Gaussian Closure Method Research Articles

A novel stochastic approach to investigate the probabilistic characteristics of the ship roll system with sinusoidal restoring force

A novel constrained optimization-oriented exponential–polynomial–closure (CO-EPC) method is developed to investigate the ship roll-motion with non-smooth damping and sinusoidal restoring moments under stochastic wave excitations. The ’non-smooth damping’ and ’sinusoidal restoring moments’ refer to the damping force and restoring force of the system, respectively, which are mathematically expressed using non-smooth and sinusoidal functions. The propagation of uncertainty in the system is governed by the Fokker–Plank–Kolmogorov (FPK) equation, as the presence of stochastic wave excitation induces Markovian behavior in the system responses. The solution of FPK must meet the non-negativity, normalization, and limitation attributes. To fulfill these essential requirements, a novel method has been developed. By deriving the moment functions of the non-smooth terms (for the damping moment part) and the sine terms (for the restoring moment part), the complicated FPK equation is solved. Through two case studies, it is found that the CO-EPC approach demonstrates superior efficiency compared to Monte Carlo simulation and produces more accurate solutions in contrast to the Gaussian closure method. These findings support the effectiveness of CO-EPC in studying the ship roll motion problem under stochastic wave excitations.

Investigation on optimization-oriented EPC method in analyzing the non-linear oscillations under multiple excitations

The optimization-oriented exponential-polynomial-closure (OEPC) method is extended and investigated to analyze stochastic nonlinear oscillators under multiple excitations, with the purpose of obtaining probabilistic solutions for the corresponding system. The presented method extends the original EPC projection procedure for solving the FPK equation by minimizing an objective function, which is defined as the spatial integration of the weighted square residual error. Using the exponential polynomial function, the parameters within it can be located by seeking the minimum of the objective function. In this paper, the novel method has been tested with four examples of strong nonlinearities raised by polynomial nonlinear terms and parametric excitations. The results provide adequate evidence that the OEPC approach delivers notably improved accuracy compared to the Gaussian closure method and demonstrates superior efficiency compared to Monte Carlo simulation. The OEPC method presents a viable alternative for investigating stochastic nonlinear oscillators.

Optimization-Oriented EPC Approach for Analyzing the Stochastic Nonlinear Oscillators with Displacement-Multiplicative and Additive Excitations

This paper presents a new approach named optimization-oriented exponential–polynomial-closure (OEPC) to study the behavior of stochastic nonlinear oscillators under both displacement-multiplicative and additive excitations that are Gaussian white noise. In addition to the original projection exponential–polynomial-closure (PEPC) solution procedure, the OEPC method provides an alternative procedure for solving the Fokker–Planck–Kolmogorov (FPK) equation. The OEPC method is formulated by constructing an objective function with the residue of FPK equation. By minimizing the objective function with a gradient-based method, the parameters in the exponential polynomial can be determined and the approximate PDF solution of the nonlinear random oscillator can be obtained. The solutions obtained through the OEPC approach are highly consistent with the available true solutions in special case or the Monte Carlo simulations (MCS). They are much more accurate than those obtained using the Gaussian closure method when the nonlinearity is strong. In addition, the OEPC method is computationally more efficient than MCS. The difference of the results from the OEPC and PEPC is compared and the advantage of the OEPC over PEPC is also shown when the system nonlinearity is strong and complicated.

Stationary Response of Nonlinear Vibration Energy Harvesters by Path Integration

Abstract A path integration procedure based on Gauss–Legendre integration scheme is developed to analyze probabilistic solution of nonlinear vibration energy harvesters (VEHs) in this paper. First, traditional energy harvesters are briefly introduced, and their nondimensional governing and moment equations are given. These moment equations can be solved through the Runge–Kutta and Gaussian closure method. Then, the path integration method is extended to three-dimensional situation, solving the probability density function (PDF) of VEH. Three illustrative examples are considered to evaluate the effectiveness of this method. The effectiveness of nonlinearity of traditional monostable VEH is studied. The bistable VEH is further studied too. At the same time, equivalent linearization method (EQL) and Monte Carlo simulation (MCS) are employed. The results indicate that three-dimensional path integration method can give satisfactory results for the global PDF, especially when solving bistable VEH problems. The results of this method have better consistency with the simulation results than those of EQL. In addition, different degrees of hardening and softening behaviors of PDFs occur when the magnitude of nonlinearity coefficient increases or the bistable VEH is considered.

Moment dynamics for gene regulation with rational rate laws.

