Stochastic Perturbation Method Research Articles

Probabilistic relative entropy in elasto‐plasticity using the iterative generalized stochastic stress‐based Finite Element Method

AbstractThe generalized iterative stochastic perturbation approach to the stress‐based Finite Element Method has been proposed in this work. This approach is completed using the complementary energy principle, Taylor expansion of the general order applicable to all random functions and parameters as well as nodal polynomial response bases determined with the use of the Least Squares Method. The main aim of this elaboration is the usage of such a probabilistic approach to determine Bhattacharyya relative entropy for some nonlinear engineering stress analysis with uncertainty. Mathematical apparatus with its numerical implementation has been used to study elastoplastic torsion of some prismatic bar with Gaussian material uncertainty and the corresponding reliability measures. This problem has been solved using the Constant Stress Triangular (CST) plane finite elements, the modified Newton–Raphson algorithm, whereas the first four probabilistic characteristics resulting from the iterative generalized stochastic perturbation method have been contrasted with these obtained with the crude Monte‐Carlo sampling and the semi‐analytical probabilistic approach.

Relative Entropy Application to Study the Elastoplastic Behavior of S235JR Structural Steel.

The main issue in this work is to study the limit functions necessary for the reliability assessment of structural steel with the use of the relative entropy apparatus. This will be done using a few different mathematical theories relevant to this relative entropy, namely those proposed by Bhattacharyya, Kullback-Leibler, Jeffreys, and Hellinger. Probabilistic analysis in the presence of uncertainty in material characteristics will be delivered using three different numerical strategies-Monte Carlo simulation, the stochastic perturbation method, as well as the semi-analytical approach. All of these methods are based on the weighted least squares method approximations of the structural response functions versus the given uncertainty source, and they allow efficient determination of the first two probabilistic moments of the structural responses including stresses, displacements, and strains. The entire computational implementation will be delivered using the finite element method system ABAQUS and computer algebra program MAPLE, where relative entropies, as well as polynomial response functions, will be determined. This study demonstrates that the relative entropies may be efficiently used in reliability assessment close to the widely engaged first-order reliability method (FORM). The relative entropy concept enables us to study the probabilistic distance of any two distributions, so that structural resistance and extreme effort in elastoplastic behavior need not be restricted to Gaussian distributions.

Impact of Momentum Perturbation on Convective Boundary Layer Turbulence

AbstractMesoscale‐to‐microscale coupling is an important tool for conducting turbulence‐resolving multiscale simulations of realistic atmospheric flows, which are crucial for applications ranging from wind energy to wildfire spread studies. Different techniques are used to facilitate the development of realistic turbulence in the large‐eddy simulation (LES) domain while minimizing computational cost. Here, we explore the impact of a simple and computationally efficient Stochastic Cell Perturbation method using momentum perturbation (SCPM‐M) to accelerate turbulence generation in boundary‐coupled LES simulations using the Weather Research and Forecasting model. We simulate a convective boundary layer (CBL) to characterize the production and dissipation of turbulent kinetic energy (TKE) and the variation of TKE budget terms. Furthermore, we evaluate the impact of applying momentum perturbations of three magnitudes below, up to, and above the CBL on the TKE budget terms. Momentum perturbations greatly reduce the fetch associated with turbulence generation. When applied to half the vertical extent of the boundary layer, momentum perturbations produce an adequate amount of turbulence. However, when applied above the CBL, additional structures are generated at the top of the CBL, near the inversion layer. The magnitudes of the TKE budgets produced by SCPM‐M when applied at varying heights and with different perturbation amplitudes are always higher near the surface and inversion layer than those produced by No‐SCPM, as are their contributions to the TKE. This study provides a better understanding of how SCPM‐M reduces computational costs and how different budget terms contribute to TKE in a boundary‐coupled LES simulation.

Eigenvibrations of Kirchhoff Rectangular Random Plates on Time-Fractional Viscoelastic Supports via the Stochastic Finite Element Method.

The present work's main objective is to investigate the natural vibrations of the thin (Kirchhoff-Love) plate resting on time-fractional viscoelastic supports in terms of the Stochastic Finite Element Method (SFEM). The behavior of the supports is described by the fractional order derivatives of the Riemann-Liouville type. The subspace iteration method, in conjunction with the continuation method, is used as a tool to solve the non-linear eigenproblem. A deterministic core for solving structural eigenvibrations is the Finite Element Method. The probabilistic analysis includes the Monte-Carlo simulation and the semi-analytical approach, as well as the iterative generalized stochastic perturbation method. Probabilistic structural response in the form of up to the second-order characteristics is investigated numerically in addition to the input uncertainty level. Finally, the probabilistic relative entropy and the safety measure are estimated. The presented investigations can be applied to the dynamics of foundation plates resting on viscoelastic soil.

