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Bayesian techniques are vital in mechanical manufacturing for uncertainty quantification and process optimization. This review explores their diverse applications, highlighting advantages in handling small data and incorporating expertise for improved decision-making in quality control, reliability, and machining. It also discusses integration with machine learning and applications in specialized areas. Future research should focus on Industry 4.0 integration and user-friendly tools, emphasizing Bayesian methods' role in intelligent manufacturing.

이 연구에서는 Eagle Ford Shale 지역의 셰일 오일 생산정을 대상으로 랜덤 포레스트 분위수 회귀기반의 궁극가채량 예측 및 불확실성 정량화를 수행하였다. 랜덤 포레스트 분위수 회귀로부터 도출된 P90 및 P10 방식은 예측구간 포함확률(PICP)이 80%를 과도하게 초과하여 불확실성 분석에적합하지 않은 한계를 보였다. 이를 개선하기 위해 수정된 적합분위수 회귀법을 통한 신뢰구간 보정 기법을 적용하였다. 보정된 모델은 PICP 80%에 해당하는 보정된 P90 및 P10 값을 도출함으로써 예측 신뢰구간의 적정성을 확보하였다. 또한 중앙값 예측은 보정 전후 변화가 거의 없어 모델의구조적 안정성이 유지됨을 확인하였다. 제안된 방법은 셰일저류층 생산 예측의 불확실성 정량화에 대한 신뢰성과 실용성을 향상시킬 수 있는 효과적인 대안이 될 수 있다.

This study uses quantile regression forest (QRF), which is based on random forest (RF), to estimate the estimated ultimate recovery of shale oil in the Eagle Ford Shale and quantify uncertainty. Conventional prediction intervals using the P90 and P10 quantiles from the RF quantile model yielded higher prediction interval coverage probability (PICP) than the target of 80%, leading to uncertainty overestimation. A calibration method based on modified conformalized quantile regression (CQR) was used for the prediction intervals to solve this problem. By generating adjusted P90 and P10 values, the calibrated model ensured a PICP of 80% and a more appropriate uncertainty representation. The median prediction remained nearly unchanged before and after calibration, confirming the structural stability of the model. The proposed method enhances the reliability and practical applicability of uncertainty quantification in forecasting production from shale reservoirs.

본 연구에서는 앙상블스무더의 불확실성평가 신뢰도를 향상하기 위해 초기앙상블을 잘 대표하는 다수의 칼만게인을 이용하는 기법을 제안하였다. 제안된 거리기반 앙상블스무더를 채널저류층에 적용한 결과, 기존의 오버슈팅과 필터발산 문제를 해결하였고 채널연결성과 이봉분포의 특징을 잘 보존하였다. 제안된 기법은앙상블스무더의 수식과 교정방식을 수정없이 사용하므로 계산속도가 빠르다. 따라서 200개의 앙상블로 45번의교정을 수행한 경우, 앙상블칼만필터의 소요시간보다 97% 이상 감소시켰다. 거리기반 앙상블스무더의 경우 유정별 생산량뿐만 아니라 누적 오일 및 물 생산량을 성공적으로 예측하였고 편향없는 불확실성을 제공하였다. 또한 군집별 대표앙상블만으로 전체앙상블과 비슷한 수준의 불확실성평가가 가능하므로 군집수에 반비례하여소요시간이 줄어든다. 따라서 제안된 기법은 빠르고 신뢰할 수 있는 불확실성평가가 가능하므로 채널저류층개발 시 의사결정을 위한 도구로 활용될 수 있다.

This paper suggests a new method of ensemble smoother(ES) for reliable uncertainty quantification. The method uses several Kalman gains rather than one representative Kalman gain. When the proposed method is applied to channelized reservoirs, the results manage typical overshooting and filter divergence problems. Also, they conserve channel connectivity and bimodal distribution of the model parameter. The proposed method can keep history matching time short since there are no modifications in the standard ES. Therefore, the time of the proposed method reduces more than 97% of that of ensemble Kalman filter(EnKF) with 45 assimilation steps and 200 total ensembles. The ES with a distance-based method provides reliable productions with reasonable uncertainty ranges. Also, prediction time of future performances can be reduced since the representative ensembles from each group estimate similar uncertainty ranges over all ensembles. Therefore, the proposed method can be applied for decision making because it gives fast and reliable uncertainty quantification for channelized reservoirs.

