Polynomial Chaos Expansion Research Articles

A probabilistic life prediction framework for vibration fatigue of turbine blade based on refined polynomial chaos expansion

AbstractVibration fatigue is a typical form of multiaxial fatigue. In this study, initial vibration fatigue tests were conducted on rocket turbine blades, and significant dispersion in fatigue life was found. Consistent with this finding, a novel probabilistic life prediction framework was proposed. This framework integrates sensitivity analysis and sequential sampling technology and introduces polynomial chaos expansion as a computationally efficient alternative to finite element analysis. And a continuous mechanics‐based damage evolution model was employed to examine the vibration fatigue life of turbine blades. The findings validate the effectiveness of the framework, as no significant difference was found between the experimental results and simulated predictions at the 95% confidence level. Furthermore, comparison with the Monte Carlo simulation indicated that this framework achieves comparable prediction accuracy, while significantly reducing the required samples by 2 orders of magnitude, which effectively addresses the fatigue problem of small sample data. This framework enables rapid and accurate multiaxial fatigue probabilistic life prediction, which holds important implications for the reliability design of reusable spacecraft.

An enhanced learning function for bootstrap polynomial chaos expansion-based enhanced active learning algorithm for reliability analysis of structure

Sparse polynomial chaos expansion (PCE) combined with the bootstrap resampling method is a viable alternative to obtain an active learning algorithm for reliability analysis. The existing learning functions in PCE-based active learning algorithms do not consider the joint probability density function (PDF) information. The present study explores a sparse PCE-based active learning algorithm based on a newly proposed learning function that maintains a balance between the misclassification probability and the joint PDF information of sample points. In doing so, the coefficients of the sparse PCE are estimated using a Bayesian compressive sensing regressor, as it is noted to be one of the best-performing regression solvers for PCE, irrespective of sampling schemes. The proposed learning function considers the weight of the joint PDF with the local accuracy measure of bootstrap PCE (bPCE) to add new samples iteratively in the existing training set. The convergence is achieved when the ten consecutive failure estimates are within a negligible discrepancy and also checks the confidence bounds of the bPCE estimates. The effectiveness of the proposed approach is demonstrated using two structural engineering examples and one well-known analytical test function and is found to be quite efficient and accurate in estimating reliability.

Uncertainty quantification with high-dimensional correlated process variations for an in-situ thermal expansion coefficient test structure

The impacts of process variations on the π-shaped thermal expansion coefficient (TEC) in-situ test structure are analyzed. Correlations among high-dimensional design parameters have to be taken into account for this type of structure, while the widely studied polynomial chaos expansion (PCE) technique is incapable of solving such problems. This paper presents a novel method for high-dimensional uncertainty quantification problems with non-Gaussian correlated variants. As a pre-processing method, the Spearman correlation coefficient and generalized lambda distribution are merged to eliminate the correlation. Then a compressed-sensing based Legendre-sparse solver is applied benefiting from the uniformly distributed parameters derived from the pre-processing and thus the cost of rebuilding correlated basis functions is eliminated. This method has demonstrated superior performance over existing methods in the analysis of TEC in-situ test structures. This analysis shows that the measurement accuracy of the test structure demonstrates good performance and is insensitive to process errors, which is consistent with experimental data. The presented technique has the potential to be applied in the design of various microelectromechanical system (MEMS) devices.

Physics-informed polynomial chaos expansions

Developing surrogate models for costly mathematical models representing physical systems is challenging since it is typically not possible to generate large training data sets, i.e. to create a large experimental design. In such cases, it can be beneficial to constrain the surrogate approximation to adhere to the known physics of the model. This paper presents a novel methodology for the construction of physics-informed polynomial chaos expansions (PCE) that combines the conventional experimental design with additional constraints from the physics of the model represented by a set of differential equations and specified boundary conditions. A computationally efficient means of constructing physically constrained PCEs, termed PC2, are proposed and compared to the standard sparse PCE. Algorithms are presented for both full-order and sparse PC2 expansions and an iterative approach is proposed for addressing nonlinear differential equations. It is shown that the proposed algorithms lead to superior approximation accuracy and do not add significant computational burden over conventional PCE. Although the main purpose of the proposed method lies in combining training data and physical constraints, we show that the PC2 can also be constructed from differential equations and boundary conditions alone without requiring model evaluations. We further show that the constrained PCEs can be easily applied for uncertainty quantification through analytical post-processing of a reduced PCE by conditioning on the deterministic space-time variables. Several deterministic examples of increasing complexity are provided and the proposed method is demonstrated for uncertainty quantification.

