Polynomial Chaos Expansion Research Articles

Robust design of a low cost flat plate collector under uncertain design parameters

The high expenses of electrical energy needed for heating water can be significantly decreased by using flat plate collectors. Therefore, the design of flat plate collectors having optimal performance has received a lot of attention in recent research projects. However, due to the existence of uncertainties in material properties, geometry and environmental conditions, it is imperative to incorporate uncertainty analysis into the design optimization to obtain reliable results. In this work, a multi-criteria robust design based on multi-objective optimization of the flat plate collector system is developed. Firstly, a determinist optimization (non-consideration of uncertain design parameters in this optimization strategy) of flat plate collector design is conducted. The obtained results showed that the determinist optimization leads to a flat plate collector design with 30% efficiency sensitivity and 27% total cost sensitivity to design parameters uncertainty. As an alternative, the multi-criteria robust design based on multi-objective optimization is developed. For this goal, a combined multi-objective colonial competitive algorithm and polynomial chaos expansion method is developed and used. This multi-criteria robust optimization considers simultaneously four objectives functions including the flat plate collector efficiency, the flat plate collector total cost, and their sensitivities to uncertain design parameters. It is noted that the multi-criteria robust design based on multi-objective optimization offers a robust flat plate collector with performances sensitivity to design parameters uncertainty less than 4.5%. Furthermore, when compared to the literature results, the performances sensitivities of flat plate collector design, obtained by the multi-criteria robust design based on multi-objective optimization, were reduced until 64%.

Uncertainty Quantification for Thermodynamic Simulations with High-Dimensional Input Spaces Using Sparse Polynomial Chaos Expansion: Retrofit of a Large Thermal Power Plant

The assessment of the future thermodynamics performance of a retrofitted heat and power production unit is prone to many uncertainties due to the large number of parameters involved in the modeling of all its components. To carry out uncertainty quantification analysis, alternatives to the traditional Monte Carlo method must be used due to the large stochastic dimension of the problem. In this paper, sparse polynomial chaos expansion (SPCE) is applied to the retrofit of a large coal-fired power plant into a biomass-fired combined heat and power unit to quantify the main drivers and the overall uncertainty on the plant’s performance. The thermodynamic model encompasses over 180 components and 1500 parameters. A methodology combining the use of SPCE and expert judgment is proposed to narrow down the sources of uncertainty and deliver reliable probability distributions for the main key performance indicators (KPIs). The impact of the uncertainties on each input parameter vary with the considered KPI and its assessment through the computation of Sobol’ indices. For both coal and biomass operations, the most impactful input parameters are the composition of the fuel and its heating value. The uncertainty on the performance and steam quality parameters is not much affected by the retrofit. Key furnace parameters exhibit a skewed probability distribution with large uncertainties, which is a strong attention point in terms of boiler operation and maintenance.

Evaluation of uncertainty in direct measurement for parameterization of pyrolysis models, Part I: Thermal analysis

A comprehensive uncertainty analysis is an important part of any engineering calculation because it allows stakeholders to assess the confidence in the conclusions that come from the calculation. Without this necessary step in hypothesis testing and model validation, the true accuracy and confidence of predictions cannot be completely quantified. An experimental and analytical approach is presented in this work to determine the kinetics and energetics of pyrolysis which form the foundation for larger scale models of material burning. A detailed uncertainty quantification which utilized the generalized polynomial chaos expansion method is presented to propagate uncertainty through the pyrolysis model representation of a thermal analysis experiment. Using this methodology, the uncertainty in the predictions was found to be comparable to the uncertainty in the experimental thermal analysis data within calibration conditions. Sensitivity analyses revealed the largest contributor to uncertainty in heat flow rate predictions was the specific heat capacity of all components. Comparison of the uncertainty quantification and sensitivity analyses to well-known results validated the method of uncertainty propagation for more complex scenarios.

Data-informed statistical finite element analysis of rail buckling

In this paper, the statistical finite element method is developed further to synthesize distributed rail response data with nonlinear finite element model predictions within and outside the measured loading range. In the data-generating model of the statistical finite element method, the distributed sensing data is decomposed into a finite element model component, a model-reality mismatch component, and a noise component. Each component is considered uncertain and is represented as a Gaussian random vector with a corresponding prior density. The finite element prior density is updated using the Bayesian statistical framework in light of the distributed fiber optic sensing data. The calculated posterior density enables one to infer the true structural response. The required finite element prior density is determined by solving a conventional stochastic forward problem using a polynomial chaos expansion of random fields and a non-intrusive pseudo-spectral projection approach. In this paper, a lab test involving a section of a rail (i.e., a beam-column structural member) instrumented with distributed fiber optic sensors and subjected to axial load causing progressive lateral buckling is considered to demonstrate the use of the statistical finite element method to improve rail response prediction. The results show improved prediction of true rail strain in linear and nonlinear regimes compared with a pure finite element model-based prediction.