This aim of this paper is mainly to investigate the performance of two typical moment closure schemes in gene regulatory master equations of rational rate laws. When the reaction rate is polynomial, the error bounds between the authentic and approximate moments obtained by schemes of Gaussian moment closure and log-normal moment closure are explicitly given. When the reaction rate is not polynomial, it is shown that the two schemes both behave well in the absence of active-inactive state switch, but in the presence of active-inactive state switch the log-normal closure scheme is far superior to the Gaussian closure scheme in capturing the asymptotic ensemble statistics. Moreover, the accuracy of the log-normal closure method is further confirmed by steady-state analytic results and the conditional Gaussian closure method. It is also disclosed that optimal negative feedback exists in suppressing protein noise in the presence of the on-off switch control.

Probabilistic analysis on parametric random vibration of a marine riser excited by correlated Gaussian white noises

A path integration method is used to discuss the stochastic response of a marine riser excited parametrically and externally by correlated Gaussian white noises. On the basis of the nonlinear vibration equation of a marine riser, the first-order vibration mode is studied and its governing equation is formulated. The response probability density function (PDF) is further developed by a path integration method. During the solution procedure, the Gaussian closure method and the short-time Gaussian approximation method are used to develop the transition probability density function. The results of different time intervals are compared with the results of Monte-Carlo simulation (MCS) to determine the adequate time interval. After that, the path integration method with a Gauss–Legendre scheme is used to obtain the stationary PDF solution of displacement and velocity. The path integration solutions (PIS) are compared with those of the equivalent linearization method (EQL) and MCS. Five demonstrative cases are analyzed to evaluate the effectiveness of the path integration method. The influences of nonlinearity and excitations are considered. It shows that the results of PIS coincide better with the results of MCS than those of EQL. The analysis shows that parametric excitation causes the softening behavior of the PDF distribution compared with those of EQL. The increase of excitation correlation results in the nonsymmetrical PDF distribution of response. The increase of nonlinearity in displacement can lead the PDF of displacement to showing a hardening behavior.

A copula-based Gaussian mixture closure method for stochastic response of nonlinear dynamic systems

Gaussian closure method is commonly used in the analysis of nonlinear stochastic systems. However, Gaussian closure may lead to unacceptable errors when system response is very much different from being Gaussian, and accuracy of the method decreases as the nonlinearity of the system increases. The need for better accuracy in strongly non-linear problems has caused the development of non-Gaussian closure schemes. In this paper, we develop a new copula-based Gaussian mixture closure method for randomly excited nonlinear systems. Our method relies on the assumption of marginal PDF of response in terms of finite Gaussian mixture model, and the derivation of joint PDF with aid of dependence modeling of Gaussian copula. By substituting the non-Gaussian PDF representation into moment equations of nonlinear system, we further develop an optimization-based closure scheme for the solution of the unknown parameters in joint PDF. In this way, PDF and thus, moments of response of highly nonlinear system can be described in a more flexible and robust way. Effectiveness of the new closure method is demonstrated by a nonlinear and a Duffing oscillator that are subjected to Gaussian white noise. The results are compared with the Gaussian closure and exact solution. It has been shown that Gaussian closure is a special case of the new closure method, and accuracy of Gaussian closure is the lower bound of that of the new closure method.

Reliability analysis of nonlinear vibro-impact systems with both randomly fluctuating restoring and damping terms

Abstract In this paper, the reliability analysis of vibro-impact system with both randomly fluctuating restoring and damping terms is studied by a modified path integration (PI) method. Specifically, the Ivanov nonsmooth transformation technique is adopted to transform the vibro-impact system into the system without barrier, then the modified PI application on vibro-impact systems is applied, which is based on the Gaussian closure method and the localized approximation of moment function. According to the first passage theory, the reliability function, the first passage probability density function (PDF) and the mean first passage time are numerically calculated. In the framework of our numerical results, the influences of different random restoring terms, random damping terms and impact conditions on the system’s reliability are discussed. The modified PI results are compared with the Monte Carlo Simulation (MCS) results, which shows that the proposed PI method can not only provide sufficiently accurate results to observe the weak influence of parameters, but also has obvious advantages in computational efficiency.

