly 10% uncertainty in the transition lengths and more than 15% uncertainty in the transitional aeroheating and skin friction. The freestream temperature and angle of attack strongly influence the transitional aeroheating. The angle of attack, freestream pressure, and freestream temperature are the major uncertainty contributors to the skin friction. The freestream pressure and angle of attack significantly impact the transition onset and length.

Uncertainty Quantification of GEKO Model Coefficients on Compressible Flows

In the present work, supersonic flows over an axisymmetric base and a 24-deg compression ramp are investigated using the generalized k - ω (GEKO) model implemented in the commercial software, ANSYS FLUENT. GEKO is a two-equation model based on the k - ω formulation, and some specified model coefficients can be tuned depending on the flow characteristics. Uncertainty quantification (UQ) analysis is incorporated to quantify the uncertainty of the model coefficients and to calibrate the coefficients. The Latin hypercube sampling (LHS) method is used for sampling independent input parameters as a uniform distribution. A metamodel is constructed based on general polynomial chaos expansion (gPCE) using ordinary least squares (OLS). The influential coefficient closure is obtained by using Sobol indices. The affine invariant ensemble algorithm (AIES) is selected to characterize the posterior distribution via Markov chain Monte Carlo sampling. Calibrated model coefficients are extracted from posterior distributions obtained through Bayesian inference, which is based on the point-collocation nonintrusive polynomial chaos (NIPC) method. Results obtained through calibrated model coefficients by Bayesian inference show superior prediction with available experimental measurements than those from original model ones.

Point-Collocation NIPC 기법 기반 천이 난류 모델 계수의 베이지안 추론 연구

In the present work, Bayesian inference based on the point-collocation Non-intrusive Polynomial Chaos was conducted in transitional flow around the flat plate using Menter’s γ-Re<SUB>θ</SUB> transition model. Three model coefficients, C<SUB>a2</SUB>, C<SUB>e1</SUB> and C<SUB>e2</SUB> were considered for random variables based on the sensitivity analysis of the present turbulence model and quantity of interest was set to the drag coefficient, C<SUB>d</SUB>. With the assumption of uniform distribution of three model coefficients within ±10%, the surrogate model was obtained based on the general Polynomial Chaos expansion. The simulation results were used for observation to calculate the likelihood function and MCMC sampling algorithm was adopted for Bayesian inference. The correlation between C<SUB>e1</SUB> and C<SUB>d</SUB> was predicted with highest value than other two model coefficients. Based on the original Bayesian inference result, the effect of number of observation data was studied. Also the 2nd order of gPC for surrogate model was adequate through comparison with results of 3rd order gPC. Finally, two prior distributions of input random variables for Bayesian inference were considered and the differences of posterior distributions of input and output were investigated.

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 Two-Phase Monte Carlo Simulation/Non-Intrusive Polynomial Chaos (MSC/NIPC) Method for Quantification of Margins and Mixed Uncertainties (QMMU) in Flutter Analysis

A two-phase Monte Carlo Simulation/Non-intrusive Polynomial Chaos (MCS/NIPC) method for quantification of margins and mixed uncertainties (aleatory and epistemic uncertainties) is proposed in this paper for the flutter speed boundary analysis. Compared with the traditional MCS/MCS method which needs lots of numerical simulations, the MCS/NIPC method can reduce the computational cost without losing accuracy due to the use of point collocation non-intrusive polynomial chaos in the inner loop. Based on the results of uncertainty quantification, a novel practical quantification of margins and mixed uncertainties (QMMU) framework is proposed considering both aleatory and epistemic uncertainties such as the parametric uncertainties in complicated models. A two-dimensional airfoil and a three-dimensional benchmark wing, the AGARD 445.6 wing, are both employed to illustrate the practical application of the proposed methods for flutter speed computation in the presence of mixed uncertainties arising from the material properties and flight conditions. Within the analysis, aleatory and mixed uncertainty quantification are conducted to prove the effectiveness and efficiency of the MCS/NIPC method. Then the QMMU metrics are accomplished for the flutter speed boundary analysis of the two airfoil models. The results demonstrate the potential of the proposed QMMU framework to analyze the stability of engineering systems.