Impact of the Spatial Velocity Inlet Distribution on the Hemodynamics of the Thoracic Aorta.

The impact of the distribution in space of the inlet velocity in the numerical simulations of the hemodynamics in the thoracic aorta is systematically investigated. A real healthy aorta geometry, for which in-vivo measurements are available, is considered. The distribution is modeled through a truncated cone shape, which is a suitable approximation of the real one downstream of a trileaflet aortic valve during the systolic part of the cardiac cycle. The ratio between the upper and the lower base of the truncated cone and the position of the center of the upper base are selected as uncertain parameters. A stochastic approach is chosen, based on the generalized Polynomial Chaos expansion, to obtain accurate response surfaces of the quantities of interest in the parameter space. The selected parameters influence the velocity distribution in the ascending aorta. Consequently, effects on the wall shear stress are observed, confirming the need to use patient-specific inlet conditions if interested in the hemodynamics of this region. The surface base ratio is globally the most important parameter. Conversely, the impact on the velocity and wall shear stress in the aortic arch and descending aorta is almost negligible.

Data-driven sparse polynomial chaos expansion for models with dependent inputs

Polynomial chaos expansions (PCEs) have been used in many real-world engineering applications to quantify how the uncertainty of an output is propagated from inputs by decomposing the output in terms of polynomials of the inputs. PCEs for models with independent inputs have been extensively explored in the literature. Recently, different approaches have been proposed for models with dependent inputs to expand the use of PCEs to more real-world applications. Typical approaches include building PCEs based on the Gram–Schmidt algorithm or transforming the dependent inputs into independent inputs. However, the two approaches have their limitations regarding computational efficiency and additional assumptions about the input distributions, respectively. In this paper, we propose a data-driven approach to build sparse PCEs for models with dependent inputs without any distributional assumptions. The proposed algorithm recursively constructs orthonormal polynomials using a set of monomials based on their correlations with the output. The proposed algorithm on building sparse PCEs not only reduces the number of minimally required observations but also improves the numerical stability and computational efficiency. Four numerical examples are implemented to validate the proposed algorithm. The source code is made publicly available for reproducibility.

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

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.

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.

Probabilistic analysis of width‐limited 3D slope in spatially variable soils: UBLA enhanced with efficiency‐improved discretization of horn‐like failure mechanism

AbstractReliability analysis of earth slopes considering soil spatial variability has garnered significant attention from researchers. However, previous studies have predominantly focused on long slopes with infinite or ample width, such as dams and subgrades, but rare research was dedicated to slopes with width restricted by boundary constraints. In this regard, this paper proposes an efficient reliability framework suitable for width‐limited slopes in spatially variable soils. The proposed framework employed spatial discretization‐based upper bound limit analysis (UBLA) to conduct the deterministic slope stability analysis. The established failure mechanism can well capture the width‐limited three‐dimensional slip surface characteristics and satisfies the kinematically admissible conditions in spatially variable soils. Two innovative strategies are proposed to boost the efficiency of the discretized mechanism. They reduce the computational time to determine safety factors from over 20 min to less than 2 min. For the uncertainty modeling and slope reliability computing procedure, the Sparse Polynomial Chaos Expansion (SPCE) and Monte Carlo Simulation (MCS) methods are combined to generate the distribution of safety factors and failure probabilities. This combination addresses the high‐dimensional stochastic problem arising from 3D spatial variability. Using the proposed framework, a parametric analysis of a 3D spatially variable slope is conducted to investigate the impact of various factors. The findings of this paper are beneficial to the risk evolution of width‐limited slopes, and the efficiency enhancement strategies have the potential to expedite the limit analysis of other geo‐structures, such as tunnels, retaining walls, and foundations.

