Non-Gaussian simulation using Hermite polynomials expansion and maximum entropy principle

Puig, Bénédicte; Akian, Jean‐Luc

doi:10.1016/j.probengmech.2003.09.002

Cited by 33 publications

(13 citation statements)

References 11 publications

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“…The method leads in most cases to good approximations of the target non-Gaussian distribution. Puig et al [149,150] examined the convergence behaviour of PC expansion and proposed an optimization technique for the determination of the underlying Gaussian autocorrelation function. Some limitations of PC approximations have been recently pointed out by Field and Grigoriu [55,83].…”

Section: Methods Based On Polynomial Chaos (Pc) Expansionmentioning

confidence: 99%

The stochastic finite element method: Past, present and future

Stefanou

2009

Computer Methods in Applied Mechanics and Engineering

869

407

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a b s t r a c tA powerful tool in computational stochastic mechanics is the stochastic finite element method (SFEM). SFEM is an extension of the classical deterministic FE approach to the stochastic framework i.e. to the solution of static and dynamic problems with stochastic mechanical, geometric and/or loading properties. The considerable attention that SFEM received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large-scale problems. This article aims at providing a state-of-the-art review of past and recent developments in the SFEM area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future.

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Section: Methods Based On Polynomial Chaos (Pc) Expansionmentioning

confidence: 99%

The stochastic finite element method: Past, present and future

Stefanou

2009

Computer Methods in Applied Mechanics and Engineering

869

407

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“…A group of nonlinear equation sets, shown as the Equation (27), can be formulated by substituting the Equation (26) into the Equations (19) and (22) (27) According to the principle of solving the equation sets, only when the number of unknown variables equals the amount of equations can they be solvable. However, Equation (27) consists of a system of nonlinear equations with the unknown variables sitting in the position of the exponential terms; therefore, it is nearly infeasible to be solved by the regular mathematical methods.…”

Section: B Analysis Of the Complete Odf(c-odf) Via The Maximum Entromentioning

confidence: 99%

“…However, Equation (27) consists of a system of nonlinear equations with the unknown variables sitting in the position of the exponential terms; therefore, it is nearly infeasible to be solved by the regular mathematical methods. Up to now, researchers still make unceasing progress in exploring the possible paths for resolving the problem of MEM [42][43][44].…”

Section: B Analysis Of the Complete Odf(c-odf) Via The Maximum Entromentioning

confidence: 99%

“…It is crucial for such an analytical method to solve the fuctional relevancy among multivariate nonlinear variables by means of the most accessible variables that are linearly independent of each other [22][23][24][25][26][27][28][29]. In order to improve the efficiency for solving this problem, a DE algorithm was introduced for the solution procedure.…”

Section: Introductionmentioning

confidence: 99%

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An Evolutionary Algorithm for the Texture Analysis of Cubic System Materials Derived by the Maximum Entropy Principle

Wang

et al. 2014

Entropy

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Based on the principle of maximum entropy method (MEM) for quantitative texture analysis, the differential evolution (DE) algorithm was effectively introduced. Using a DE-optimized algorithm with a faster but more stable convergence rate of iteration, more reliable complete orientation distribution functions (C-ODF) have been obtained for deep-drawn IF steel sheets and the recrystallized aluminum foils after cold-rolling, which are both designated as the macroscopic cubic-orthogonal symmetry. With special reference to the data processing, no more other assumptions are required for DE-optimized MEM except that the system entropy approaches the maximum.

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“…Since Equations (14)- (19) do not require solving additional matrix equations, the embedded Monte Carlo simulation can be efficiently conducted for any sample size. Furthermore, statistical moments obtained from Equations (14)- (19) can be employed in reconstructing probability density function using the maximum entropy approach [21].…”

Section: Monte Carlo Simulationmentioning

confidence: 99%

Stochastic dynamic systems with complex‐valued eigensolutions

Rahman

2007

Numerical Meth Engineering

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SUMMARYA dimensional decomposition method is presented for calculating the probabilistic characteristics of complex-valued eigenvalues and eigenvectors of linear, stochastic, dynamic systems. The method involves a function decomposition allowing lower-dimensional approximations of eigensolutions, Lagrange interpolation of lower-dimensional component functions, and Monte Carlo simulation. Compared with the commonly used perturbation method, neither the assumption of small input variability nor the calculation of the derivatives of eigensolutions is required by the method developed. Results of numerical examples from linear stochastic dynamics indicate that the decomposition method provides excellent estimates of the moments and/or probability densities of eigenvalues and eigenvectors for various cases including large statistical variations of input.

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Non-Gaussian simulation using Hermite polynomials expansion and maximum entropy principle

Cited by 33 publications

References 11 publications

The stochastic finite element method: Past, present and future

The stochastic finite element method: Past, present and future

An Evolutionary Algorithm for the Texture Analysis of Cubic System Materials Derived by the Maximum Entropy Principle

Stochastic dynamic systems with complex‐valued eigensolutions

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