Response statistics of nonlinear, compliant offshore structures by the path integral solution method

Næss, Arvid; Johnsen, John

doi:10.1016/0266-8920(93)90003-e

Cited by 163 publications

(57 citation statements)

References 23 publications

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“…An alternative computational approach, the path integral solution (PIS) method, has been developed to provide the response pdf for general nonlinear systems at a specific time instant given the pdf of an earlier time instant. Many studies have been focused on the application of step-by-step PIS method numerically [10][11][12] and analytically [13][14][15] reporting its effectiveness on capturing the response statistics. On the other hand, for non-Markovian systems subjected to correlated excitations the joint response-excitation pdf method provides a computational framework for the full statistical solution [16][17][18].…”

Section: Introductionmentioning

confidence: 99%

A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise

Joo

Sapsis

2016

Probabilistic Engineering Mechanics

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We develop a moment-equation-copula-closure method for the inexpensive approximation of the steady state statistical structure of strongly nonlinear systems which are subjected to correlated excitations. Our approach relies on the derivation of moment equations that describe the dynamics governing the two-time statistics. These are combined with a non-Gaussian pdf representation for the joint response-excitation statistics, based on copula functions that has i) single time statistical structure consistent with the analytical solutions of the Fokker-Planck equation, and ii) two-time statistical structure with Gaussian characteristics. Through the adopted pdf representation, we derive a closure scheme which we formulate in terms of a consistency condition involving the second order statistics of the response, the closure constraint. A similar condition, the dynamics constraint, is also derived directly through the moment equations. These two constraints are formulated as a low-dimensional minimization problem with respect to the unknown parameters of the representation, the minimization of which imposes an interplay between the dynamics and the adopted closure. The new method allows for the semi-analytical representation of the two-time, non-Gaussian structure of the solution as well as the joint statistical structure of the response-excitation over different time instants. We demonstrate its effectiveness through the application on bistable nonlinear single-degree-offreedom energy harvesters with mechanical and electromagnetic damping, and we show that the results compare favorably with direct Monte-Carlo simulations.

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Section: Introductionmentioning

confidence: 99%

A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise

Joo

Sapsis

2016

Probabilistic Engineering Mechanics

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“…If S is a nonlinear dynamic system, analytical and numerical solutions for the output distribution are limited to small dimensional systems. The path integral method [19] can be applied to propagate the density of the state vector in time, provided that the state can be expressed as a diffusion process with known functional form. The Fokker-Planck equation can also be used, but numerical solutions are required for most problems and the method is typically limited to state vectors with dimension four or less [24,26].…”

Section: Introductionmentioning

confidence: 99%

Reliability of dynamic systems under limited information

Field

Grigoriu

2009

Probabilistic Engineering Mechanics

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A method is developed for reliability analysis of dynamic systems under limited information. The available information includes one or more samples of the system output; any known information on features of the output can be used if available. The method is based on the theory of non-Gaussian translation processes and is shown to be particularly suitable for problems of practical interest. For illustration, we apply the proposed method to a series of simple example problems and compare with results given by traditional statistical estimators in order to establish the accuracy of the method. It is demonstrated that the method delivers accurate results for the case of linear and nonlinear dynamic systems, and can be applied to analyze experimental data and/or mathematical model outputs. Two complex applications of direct interest to Sandia are also considered. First, we apply the proposed method to assess design reliability of a MEMS inertial switch. Second, we consider re-entry body (RB) component vibration response during normal re-entry, where the objective is to estimate the time-dependent probability of component failure. This last application is directly relevant to re-entry random vibration analysis at Sandia, and may provide insights on test-based and/or model-based qualification of weapon components for random vibration environments.

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“…The execution time for both approaches is recorded seeking to quantify the acceleration of the code's execution. The necessary discretization is performed with a 101×101 mesh that spans along proper ranges estimated by MC simulations and the time step is chosen to be ∆t=0.0035 secs, according to the scheme found in [7]. …”

Section: Linear Systemmentioning

confidence: 99%

“…It can be handled by conventional methods such as cell mapping technique [6] or Path Integration (PI) approach. Despite some advantage of one method over another, finding a joint response PDF of a generally nonlinear multi-degree-of-freedom (MDOF) system is computationally challenging and expensive, therefore mostly low dimensional 1 − 2 DOF systems resulted in 2 − 4D FPK equation (not counting time) have been analyzed, although the amount of real time required was tremendous [7][8][9][10]. The fact that the higher the order of a system the heavier the computational costs generated a term the curse of dimensionality.…”

Section: Introductionmentioning

confidence: 99%

“…The PI methodology in some way helped to explain the controversial and entangled result of the two-slit experiment [15]. Historical development of the PI method, starting from a formula derived by Onsager and Machlup [16] for the Ornstein-Uhlenbeck process to the numerical implementation of the PI method [7,17] can be found in [18]. Despite the lack of rigorous mathematical substantiation, unlike the well defined Wiener path integral [19], the PI method has been widely used in various areas of engineering, physics and finance [20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

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GPU computing for accelerating the numerical Path Integration approach

Alevras

Yurchenko

2016

Computers & Structures

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The paper discusses a novel approach of accelerating the numerical Path Integration method, used for generating a stationary joint response probability density function of a dynamic system subjected to a random excitation, by the GPU computing. The paper proposes the parallelization of nested loops technique and demonstrates the advantages of GPU computing. Two, three and four dimensional in space problems are investigated as a part of the pilot project and the achieved maximum accelerations are reported. Three degree-of-freedom system (6D) is approached by the Path Integration technique for the first time. The application of the proposed GPU methodology for problems of stochastic dynamics and reliability are discussed.

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Response statistics of nonlinear, compliant offshore structures by the path integral solution method

Cited by 163 publications

References 23 publications

A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise

A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise

Reliability of dynamic systems under limited information

GPU computing for accelerating the numerical Path Integration approach

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