Uncertainty and sensitivity analysis of inflow parameters for HyShot II scramjet numerical simulaiton

The flowfield inside a combustor of scramjet such as Hyshot II system is highly complicated and many factors have effect on its numerical prediction. Uncertainty and sensitivity analysis of aerodynamics prediction on Hyshot II intake configuration is performed in this work. The inflow conditions of intake configuration are of vital crucial since they directly determine the inflow conditions of combustor configuration, which in turn have a remarkable influence on the combustion and mixing process. Meanwhile, the varied flight conditions may also introduce great and direct impact on the performance in actual flight tests. It is significant to evaluate the uncertainties and sensitivities and to identify the critical factors of inflow conditions for the intake configuration. Five input parameters are selected as the epistemic uncertain variables. The flowfield prediction is conducted via using Reynolds-averaged Navier-Stokes simulations as the numerical study approach with standard Menter's shear stress transport turbulence model and species transport process for Hyshot II configuration. Stochastic expansions based on point collocation non-intrusive polynomial chaos, which require 42 deterministic simulations in this study, are employed to propagate and estimate the uncertainties. Meanwhile, Sobol indices are adopted to rank and indicate the individual contribution and to identify the critical inflow parameters. The results demonstrate that freestream Mach number and temperature are the dominated contributing parameters among the performance parameters. In addition, as observed from the wall-normal profile of related variables extracted at 355 mm in x coordinate direction in the simulation results, the freestream pressure plays a dominant role in determining the pressure distribution and the freestream temperature is the most prominent contributor to the distribution of temperature and velocity.

Quantification of parametric uncertainty in k–ω–γ transition model for hypersonic flow heat transfer

The uncertain empirical values of closure coefficients in k–ω–γ transition model flaws the predictive capability. The current work aims to examine the uncertainty of k–ω–γ transition model due to uncertain closure coefficients for hypersonic flow over a flat plate and a straight cone, with introducing the stochastic expansions based on the point-collocation non-intrusive polynomial chaos method into computational fluid dynamics simulations for efficient and accurate uncertainty propagation. Specifically, five model coefficients are regarded as epistemic uncertain probabilistic variables on intervals. The output quantities of interests include the surface heat flux, skin friction coefficient, transition onsets and transition lengths. To better identify the key uncertainty contributor, Sobol indices are utilized as a variable-based global sensitivity analysis approach. Results show that the performance of the k–ω–γ transition model is highly susceptible to its closure coefficients evidenced by that small changes (10%) of model coefficients could yield more than 30% uncertainty in the transition onsets and roughly 17% uncertainty in the transition lengths. The largest uncertainty source contributing to the uncertainty in heat flux, Cf and transition onsets is C3 for hypersonic transitional flows. Besides, C4 dominates the uncertainty in transition lengths. In the future model improvement, attention should be paid to improve apprehension of rational values or constitutions of C2, C3 and C4.

Uncertainty and sensitivity study on blunt body's drag and heat reduction with combination of spike and opposing jet

Uncertainty and sensitivity study of three main parameters including length-to-diameter ratio, nozzle radius and total pressure ratio on the drag and heat reduction for the blunt body with spike and opposing jet is conducted. According to the simulation results obtained from the computational fluid dynamics, the relationship between input parameters and aerodynamic force/heat is established via the point-collocation non-intrusive polynomial chaos method. Uncertainties of pressure and heat flux caused by changes of input parameters are quantified. Sensitivity analysis is carried out to determine key parameters affecting pressure and heat flux changes. The results indicate that changes of peak pressure and peak heat flux are resulted from significant change in reattachment shock strength which varies with input parameters. Sensitivity analysis reveals that the length-to-diameter ratio has huge influence on the uncertainty in the recirculation zone, while the nozzle radius dominates the uncertainty in the reattachment zone. Moreover, it is found that the nozzle radius has an enormous effect on peak pressure and peak heat flux, and a significant negative correlation between them is identified.

Comparison of the Point-Collocation Non-Intrusive Polynomial (NIPC) and Non-Intrusive Spectral Projection (NISP) Methods for the γ-Rθ Transition Model

In the present work, a comparative study of two major non-intrusive polynomial chaos methods, Point-Collocation Non-Intrusive Polynomial Chaos (NIPC) and Non-Intrusive Spectral Projection (NISP), was conducted for the transitional transitional model. Three multiple model coefficients, Ca2, Ce1, and Ce2 were considered with multiple random inputs with the assumption of uniform distributions with ±10% deviation. The target transitional flows were one around a flat plate and Aerospatiale A-airfoil. Deterministic solutions were obtained by employing the open source software OpenFOAM. The results of two methods were compared to the results of Monte Carlo simulation with 500 runs. The order convergence of the mean value and the standard deviation (STD) were compared in terms of the quantities of interest, drag and lift coefficients. Further, the most effective model coefficient for each transitional flow could be found through the calculation of the Sobol index.

