Transient probabilistic systems

Liu, Wing Kam; Besterfield, Glen; Belytschko, Ted

doi:10.1016/0045-7825(88)90067-9

Cited by 85 publications

(19 citation statements)

References 18 publications

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“…The alternative methods to reduce the computational efforts are the non-statistical approaches, such as the random perturbation method [5] and the orthogonal polynomial expansion method [6]. The accuracy of the random perturbation method is greatly limited by the degree of variance of the source random variables and the secular term problem [7]. On the other hand, there are no such limitations in the orthogonal polynomial expansion method [8].…”

Section: Introductionmentioning

confidence: 99%

Dynamic response and reliability analysis of structures with uncertain parameters

Chen

2004

Numerical Meth Engineering

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SUMMARYAn original approach for dynamic response and reliability analysis of stochastic structures is proposed. The probability density evolution equation is established which implies that incremental rate of the probability density function is related to the structural response velocity. Therefore, the response analysis of stochastic structures becomes an initial-value partial differential equation problem. For the dynamic reliability problem, the solution can be derived through solving the probability density evolution equation with an initial value condition and an absorbing boundary condition corresponding to specified failure criterion. The numerical algorithm for the proposed method is suggested by combining the precise time integration method and the finite difference method with TVD scheme. To verify and validate the proposed method, a SDOF system and an 8-storey frame with random parameters are investigated in detail. In the SDOF system, the response obtained by the proposed method is compared with the counterparts by the exact solution. The responses and the reliabilities of a frame with random stiffness, subject to deterministic excitation or random excitation, are evaluated by the proposed method as well. The mean, the standard deviation and the reliabilities are compared, respectively, with the Monte Carlo simulation. The numerical examples verify that the proposed method is of high accuracy and efficiency. Moreover, it is found that the probability transition of structural responses is like water flowing in a river with many whirlpools, showing complexity of probability transition process of the stochastic dynamic responses.

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Section: Introductionmentioning

confidence: 99%

Dynamic response and reliability analysis of structures with uncertain parameters

Chen

2004

Numerical Meth Engineering

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“…Equation (77) can be computed in a coarse way 4 for s = 4. However, a refined computational scheme can be carried out through…”

Section: The Strategy For Selecting Representative Points Via Tangentmentioning

confidence: 99%

Strategy for selecting representative points via tangent spheres in the probability density evolution method

Chen¹,

Li²

2007

Numerical Meth Engineering

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SUMMARYA strategy of selecting efficient integration points via tangent spheres in the probability density evolution method (PDEM) for response analysis of non-linear stochastic structures is studied. The PDEM is capable of capturing instantaneous probability density function of the stochastic dynamic responses. The strategy of selecting representative points is of importance to the accuracy and efficiency of the PDEM. In the present paper, the centers of equivalent non-overlapping tangent spheres are used as the basis to construct a representative point set. An affine transformation is then conducted and a hypersphere sieving is imposed for spherically symmetric distributions. Construction procedures of centers of the tangent spheres are elaborated. The features of the point sets via tangent spheres, including the discrepancy and projection ratio, are observed and compared with some other typical point sets. The investigations show that the discrepancies of the point sets via tangent spheres are in the same order of magnitude as the point sets by the number theoretical method. In addition, it is observed that rotation transformation could greatly improve the projection ratios. Numerical examples show that the proposed method is accurate and efficient for situations involving up to four random variables.

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“…However, the Taylor expansion based SFEM and the perturbation-based SFEM are only suitable for the problems with small parametric variation. Moreover, the secular terms issue will result in large error when the perturbation-based SFEM is employed in dynamic problems [14]. In contrast, the alternative approach based on orthogonal polynomials expansion can avoid such limitations [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Probability density evolution method for dynamic response analysis of structures with uncertain parameters

Chen

2004

Computational Mechanics

267

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Probability density evolution method is proposed for dynamic response analysis of structures with random parameters. In the present paper, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability. With the state equation expression, the PDEE is further reduced to a onedimensional partial differential equation. The numerical algorithm is studied through combining the precise time integration method and the finite difference method with TVD schemes. The proposed method can provide the probability density function (PDF) and its evolution, rather than the second-order statistical quantities, of the stochastic responses. Numerical examples, including a SDOF system and an 8-story frame, are investigated. The results demonstrate that the proposed method is of high accuracy and efficiency. Some characteristics of the PDF and its evolution of the stochastic responses are observed. The PDFs evidence heavy variance against time. Usually, they are much irregular and far from well-known regular distribution types. Additionally, the coefficients of variation of the random parameters have significant influence on PDF and second-order statistical quantities of responses of the stochastic structure.

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Transient probabilistic systems

Cited by 85 publications

References 18 publications

Dynamic response and reliability analysis of structures with uncertain parameters

Dynamic response and reliability analysis of structures with uncertain parameters

Strategy for selecting representative points via tangent spheres in the probability density evolution method

Probability density evolution method for dynamic response analysis of structures with uncertain parameters

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