Brute Force Monte Carlo Simulation Research Articles

Effect of sample size on the maximum value distribution of fatigue driving forces in metals and alloys

An analytical framework is presented to predict the effects of sample size on the maximum value distribution (MVD) of the driving forces for fatigue crack formation in metals and alloys. The distribution of the maximum driving force for fatigue crack formation over the domain follows the generalized extreme value theory in the limit as the domain size increases to infinity. However, a simulation-based analysis of microstructure influences on fatigue resistance for polycrystalline metals and alloys is very costly, and reaching those limits is intractable. This work models the MVD of Fatigue Indicator Parameters (FIPs), which serve as surrogate measures for the driving force for fatigue crack formation, at finite sample sizes prior to their convergence to a limiting extreme value distribution. Large-scale crystal plasticity finite element (CPFE) simulations of FCC Al 7075-T6 with microstructure realizations of various sizes are incorporated to calibrate and evaluate the developed framework, and a total of ∼6.5 million grains of Al 7075-T6 are examined. The calibrated analytical solution agrees well with the brute force Monte Carlo simulation results extracted from the CPFE simulations. Furthermore, the developed formulation can predict the MVD of FIPs for different sample sizes using a size-dependent parameter, and it is capable of accurately extrapolating the MVD of FIPs for much larger microstructure sample sizes than the size used for its calibration.

A Computationally Efficient Method for Estimating Multi‐Model Process Sensitivity Index

AbstractIdentification of important processes of a hydrologic system is critical for improving process‐based hydrologic modeling. To identify important processes while jointly considering parametric and model uncertainty, Dai et al. (2017), https://doi.org/10.1002/2016WR019715, developed a multi‐model process sensitivity index. Numerical evaluation of the index using a brute force Monte Carlo (MC) simulation is computationally expensive, because it requires a nested structure of parameter sampling and the number of model simulations is on the order of (N being the number of parameter samples). To reduce computational cost, we develop a new method (here denoted as quasi‐MC for brevity) that uses triple sets of parameter samples (generated using quasi‐MC sequence) to remove the nested structure of parameter sampling in a theoretically rigorous way. The quasi‐MC method reduces the number of model simulations from the order of to 2N. The performance of the method is assessed against the brute force MC approach and the recent binning method developed by Dai et al. (2017), https://doi.org/10.1002/2016WR019715, through two synthetic cases of groundwater flow and solute transport modeling. Due to its rigorous theoretical foundation, the quasi‐MC method overcomes the limitations imposed by the inherently empirical nature of the binning method. We find that the quasi‐MC method outperforms both the brute force MC and the binning method in terms of computational requirements and theoretical aspects, thus strengthening its potential for the assessment of process sensitivity indices subject to various sources of uncertainty.

Interval uncertainty analysis of vibration response of hydroelectric generating unit based on Chebyshev polynomial

The uncertainty of the hydroelectric generating unit (HGU) is unavoidable in practical engineering. However, up to now, the research about influences of the uncertain parameters on the system responses is still very scarce, especially in the case that the probability distribution of the uncertain parameter is not clear. So, this paper will aim to study the influence of interval uncertain parameters on the vibration response characteristics of HGU. Firstly, a nonlinear mathematical model of rotor-shaft-bearing system of the HGU is established under the action of interval uncertain parameters. Then, based on Chebyshev polynomial and interval analysis theory, three feasible analysis methods of interval uncertainty for the HGU are proposed, which are also compared with the brute-force Monte Carlo Simulation (MCS) method. And it is proved the proposed Chebyshev-MCS method has higher accuracy and lower cost obtaining the vibration response boundary of the HGU. Next, using Chebyshev-MCS method, the uncertain response characteristics of the system under single parameter, multiple parameters and shock excitation are further analyzed, respectively. Meanwhile, the upper and lower boundaries of the vibration response under different degrees of uncertainty and rotational speeds are obtained. Besides, the coupling effects of several uncertain parameters on the vibration response under the rated rotational speed condition are also presented. And the influence of uncertain parameters on the transient characteristics of the system under shock excitation is also gotten. More importantly, the proposed method and obtained conclusions are helpful to the manufacture, installation, operation of the HGU.