Uncertainty and sensitivity analysis of flow parameters for transition models on hypersonic flows

The impossibility to reproduce the experimental conditions, which is difficult to perfectly control and inevitably fluctuate, raises the uncertainties in numerical simulation. The objective of this work is to quantify the uncertainty and sensitivity of hypersonic transition modeling due to uncertainty in the flow parameters. Two representative transition models, e.g. γ-Reθ and k-ω-γ, are considered. And five flow parameters including freesteam Mach number, pressure, temperature, turbulence intensity and wall temperature are selected as aleatory uncertain variables uniformly distributing among the bound. Then the stochastic expansions based on point-collocation non-intrusive polynomial chaos (NIPC) method is utilized to represent and propagate the uncertainties in the output quantities of interests (QoIs) for hypersonic flow over the flat plate and the straight cone. Benefiting from the efficiency of the NIPC method, accurate uncertainty results are obtained with only 42 evaluations for each problem. Furthermore, Sobol indices are utilized to provide a relative ranking of each input uncertainty to the overall uncertainty in the output. The uncertainty results quantitatively verify that the transition zone is great sensitive to changes in flow parameters. Besides, for both the heat flux and skin friction coefficient, the importance ranking of the flow parameters and their trends are roughly resemble between the γ-Reθ and k-ω-γ model. This signifies that their sensitivity with respect to the input parameters is mainly associated with flow features rather than transition models. However, the contrary is the case for uncertainty in transition onset and length whose sensitivity to uncertainties of input parameters depends strongly on transition models. The present investigation shows that the freestream pressure is the greatest contributor to uncertainty in most cases. The freestream turbulence intensity, whose effect can be nearly negligible in heat flux and Cf, plays a predominant role on transition length.

Uncertainty and sensitivity analysis of flow parameters on aerodynamics of a hypersonic inlet

The performance of the inlet is crucial to the cruise flight of a hypersonic air-breathing propulsion vehicle. The objective of this work is to investigate the uncertainty and sensitivity of pressure field and the performance parameters for a hypersonic inlet due to the uncertainty of five flow parameters, including freestream Mach number, Reynolds number, angle of attack, temperature and wall temperature. The steady Reynolds Averaged Navier-Stokes equations are solved to predict the inlet start and unstart flows within the hysteresis loop. Then, a point-collocation non-intrusive polynomial chaos method (NIPC) is utilized to quantify the uncertainty and sensitivity in the output quantities of interest. The uncertainty analysis in pressure field shows that Mach number and angle of attack of freestream make dominant contributions to the total uncertainty, and the Mach number has remarkable impacts in the isolator. In the start flow, the angle of attack exerts its prominent influence in the post-shock regions, while Mach number mainly dominates these regions ahead of and around the shocks. The reason may be interpreted as the much greater pressure derivatives with respect to angle of attack in the post-shock regions. Significant discrepancies are presented for the unstart flow. The reflected shock waves in the unstart flow are less sensitive to the variations of flow parameters. The external flow field, separation bubble and reflected shocks are significantly affected by angle of attack. Besides, the uncertainties of the performance parameters in the start flow are about twice those in the unstart flow. The sensitivity analysis further reveals that Mach number is the major contributor to the total uncertainty of performance parameters. The correlation coefficients via linear regression method clearly illustrate the relationships between the five input parameters and the performance parameters.

Point-Collocation Non-intrusive Polynomial Chaos 기법을 활용한 평판 주위 난류 유동장 해석

In the present work, uncertainty quantification based on the point-collocation Non-intrusive Polynomial Chaos was studied in turbulent flow around the flat plate using Menter’s SST k-ω model. Two random variables of β* , model coefficient of Menter’s SST k-ω model and USUB∞/SUB, the freestream velocity are considered. β*, model coefficient was assumed to be uniformly distributed and the freestream velocity USUB∞/SUB, normally distributed with a certain value of variance. First, the uncertainty quantification with the single random variable of each one was simulated for sensitivity analysis of each parameter on the outputs such as drag coefficient and skin friction one. When the normal distribution of freestream velocity is considered, the drag coefficient showed the same type of distribution with 0.9% standard deviation. The uniform distribution of β* with the internals of 0.0784 and 0.1024 had an effect on 5% variance on drag coefficient. When two random variables are considered simultaneously, the distribution of drag coefficient showed closer type with one of model coefficient β* .The convergence rate of mean value was late than that with single random variable and standard deviation was not converged until the 5th order of gPC.