A quasi-Gaussian approximation method for the Duffing oscillator with colored additive random excitation

Statistical analysis of stochastic dynamical systems is of considerable importance for engineers as well as scientists. Engineering applications require approximate statistical methods with a trade-off between accuracy and simplicity. Most exact and approximate methods available in the literature to study stochastic differential equations (SDEs) are best suited for linear or lightly nonlinear systems. When a system is highly nonlinear, e.g., a system with multiple equilibria, the accuracy of conventional methods degrades. This problem is addressed in this article, and a novel method is introduced for statistical analysis of special types of essentially nonlinear SDEs. In particular, second-order dynamical systems with nonlinear stiffness and additive random excitations are considered. The proposed approximate method can estimate second-order moments of the state vector (namely position and velocity), not only in the case of white noise excitation, but also when the excitation is a correlated noise. To illustrate the efficiency, a second-order dynamical system with bistable Duffing-type nonlinearity is considered as the case study. Results of the proposed method are compared with the Gaussian moment closure approximation for two types of colored noise excitations, one with first-order dynamics and the other with second-order dynamics. In the absence of exact closed-form solutions, Monte Carlo simulations are considered as the reference ideal solution. Results indicate that the proposed method gives proper approximations for the mean square value of position, for which the Gaussian moment closure method cannot provide good estimations. On the other hand, both methods provide acceptable estimations for the mean square value of velocity in terms of accuracy. Such nonlinear SDEs especially arise in energy-harvesting applications, when the ambient vibration can be modeled as a wideband random excitation. In such conditions, linear energy harvesters are no longer optimal designs, but nonlinear broadband harvesting techniques are hoped to show much better performance.

Periodic Response of a Duffing Oscillator Under Combined Harmonic and Random Excitations

This paper presents a solution procedure to investigate the periodic response of a Duffing oscillator under combined harmonic and random excitations. The solution procedure consists of an implicit harmonic balance method and a Gaussian closure method. The implicit harmonic balance method, previously developed for the case of harmonic excitation, is extended to the present case of combined harmonic and random excitations with the help of the Gaussian closure method. The amplitudes of the periodic response and the steady variances can be automatically found by the proposed solution procedure. First, the response process is separated into the mean part and the random process part. Then the Gaussian closure method is adopted to reformulate the original equation into two coupled differential equations. One is a deterministic equation about the mean part and the other is a stochastic equivalent linear equation. In terms of these two coupled equations, the implicit harmonic balance method is used to obtain a set of nonlinear algebraic equations relating to amplitudes, frequency, and variance. The resulting equations are not explicitly determined and they can be implicitly solved by nonlinear equation routines available in most mathematical libraries. Three illustrative examples are further investigated to show the effectiveness of the proposed solution procedure. Furthermore, the proposed solution procedure is particularly convenient for programming and it can be extended to obtain the periodic solutions of the response of multi degrees-of-freedom systems.

The stationary solution of a random dynamical model

This paper studies the stationary probability density function (PDF) solution of a nonlinear business cycle model subjected to random shocks of Gaussian white-noise type. The PDF solution is controlled by a Fokker–Planck–Kolmogorov (FPK) equation, and we use exponential polynomial closure (EPC) method to derive an approximate solution for the FPK equation. Numerical results obtained from EPC method, better than those from Gaussian closure method, show good agreement with the probability distribution obtained with Monte Carlo simulation including the tail regions.

Numerical path integration method based on bubble grids for nonlinear dynamical systems

A new numerical path integration method based on bubble grids for nonlinear dynamical systems is presented in this paper. The ordinary differential equations for the first and second order moments are derived on the basis of the Gaussian closure method. Then the probability density values on the bubble nodes in the computational domain can be calculated via the obtained method. The good performance of the resulting method is finally shown in the numerical examples by using some specific nonlinear dynamical systems: Duffing oscillator subjected to harmonic and stochastic excitations, and Duffing–Rayleigh oscillator subjected to harmonic and stochastic excitations.

Numerical meshfree path integration method for non-linear dynamic systems

We present a new numerical meshfree path integration (MPI) method for non-linear dynamic systems. The obtained MPI method can be performed in the irregular computational domain and the probability density values of the random nodes in the domain can be calculated via the MPI method and the ordinary differential equations for the first and second-order moments on the basis of Gaussian closure method. The piecewise linear interpolation based on adaptive least squares is utilized as a post processor to approximate the probability density values on arbitrary positions. The good performance of the resulting method is finally shown in the numerical examples by using three specific non-linear dynamic systems: Duffing oscillator subjected to both harmonic and stochastic excitations, Duffing–Rayleigh oscillator subjected to both harmonic and stochastic excitations, and CHEN system driven by three different Gaussian white noises.