Polynomial chaos enhanced by dynamic mode decomposition for order-reduction of dynamic models

Thanks to their low computational cost, reduced-order models (ROMs) are indispensable in ensemble-based simulations used, e.g., for uncertainty quantification, inverse modeling, and optimization. Since data used to train a ROM are typically obtained by running a high-fidelity model (HFM) multiple times, a ROM’s efficiency rests on the computational cost associated with the data generation and training phase. One such ROM, a polynomial chaos expansion (PCE), often provides a robust description of an HFM’s response surface in the space of model parameters. To reduce the data-generation cost, we propose to train a PCE on multi-fidelity data, part of which come from the dynamic HFM and the remainder from dynamic mode decomposition (DMD); the latter is used to interpolate the HFM data in time. Our numerical experiments demonstrate the accuracy of the proposed method and provide guidelines for the optimal use of DMD for interpolation purposes.

A fast and robust constraint-based online re-optimization approach for automated online adaptive intensity modulated proton therapy in head and neck cancer

Objective. In head-and-neck cancer intensity modulated proton therapy, adaptive radiotherapy is currently restricted to offline re-planning, mitigating the effect of slow changes in patient anatomies. Daily online adaptations can potentially improve dosimetry. Here, a new, fully automated online re-optimization strategy is presented. In a retrospective study, this online re-optimization approach was compared to our trigger-based offline re-planning (offlineTB re-planning) schedule, including extensive robustness analyses. Approach. The online re-optimization method employs automated multi-criterial re-optimization, using robust optimization with 1 mm setup-robustness settings (in contrast to 3 mm for offlineTB re-planning). Hard planning constraints and spot addition are used to enforce adequate target coverage, avoid prohibitively large maximum doses and minimize organ-at-risk doses. For 67 repeat-CTs from 15 patients, fraction doses of the two strategies were compared for the CTVs and organs-at-risk. Per repeat-CT, 10.000 fractions with different setup and range robustness settings were simulated using polynomial chaos expansion for fast and accurate dose calculations. Main results. For 14/67 repeat-CTs, offlineTB re-planning resulted in <50% probability of D 98% ≥ 95% of the prescribed dose (D pres) in one or both CTVs, which never happened with online re-optimization. With offlineTB re-planning, eight repeat-CTs had zero probability of obtaining D 98% ≥ 95%D pres for CTV7000, while the minimum probability with online re-optimization was 81%. Risks of xerostomia and dysphagia grade ≥ II were reduced by 3.5 ± 1.7 and 3.9 ± 2.8 percentage point [mean ± SD] (p < 10−5 for both). In online re-optimization, adjustment of spot configuration followed by spot-intensity re-optimization took 3.4 min on average. Significance. The fast online re-optimization strategy always prevented substantial losses of target coverage caused by day-to-day anatomical variations, as opposed to the clinical trigger-based offline re-planning schedule. On top of this, online re-optimization could be performed with smaller setup robustness settings, contributing to improved organs-at-risk sparing.

Robust, planning-based targeted locoregional tumour heating in small animals.