Space Imaging Sensor Power Supply Filtering: Improving EMC Margin Assessment with Clustering and Sensitivity Analyses

This work is dedicated to the assessment of the filtering performances of an optoelectronic sensor for space applications. Particular care is taken concerning the power supply subsystem (here voltage shifter integrated circuit), which is part of the electromagnetic compatibility (EMC) compliance of an imaging equipment embedded on spacecrafts. The proposed methodology aims at two major targets: First, evaluating the Filter Effectiveness (FE) subject concerning varying parameters (including filter topology, parasitic effects and source/load impedance variations); second, quantifying the relative importance of representative equivalent electrical components through sensitivity analyses (nominal and parasitic values). The latter point is of utmost importance considering the expected versatility of such systems, such as manufacturing tolerances, for instance. Nominal values and/or components are often badly defined for confidentiality reasons, lack of knowledge or pure ignorance of inputs. An analytical deterministic formulation (here through the transfer matrix approach) is proposed and completed with an original stochastic strategy (Reduced Order Clustering, ROC). This ensures the reliable assessment of both statistical filter performances and most influential parameters, jointly with computational resources saving relatively to brute force Monte Carlo simulations.

Shear capacity distribution of reinforced concrete beams: An information theoretic entropy approach

Using the concept of information theoretic entropy, the probability density function (pdf) of shear capacity of the reinforced concrete beam with stirrup reinforcement is determined. Entropy, expressed in terms of Shannon functional, is maximized subjected to the statistical moment and normalization constraints of pdf of shear capacity. The statistical moments of shear capacity distribution are obtained using second-order approximation of shear capacity equation. The pdf so determined has strong statistical mechanics interpretation of maximum entropy principle. Also, a procedure for goodness-of-fit test has been proposed, for the given data, using the information theoretic entropy as a measure of goodness-of-fit. In the present investigation, beams of three different ranges of shear span to effective depth ratios are considered. The mechanics-based shear capacity equations, presented earlier by authors along with associated modelling errors, are used for estimating the statistical moments of shear capacity distribution. The computationally efficient approach of determination of maximum entropy distribution presented in this article can be viewed as an alternate to the process of determination of pdf using brute force Monte Carlo simulation approach.

An effective Kriging-based approximation for structural reliability analysis with random and interval variables

Aleatory and epistemic uncertainties usually coexist within a mechanistic model, which motivates the hybrid structural reliability analysis considering random and interval variables in this paper. An introduction of the interval variable requires one to recursively evaluate embedded optimizations for the extremum of a performance function. The corresponding structural reliability analysis, hence, becomes a rather computationally intensive task. In this paper, physical characteristics for potential optima of the interval variable are first derived based on the Karush-Kuhn-Tucker condition, which is further programmed as a simulation procedure to pair qualified candidate samples. Then, an outer truncation boundary provided by the first-order reliability method is used to link the size of a truncation domain with the targeted failure probability, whereas the U function is acted as a refinement criterion to remove inner samples for an increased learning efficiency. Given new samples detected by the revised reliability-based expected improvement function, an adaptive Kriging surrogate model is determined to tackle the hybrid structural reliability analysis. Several numerical examples in the literature are presented to demonstrate applications of this proposed algorithm. Compared to benchmark results provided by the brute-force Monte Carlo simulation, the high accuracy and efficiency of this proposed approach have justified its potentials for the hybrid structural reliability analysis.