Stochastic optimal control for autonomous driving applications via polynomial chaos expansions

AbstractModel‐based methods in autonomous driving and advanced driving assistance gain importance in research and development due to their potential to contribute to higher road safety. Parameters of vehicle models, however, are hard to identify precisely or they can change quickly depending on the driving conditions. In this paper, we address the problem of safe trajectory planning under parametric model uncertainties motivated by automotive applications. We use the generalized polynomial chaos expansions for efficient nonlinear uncertainty propagation and distributionally robust inequalities for chance constraints approximation. Inspired by the tube‐based model predictive control, an ancillary feedback controller is used to control the deviations of stochastic modes from the nominal solution, and therefore, decrease the variance. Our approach allows reducing conservatism related to nonlinear uncertainty propagation while guaranteeing constraints satisfaction with a high probability. The performance is demonstrated on the example of a trajectory optimization problem for a simplified vehicle model with uncertain parameters.

Application of polynomial chaos expansion in sensitivity analysis of towed cable parameters of the underwater towing system

The key to achieving the optimal design of towed cables, maintaining numerical simulation accuracy, and achieving precise control of the towed body lies in sensitivity analysis. However, the traditional global sensitivity analysis method presents challenges such as high calculation costs and low accuracy. To address these issues, this paper introduces polynomial chaos expansion (PCE) to quantitatively analyze the impact of uncertainties in physical and environmental parameters on the position and attitude of the towed cable. Latin hypercube sampling is employed to obtain sample sets of input parameters, and these samples are applied to the lumped mass method to calculate the end position coordinates of the towed cable, which serves as the output response. PCE is utilized to quantitatively compute the Sobol global sensitivity index of the towed cable parameters. The accuracy of the PCE model is verified, and the optimal degree of basis functions is selected using the bias-variance trade-off. The advantages of PCE are demonstrated by comparing it with the Monte Carlo and Morris methods. The results indicate that PCE accurately calculates the global sensitivity index of towed cable parameters even with a limited sample size. Under the condition of a fixed cable length, the position and attitude of the towed cable are sensitive to the current rate, liquid density, cable diameter, normal drag coefficient, and specific gravity. The feasibility and efficiency of PCE applied to the sensitivity analysis of towed cable parameters is verified, and recommendations for the engineering application of towed cables are summarized.

Experimental and numerical investigations on the stress waves propagation in strain-hardening fiber-reinforced cementitious composites: Stochastic analysis using polynomial chaos expansions

The propagation of stress waves can result in spalling fractures in materials, which poses a threat to the safety of engineering structures. Hence, in-depth studies are necessary to mitigate such risks. Ultra-high toughness cementitious composites (UHTCC) with significant tensile strain-hardening properties are advanced engineering materials with great potential for impact engineering. The study used a combination of experimental and numerical methods to analyse the propagation and attenuation behaviour of stress waves in the UHTCC. The numerical model was developed using the Smooth Particle Galerkin (SPG) method and the Karagozian & Case concrete/cementitious (KCC) material constitutive model. Additionally, a quantitative global sensitivity analysis (GSA) using a polynomial chaos expansion (PCE) surrogate model and Sobol's method was conducted to analyse the stochastic input parameters of the numerical model. Results show that compressive stress waves undergo significant attenuation during propagation in UHTCC, and the developed numerical model accurately simulated the stress wave propagation process Furthermore, the PCE-Sobol GSA revealed that the selected stress wave propagation responses are most sensitive to the tensile strength of UHTCC. This study is expected to deepen the understanding of the propagation behaviour of stress waves in UHTCC and pave the way for promising applications of UHTCC in impact engineering.

Reliability-constrained optimal attitude-vibration control for rigid-flexible coupling satellite using interval dimension-wise analysis

This study proposes an uncertain optimal attitude–vibration control method for rigid–flexible coupling satellites with reliability constraints based on the interval dimension-wise analysis. To model the complex uncertain control system, multi–source uncertainties in rigid–flexible coupling satellites are quantified to interval parameters. Based on the nominal rigid–flexible coupling satellite dynamic model, the interval attitude–vibration model is derived and the corresponding coupled state equation is established. Considering polynomial chaos expansions in the surrogate model, an interval dimension-wise analysis (IDWA) method is proposed to predict the uncertain state response of the uncertain attitude–vibration control system, which can help solve the overestimation problem of the conventional perturbation method. An interval linear quadratic regulator (LQR) is proposed by combining the IDWA method and deterministic LQR, to accurately optimize the uncertain control input and cost function. With regard to harsh deterministic state response constraints, this study proposes a non-probabilistic time-dependent reliability method, which allows for an effective evaluation by flexibly characterizing the relationship between the uncertain state response and threshold. The reliability-constrained multi–objective optimal attitude–vibration control problem can be solved by considering the nominal cost function and uncertain state response as two optimization objectives. The effectiveness of the proposed method is verified using a rigid–flexible coupling satellite.