Stochastic nonlinear eigenvibrations of thin elastic plates resting on time-fractional viscoelastic supports

– The main objective of this work is to investigate natural vibrations of the thin (Kirchhoff-Love) plate resting on time-fractional viscoelastic supports using the Stochastic Finite Element Method (SFEM). The mechanical behavior of the supports is deterministic and is described by the fractional order derivatives of the Riemann-Liouville type. A subspace iteration method in conjunction with the continuation method is the basis of the numerical solution carried out using the displacement formulation of the Finite Element Method. The probabilistic analysis includes independent Monte-Carlo simulation, the semi-analytical approach, and the iterative generalized stochastic perturbation method of the first two probabilistic moments. Moreover, a corresponding relative entropy and the entropy-based reliability index are contrasted with the First Order Reliability Method (FORM). It has been found that the relative entropy for analyzed eigen-vibrations is some interesting alternative to the FORM methodology in reliability assessment.

Uncertainty analysis of the control rod drop based on the adaptive collocation stochastic perturbation method

There are many uncertain factors to affect the control rod drop from its design and manufacture to the chain reaction in the reactor. They have to be considered in the simulation to realistically predict the dynamic behavior of control rod drop. Different from the previous method by using the nominal and extreme values of the uncertain factors, a statistical method is applied in this paper to analyze them in the process of the control rod drop. The uncertain factors obey uniform distribution, based on which a fifth-order adaptive collocation stochastic perturbation (ACSP) method is established in this paper, which is featured with simple format, high precision, and high calculation efficiency. The mean and standard deviation of the responses (such as the final time of the control rod drop) in two different types of reactors are calculated by using the ACSP method. On this basis, the response intervals and corresponding probabilities of the control rod drop are obtained. Besides, influences of each random variable on the responses are analyzed by the sensitivity analysis. In addition, the linear correlation degree and correlation between the two are discussed through the correlation coefficient of random variables and the responses. The uncertainty analysis of control rod drop in this paper not only provides a relatively strong theoretical support for scientifically dealing with uncertain factors, but also offers specific help for the optimal design of control rod assembly.

An adaptive divided-difference perturbation method for solving stochastic problems

Most stochastic problems are based on the independent and symmetrically distributed random variables. Using this principle, in this study, we propose an efficient calculation scheme to obtain the statistical moments based on high-order stochastic perturbation (SP) expansion. This solves the long-existing problem of the SP method, namely, the low calculation efficiency of statistical moments using high-order perturbation expansion, and the complexity of the calculation format. The scheme involves adopting the divided-difference method to approximate the partial derivative items of the SP method. The proposed method, which is similar to the collocation point method, only requires the deterministic calculation result for each node in the divided-difference method. In addition, as the adoption of high-order perturbation expansion is limited in multidimensional random problems due to the curse of dimensionality, an adaptive partial derivative items selection method is proposed in this paper to efficiently and reasonably ignore the expansion items that have an insignificant influence on statistical moments. This considerably improves the calculation efficiency of the statistical moments. Finally, the adaptive divided-difference perturbation (ADDP) method proposed in this paper is compared with the quasi-Monte Carlo method and the adaptive sparse grid methods using three numerical examples. The results prove that the ADDP method not only has high efficiency but also leads to significant improvements in the accuracy of statistical moments.

Data Level Privacy Preserving: A Stochastic Perturbation Approach Based on Differential Privacy

With the great amount of available data, especially collecting from the ubiquitous Internet of Things (IoT), the issue of privacy leakage arises increasingly concerns recently. To preserve the privacy of IoT datasets, traditional methods usually calibrate random noises on the data values to achieve differential privacy (DP). However, the amount of the calibrating noises should be carefully designed and a heedless value will definitely degrade the availability of datasets. Thus, in this work, we propose a stochastic perturbation method to sanitize the dataset, where the perturbation is obtained from the rest samples in the same dataset. In addition, we derive the expression of the utility level based on its unique framework and prove that the proposed algorithm can achieve the <inline-formula><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -DP. To show the effectiveness of the proposed algorithm, we conduct extensive experiments on real-life datasets by various functions, such as query answers and machine learning tasks. By comparing with the state-of-the-art methods, our proposed algorithm can achieve a better performance under the same privacy level.