Large-scale reactor simulation often requires the use of Monte Carlo calculation techniques to estimate important reactor parameters. One drawback of these Monte Carlo calculation techniques is they inevitably result in some uncertainty in calculated quantities. The present study includes parametric uncertainty quantification (UQ) and sensitivity analysis (SA) on the Advanced Test Reactor Critical (ATRC) facility housed at Idaho National Laboratory (INL) and addresses some complications due to Monte Carlo uncertainty when performing these analyses. This approach for UQ/SA includes consideration of Monte Carlo code uncertainty in computed sensitivities, consideration of uncertainty from directly measured parameters and a comparison of results obtained from brute-force Monte Carlo UQ versus UQ obtained from a surrogate model. These methodologies are applied to the uncertainty and sensitivity of k_eff for two sets of uncertain parameters involving fuel plate geometry and fuel plate composition. Results indicate that the less computationally-expensive method for uncertainty quantification involving a linear surrogate model provides accurate estimations for k_eff uncertainty and the Monte Carlo uncertainty in calculated k_eff values can have a large effect on computed linear model parameters for parameters with low influence on k_eff.

The coolant temperature feedback coefficient is an important parameter of reactor core power control system. To study the coolant temperature feedback coefficient influence on the core power control system of small PWR, the core power control system is built with the nonlinear model and fuzzy control theory. Then, the uncertainty quantification method of reactor core parameters is established based on the Latin hypercube sampling method and the Bootstrap method. Finally, under the conditions of reactivity step perturbation and coolant inlet temperature step perturbation, uncertainty analysis for two cases is carried out. The result shows that with fuzzy controller and fuzzy PID controller, the uncertainty of the coolant temperature feedback coefficient affects the core power control system, and the maximum uncertainties of core relative power, coolant temperature deviation, fuel temperature deviation and total reactivity are acceptable.

In the last decade, the best estimate plus uncertainty methodologies in nuclear technology and nuclear power plant design have become a trending topic in the nuclear field. Since BEPU was allowed for licensing purposes by the most important regulator bodies, different uncertainty assessment methods have become popular, overall non-parametric methods. While non-parametric tolerance regions can be well stated and used in uncertainty quantification for licensing purposes, the propagation of the uncertainty through different codes (multi-scale, multiphysics) in cascade needs a better depiction of uncertainty than the one provided by the tolerance regions or a probability distribution. An alternative method based on the parametric or distributional probability boxes is used to perform uncertainty quantification and propagation regarding statistic uncertainty from one code to another. This method is sample-size independent and allows well-defined tolerance intervals for uncertainty quantification, manageable for uncertainty propagation. This work characterizes the distributional p-boxes behavior on uncertainty quantification and uncertainty propagation through nested random sampling.

Uncertainties are calculated for pressurized water reactor (PWR) spent nuclear fuel (SNF) characteristics. The deterministic code STREAM is currently being used as an SNF analysis tool to obtain isotopic inventory, radioactivity, decay heat, neutron and gamma source strengths. The SNF analysis capability of STREAM was recently validated. However, the uncertainty analysis is yet to be conducted. To estimate the uncertainty due to nuclear data, STREAM is used to perturb nuclear cross section (XS) and resonance integral (RI) libraries produced by NJOY99. The perturbation of XS and RI involves the stochastic sampling of ENDF/B-VII.1 covariance data. To estimate the uncertainty due to modeling parameters (fuel design and irradiation history), surrogate models are built based on polynomial chaos expansion (PCE) and variance-based sensitivity indices (i.e., Sobol' indices) are employed to perform global sensitivity analysis (GSA). The calculation results indicate that uncertainty of SNF due to modeling parameters are also very important and as a result can contribute significantly to the difference of uncertainties due to nuclear data and modeling parameters. In addition, the surrogate model offers a computationally efficient approach with significantly reduced computation time, to accurately evaluate uncertainties of SNF integral characteristics.