Parametric study on the heat transfer of a blunt body with counterflowing jets in hypersonic flows

A quantified parametric study for the heat transfer acting on a hypersonic blunt body with counterflowing jets is presented. Three-dimensional turbulent Navier-Stokes equations are solved to simulate freestream-jet interactive flowfields. The freestream and jet controlling parameters are treated as input sources of variation, and a point-collocation non-intrusive polynomial chaos (NIPC) method is utilized to quantify the variations in the output surface heat flux and total surface heat load acting on the blunt body by identifying the maximum and minimum of surrogate response values predicted by the NIPC. All of the sample cases are confirmed to form steady jet structures. Furthermore, through a global sensitivity analysis, Sobol indices evaluated by the NIPC, are used to rank the contributions of each input parameter to the variation in output quantities of interest. It is found that the designed upstream injection of baseline case effectively reduces the heat transfer to body surface compared with the no-jet case. The variations of input parameters induce remarkable variations of output heat flux and total heat load. The sensitivity analysis indicates that the jet-to-freestream total-pressure ratio is the top contributor to variations in heat flux, followed by the freestream Mach number. The jet total temperature is mainly important on the front part of forebody, while the contributions of jet Mach number and freestream temperature slightly increase downstream. The freestream density has the smallest effect. The sensitivity of total heat load to input parameters coincides with that of heat flux. This parametric study is expected to illustrate the significance of flow-controlling parameters to the heat transfer over blunt body and provide insight for aerothermal management by using counterflowing jets in hypersonic flows.

Uncertainty analysis of laminar and turbulent aeroheating predictions for Mars entry

Uncertainty analysis of laminar and turbulent aeroheating predictions of a 70° spherically blunted cone at peak heating condition during Mars entry is performed. Due to the difficulty in obtaining accurate chemical kinetic model, numerical aeroheating predictions possess significant uncertainties. In this study, 14 rates in Park 8-species 14-reaction model are treated as epistemic uncertain variables represented with intervals, meanwhile stochastic expansion based on point collocation non-intrusive polynomial chaos expansion, is utilized to represent and propagate the uncertainties. In particular, Sobol indices are used to rank the relative contribution of each rate. 240 CFD evaluations are employed to obtain the laminar and turbulent uncertainty results respectively. The results show that the top contributing parameters to turbulent heat flux are similar to those observed in the laminar case in the windside region. However, in the leeside region, the key parameters, which produce significant uncertainties in laminar and turbulent cases, are evidently different. The maximum uncertainty in laminar aeroheating prediction is about 6%. In the large region of leeside flank, the uncertainty in turbulent aeroheating prediction is above 7%. Moreover, the uncertainty interval is as wide as 32.4W/cm2, which is about 15% of the mean value.

Uncertainty Analysis of Radiative Heating Predictions for Titan Entry

The objective of this study was to investigate the uncertainty in shock layer radiative heating predictions on the surface of a hypersonic inflatable aerodynamic decelerator during Titan entry at peak radiative heating conditions. Computational fluid dynamics simulations of planetary entry flows and radiative heating predictions possess a significant amount of uncertainty due to the complexity of the flow physics and the difficulty in obtaining accurate experimental results of molecular-level phenomena. Sources of uncertainty considered include flowfield chemical rate models, molecular band emission, and the excitation/deexcitation rates of molecules modeled with a non-Boltzmann approach. Because of the computational cost of the numerical models, uncertainty quantification was performed with a surrogate modeling approach based on a sparse approximation of the point-collocation nonintrusive polynomial chaos expansion. Accurate uncertainty results were obtained with only 500 evaluations of the computational model. Results showed that epistemic uncertainty intervals of surface radiative heating predictions were as wide as during Titan entry, indicating the significant effect of uncertainty. A global nonlinear sensitivity analysis showed that the top uncertainty source contributing to the uncertainty in radiative heating was the flowfield chemistry modeling throughout the shock layer.

Quantification of margins and mixed uncertainties using evidence theory and stochastic expansions

The objective of this paper is to implement Dempster–Shafer Theory of Evidence (DSTE) in the presence of mixed (aleatory and multiple sources of epistemic) uncertainty to the reliability and performance assessment of complex engineering systems through the use of quantification of margins and uncertainties (QMU) methodology. This study focuses on quantifying the simulation uncertainties, both in the design condition and the performance boundaries along with the determination of margins. To address the possibility of multiple sources and intervals for epistemic uncertainty characterization, DSTE is used for uncertainty quantification. An approach to incorporate aleatory uncertainty in Dempster–Shafer structures is presented by discretizing the aleatory variable distributions into sets of intervals. In view of excessive computational costs for large scale applications and repetitive simulations needed for DSTE analysis, a stochastic response surface based on point-collocation non-intrusive polynomial chaos (NIPC) has been implemented as the surrogate for the model response. The technique is demonstrated on a model problem with non-linear analytical functions representing the outputs and performance boundaries of two coupled systems. Finally, the QMU approach is demonstrated on a multi-disciplinary analysis of a high speed civil transport (HSCT).