Objective.To improve hyperthermia in clinical practice, pre-clinical hyperthermia research is essential to investigate hyperthermia effects and assess novel treatment strategies. Translating pre-clinical hyperthermia findings into clinically viable protocols requires laboratory animal treatment techniques similar to clinical hyperthermia techniques. The ALBA micro8 electromagnetic heating system (Med-logix SRL, Rome, Italy) has recently been developed to provide the targeted locoregional tumour heating currently lacking for pre-clinical research. This study evaluates the heat focusing properties of this device and its ability to induce robust locoregional tumour heating under realistic physiological conditions using simulations.Approach.Simulations were performed using the Plan2Heat treatment planning package (Amsterdam UMC, the Netherlands). First, the specific absorption rate (SAR) focus was characterised using a homogeneous phantom. Hereafter, a digital mouse model was used for the characterisation of heating robustness in a mouse. Device settings were optimised for treatment of a pancreas tumour and tested for varying circumstances. The impact of uncertainties in tissue property and perfusion values was evaluated using polynomial chaos expansion. Treatment quality and robustness were evaluated based on SAR and temperature distributions.Main results.The SAR distributions within the phantom are well-focused and can be adjusted to target any specific location. The focus size (full-width half-maximum) is a spheroid with diameters 9 mm (radially) and 20 mm (axially). The mouse model simulations show strong robustness against respiratory motion and intestine and stomach filling (∆T90≤0.14°C).Mouse positioning errors in the cranial-caudal direction lead to∆T90≤0.23°C. Uncertainties in tissue property and perfusion values were found to impact the treatment plan up to 0.56 °C (SD), with a variation onT90of 0.32 °C (1 SD).Significance.Our work shows that the pre-clinical phased-array system can provide adequate and robust locoregional heating of deep-seated target regions in mice. Using our software, robust treatment plans can be generated for pre-clinical hyperthermia research.

Global sensitivity of EEG source analysis to tissue conductivity uncertainties.

To reliably solve the EEG inverse problem, accurate EEG forward solutions based on a detailed, individual volume conductor model of the head are essential. A crucial-but often neglected-aspect in generating a volume conductor model is the choice of the tissue conductivities, as these may vary from subject to subject. In this study, we investigate the sensitivity of EEG forward and inverse solutions to tissue conductivity uncertainties for sources distributed over the whole cortex surface. We employ a detailed five-compartment head model distinguishing skin, skull, cerebrospinal fluid, gray matter, and white matter, where we consider uncertainties of skin, skull, gray matter, and white matter conductivities. We use the finite element method (FEM) to calculate EEG forward solutions and goal function scans (GFS) as inverse approach. To be able to generate the large number of EEG forward solutions, we employ generalized polynomial chaos (gPC) expansions. For sources up to a depth of 4 cm, we find the strongest influence on the signal topography of EEG forward solutions for the skull conductivity and a notable effect for the skin conductivity. For even deeper sources, e.g., located deep in the longitudinal fissure, we find an increasing influence of the white matter conductivity. The conductivity variations translate to varying source localizations particularly for quasi-tangential sources on sulcal walls, whereas source localizations of quasi-radial sources on the top of gyri are less affected. We find a strong correlation between skull conductivity and the variation of source localizations and especially the depth of the reconstructed source for quasi-tangential sources. We furthermore find a clear but weaker correlation between depth of the reconstructed source and the skin conductivity. Our results clearly show the influence of tissue conductivity uncertainties on EEG source analysis. We find a particularly strong influence of skull and skin conductivity uncertainties.

Multi-objective design optimization of a transonic axial fan stage using sparse active subspaces

In this paper, a multi-objective optimization strategy for efficient design of turbomachinery blades using sparse active subspaces is implemented for a turbofan stage design. The proposed strategy utilized sparse polynomial chaos expansion on a limited dataset to generate a function from which the differential and the covariance matrix can be obtained. Active subspace was used to compute the active variables via singular value decomposition and a hybrid polynomial correlated function expansion was used to construct a surrogate model on the active subspace. Coupled with freeform method and multi-objective genetic algorithm, an automated optimization loop was run at a single operating condition. An improvement in stage efficiency and total pressure ratio of 2.97% and 1.15% was achieved for the optimum design compared with the baseline. Additionally, total pressure loss coefficient decreased by 5.88%, exit flow angle by 34.65% and shock strength by 5.32%. The coupled effect of change in stagger angle, forward sweep, forward lean, and chord length reduced the recirculation at the hub, and blockage at the shroud due the tip leakage flow by decreasing the blade loading. The threshold value hyperparameter was found to be the most influential and must be accurately determined.