HPOSS: A hierarchical portfolio optimization stacking strategy to reduce the generalization error of ensembles of models.

Surrogate models are frequently used to replace costly engineering simulations. A single surrogate is frequently chosen based on previous experience or by fitting multiple surrogates and selecting one based on mean cross-validation errors. A novel stacking strategy will be presented in this paper. This new strategy results from reinterpreting the model selection process based on the generalization error. For the first time, this problem is proposed to be translated into a well-studied financial problem: portfolio management and optimization. In short, it is demonstrated that the individual residues calculated by leave-one-out procedures are samples from a given random variable ϵi, whose second non-central moment is the i-th model's generalization error. Thus, a stacking methodology based solely on evaluating the behavior of the linear combination of the random variables ϵi is proposed. At first, several surrogate models are calibrated. The Directed Bubble Hierarchical Tree (DBHT) clustering algorithm is then used to determine which models are worth stacking. The stacking weights can be calculated using any financial approach to the portfolio optimization problem. This alternative understanding of the problem enables practitioners to use established financial methodologies to calculate the models' weights, significantly improving the ensemble of models' out-of-sample performance. A study case is carried out to demonstrate the applicability of the new methodology. Overall, a total of 124 models were trained using a specific dataset: 40 Machine Learning models and 84 Polynomial Chaos Expansion models (which considered 3 types of base random variables, 7 least square algorithms for fitting the up to fourth order expansion's coefficients). Among those, 99 models could be fitted without convergence and other numerical issues. The DBHT algorithm with Pearson correlation distance and generalization error similarity was able to select a subgroup of 23 models from the 99 fitted ones, implying a reduction of about 77% in the total number of models, representing a good filtering scheme which still preserves diversity. Finally, it has been demonstrated that the weights obtained by building a Hierarchical Risk Parity (HPR) portfolio perform better for various input random variables, indicating better out-of-sample performance. In this way, an economic stacking strategy has demonstrated its worth in improving the out-of-sample capabilities of stacked models, which illustrates how the new understanding of model stacking methodologies may be useful.

A discrete adjoint framework coupled with adaptive PCE for robust aerodynamic optimization of turbomachinery under flow uncertainty

Flow uncertainty is commonly encountered in turbomachinery. To mitigate the negative effects caused by the flow uncertainty, a framework coupled with adaptive polynomial chaos expansion (PCE) and discrete adjoint method for robust aerodynamic design optimization (RADO) of turbomachinery blades is developed. The optimization framework relies on gradient algorithms, which effectively avoids “dimensional curse” problem. Adaptive PCE method improves the efficiency of uncertainty quantification and only once PCE model need to be constructed in per optimization step, making the framework highly efficient. The PCE matrix and chain rule are employed to determine the gradient of the RADO objective function. This framework is employed for optimizing turbine blade, considering both inviscid and viscous flow. Monte Carlo simulation and finite difference PCE are used to validate the reliability of uncertainty quantification and gradient propagation methods. A comparative analysis of the optimization results obtained by both RADO and deterministic aerodynamic design optimization (DADO) is demonstrated. The higher robustness brought by RADO indicates the practicality of this framework in robust optimization of turbomachinery blades.

A hybrid methodology for the frequency response function variability due to joint uncertainty

In vibroacoustic engineering, the most probable cause of dynamic uncertainty are the joints since it is not easy to strictly control the properties of manufactured joints. Although uncertainty in joints is localized in a complex structure, it may affect the dynamic response of the whole structure especially at higher frequencies. Generally, uncertain industrial structures are modelled numerically by FE whereas the uncertainty is modelled by performing Monte Carlo Simulations (MCS). These combined approaches are named FE-MCS. Application of FE-MCS to analyse local uncertainty in a complex structure is computationally slow, as FE and MCS requires a high number of elements and sampling, respectively. A possible solution is to introduce a combined hybrid Wave Finite Element and FE (shortly hybrid WFE) model, treating the uniform structures as waveguides joined by a local FE joint representation. Then, Polynomial Chaos Expansion (PCE) can be applied to introduce and model the uncertainty. The methodology is developed herein and tested on two right angled beams forming a L-shaped joint. The joint thickness is assumed to have a uniform distribution as an uncertain parameter. The scattering coefficients and frequency response function for both beams, are selected as the resulting uncertain variables. The results are subsequently verified with FE-MCS simulations using 200 samples and a limited number of experiments. It is clearly shown that the methodology introduced is an efficient tool for the structures possessing local uncertainty in terms of computational load as well as producing good frequency response function predictions when compared to both FE-MCS simulations and experimental validation.