Efficient characterization of dynamic response variation using multi-fidelity data fusion through composite neural network

Uncertainties in a structure is inevitable, which generally lead to variation in dynamic response predictions. For a complex structure, brute force Monte Carlo simulation for response variation analysis is infeasible since one single run may already be computationally costly. Data driven meta-modeling approaches have thus been explored to facilitate efficient emulation and statistical inference. The performance of a meta-model hinges upon both the quality and quantity of training dataset. In actual practice, however, high-fidelity data acquired from high-dimensional finite element simulation or experiment are generally scarce, which poses significant challenge to meta-model establishment. In this research, we take advantage of the multi-level response prediction opportunity in structural dynamic analysis, i.e., acquiring rapidly a large amount of low-fidelity data from reduced-order modeling, and acquiring accurately a small amount of high-fidelity data from full-scale finite element analysis. Specifically, we formulate a composite neural network fusion approach that can fully utilize the multi-level, heterogeneous datasets obtained. It implicitly identifies the correlation of the low- and high-fidelity datasets, which yields improved accuracy when compared with the state-of-the-art. Comprehensive investigations using frequency response variation characterization as case example are carried out to demonstrate the performance.

High-Density SRAM Read Access Yield Estimation Methodology

As high-density SRAMs must be designed to ensure a substantially small failure rate, the accurate yield estimation with practically acceptable runtime of circuit simulations is highly challenging. Here, a read access yield estimation method for high-density static random access memory (SRAM) is proposed. Instead of performing SPICE runs for the entire SRAM circuit, the proposed method partitions the SRAM into three parts—the control signal generation circuit, bitcell array, and sense amplifier (SA)—that determine three key parameters: word-line to SA enable delay, bit-line voltage difference, and SA offset voltage. Subsequently, the proposed method derives the probability density of these key parameters from each of the three partitioned circuits. Here, different methods are applied to derive the probability of the key parameters, considering the respective characteristics of each circuit part and parameter. According to our experimental results, the proposed method can accelerate the yield estimation by 500– $3000\times $ , compared with the brute-force Monte Carlo simulation method, and 10– $100\times $ compared with the other state-of-art methods. In addition, the proposed method can accelerate the circuit optimization procedure accompanied by multiple circuit revisions, that is, the circuit revisions can be reflected with SPICE runs only for the revised circuit part, unlike the previous methods that require SPICE runs for the entire SRAM.

Particles confined in arbitrary potentials with a class of finite-range repulsive interactions.

In this paper, we develop a large N field theory for a system of N classical particles in one dimension at thermal equilibrium. The particles are confined by an arbitrary external potential, V_{ex}(x), and repel each other via a class of pairwise interaction potentials V_{int}(r) (where r is distance between a pair of particles) such that V_{int}∼|r|^{-k} when r→0. We consider the case where every particle is interacting with d (finite-range parameter) number of particles to its left and right. Due to the intricate interplay between external confinement, pairwise repulsion, and entropy, the density exhibits markedly distinct behavior in three regimes k>0,k→0, and k<0. From this field theory, we compute analytically the average density profile for large N in these regimes. We show that the contribution from interaction dominates the collective behavior for k>0 and the entropy contribution dominates for k<0, and both contribute equivalently in the k→0 limit (finite-range log-gas). Given the fact that this family of systems is of broad relevance, our analytical findings are of paramount importance. These results are in excellent agreement with brute-force Monte Carlo simulations.

Analysis of Parameter Variability in an Integrated Wireless Power Transfer System via Partial Least-Squares Regression

This article deals with the application of the partial least-squares (PLS) regression to the uncertainty quantification of an integrated wireless power transfer with 30 random variables. It considers the development of surrogate models using a limited set of training samples in order to estimate statistical quantities of the converter efficiency with a relatively low computational cost compared with the standard brute-force Monte Carlo (MC) simulation. The strength, the performance, and the features of the proposed modeling approach are then compared with the ones of an advanced probabilistic surrogate model combining the least-squares support vector machine (LS-SVM) and the Gaussian process regression (GPR). The MC simulation is considered as a reference.

