An APDM-based method for the analysis of systems with uncertainties

Settineri, D.; Falsone, G.

doi:10.1016/j.cma.2014.06.014

Cited by 15 publications

(7 citation statements)

References 28 publications

(56 reference statements)

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“…It is well known that a variety of computer methods has been implemented accounting for uncertainty analysis in computational mechanics of both solids and fluids. There are a number of simulation techniques available , Bayesian approaches, spectral methods , fuzzy sets theories , polynomial chaos, Karhunen–Loeve , and even Taylor expansions known as stochastic perturbation techniques as well as the approximated deformation principal modes worked out recently . A final choice of the solution method can be made considering complexity of the common application with the finite element method (FEM) (or some other discrete numerical apparatus), time, and computer power consumption, expected overall accuracy and, finally, also availability of reliable determination of higher‐order statistics, whose precise determination is still an open problem in computational mechanics.…”

Section: Introductionmentioning

confidence: 99%

On the dual iterative stochastic perturbation-based finite element method in solid mechanics with Gaussian uncertainties

Kamiński

2015

Int. J. Numer. Meth. Engng

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SUMMARYThe main idea is a dual mathematical formulation and computational implementation of the iterative stochastic perturbation-based finite element method for both linear and nonlinear problems in solid mechanics. A general-order Taylor expansion with random coefficients serves here for the iterative determination of the basic probabilistic characteristics, where linearization procedure widely applicable in stochastic perturbation technique is replaced with the iterative one. The expected values and, in turn, the variances are derived first, and then, they are substituted into the equations for higher central probabilistic moments and additional probabilistic characteristics. The additional formulas for up to the fourth-order probabilistic characteristics are derived thanks to the 10th-order Taylor expansion. Computational implementation of this idea in the stochastic finite element method is provided by using the direct differentiation method and, independently, the response function method with polynomial basis. Numerical experiments include the simple tension of the elastic bar, nonlinear elasto-plastic analysis of the aluminum 2D truss, and solution to the homogenization problem of periodic fiber-reinforced composite with random elastic properties. The expected values, coefficients of variation, skewness, and kurtosis of the structural response determined via this iterative scheme are contrasted with these estimated by the Monte Carlo simulation as well as with the results of the semi-analytical probabilistic technique following the response function method itself. Although the entire methodology is illustrated here by using the Gaussian variables where all odd-order terms simply vanish, it can be extended towards non-Gaussian processes as well and completed with all the perturbation orders.

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Section: Introductionmentioning

confidence: 99%

On the dual iterative stochastic perturbation-based finite element method in solid mechanics with Gaussian uncertainties

Kamiński

2015

Int. J. Numer. Meth. Engng

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“…A solution of the structural problem including randomness can be accompanied by a verification of the reliability indices calculated, for example, according to Cornell [3]or using more sophisticated indicators [4][5][6][7][8]-by using a simple limit state function g defined as the difference between structural responses fa(b) and their given thresholds fmax (Figure 1). Then, one may apply polynomial chaos [9], Monte-Carlo simulation [10], approximated principal deformation modes [11] and/or the probability transformation method [12], for instance, to compute the basic probabilistic moments of the limit state functions. Independently of the method choice, the determination of structural responses-with respect to some input parameter subjected to uncertain fluctuations-is required.…”

Section: Introductionmentioning

confidence: 99%

On Selecting Composite Functions Based on Polynomials for Responses Describing Extreme Magnitudes of Structures

Pokusiński

Kamiński

2021

Applied Sciences

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The main aim of this work was to investigate a numerical error in determining limit state functions, which describe the extreme magnitudes of steel structures with respect to random variables. It was assisted here by the global version of the response function method (RFM). Various approximations of trial points generated on the basis of several hundred selected reference composite functions based on polynomials were analyzed. The final goal was to find some criterion—between approximation and input data—for the selection of the response function leading to relative a posteriori errors less than 1%. Unlike the classical problem of curve fitting, the accuracy of the final values of probabilistic moments was verified here as they can be used in further reliability calculations. The use of the criterion and the associated way of selecting the response function was demonstrated on the example of steel diagrid grillages. It resulted in quite high correctness in comparison with extended FEM tests.

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“…Recently the Surface Reference Method (SRM) has been presented [15]; the key step is the projection of every point of the parameter space on a given surface centred on a reference point (where the approximated solution is derived by the APDM) along the direction toward the space origin. The approach provides accurate results even for broad range of variation of the parameters.…”

Section: Introductionmentioning

confidence: 99%

Explicit solution for linear systems with parametric stiffness and application to uncertain structures

Settineri

Impollonia

2015

Meccanica

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The paper proposes a novel approximate explicit solution for the static response of structural systems discretized by finite elements with parametric stiffness. The solution exploits the following properties: (a) if the system comprises one parameter only, i.e. a single finite element has parametric stiffness, then the exact explicit solution can always be obtained; (b) if all element stiffnesses are considered as parameters and the exact solution is available at a given point of the parameter space (i.e. for a given realization of the parameters), then it is also known along the line passing through that point and the origin of the parameter space. The joint use of the two properties allows an accurate explicit relationship for nodal displacements which is suitable for treating different structural problems such as optimization, reanalysis, inverse analysis and reliability. The latter will be investigated in the numerical applications where Monte Carlo samples are produced by the explicit relationship rather than by the re-analysis of the structural problem with an evident reduction of computational burden.

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An APDM-based method for the analysis of systems with uncertainties

Cited by 15 publications

References 28 publications

On the dual iterative stochastic perturbation-based finite element method in solid mechanics with Gaussian uncertainties

On the dual iterative stochastic perturbation-based finite element method in solid mechanics with Gaussian uncertainties

On Selecting Composite Functions Based on Polynomials for Responses Describing Extreme Magnitudes of Structures

Explicit solution for linear systems with parametric stiffness and application to uncertain structures

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