Numerical convergence and error analysis for the truncated iterative generalized stochastic perturbation-based finite element method

The main aim of this work is to present a numerical analysis of convergence and accuracy of the selected generalized perturbation-based schemes in linear and nonlinear problems of solid mechanics. An algorithm for determining the basic probabilistic characteristics has been developed using the iterative generalized stochastic higher-order perturbation method adjacent to symmetrically truncated Gaussian random variables. It has been confirmed that usage of a sufficiently high order of the Truncated Iterative Stochastic Perturbation Technique (TISPT) allows for achieving any desired accuracy in determining up to the fourth-order probabilistic characteristics of static structural response. The semi-analytical probabilistic approach is the reference solution in this study, which is based upon the same composite response functions determined with the use of the Least Squares Method created using specific series of FEM experiments. The entire methodology has been provided for the given extreme value of the coefficient of variation αmax of the input uncertainty source. On the other hand, a selection procedure of the stochastic perturbation method order to achieve 1% numerical accuracy in all up to the fourth-order probabilistic moments has been proposed. Computational experiments include simply supported elastic Euler–Bernoulli beam, a set of steel diagrid structures, nonlinear tension of steel round bar as well as homogenization procedure of some particulate composite.

The efficient calculation methods for stochastic nonlinear transient heat conduction problems

This paper proposes an efficient model reduction strategy in conjunction with non-intrusive statistical moment methods for the stochastic nonlinear transient heat conduction problem. In the proposed strategy, the deterministic reduced basis vectors (RBVs) are adopted to approximate the random temperature response, and the thermal conductivity of the random nonlinear heat conduction matrix in the RBVs space is decoupled from the geometric characteristic matrix (GCM). The computational efficiency of solving the reduced random nonlinear heat conduction models can be enhanced by implementing these two technologies. The large coefficient of variation and the small coefficient of variation stochastic heat conduction problems, respectively, are addressed by the two non-intrusive statistical methods, i.e., the quasi-Monte Carlo method (QMCM) based on number theory and the modified stochastic perturbation method (MSPM). These two methods necessitate deterministic analysis and are easily integrated into the model reduction method. In four numerical instances, the selection of RBVs, the influence of the nonlinear term on the random temperature response, and the accuracy and efficiency of the two non-intrusive statistical moment methods are discussed. The numerical results highlight that the model reduction strategy based on the decoupling of thermal conductivity and GCM, in conjunction with the QMCM and the MSPM, enables the analysis of stochastic nonlinear transient heat conduction problems with a large coefficient of variation and a small coefficient of variation, respectively.

Stochastic perturbation method for optimal control problem governed by parabolic PDEs with small uncertainties

In this paper, we investigate the first-order and second-order perturbation approximation schemes for an optimal control problem governed by parabolic PDEs with small uncertainties. First, we propose the finite dimensional noise assumption for the random coefficient, and expand the state and co-state variables up to a certain order with respect to a small parameter by using the perturbation technique. Then, we insert all the expansions into the known deterministic parametric optimality system, and obtain the first-order and second-order optimality systems. These two systems are discretized by the finite element method in space and the backward Euler method in time to establish the first-order and second-order approximate schemes. Further, a priori error estimates are derived for the state, co-state and control variables of the first-order and second-order approximation, respectively. Finally, some numerical experiments are presented to verify the theoretical results.

Stability of rectangular Kirchhoff plates using the Stochastic Boundary Element Methods

The main objective in this work is to study an application of the Stochastic Boundary Element Methods implemented due to three different probabilistic approaches to analyze stability of the rectangular thin elastic and isotropic plates. This is completed with the use of polynomial approximations applied for the Least Squares Method recovery of critical forces resulting from some material and geometrical random imperfections in the plate, e.g. Young's modulus, Poisson's ratio, thickness. A deterministic core for solving the plate stability problem is the Boundary Element Method in direct approach and modified formulation of the boundary and domain integral equations. Probabilistic approaches employed here include traditional Monte-Carlo simulation, the semi-analytical approach as well as the iterative generalized stochastic perturbation method. Stochastic response in the form of up to the fourth order characteristics are studied numerically in addition to the input uncertainty level.