Adaptive importance sampling based neural network framework for reliability and sensitivity prediction for variable stiffness composite laminates with hybrid uncertainties

In this work, we propose to leverage the advantages of both the Artificial Neural Network (ANN) based Second Order Reliability Method (SORM) and Importance sampling to yield an Adaptive Importance Sampling based ANN, with specific application towards failure probability and sensitivity estimates of Variable Stiffness Composite Laminate (VSCL) plates, in the presence of multiple independent geometric and material uncertainties. The performance function for the case studies is defined based on the fundamental frequency of the VSCL plate. The accuracy in both the reliability estimates and sensitivity studies using the proposed method were found to be in close agreement with that obtained using the ANN based brute-force Monte Carlo Simulations (MCS) method, with a significant computational savings of 95%. Moreover, the importance of taking into account the randomness in ply thickness for failure probability estimates is also highlighted quantitatively under the sensitivity studies section.

Arbitrary polynomial chaos expansion method for uncertainty quantification and global sensitivity analysis in structural dynamics

Uncertainty quantification (UQ) and global sensitivity analysis (GSA) of dynamic characteristics of complex systems subjected to uncertainty are jointly investigated in this paper. An efficient approach based on arbitrary polynomial chaos expansion (aPCE) is presented for analytical, unified implementation of UQ and GSA in structural dynamics. For UQ of dynamic characteristics, statistical moments and probability distributions of dynamic characteristics are analytically derived. Specifically, the aPCE is used to analytically calculate the statistical moments, and then the maximum entropy principle (MEP) is adopted to derive the closed-form expressions of the probability distributions using the obtained statistical moments. As an extension of UQ, GSA, which aims to assess the quantitative contributions of different structural parameters to the resultant variations of dynamic characteristics, is also analytically achieved by simply post-processing the aPCE coefficients. The present aPCE UQ and GSA method is highly computationally efficient for large-scale, complex structures, and it is also generally applicable independent of parameter distributions. The proposed aPCE-based approach for UQ and GSA is validated through a numerical truss bridge by the brute-force Monte Carlo simulation (MCS), and then is applied to a long-span steel arch bridge.

Reliability sensitivity analysis with subset simulation: application to a carbon dioxide storage problem

Reliability sensitivity analysis (RSA) is a sensitivity analysis to measure the effect of modelling parameters on the predicted reliability of a system. It can be used for reliability-based design, safety management, etc. The output-classification-based version of RSA compares the failure-conditional probability density function (PDF) of model parameters with their unconditional PDF to measure sensitivity. The main challenge is to estimate failure-conditional PDFs. Usually, these PDFs can be estimated through the failure samples obtained by Monte Carlo simulation. However, practical systems usually have a small failure probability. For such cases, the brute-force Monte Carlo simulation requires a larger number of samples to obtain enough failure samples. Therefore, the computational cost is very high. In this paper, we propose to use subset simulation to estimate the output-classification-based reliability sensitivity index. Subset simulation introduces a series of intermediate failure events which are easier to sample from, and then iteratively samples in each constrained failure region until the target failure event is reached. Compared to brute-force Monte Carlo simulation, subset simulation samples in a direction towards the target failure domain. Therefore, the failure samples can be obtained more efficiently. We apply subset simulation to perform RSA for a carbon dioxide storage benchmark problem. We show that subset simulation can estimate the output-classification-based reliability sensitivity index more efficiently compared to brute-force Monte Carlo simulation.

An Effective Approach for Reliability-Based Sensitivity Analysis with the Principle of Maximum Entropy and Fractional Moments.

The reliability-based sensitivity analysis requires to recursively evaluate a multivariate structural model for many failure probability levels. This is in general a computationally intensive task due to irregular integrations used to define the structural failure probability. In this regard, the performance function is first approximated by using the multiplicative dimensional reduction method in this paper, and an approximation for the reliability-based sensitivity index is derived based on the principle of maximum entropy and the fractional moment. Three examples in the literature are presented to examine the performance of this entropy-based approach against the brute-force Monte-Carlo simulation method. Results have shown that the multiplicative dimensional reduction based entropy approach is rather efficient and able to provide reliability estimation results for the reliability-based sensitivity analysis of a multivariate structural model.

