Forward Kinematic Analysis of Non-Deterministic Articulated Multibody Systems With Kinematically Closed-Loops in Polynomial Chaos Expansion Scheme

2015 ◽  
Author(s):  
Sahand Sabet ◽  
Mohammad Poursina
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Kinematic Analysis ◽  
Multibody Systems ◽  
Polynomial Chaos ◽  
Uncertain Parameters ◽  
Chaos Expansion ◽  
Closed Loops ◽  
Base Functions ◽  
Forward Kinematic

This paper presents the method of polynomial chaos expansion (PCE) for the forward kinematic analysis of nondeterministic multibody systems with kinematically closed-loops. The PCE provides an efficient mathematical framework to introduce uncertainty to the system. This is accomplished by compactly projecting each stochastic response output and random input onto the space of appropriate independent orthogonal polynomial base functions. This paper presents the detailed formulation of the kinematics of a constrained multibody system at the position, velocity, and acceleration levels in the PCE scheme. This analysis is performed by projecting the governing kinematic constraint equations of the system onto the space of appropriate polynomial base functions. Furthermore, forward kinematic analysis is conducted at the position, velocity, and acceleration levels for a non-deterministic four-bar mechanism with single and multiple uncertain parameters in the length of linkages of the system. Time efficiency and accuracy of the intrusive PCE approach are compared with the traditionally used Monte Carlo method. The results demonstrate the drastic increase in the computational time of Monte Carlo method when analyzing complex systems with a large number of uncertain parameters while the intrusive PCE provides better accuracy with much less computation complexity.

Uncertainty Analysis of Nondeterministic Multibody Systems

2016 ◽  
Author(s):  
Sahand Sabet ◽  
Mohammad Poursina
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Kinematic Analysis ◽  
Multibody Systems ◽  
Computation Time ◽  
Open Loop ◽  
Mathematical Framework ◽  
Uncertain Parameters ◽  
Response Output ◽  
Forward Kinematic

This paper presents the method of polynomial chaos expansion (PCE) for the forward kinematic analysis of non-deterministic multibody systems. Kinematic analysis of both open-loop and closed-loop systems are presented. The PCE provides an efficient mathematical framework to introduce uncertainty to the system. This is accomplished by compactly projecting each stochastic response output and random input onto the space of appropriate independent orthogonal polynomial basis functions. This paper presents the detailed formulation of the kinematics of constrained multibody systems at the position, velocity, and acceleration levels in the PCE scheme. This analysis is performed by projecting the governing kinematic constraint equations of the system onto the space of appropriate polynomial base functions. Furthermore, forward kinematic analysis is conducted at the position, velocity, and acceleration levels for a non-deterministic four-bar mechanism with single and multiple uncertain parameters and a SCARA robot. Also, the convergence of the PCE and Monte Carlo methods is analyzed in this paper. Time efficiency and accuracy of the intrusive PCE approach are compared with the traditionally used Monte Carlo method. The results demonstrate the drastic increase in the computation time of Monte Carlo method when analyzing complex systems with a large number of uncertain parameters while the intrusive PCE provides better accuracy with much less computational complexity.

On the convergence of the quasi-regression method: polynomial chaos and regularity

2017 ◽  
Vol 54 (2) ◽  
pp. 424-443
Author(s):  
Je Guk Kim
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Rate Of Convergence ◽  
Polynomial Chaos ◽  
Polynomial Chaos Expansion ◽  
Regression Method ◽  
Basis Functions ◽  
Chaos Expansion ◽  
Value Functions ◽  
Weaker Conditions

Abstract We present an analysis of convergence of a quasi-regression Monte Carlo method proposed by Glasserman and Yu (2004). We show that the method surely converges to the true price of an American option even under multiple underlyings via polynomial chaos expansion and weaker conditions than those used in Glasserman and Yu (2004). Further, we show the number of simulation paths grows exponentially in the number of basis functions to obtain convergence in implementing the method. Finally, we propose a rate of convergence considering regularity of value functions.

UNCERTAINTY QUANTIFICATION IN STOCHASTIC SYSTEMS USING POLYNOMIAL CHAOS EXPANSION

2010 ◽  
Vol 02 (02) ◽  
pp. 305-353 ◽  
Author(s):  
K. SEPAHVAND ◽  
S. MARBURG ◽  
H.-J. HARDTKE
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Uncertainty Quantification ◽  
Stochastic Systems ◽  
Polynomial Chaos ◽  
Random Variables ◽  
Polynomial Chaos Expansion ◽  
Uncertain Parameters ◽  
Chaos Expansion ◽  
Generalized Polynomial