Comparison between Multi-Physics and Stochastic Approaches for the 20 July 2021 Henan Heavy Rainfall Case

In this study, three model perturbation schemes, the stochastically perturbed parameter scheme (SPP), stochastically perturbed physics tendency (SPPT), and multi-physics process parameterization (MP), were used to represent the model errors in the regional ensemble prediction systems (REPS). To study the effects of different model perturbation schemes on heavy rainfall forecasting, three sensitive experiments using three different combinations (EXP1: MP, EXP2: SPPT + SPP, and EXP3: MP + SPPT + SPP) of the model perturbation schemes were set up based on the Weather Research and Forecasting (WRF)-V4.2 model for a heavy rainfall case that occurred in Henan, China during 20–22 July 2021. The results show that the model perturbation schemes can provide forecast uncertainties for this heavy rainfall case. The stochastic physical perturbation method could improve the heavy rainfall forecast skill by approximately 5%, and EXP3 had better performance than EXP1 or EXP2. The spread-to-root mean square error ratios (spread/RMSE) of EXP3 were closer to 1 compared with those of the EXP1 and EXP2; particularly for the meridional wind above 10 m, the spread/RMSE was 0.94 for EXP3 and approximately 0.85 for EXP1 and EXP2. EXP3 exhibited better performance in Brier score verification. EXP3 had a 5% lower Brier score than EXP1 and EXP2, when the rainfall threshold was 25 mm. The growth of the initial ensemble variances of different model perturbation schemes were explored, and the results show that the perturbation energy of EXP3 developed faster, with a magnitude of 27.22 J/kg, whereas those of EXP1 and EXP2 were only 19.18 J/kg and 20.81 J/kg, respectively. The weak initial perturbation associated with the wind shear north of the heavy rainfall location can be easily developed by EXP3.

A stochastic and non‐linear representation of model uncertainty in a convective‐scale ensemble prediction system

AbstractAccurately addressing model uncertainties with a consideration of the enhanced effect of non‐linearities in a high‐resolution convective‐scale system is a crucial issue for performing convection‐allowing ensemble prediction systems (CAEPSs). In this study, a conditional non‐linear–stochastic perturbation method is developed to simultaneously consider both a stochastic and a non‐linear representation of model uncertainties associated with physics parameterization in the Global and Regional Assimilation and Prediction Enhanced System (GRAPES)‐CAEPS with a horizontal resolution of 3 km. The non‐linear forcing singular vector (NFSV) for a non‐linear representation of model uncertainties and the Stochastically Perturbed Parameterization Tendencies (SPPT) scheme for a stochastic representation of model uncertainties, are applied. Two experiments were carried out over South China for a month (May 1–30, 2020), one with a SPPT scheme and the other with a non‐linear–stochastic perturbation using a combination of SPPT and NFSV schemes. The combination of SPPT and NFSV schemes is compared with the SPPT scheme alone to investigate whether the conditional non‐linear–stochastic perturbation method that combines non‐linear and stochastic schemes can represent model uncertainty better than the traditional stochastic SPPT approach. The results show that combining the NFSV and SPPT schemes improves the overall probabilistic skill and has an advantage over the SPPT scheme, which may imply that adding additional state‐independent non‐linear noise contributes to a more comprehensive characterization of model error for representing model uncertainties in CAEPSs. This discovery sheds light on the design and development of model perturbation strategies for convective‐scale ensembles in the future.

Statistical analyses of the energy demand and thermal comfort for multiple uncertain input parameters, performed using transformed variable and perturbation method

In the calculations of buildings’ thermal comfort, the input parameters are usually considered as strictly determined values. Numerous of them may be characterized by certain probability density functions. In the energy related problems, the uncertainty analyses are usually performed using the Monte Carlo method. However, this method requires multiple calculations and, therefore, may be very time-consuming. In the proposed work, two approaches are applied for the probabilistic studies: the stochastic perturbation method and the transformed random variables method. The stochastic analysis is based on the response functions and their derivatives with respect to all random input parameters. The relation between the thermal comfort and the input (random) variables have been calculated using the Energy Plus software. Afterwards, the response functions were estimated using the polynomial regression. The expected value and central moments of the response functions were calculated by means of the perturbation method and the transformed random variable theorem. The latter method allowed to obtain, using the same response functions, the implicit form of probability distributions function of the output parameter.

Combined stochastic and discrete simulation to optimise the economics of mixed single-horizontal and multilateral well offshore oil developments

Multilateral wells promise cost savings to oil and fields as they have the potential to reduce overall drilling distances and minimize the number of slots required for the surface facility managing the well. However, drilling a multilateral well does not always increase the flow rate when compared to two single-horizontal wells due to competition in production inside the mother-bore. Here, a holistic approach is proposed to find the optimum balance between single and multilateral wells in an offshore oil development. In so doing, the integrated approach finds the highest Net Present Value (NPV) configuration of the field considering drilling, subsurface, production and financial analysis. The model employs stochastic perturbation and Markov Chain Monte-Carlo methods to solve the global maximising-NPV problem. In addition, a combination of Mixed-Integer Linear Programming (MILP), an improved Dijkstra algorithm and a Levenberg-Marquardt optimiser is proposed to solve the rate allocation problem. With the outcome from this analysis, the model suggests the optimum development including number of multilateral and single horizontal wells that would result in the highest NPV. The results demonstrate the potential for modelling to find the optimal use of petroleum facilities and to assist with planning and decision making.