Combined stochastic lateral and in-plane loading of a stiffened ship panel leading to collapse

With harshening ocean environments and the push to extend the service life of vessels and platforms, lifetime structural design performance is an increasingly important factor to differentiate between design options. But despite many advancements in structural modeling, reliability analysis may still be too time consuming and computationally expensive to be employed for local design choices. Reliability analysis for stiffened ship panels is additionally complicated as the collapse mechanism is due to combined stochastic lateral and in-plane loading effects [1]. This paper employs the non-linear Design Loads Generator (NL-DLG) process [2] to efficiently estimate the probability of stiffened ship panel collapse while retaining the wave profiles which lead to lifetime responses. This information allows a direct comparison between panel options and highlights critical panel parameters which are strongly related to design robustness. The resulting ensemble of short NL-DLG wave profiles lead to similar statistical characteristics of the panel designs as do brute-force Monte Carlo Simulations, but reduce the required simulation time by a factor of nearly 87,000 for the same number of simulations. These directed wave profiles offer naval architects the opportunity to efficiently examine rare complex structural responses by linking high-fidelity structural and hydrodynamic models. The potential of the NL-DLG process is that relevant information about complex marine structures, even those with limit surfaces excited by combined non-Gaussian loading, can be obtained efficiently and in earlier stages of the design process via low-order surrogate models.

REIF: A novel active-learning function toward adaptive Kriging surrogate models for structural reliability analysis

Structural reliability analysis is typically evaluated based on a multivariate function that describes underlying failure mechanisms of a structural system. It is necessary for a surrogate model to mimic the true performance function as the brute-force Monte-Carlo simulation is computationally intensive for rare failure probabilities. To this end, the paper presents an effective active-learning based Kriging method for structural reliability analysis. The reliability-based expected improvement function (REIF) is first derived based on the folded-normal distribution. To account for the modulating effect of the joint probability density function of input random variables on the scattering geometry of candidate samples, an improvement of the REIF active-learning function, i.e., the REIF2 is further presented. Then, the low-discrepancy samples and adaptively truncated sampling regions are combined together to initiate efficient active-learning iterations. The truncated sampling region is directly related to a structural failure probability result, rather than subjectively fixed by an analyst. Numerical validity of the proposed active-learning functions in conjunction with adaptively truncated sampling region and low-discrepancy samples is demonstrated by several structural reliability examples in the literature.

Modeling and simulation of steady heat transfer analysis with material uncertainty: Generalized nth order perturbation isogeometric stochastic method

The contribution herein proposes a novel generalized nth order perturbation isogeometric method (GNP-IGA) for efficient steady heat transfer stochastic analysis with material uncertainty. While various space discretizations may be employed for spatial modeling, first, the isogeometric analysis (IGA) technique is employed and is the selected choice to exactly represent the structural geometry, as well as to provide more accurate results. Secondly, the randomness of thermal conductivity coefficient is considered in the steady heat transfer analysis, and its stochastic field is modeled by a generalized nth order perturbation method. Namely, we derive and expand the IGA based random-input parameters (thermal conductivity coefficient) and all state functions included in the governing equations around their expectations via a generalized nth order Taylor series using a small perturbation parameter . More importantly, we can satisfy accuracy requirements of the probabilistic moments of the stochastic solution by expanding Taylor series to nth order. As comparison, the brute force Monte Carlo simulations are conducted in various numerical examples and primarily serve as a benchmark; both, a simple square plate and a real engineering application are demonstrated. The results verify the proposed GNP-IGA is effective and significantly efficient for steady heat transfer stochastic analysis with material uncertainty. In addition, the larger the problem’s scale (DOFs) and/or the number of samples are, the more efficiency the GNP-IGA method will inherit.