This paper addresses the uncertainty quantification and sensitivity analysis of a depleted light-water fuel assembly of the Turkey Point-3 benchmark. The uncertainty of the fuel assembly decay heat and isotopic densities is quantified with respect to three different groups of diverse parameters: nuclear data, assembly design, and reactor core operation. The uncertainty propagation is conducted using a two-step analysis code system comprising the lattice code STREAM, nodal code RAST-K, and spent nuclear fuel module SNF through the random sampling of microscopic cross-sections, fuel rod sizes, number densities, reactor core total power, and temperature distributions. Overall, the statistical analysis of the calculated samples demonstrates that the decay heat uncertainty decreases with the cooling time. The nuclear data and assembly design parameters are proven to be the largest contributors to the decay heat uncertainty, whereas the reactor core power and inlet coolant temperature have a minor effect. The majority of the decay heat uncertainties are delivered by a small number of isotopes such as ²⁴¹Am, ¹³⁷Ba, ²⁴⁴Cm, ²³⁸Pu, and ⁹⁰Y.

In order to study the sensitivity and the uncertainty of the Moroccan research reactor TRIGA Mark II, a model of this reactor has been developed in our ERSN laboratory for use with the N-Particle MCNP Monte Carlo transport codes (version 6). In this article, the sensitivities of the effective multiplication factor of this reactor are evaluated using the ENDF/B-VII.0, ENDF/B-VII.1 and JENDL-4.0 libraries and in 44 energy groups, for the cross sections of the fuel (U-235 and U-238) and the moderator (H-1 and O-16). However, the quantification of the uncertainty of the nuclear data is performed using the nuclear code NJOY99 for the generation and processing of covariance matrices. On the one hand, the highest uncertainty deviations, calculated using the ENDFB-VII.1 and JENDL4.0 evaluations, are 2275, 386 and 330 pcm respectively for the reactions U₂₃₅(n, f), $ U_{235}(n\bar{\nu})$ and H₁(n, γ). On the other hand, these differences are very small for the neutron reactions of O-16 and U-238. Regarding the neutron spectra, in CT-mid plane, they are very close for the three evaluations (ENDF/B-VII.0, ENDF/B-VII.1 and JENDL-4.0). These spectra present two peaks (thermal and fission) around the energies 0.05 eV and 1 MeV.

An analysis of the uncertainty quantification related to LBLOCA using the Monte-Carlo calculation has been performed and compared with the tolerance level determined by the Wilks' formula. The uncertainty range and distribution of each input parameter associated with the LOCA phenomena were determined based on previous PIRT results and documentation during the BEMUSE project. Calulations were conducted on 3,500 cases within a 2-week CPU time on a 14-PC cluster system. The Monte-Carlo exercise shows that the 95% upper limit PCT value can be obtained well, with a 95% confidence level using the Wilks' formula, although we have to endure a 5% risk of PCT under-prediction. The results also show that the statistical fluctuation of the limit value using Wilks' first-order is as large as the uncertainty value itself. It is therefore desirable to increase the order of the Wilks' formula to be higher than the second-order to estimate the reliable safety margin of the design features. It is also shown that, with its ever increasing computational capability, the Monte-Carlo method is accessible for a nuclear power plant safety analysis within a realistic time frame.

Probabilistic safety assessment is widely used to quantify the risks of nuclear power plants and their uncertainties. When the lognormal distribution describes the uncertainties of basic events, the uncertainty of the top event in a fault tree is approximated with the sum of lognormal random variables after minimal cutsets are obtained, and rare-event approximation is applied. As handling complicated analytic expressions for the sum of lognormal random variables is challenging, several approximation methods, especially Monte Carlo simulation, are widely used in practice for uncertainty analysis. In this study, a theoretical approach for analyzing the sum of lognormal random variables using an efficient numerical integration method is proposed for uncertainty analysis in probability safety assessments. The change of variables from correlated random variables with a complicated region of integration to independent random variables with a unit hypercube region of integration is applied to obtain an efficient numerical integration. The theoretical advantages of the proposed method over other approximation methods are shown through a benchmark problem. The proposed method provides an accurate and efficient approach to calculate the uncertainty of the top event in probabilistic safety assessment when the uncertainties of basic events are described with lognormal random variables.

In order to properly quantify fission reactor neutronics parameter uncertainties, we have to use covariance data and sensitivity profiles consistently. In the present paper, we establish two consistent methodologies for uncertainty quantification: a self-shielded cross section-based consistent methodology and an infinitely-diluted cross section-based consistent methodology. With these methodologies and the covariance data of uranium-238 nuclear data given in JENDL-3.3, we quantify uncertainties of infinite neutron multiplication factors of light water reactor and fast reactor fuel cells. While an inconsistent methodology gives results which depend on the energy group structure of neutron flux and neutron-nuclide reaction cross section representation, both the consistent methodologies give fair results with no such dependences.