A comprehensive study on seismic dynamic responses of stochastic structures using sparse grid-based polynomial chaos expansion

The seismic response is a critical aspect of structural engineering, as it directly influences the design and safety evaluation of such systems. Structural seismic response becomes more complicated when unavoidable uncertainties in the structure are considered. Therefore, solving and analyzing the stochastic dynamic response of structures efficiently and accurately is of great significance. Various uncertainty quantification methods such as the stochastic finite element method, Monte Carlo method, and polynomial chaos expansion method have made significant progress in this field. However, these methods do not provide a comprehensive study of the seismic response of stochastic structures and efficiency problems rapidly emerge as the number of random variables increases. This paper delivers a comprehensive study of the seismic dynamic response of space truss and reticulated shell structure considering uncertainties in material properties, geometry, and loading. To this end, the finite element model of the structure was first modeled and the deterministic solutions were obtained. Then, polynomial surrogates are modeled for the natural frequency, mode participation factor, mode superposition response, and transient response of the structure under stochastic uncertainties. The sparse grid technique is used to select sparse quadrature points, which overcomes the curse of dimensionality caused by full grid integration. The results demonstrate that the sparse grid-based polynomial chaos expansion is more efficient. Uncertainty quantification reveals the uncertainty of modal analysis, response spectrum analysis, and transient response analysis of space truss and reticulated shell under seismic excitation. In summary, this study provides valuable insights and an efficient and accurate uncertainty quantification tool for the seismic dynamic analysis of stochastic structures, ultimately contributing to safer and more robust structural designs.

Analytical study of quasi-zero stiffness vibration isolator with uncertain parameters based on arbitrary polynomial chaos expansions method

Analytical study of quasi-zero stiffness vibration isolator with uncertain parameters based on arbitrary polynomial chaos expansions method

Polynomial chaos expansions for modelling the statistics of acoustic propagation in random waveguides

The use of Polynomial Chaos expansions to model the transfer of the statistical properties of the sound speed in ocean waveguides to those of acoustic waves passing through them is described. A perturbational framework approximating the interaction of acoustic normal modes with fluctuations around a background sound speed is used, with the fluctuations being represented vertically by empirical orthogonal functions and horizontally by correlation functions. The Polynomial Chaos expansions are derived to second order in the uncorrelated random variable representing the sound speed fluctuations for the generalized phase of the complex modal amplitudes, and to third order for the modal amplitudes themselves. The ability of the second order polynomial chaos expansion for the generalized model phase to accurately model acoustic propagation statistics is closely coupled to the accuracy of the adiabatic approximation in light scattering regimes. The weights of the Polynomial Chaos expansions are obtained using both direct (theoretical) and indirect (least-squares fit) methods. Divergence between these two sets of weights increases with combinations of increasing strength of sound speed fluctuations, increasing frequency, or increasing range, indicating where a higher order expansion may be necessary. Modelling results for the first and second moments of the acoustic intensity and the Scintillation Index in random waveguides are presented and compared to Monte Carlo results.

Stochastic simulation of magnetically controlled convection in porous media with random porosity via Karhunen-Loève expansion and intrusive polynomial chaos expansion

Understanding and quantifying uncertainty factors are crucial for accurately predicting thermomagnetic convection phenomena. This study presented a mathematical model and algorithm framework for uncertainty analysis of thermomagnetic convection in porous media with random porosity. The proposed method combined Karhunen-Loève expansion and intrusive polynomial chaos expansion to represent the input random parameters and output response, respectively. The Galerkin projection method was employed to decouple the stochastic governing equations of thermomagnetic convection into deterministic governing equations that can be efficiently solved using the finite volume method. By solving the polynomial chaos expansion of the output response and employing the stochastic projection method to solve the associated decoupled governing equations, the temporal evolution and statistical characteristics of the output response were obtained. The study revealed that porosity uncertainty in the porous medium affects the thermomagnetic convection of paramagnetic fluid, exhibiting significant chaos effects. Monte Carlo simulations were performed to validate the accuracy of the proposed approach and demonstrate its computational efficiency compared to traditional Monte Carlo methods. This research provides valuable insights into the uncertainty analysis of thermomagnetic convection and offers a promising methodology for analyzing heat transfer in various porous media applications.