Stochastic energy-demand analyses with random input parameters for the single-family house

In the analyses of the uncertainty propagation of buildings’ energy-demand, the Monte Carlo method is commonly used. In this study we present two alternative approaches: the stochastic perturbation method and the transformed random variable method. The energy-demand analysis is performed for the representative single-family house in Poland. The investigation is focused on two independent variables, considered as uncertain, the expanded polystyrene thermal conductivity and external temperature; however the generalization on any countable number of parameters is possible. Afterwards, the propagation of the uncertainty in the calculations of the energy consumption has been investigated using two aforementioned approaches. The stochastic perturbation method is used to determine the expected value and central moments of the energy consumption, while the transformed random variable method allows to obtain the explicit form of energy consumption probability density function and further characteristic parameters like quantiles of energy consumption. The calculated data evinces a high accordance with the results obtained by means of the Monte Carlo method. The most important conclusions are related to the computational cost reduction, simplicity of the application and the appropriateness of the proposed approaches for the buildings’ energy-demand calculations.

Probabilistic analysis of random nonlinear oscillators subject to small perturbations via probability density functions: theory and computing

We study a class of single-degree-of-freedom oscillators whose restoring function is affected by small nonlinearities and excited by stationary Gaussian stochastic processes. We obtain, via the stochastic perturbation technique, approximations of the main statistics of the steady state, which is a random variable, including the first moments, and the correlation and power spectral functions. Additionally, we combine this key information with the principle of maximum entropy to construct approximations of the probability density function of the steady state. We include two numerical examples where the advantages and limitations of the stochastic perturbation method are discussed with regard to certain general properties that must be preserved.

Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

We combine the stochastic perturbation method with the maximum entropy principle to construct approximations of the first probability density function of the steady-state solution of a class of nonlinear oscillators subject to small perturbations in the nonlinear term and driven by a stochastic excitation. The nonlinearity depends both upon position and velocity, and the excitation is given by a stationary Gaussian stochastic process with certain additional properties. Furthermore, we approximate higher-order moments, the variance, and the correlation functions of the solution. The theoretical findings are illustrated via some numerical experiments that confirm that our approximations are reliable.

Efficacy of the Cell Perturbation Method in Large-Eddy Simulations of Boundary Layer Flow over Complex Terrain

A challenge to simulating turbulent flow in multiscale atmospheric applications is the efficient generation of resolved turbulence motions over an area of interest. One approach is to apply small perturbations to flow variables near the inflow planes of turbulence-resolving simulation domains nested within larger mesoscale domains. While this approach has been examined in numerous idealized and simple terrain cases, its efficacy in complex terrain environments has not yet been fully explored. Here, we examine the benefits of the stochastic cell perturbation method (CPM) over real complex terrain using data from the 2017 Perdigão field campaign, conducted in an approximately 2-km wide valley situated between two nearly parallel ridges. Following a typical configuration for multiscale simulation using nested domains within the Weather Research and Forecasting (WRF) model to downscale from the mesoscale to a large-eddy simulation (LES), we apply the CPM on a domain with horizontal grid spacing of 150 m. At this resolution, spurious coherent structures are often observed under unstable atmospheric conditions with moderate mean wind speeds. Results from such an intermediate resolution grid are often nested down for finer, more detailed LES, where these spurious structures adversely affect the development of turbulence on the subsequent finer grid nest. We therefore examine the impacts of the CPM on the representation of turbulence within the nested LES domain under moderate mean flow conditions in three different stability regimes: weakly convective, strongly convective, and weakly stable. In addition, two different resolutions of the underlying terrain are used to explore the role of the complex topography itself in generating turbulent structures. We demonstrate that the CPM improves the representation of turbulence within the LES domain, relative to the use of high-resolution complex terrain alone. During the convective conditions, the CPM improves the rate at which smaller-scales of turbulence form, while also accelerating the attenuation of the spurious numerically generated roll structures near the inflow boundary. During stable conditions, the coarse mesh spacing of the intermediate LES domain used herein was insufficient to maintain resolved turbulence using CPM as the flow develops downstream, highlighting the need for yet higher resolution under even weakly stable conditions, and the importance of accurate representation of flow on intermediate LES grids.