This paper presents an advanced computational method for the prediction of the responses in the frequency domain of general linear dissipative structural-acoustic and fluid-structure systems, in the low-and medium-frequency domains and this includes uncertainty quantification. The system under consideration is constituted of a deformable dissipative structure that is coupled with an internal dissipative acoustic fluid. This includes wall acoustic impedances and it is surrounded by an infinite acoustic fluid. The system is submitted to given internal and external acoustic sources and to the prescribed mechanical forces. An efficient reduced-order computational model is constructed by using a finite element discretization for the structure and an internal acoustic fluid. The external acoustic fluid is treated by using an appropriate boundary element method in the frequency domain. All the required modeling aspects for the analysis of the medium-frequency domain have been introduced namely, a viscoelastic behavior for the structure, an appropriate dissipative model for the internal acoustic fluid that includes wall acoustic impedance and a model of uncertainty in particular for the modeling errors. This advanced computational formulation, corresponding to new extensions and complements with respect to the state-of-the-art are well adapted for the development of a new generation of software, in particular for parallel computers.

The choice of input distribution in quantitative risk assessments modeling is of great importance to get unbiased overall estimates, although it is difficult to characterize them in situations where data available are too sparse or small. The present study is particularly concerned with accommodation of uncertainties commonly encountered in the practice of modeling. The authors applied parametric and non-parametric bootstrap simulation methods which consist of re-sampling with replacement, in together with the classical Student-t statistics based on the normal distribution. The implications of these methods were demonstrated through an empirical analysis of trade volume from the amount of chicken and pork meat imported to Korea during the period of 1998-2005. The results of bootstrap method were comparable to the classical techniques, indicating that bootstrap can be an alternative approach in a specific context of trade volume. We also illustrated on what extent the bias corrected and accelerated non-parametric bootstrap method produces different estimate of interest, as compared by non-parametric bootstrap method.

A data-based model, such as an AAKR model is widely used for monitoring the drifts of sensors in nuclear power plants. However, since a training dataset and a test dataset for a data-based model cannot be constructed with the data from all the possible states, the model uncertainty cannot be good enough to represent the uncertainty of estimations. In fact, the errors of estimation grow much bigger if the incoming data come from inexperienced states. To overcome this limitation of the model uncertainty, a new measure of uncertainty for a data-based model is developed and the predicted uncertainty is introduced. The predicted uncertainty is defined in every estimation according to the incoming data. In this paper, the AAKR model is used as a data-based model. The predicted uncertainty is similar in magnitude to the model uncertainty when the estimation is made for the incoming data from the experienced states but it goes bigger otherwise. The characteristics of the predicted model uncertainty are studied and the usefulness is demonstrated with the pressure signals measured in the flow-loop system. It is expected that the predicted uncertainty can quite reduce the false alarm by using the variable threshold instead of the fixed threshold.

본 논문에서는 랜덤 필드 입력을 갖는 구조 해석 문제에서의 순방향 불확실성 정량화(Forward Uncertainty Quantification, UQ)를 위한 차수축소 모델링 기법을 제안한다. 제안된 기법은 재료 물성의 공간적 불확실성을 효율적으로 표현하기 위해 Karhunen-Loève(KL) 전개를 활용하고, 공간 변수와 확률 변수를 분리함으로써 고차원 문제를 효율적으로 해석할 수 있도록 Proper Generalized Decomposition(PGD)을 결합하였다. 이와 같은 접근은 전체 매개변수 공간에 대한 오프라인 계산을 가능하게 하며, 새로운 샘플에 대해서는 빠른 온라인 평가를 제공한다. 본 연구에서는 구조 해석을 포함한 수치 예제를 통해 제안된 방법을 검증하였으며, 평균 및 분산과 같은 통계적 모멘트와 위험가치(Value at Risk) 계산을 통해 불확실성 정량화의 성능을 평가하였다. 실험 결과, 제안된 방법은 전통적인 유한요소 해석과 몬테카를로 시뮬레이션(Monte Carlo Simulation, MCS)에 비해 높은 정확도를 유지하면서도 계산 비용을 크게 절감하는 것으로 나타났다.

