Chaos Expansion Method Research Articles

On a Wick-type stochastic parabolic equations with random potentials

Stochastic parabolic equations with random potentials, where the Wick product is used to give sense to the product of generalized stochastic processes, are considered. The existence and uniqueness of solutions are proved via the chaos expansion method from white noise analysis and explicit estimates on the coefficients in the chaos expansion representation of the solutions are provided.

Probabilistic structural reliability analysis of a horizontal axis tidal turbine blade by considering the moisture effects on the blade material

The use of composite materials in marine structures like the horizontal axis tidal turbine blade introduces inherent uncertainties in the material properties of the blade. Further, the blade material ages because of the absorption of seawater into the material. Hence, whilst performing the static structural analysis of the blade, the probabilistic nature of the material properties, as well as its degradation due to ageing, must be considered. In this study, the probabilistic structural response of a 0.5 m blade was estimated by considering the experimentally determined statistical distributions of the dry and aged material properties. The polynomial chaos expansion (PCE) method was used to build a surrogate model to mimic the static analysis of the blade, which was used in conjunction with the Monte Carlo simulation to estimate the distributions of the structural response of the blade and the probability of failure of the dry and aged blade. Stochastic convergence was used to check the accuracy of the surrogate model built via the PCE method. The study revealed a statistically significant increase in the probability of failure of the blade due to the ageing induced degradation of the blade material. Further, the need for a probabilistic structural analysis was also established.

Polynomial chaos expansion and response surface method for nonlinear modelling of reference evapotranspiration

ABSTRACTThe feasibility of polynomial chaos expansion (PCE) and response surface method (RSM) models is investigated for modelling reference evapotranspiration (ET0). The modelling results of the proposed models are validated against the M5 model tree and multi-layer perceptron neural network (MLPNN) methods. Two meteorological stations, Isparta and Antalya, in the Mediterranean region of Turkey, are inspected. Various input combinations of daily air temperature, solar radiation, wind speed and relative humidity are constructed as input attributes for the ET0. Generally, the modelling accuracy is increased by increasing the number of inputs. Including wind speed in the model inputs considerably increases their accuracy in modelling ET0. Mean absolute error (MAE), root mean square error (RMSE), agreement index (d) and Nash-Sutcliffe efficiency (NSE) are used as comparison criteria. The PCE is the most accurate model in estimating daily ET0, giving the lowest MAE (0.036 and 0.037 mm) and RMSE (0.047 and 0.050 mm) and the highest d (0.9998 and 0.9999) and NSE (0.9992 and 0.9996) with the four-input PCE models for Isparta and Antalya, respectively.

Stochastic Modeling of Electrical Field in Potato Tuber using Polynomial Chaos Expansion

The paper deals with numerical modeling of objects with a natural origin. The stochastic approach based on description using random variables allows processing such challenges. The Monte-Carlo methods are known a tool for simulations containing stochastic parameters however, they require significant computational power to obtain stable results. Authors compare Monte- Carlo with more advanced Polynomial Chaos Expansion (PCE) method. Both statistical tools have been applied for simulation of the electric field used in ohmic heating of potato tuber probes. Results indicate that PCE is remarkably faster, however, it simplifies some probabilistic features of the solution.

Stochastic elastic equation driven by multiplicative multi-parameter fractional noise

In this paper, we consider the stochastic elastic equation driven by multiplicative multiparameter fractional noise. By using the Wiener chaos expansion and undetermined coefficient methods, we obtain the existence and uniqueness of the solution in a distribution space. The asymptotic behavior and the Hölder index of the solution are also estimated.

Corrigendum and addendum to "Chaos expansion methods in Malliavin calculus: A survey of recent results"

Corrigendum and addendum to "Chaos expansion methods in Malliavin calculus: A survey of recent results"

Time response of structures with uncertain parameters under stochastic inputs based on coupled polynomial chaos expansion and component mode synthesis methods

ABSTRACTThis article presents a numerical procedure to compute the stochastic dynamic response of large finite element models with uncertain parameters based on polynomial chaos and component mode synthesis methods. Polynomial chaos expansions at higher orders are used to derive the statistical solution of the dynamic response as well as the Monte Carlo simulation procedure. Based on various component mode synthesis methods, the size of the model is reduced. These methods are coupled with polynomial chaos expansion and the explicit mathematical formulations are given. Numerical results illustrating the accuracy and efficiency of the proposed coupled methodological procedures are presented.

Chaos expansion methods in Malliavin calculus: A survey of recent results

We present a review of the most important historical as well as recent results of Malliavin calculus in the framework of the Wiener-It^ o chaos expansion.

A fast and more universal algorithm for an uncertain acoustic filed in shallow-water

Non-intrusive polynomial chaos expansion (NPCE) method is a fast algorithm with the best performances for an uncertain acoustic filed currently, in which the selection of collocation points is an important factor for the computational accuracy, and some special processing methods such as piecewise probabilistic collocation method, should be adopted when the outputs of acoustic field vary severely with uncertain ocean environmental parameters. A new fast algorithm for uncertain acoustic filed in shallow-water is proposed based on Kriging model. The theoretical description of the new algorithm is given, and numerical simulations are conducted to verify the performances of the proposed algorithm. The physical interpretations are given in detail. The results demonstrate that the proposed algorithm is more accurate than the NPCE method under the same conditions, and any special processing method does not need to be adopted when the outputs of acoustic field vary severely with the uncertain ocean environmental parameters. The weakness of NPCE method can be overcome by the proposed algorithm, which is that the computational cost increases with the stochastic polynomial expanding to a higher order for enhancing the computational accuracy. The selection of sample point of the proposed algorithm is simpler and easier than that of NPCE method, and the calculation errors can be given directly. Thus, the proposed algorithm is more universal than NPCE method.

SRSM-Based Stochastic Model Updating Method

With the combination of stochastic response surface model (SRSM) and Monte Carlo simulation (MCS)-based inverse error propagation method, a computational efficiency stochastic model updating approach has been proposed. The SRSM of original finite element model is determined using the Hermite polynomial chaos expansion and regression-based efficient collocation method. The efficiency of this method is demonstrated as a large number of computational demanding full model simulations are no longer essential, and instead, the updating of parameter mean values and variances is implemented on explicit SRSM. The effectiveness of this approach is validated through a bolt-jointed double-hat structure numerical example. The proposed method can be applied on stochastic uncertainty quantification of complex engineering structures.

Reliability assessment using stochastic local regression

A primary challenge of stochastic analysis is to discover rigorous ways to estimate the low probability of failure which is critical to reliability constraints. In this paper, a new framework is proposed for the improved estimation of the low failure probability. Combining the significant advantages of the polynomial chaos expansion, Karhunen-Loeve transform and local regression method will result in a new simulation-based modelling technique that enables the accuracy of the structural integrity prediction. The proposed procedure can allow for realistic modelling of sophisticated statistical variations and facilitate in order to achieve improved reliability by eliminating unnecessary conservative approximations. Several specific examples including a three-bar truss and an unmanned undersea vehicle are depicted to illustrate how the method is used to provide a quantitative basis for developing robust designs associated with the low probability of failure.