Learnable quantile polynomial chaos expansion: An uncertainty quantification method for interval reliability analysis

Various uncertainties like random input and data noise in aerospace engineering affect flight vehicle safety, making uncertainty quantification important for improving system reliability. Polynomial chaos expansion (PCE) is a versatile method for quantifying stochastic systems’ uncertainty caused by random input, while it cannot quantify data uncertainty caused by noise. To solve this problem, a novel uncertainty quantification method, learnable quantile PCE (LQPCE), is proposed for interval reliability analysis. Unlike the traditional PCE, the multi-dimensional orthonormal basis is constructed for combination input rather than only random variables, and the expansion coefficients are parameterized to learnable weights, ensuring the LQPCE model can capture data noise’s features. The LQPCE Pinball loss function is constructed to learn parameterized expansion coefficients by quantile level iterative sampling, achieving accurate prediction and data uncertainty quantification. The quantified data uncertainty is used for interval reliability analysis, calculating more reliable results about the influence of input uncertainty on stochastic systems’ reliability. Three numerical examples and one engineering application validate the LQPCE method’s effectiveness. The results show that the LQPCE method can quantify data uncertainty accurately and obtain credible interval reliability.

Investigation on uncertainty quantification of transonic airfoil using compressive sensing greedy reconstruction algorithms

Uncertainty quantification constructs the stochastic responses of system output under uncertainties. Traditional uncertainty quantification methods such as Monte Carlo and Full-Rank Polynomial Chaos Expansion need large amounts of samples. Compressive Sensing has the advantages of requiring few observation samples and sparse reconstruction. By combining Compressive Sensing with uncertainty quantification, the samples in demand can be reduced. Nevertheless, the sparse reconstruction algorithms directly affect predicting accuracy, especially in a practical aerodynamic scenario. To compare the effect of two l0-minimization based greedy reconstruction algorithms, Orthogonal Matching Pursuit and Subspace Pursuit, this study adopts them into uncertainty quantification of RAE2822 airfoil considering geometrical uncertainties. The Polynomial Chaos Expansion is used to realize sparse representation and Structured Random Observation Matrix is constructed by randomly selecting from Gaussian quadrature set. The two greedy reconstruction methods are compared to Monte Carlo and Full-Rank Probabilistic Collocations on convergence, required samples and accuracy. The results show that both methods could achieve similar accuracy as Monte Carlo with much fewer samples. Besides, Orthogonal Matching Pursuit is able to precisely reconstruct original signal with proper sparsity estimation, while Subspace Pursuit provides more robust prediction for different cases. This study presents an efficient approach that integrates compressive sensing into uncertainty quantification of airfoils.

Sequential simulated annealing for life-cycle optimization of nonlinear stochastic systems via arbitrary polynomial chaos expansion

Quantifying the uncertainties of engineering systems modeled as nonlinear oscillators subject to random excitation is a theoretically complex and computationally demanding task. Consequently, finding an optimal design of such systems considering their life-cycle performance is prohibitive with Monte Carlo simulation methods. In this paper, an efficient performance-based design optimization approach is developed for finding the optimal parameters of engineering systems modeled as nonlinear/hysteretic oscillators subject to stationary and non-stationary excitation. This novel approach utilizes an arbitrary polynomial chaos expansion to estimate the expected cost of failure without performing a computationally expensive integration over the hazard levels. Moreover, we introduce a novel sequential heuristic optimization scheme based on simulated annealing to minimize the total expected cost over the structure life-cycle. Three examples are included in the paper to assess the developed optimization scheme. First, we use the developed framework to optimize a linear single-degree-of-freedom oscillator subject to broadband excitation. Second, a multi-degree-of-freedom oscillator with cubic nonlinearity in damping and stiffness, subject to stationary broadband excitation, is optimized to show the influence of the problem dimensionality in the optimization process. In the last example, a multi-story reinforced concrete shear building modeled as a multi-degree-of-freedom Bouc-Wen oscillator with stiffness and strength degradation and subject to multi-hazards modeled as stationary (wind excitation) and non-stationary (earthquake) stochastic processes, is optimized.