This study introduces a reduced-order modeling approach for forward uncertainty quantification (UQ) in structural analysis involving random field inputs. The proposed method integrates the Karhunen-Loève (KL) expansion for efficiently representing the spatial uncertainty of material properties with the proper generalized decomposition (PGD) to solve high-dimensional problems by separating spatial and random variables. This enables offline computation across the full parameter space and rapid online evaluation for new input samples. The proposed method is further validated via numerical examples involving structural analysis, focusing on forward UQ that measures statistical moments and value at risk. The results demonstrate that the proposed method substantially reduces computational cost (by 98.65%) while maintaining accuracy (at 99.99%), compared to the full element method with crude Monte Carlo simulation (MCS).

고속 추진체계의 개발 초기는 자료의 부족, 비용 제약, 지상에서 실제 비행환경 모사의 어려움 등으로 불확실 요소들을 확률분포의 형태로 모델링하기 어려운 실정이다. 이러한 이유로 본 연구에서는 이중연소 램제트를 대상으로 전문가들의 경험에 의한 연소효율 정보를 수집하여 이를 에비던스 이론으로 모델링하여 불확실성을 정량화 하였다. 정량화한 불확실성 정보를 이용하여 흡입구와 연소기의 출구면적에 대하여 추력여유와 열질식의 불확실성을 고려한 신뢰성 최적설계를 수행하였다. 한정된 불확실 정보를 가지고 엔진의 개념설계가 가능함을 확인할 수 있었다.

At the early stages of development of high-speed propulsion systems, associated uncertainties cannot be easily modeled into probabilistic distributions, owing to the lack of test data, cost, and difficulty of simulating real-flight environments on the ground. To tackle this issue, in this research, the combustion efficiencies of dual-combustion ramjet engines are assumed to have been provided by experts and quantified by evidence theory. Using quantified uncertainty, the inlet area and combustor exit are optimized while satisfying reliability margins of thrust and thermal choking. The result shows a reasonable design of the engine under uncertain circumstances.

고속추진체계의 시험평가에는 많은 비용과 시간이 필요하므로 시험자료의 양은 항상 부족하고, 설사 있더라도 지상시험 환경이 실제 비행조건과 일치하는 경우가 드물다. 이러한 이유로 설계자들은 설계결과에 대한 불확실성을 정량적인 확률로 제시하는데 어려움을 가지고 있다. 본 논문에서는 Evidence 기법을 이용하여, 시험자료 대신 개발자들의 경험과 공학적인 지식을 바탕으로 불확실성을 모델링하는 방법을 연구하였다. 연소효율은 이중연소램제트 엔진의 초기설계단계에서 가장 예측하기 어려운 변수중의 하나이다. 유사분야의 경험을 가진 설계자들이 이 값을 제시하는 것으로 가정하여 이중연소램제트 엔진의 설계결과에 대한 불확실성을 산출하였다. 나아가 흡입구와 연소기 출구면적으로 설계변수로, 추력성능과 thermal choking의 가능성을 제약함수로 하는 신뢰성 최적설계를 수행함으로써 시스템의 안전성을 확보하면서 최적의 성능을 얻을 수 있는 설계기법을 탐색하였다.

At the early stage of the development of high speed propulsion systems, the designers suffer from the lack of both the quantity and the quality of test data. In that situation, the associated uncertainties could not be modeled as probabilistic distribution since probabilistic modelling requires large amount of data. In this paper, instead, the information provided by experts based on their experience and engineering knowledge was used to model uncertainty using the evidence theory. In designing the DCR(Dual Combustion Ramjet) engine, the combustion efficiencies, not well understood and little data existing, are assumed to have been provided by experts. And the uncertainties are quantified by Evidence theory. The quantified uncertainties are incorporated into the optimization. The design variables, area of inlet and area of combustor exit, have been found while satisfying reliability margins of thrust and thermal choking. The results show a reasonable design of the engine under the uncertain circumstances.

Uncertainty quantification of a simple cantilevered piezoelectric energy harvester (PEH) is presented in this paper. Unlike most of existing studies where only substrate and piezoelectric layers are considered, the PEH system employed in this study has an additional adhesive layer between the substrate and piezoelectric layers. A deterministic electromechanical model of the PEH is constructed using the Kane's method and the stochastic equations of the PEH system are derived employing the polynomial chaos expansion method. The thickness of adhesive and piezoelectric layers, piezoelectric coupling coefficient, and input frequency are considered as random parameters. The results show that the uncertainty of adhesive layer thickness should be considered for the reliability design of the PEH system.