A domain decomposition method for stochastic analysis of acoustic fields with hybrid and localized uncertainties

An efficient domain decomposition method (DDM) is proposed for the dynamic analysis of stochastic acoustic fields with hybrid and localized uncertainties. The hybrid and localized uncertainties refer to the parameters that are associated with local properties of the acoustic fields and meanwhile are subjected to different kinds of randomness. To take advantage of the locally distributed feature of uncertain parameters, the full acoustic domain is divided into several sub-domains, along with each localized uncertain parameter being assigned to one specific sub-domain. In each sub-domain, the deterministic Helmholtz equation is transformed to a weak integral form and the discretized governing equation is obtained by employing Chebyshev orthogonal polynomials as admissible functions. The random or interval perturbation technique is applied to the individual governing equation according to the respective uncertainty type, whereby the stochastic governing equation is established. The original acoustic field is eventually recovered by the introduction of penalty functions to impose sound pressure continuity on the interfaces of sub-domains, and the (intervals of) sound pressure, together with its expectation and variance, can be subsequently obtained. The accuracy and efficiency of the proposed method are verified in several numerical examples by comparisons with the results given by brute force Monte Carlo simulations, and the DDM-based independent way of modelling and analysis proves to be quite effective and flexible for uncertainty quantification in acoustic fields.

An efficient approach for dynamic global sensitivity analysis of stochastic train-track-bridge system

Abstract The parameters in time-varying train-track-bridge system (TTBS) are inevitably subjected to uncertainty, leading to variability in its dynamic responses. This work provides an investigation as to how uncertainty in the parameters influences the dynamic responses of time-varying TTBS, which refers to dynamic sensitivity analysis in the context of stochastic dynamic system. A powerful global sensitivity analysis (GSA), which maintains numerous attractive advantages over local sensitivity analysis (LSA), is proposed to measure the impact of uncertain system parameters on dynamic responses in a quantitative manner, thereby ranking them in order of importance. Furthermore, an analytical Gaussian process model (GPM)-based approach is developed to greatly alleviate the computational cost involved in dynamic GSA of time-varying TTBS. The proposed analytical GPM-based approach possesses the capacity to calculate the sensitivity indices of individual parameters as well as parameter clusters. The effectiveness of the proposed GPM-based dynamic GSA methodology is evaluated by comparison with the brute-force Monte Carlo simulation (MCS) on a mass-spring-damper system. The perfect agreement between the analytical GPM-obtained and MCS-derived time-varying sensitivity indices verifies the feasibility and reliability of the proposed methodology. An illustrative example is further provided to fully demonstrate the application of the developed GPM-based approach to dynamic GSA of time-varying TTBS. The dynamic GSA results enable us to gain insight into the temporal evolution of the global influences of individual parameters and groups of parameters on dynamic responses of TTBS.

Understanding the mechanics of creep deformation to develop a surrogate model for contact assessment in CANDU® fuel channels

A key element of the fuel channel life cycle management in CANDU® reactors is to prevent contact between the pressure tube (PT) and the calandria tube (CT) and to avoid the development of hydride blisters that lead to delayed hydride cracking of the PT. The PT-CT contact is the result of in-reactor deformation due to irradiation induced creep of the fuel channel assembly, which in turn is affected by uncertainties associated with various parameters like material properties, dimensional changes in the channel and boundary conditions (e.g., end slopes) of the channel. To account for these uncertainties, probabilistic assessment methods are developed to evaluate the risk of PT-CT contact and demonstrate compliance with provisions of the CSA Standard N285.8. Currently, a simulation is based on probabilistic assessments in which input parameters to a finite element model (FEM) of creep deformation are randomly sampled from their respective distributions. A simulation model involves numerous repetitive solutions of the FEM model to determine the probability distribution of PT-CT gap and the time to contact. Since the creep deformation analysis using FEM is computationally involved, this brute force Monte Carlo simulation method is not an efficient way to carry out the probabilistic assessment of the reactor core. This paper proposes a new line of thinking for probabilistic assessments of PT-CT contact in fuel channels, which is based on replacing a full FEM model by a surrogate model of a much simpler analytical form. The surrogate model not only simplifies the creep deformation analysis, but also provides a more logical basis for probabilistic assessments. This paper presents an insightful analysis of creep deformation and shows that a simple surrogate model can be developed to predict the PT-CT gap as a linear function of two primary random variables, namely, a creep factor and end slopes. This simplified representation has a far reaching effect on the probabilistic assessment of fuel channels.