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.

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.

Dynamic reliability analysis of angular contact ball bearings in positioning precision

Bearings are the core components in various kinds of rotating machineries. With the increasing demand of reliable bearings in precision equipment, it is of great significance to analyze the motion error of bearings. In this article, a nonlinear dynamic model of angular contact ball bearings installed in pairs is constructed to describe its response characteristics. The definition of a failure mode in matched bearings is that the dynamic response of each inner race is larger than the allowed axial runout. This article introduces a feasible approach to evaluate the reliability of positioning precision of matched bearings with random geometric parameters. The statistical moments of dynamic response are calculated using stochastic perturbation method. The probability distribution function of state function relating to positioning precision is approached by Edgeworth series, from which the reliability and sensitivity are obtained. A pair of 7206B bearings is taken as an application instance of the proposed method. Monte Carlo simulation is employed to provide a benchmark on which to verify the precision and efficiency of the proposed method. In addition, the effects of mean values and variances of random geometric parameters on the positioning precision are analyzed, respectively.

Component mode synthesis and stochastic perturbation method for dynamic analysis of large linear finite element with uncertain parameters

In this paper, a method to calculate the first two moments (mean and variance) of the stochastic time response as well as the frequency functions of large FE models with probabilistic uncertainties in the physical parameters is proposed. This method is based on coupling of second order perturbation method and component mode synthesis methods. Various component mode synthesis methods are used to optimally reduce the size of the model. The analysis of dynamic response of stochastic finite element system can be done in the frequency domain using the frequency transfer functions and in the time domain by a direct integration of the equations of motion, using numerical procedures. The statistical first two moments of dynamic response of the reduced system are obtained by the second order perturbation method. Numerical applications have been developed to highlight effectiveness of the method developed to analyze the stochastic response of large structures.

Energy Fluctuations in the Homogenized Hyper-Elastic Particulate Composites with Stochastic Interface Defects

The principle aim of this study is to analyze deformation energy of hyper-elastic particulate composites, which is the basis for their further probabilistic homogenization. These composites have some uncertain interface defects, which are modelled as small semi-spheres with random radius and with bases positioned on the particle-matrix interface. These defects are smeared into thin layer of the interphase surrounding the reinforcing particle introduced as the third component of this composite. Matrix properties are determined from the experimental tests of Laripur LPR 5020 High Density Polyurethane (HDPU). It is strengthened with the Carbon Black particles of spherical shape. The Arruda–Boyce potential has been selected for numerical experiments as fitting the best stress-strain curves for the matrix behavior. A homogenization procedure is numerically implemented using the cubic Representative Volume Element (RVE). Spherical particle is located centrally, and computations of deformation energy probabilistic characteristics are carried out using the Iterative Stochastic Finite Element Method (ISFEM). This ISFEM is implemented in the algebra system MAPLE 2019 as dual approach based upon the stochastic perturbation method and, independently, upon a classical Monte-Carlo simulation, and uniform uniaxial deformations of this RVE are determined in the system ABAQUS and its 20-noded solid hexahedral finite elements. Computational experiments include initial deterministic numerical error analysis and the basic probabilistic characteristics, i.e., expectations, deviations, skewness and kurtosis of the deformation energy. They are performed for various expected values of the defects volume fraction. We analyze numerically (1) if randomness of homogenized deformation energy can correspond to the normal distribution, (2) how variability of the interface defects volume fraction affects the deterministic and stochastic characteristics of composite deformation energy and (3) whether the stochastic perturbation method is efficient in deformation energy computations (and in FEM analysis) of hyper-elastic media.

Stochastic Finite Element Method Elasto-Plastic Analysis of the Necking Bar With Material Microdefects

The main aim of this work is to study a significance of structural microdefects and their uncertainty in structural steel on its elastoplastic large deformations subjected to tensile test with the use of the generalized stochastic perturbation method. Elastoplastic behavior of the macroscopically homogeneous material is defined by the Gurson–Tvergaard–Needleman (GTN) constitutive model, where Young's modulus and this model constants q1 and q2 are consecutively randomized according to the Gauss probability distribution. The stochastic finite element method (SFEM) analysis has been carried out in the system abaqus for the problem of necking under tension to compute the first four probabilistic moments and coefficients of displacements, deformations, and stresses. The tenth-order perturbation scheme has been implemented via statistically optimized least-squares method (LSM) determination of the structural nodal polynomial response functions. A comparison with Monte Carlo simulation (MCS) as well as the semi-analytical integral technique based on the same polynomial bases confirms applicability of the method proposed for the input uncertainty not larger than 0.10. Further numerical experiments with this constitutive law including stochastic nucleation and/or coalescence would be necessary to better understand deformations and stresses of stochastic porous plastic materials. This model may find its applications in various stress states of the plastic materials with voids as well as in numerical simulations of the composite materials with imperfect interphases, for instance, where some parameters exhibit initial Gaussian statistical scattering.

Task-level time-optimal collision avoidance trajectory planning for grinding manipulators

A computational framework that can plan the task-level time-optimal collision avoidance trajectory (TOCAT) of grinding manipulators is constructed based on the improved simulated annealing algorithm. When the workpiece surface has a plurality of discrete non-connected areas that need to be polished by grinding manipulators, the planning of TOCAT for a given grinding task is crucial, because it has a direct impact on the processing efficiency and intelligence of the automatic grinding system. Although many planning algorithms can be used to plan collision avoidance trajectories between any two points, the planning of the task-level TOCAT with multiple collision avoidance sub-trajectories is more difficult because it involves the permutation of the collision avoidance sub-trajectories for connecting several grinding areas. This paper proposes a task-level TOCAT planning framework based on the improved simulated annealing algorithm. Its key point is to plan the time-optimal sub-trajectory between any two points based on the trajectory evaluation mechanism. Its innovation lies in that the simulated annealing algorithm generates new solutions based on the combined stochastic perturbation method. The experimental results show that this framework can effectively solve the task-level TOCAT planning problem in multiple grinding areas, and the duration of the task-level collision avoidance trajectory is not only less discrete but also approximately globally optimal.