Uncertainty quantification for structural response field with ultra-high dimensions

The structural response field is crucial for understanding mechanical behavior, especially under uncertain conditions. However, current uncertainty quantification predominantly address one-dimensional output under multi-dimensional input. For the structural full field response with ultra-high dimension, quantifying corresponding uncertainties becomes a formidable task. This paper introduces an efficient uncertainty quantification method for structural response field. Initially, the manifold learning is introduced to transform uncertainty quantification of ultra-high-dimensional responses into the quantification of low-dimensional manifold projections. Subsequently, polynomial chaos expansions facilitate uncertainty propagation, allowing for the effective evaluation of statistical moments for each projection. Leveraging the established manifold structure enables obtaining analytical solutions for statistical moments of full field responses without additional computational costs. In the absence of prior information, the derivative λ-PDF is utilized to model uncertainty of response at any position, so as to realize the visualization of uncertainty. The proposed uncertainty quantification for structural response field provides a non-intrusive analysis framework, which can conveniently deal with structural multi-field coupling problems. Finally, three engineering examples involving multi field coupling are presented to demonstrate the effectiveness and practicability of proposed uncertainty quantification of structural response field.

Contaminant source identification in an aquifer using a Bayesian framework with arbitrary polynomial chaos expansion

Contaminant source identification in an aquifer using a Bayesian framework with arbitrary polynomial chaos expansion

Classifier-based adaptive polynomial chaos expansion for high-dimensional uncertainty quantification

A novel approach for the construction of polynomial chaos expansion (PCE) is proposed to facilitate high-dimensional uncertainty quantification (UQ). The current PCE techniques are well-known for UQ; however, they are affected by the curse of dimensionality that leads to over-fitting and tractability issues. Although L1-minimization can be used to deal with over-fitting, it is still ineffective for problems with a large number of independent random inputs or requiring a very high-order PCE. Therefore, a classifier-based PCE (CAPCE) is presented to mitigate the factorial growth of basis terms and prevent over-fitting. The adaptive framework includes four basis selection strategies – enrichment, screening, discovery, and recycling – in the inner loop and the enrichment of the training samples in the outer loop. Mainly, an adaptive classifier is trained on the dominant and discarded basis terms obtained from L1-minimization during discovery. It then allows the judicious selection of new higher-order basis candidates for L1-solver in the next iteration, thereby alleviating the effect of the curse of dimensionality. The proposed framework has been tested with analytical problems of varying sizes of independent random inputs and an engineering composite laminate problem. The comparison of CAPCE with the available efficient PCE techniques demonstrated improvements in accuracy using fewer samples due to a faster convergence rate.

Polynomial chaos expansion for sensitivity analysis of two types of transmission line models

This paper presents two sensitivity analyses, one of which is for the time-domain transmission line (TL) model using the latest weight-based algorithm that is a combination of the coherence-optimal sampling (COS) and the l1-minimum problem (l1-M) and the other is for the high-frequency field line coupling model based on the algorithm that combines the Latin hypercube sampling (LHS) with the l1-M for the first time, where the COS and the LHS are employed to draw samples and the polynomial chaos expansion (PCE) coefficients are obtained by solving the l1-M. Both algorithms are within the PCE framework. The results show that for the time-domain TL model, the capacitance has the greatest impact on the voltage response, followed by the resistance. The height and the radius have the least impact, while there is little difference between them. For the high-frequency field line coupling model, the azimuth angle has the greatest impact on the current in the TL, followed by the elevation angle. The terminal load, the conductivity, the field amplitude, the relative permittivity, and the TL radius have the least effect, and there is little difference among them. Additionally, both algorithms only require no more than 0.60% of the total Monte Carlo (MC) computation time when yielding solutions with almost identical accuracy. Compared to performing a sensitivity analysis using the MC method, the use of two algorithms significantly improves efficiency.