A sensitivity analysis framework for SMA-based self-centering systems considering uncertainty in hysteretic material parameters

Shape memory alloys (SMAs) are widely utilized as the core component to develop self-centering (SC) structures owing to their excellent super-elasticity. Seismic performance of SMA-based structures is significantly influenced by the hysteretic parameters of the exploited SMAs. Uncertainties in the hysteretic parameters of SMAs could cause variation in seismic performance however have not received sufficient attentions. This paper proposes a sensitivity analysis framework to realistically and efficiently quantify the sensitivity of SMA-based structures to hysteretic model parameters. The experimental data from SMA bars are analyzed through the Markov chain Monte Carlo (MCMC) procedure. The probabilistic distributions of SMA hysteretic parameters are then sampled using the linear moment theory. Polynomial chaos expansions (PCE) based surrogate model is then used to evaluate sensitivity of response to different parametric uncertainties through the Sobol indices. The proposed framework is then applied to a total of 120 single-degree-of-freedom (SDOF) systems to demonstrate its efficacy to generalize observations and findings. The results show that the proposed framework provided more effective and realistic sensitivity analysis of seismic responses of SMA-based SDOF systems to account for uncertainty in the hysteretic model parameters. The Sobol indices also indicate that the sensitivity of different seismic response quantities varies for different hysteretic model parameters of SMAs.

Robustness analysis of CTV and OAR dose in clinical PBS-PT of neuro-oncological tumors: prescription-dose calibration and inter-patient variation with the Dutch proton robustness evaluation protocol.

Objective. The Dutch proton robustness evaluation protocol prescribes the dose of the clinical target volume (CTV) to the voxel-wise minimum (VWmin) dose of 28 scenarios. This results in a consistent but conservative near-minimum CTV dose (D98%,CTV). In this study, we analyzed (i) the correlation between VWmin/voxel-wise maximum (VWmax) metrics and actually delivered dose to the CTV and organs at risk (OARs) under the impact of treatment errors, and (ii) the performance of the protocol before and after its calibration with adequate prescription-dose levels.Approach. Twenty-one neuro-oncological patients were included. Polynomial chaos expansion was applied to perform a probabilistic robustness evaluation using 100,000 complete fractionated treatments per patient. Patient-specific scenario distributions of clinically relevant dosimetric parameters for the CTV and OARs were determined and compared to clinical VWmin and VWmax dose metrics for different scenario subsets used in the robustness evaluation protocol.Main results. The inclusion of more geometrical scenarios leads to a significant increase of the conservativism of the protocol in terms of clinical VWmin and VWmax values for the CTV and OARs. The protocol could be calibrated using VWmin dose evaluation levels of 93.0%-92.3%, depending on the scenario subset selected. Despite this calibration of the protocol, robustness recipes for proton therapy showed remaining differences and an increased sensitivity to geometrical random errors compared to photon-based margin recipes.Significance. The Dutch proton robustness evaluation protocol, combined with the photon-based margin recipe, could be calibrated with a VWmin evaluation dose level of 92.5%. However, it shows limitations in predicting robustness in dose, especially for the near-maximum dose metrics to OARs. Consistent robustness recipes could improve proton treatment planning to calibrate residual differences from photon-based assumptions.

Stochastic modeling and statistical calibration with model error and scarce data

This paper introduces a procedure to assess the predictive accuracy of stochastic models subject to model error and sparse data. Model error is introduced as uncertainty on the coefficients of appropriate polynomial chaos expansions (PCE). The error associated with finite sample size allows us to conceive of these coefficients as statistics of the data that we describe as random variables whose influence on output quantities of interest is evaluated through the extended polynomial chaos expansion (EPCE). A Bayesian data assimilation scheme is introduced to update these expansions by considering the resulting nested chaos expansion as a hierarchical probabilistic model. Stochastic models of quantities of interest (QoI) are thus constructed and efficiently evaluated. The Metropolis–Hastings Markov chain Monte Carlo procedure is used to sample the posterior. Two illustrative analytical and numerical problems are used to demonstrate the proposed approach.