In recent years, extensive research has been reported about a method which is called the generalized polynomial chaos expansion. In contrast to the sampling methods, e.g., Monte Carlo simulations, polynomial chaos expansion is a nonsampling method which represents the uncertain quantities as an expansion including the decomposition of deterministic coefficients and random orthogonal bases. The generalized polynomial chaos expansion uses more orthogonal polynomials as the expansion bases in various random spaces which are not necessarily Gaussian. A general review of uncertainty quantification methods, the theory, the construction method, and various convergence criteria of the polynomial chaos expansion are presented. We apply it to identify the uncertain parameters with predefined probability density functions. The new concepts of optimal and nonoptimal expansions are defined and it demonstrated how we can develop these expansions for random variables belonging to the various random spaces. The calculation of the polynomial coefficients for uncertain parameters by using various procedures, e.g., Galerkin projection, collocation method, and moment method is presented. A comprehensive error and accuracy analysis of the polynomial chaos method is discussed for various random variables and random processes and results are compared with the exact solution or/and Monte Carlo simulations. The method is employed for the basic stochastic differential equation and, as practical application, to solve the stochastic modal analysis of the microsensor quartz fork. We emphasize the accuracy in results and time efficiency of this nonsampling procedure for uncertainty quantification of stochastic systems in comparison with sampling techniques, e.g., Monte Carlo simulation.

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

2019 ◽  
Vol 29 ◽  
pp. 01008
Author(s):  
Bartosz Sawicki ◽  
Artur Krupa
Keyword(s):  
Monte Carlo ◽  
Potato Tuber ◽  
Polynomial Chaos ◽  
Ohmic Heating ◽  
Stochastic Approach ◽  
Polynomial Chaos Expansion ◽  
Chaos Expansion ◽  
Computational Power ◽  
Electrical Field ◽  
Stochastic Parameters

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.

Uncertainty Quantification of Hot Gas Ingestion for a Gas Turbine Nozzle Using Polynomial Chaos

2015 ◽  
Author(s):  
Andrea Panizza ◽  
Alessio Bonini ◽  
Luca Innocenti
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Gas Turbine ◽  
Polynomial Chaos ◽  
Thermal Design ◽  
Efficient Estimation ◽  
Critical Parameters ◽  
Accurate Estimation ◽  
Stochastic Expansion ◽  
Hot Gas

One of the most critical parameters in the design process of cooled hot gas components, is the Back Flow Margin (BFM). This dimensionless parameter quantifies the margin to hot gas ingestion through a cooled component wall. A correct evaluation of this parameter is crucial in order to avoid component failure. In presence of combustion chambers that exhibit low pressure losses, BFM becomes one of the most restrictive requirements in the thermal design of cooled components. In this work, a conceptual BFM assessment of the first nozzle of an HP gas turbine is described. The component is subject to the highest thermal load; complex cooling systems are required to ensure an acceptable metal temperature and to match life time requirement. Due to manufacturing tolerances and fluid dynamic uncertainties, hot gas ingestion events are possible also for a nozzle that exhibits BFM higher than zero in nominal conditions, even if with a low probability. Here, the cooling scheme of the nozzle is modeled using an in-house fluid network tool that allows a quick and accurate computation of the equivalent cooling scheme and thus the occurrence of hot gas ingestion, corresponding to a negative flow rate in one of the cooling sub-models. However, as the probability of hot gas ingestion is rather small, an accurate estimation of this event based on the standard Monte Carlo method requires a huge number of runs. A more efficient estimation of this probability can be obtained using stochastic expansion methods, such as the Polynomial Chaos Expansion. Pseudospectral approximations based on either a tensor-product expansion or the Sparse Pseudospectral Approximation Method (SPAM) are used, in order to estimate the probability of hot gas ingestion and the sensitivity to random parameters. The results are compared with those coming from Monte Carlo method, showing the superior accuracy of the stochastic expansion methods.

scholarly journals Multi-Dimensional Harmonic Balance With Uncertainties Applied to Rotor Dynamics

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
J. Didier ◽  
J.-J. Sinou ◽  
B. Faverjon
Keyword(s):  
Monte Carlo ◽  
Dynamic Systems ◽  
Harmonic Balance ◽  
Polynomial Chaos ◽  
Harmonic Balance Method ◽  
Numerical Approach ◽  
Rotor Dynamics ◽  
Geometrical Parameters ◽  
Chaos Expansion ◽  
Response Functions

This paper describes the coupling of a Multi-Dimensional Harmonic Balance Method (MHBM) with a Polynomial Chaos Expansion (PCE) to determine the dynamic response of quasi-periodic dynamic systems subjected to multiple excitations and uncertainties. The proposed method will be applied to a rotor system excited at its support. Uncertainties considered include both material and geometrical parameters as well as excitation sources. To demonstrate the effectiveness and validity of the proposed numerical approach, the results that include mean, variation of the response, envelopes of the Frequency Response Functions and orbits will be systematically compared to a classical Monte Carlo approach.

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