Exponential closure method for some randomly excited non-linear systems

The probability density function (PDF) of the responses of non-linear stochastic system excited by white noise is approximated with the exponential function of polynomial in state variables. Special measure is taken to satisfy FPK equation in the weak sense of integration with the assumed PDF. Gaussian closure method is a special case of the proposed method. Examples are given to show the application of the method to the systems with additive random excitations and those with both additive and multiplicative random excitations. The PDFs obtained with the proposed method and conventional Gaussian closure method are compared with obtainable exact ones. Numerical results showed that the PDFs obtained with the proposed method can be very close to the exact ones regardless of the degree of system non-linearity. In some cases, even exact solution can be obtained with the proposed method.

A consistent method for the solution to reduced FPK equation in statistical mechanics

The solution of reduced Fokker–Planck–Kolmogorov (FPK) equation resulted from the problems in statistical mechanics is formulated as an exponential function of polynomials in state variables. Special measure is taken to satisfy the FPK equation in the weak sense of integration with the assumed function. Examples are given to show the application of the method to the non-linear systems with additive random excitations and those with both additive and multiplicative random excitations. The PDF solutions obtained with the proposed method and conventional Gaussian closure method are compared with the exact solutions in special cases. Numerical results showed that the solutions obtained with the proposed method are much close to the exact solutions even for highly non-linear system and the system with both additive and multiplicative excitations. The convergence of the approximate PDF solutions obtained with the method is also shown with numerical results.

Multi-Gaussian closure method for randomly excited non-linear systems

The probability density function (PDF) of the response of a non-linear stochastic system excited by white noise is assumed to be a linear superposition of basic functions. The Gaussian PDFs are used as the basic functions which coefficients are the reciprocal of the number of the basic functions. Gaussian closure method is a special case of the proposed method. Examples are given to show the application of the method to the systems with additive random excitations and those with both additive and multiplicative random excitations. The PDFs and moments obtained with the proposed method and conventional Gaussian closure method are compared with the exact ones. Numerical results showed some advantages of the proposed method over Gaussian closure method.

Crossing Rate Analysis with a Non-Gaussian Closure Method for Nonlinear Stochastic Systems

The mean upcrossing rate of the stationary responses of nonlinear stochastic system excited by white noise is analyzed based on the assumption that the probability density function (PDF) of the responses is a linear superposition of basic functions. The Gaussian PDFs are used as the basic functions of which the coefficients are the reciprocal of the number of the basic functions. The Gaussian closure method is a special case of the proposed method. Based upon the approximate PDF, the explicit expression for the mean upcrossing rate (MCR) is given. Numerical results show that the approximate MCRs approach the exact ones in the tails of the MCR curves as the number of the basic functions increases.

A finite-element method for analysis of a non-linear system under stochastic parametric and external excitation

A finite-element method approach is developed to solve the Fokker-Planck-Kol-mogorov equation for the probability density function of stationary response of a general non-linear system subject to both parametric and external Gaussian white noise excitation. N-dimensional shape functions are expanded by a one-dimensional shape function in global coordinate. As the domain of the density function is usually infinite, adaptive grid generation is adopted for the estimation of a finite range of the finite-element mesh. Several examples with existing exact solutions are used to illustrate the validity of the method. Results are also compared with those obtained by the Gaussian closure method, the cumulant-neglect method, equivalent external excitation and Monte Carlo simulation.

Internal viscosity in polymer kinetic theory: shear flows

The Gaussian closure method and Brownian dynamics simulations have been used to calculate the shear material properties of a dilute solution of Hookean dumbbells with internal viscosity. Results for the zero-shear-rate material properties and small amplitude oscillatory shear material properties have been found analytically, and numerical results for the steady state shear material properties are also presented. Two interpretations of the stress tensor are investigated and results are compared. Brownian dynamics simulations are used to obtain material properties of the Hookean dumbbell with internal viscosity without approximations. These simulation results are compared with the perturbation solution of Booij and van Wiechen as well as with a new Gaussian closure solution. Also presented are the contracted distribution functions as derived from the Gaussian closure method and from Brownian dynamics simulations.

Non-Gaussian stochastic response of nonlinear compliant platforms

A moment function method is presented to estimate the stochastic response of compliant offshore platforms with nonlinearity in stiffness based on non-Gaussian closure. For guyed towers with clump weight, the nonlinearity in stiffness is of the softening type. The random wave loading is expressed in terms of a rational spectrum, making the system Markovian. Using Ito's rule for stochastic differentiation, differential equations for moments up to the fourth order are developed. The system of equations is closed by considering the fifth and sixth cumulants to be zero. For stationary response, differential equations become algebraic equations. The moments are obtained by solving the system of nonlinear algebraic equations. It is observed that the Gaussian closure method is inadequate for defining the complete probabilistic characteristics of the response.