ORTHOGONAL POLYNOMIAL EXPANSIONS FOR SOLVING RANDOM EIGENVALUE PROBLEMS

Sharif Rahman · Vaibhav Yadav

46 citations0/0 files storedInternational Journal for Uncertainty Quantification

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Probabilistic and Robust Engineering Design Mathematical functions and polynomials Model Reduction and Neural Networks

Abstract

This paper examines two stochastic methods stemming from polynomial dimensional decomposition (PDD) and polynomial chaos expansion (PCE) for solving random eigenvalue problems commonly encountered in dynamics of mechanical systems.Although the infinite series from PCE and PDD are equivalent, their truncations endow contrasting dimensional structures, creating significant differences between the resulting PDD and PCE approximations in terms of accuracy, efficiency, and convergence properties.When the cooperative effects of input variables on an eigenvalue attenuate rapidly or vanish altogether, the PDD approximation commits a smaller error than does the PCE approximation for identical expansion orders.Numerical analyses of mathematical functions or simple dynamic systems reveal markedly higher convergence rates of the PDD approximation than the PCE approximation.From the comparison of computational efforts, required to estimate with the same precision the frequency distributions of dynamic systems, including a piezoelectric transducer, the PDD approximation is significantly more efficient than the PCE approximation.

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References

1991Stochastic Finite Elements: A Spectral Approach 2002The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations 2007Global sensitivity analysis using polynomial chaos expansions 2004A generalized dimension‐reduction method for multidimensional integration in stochastic mechanics 2002An importance quantification technique in uncertainty analysis for computer models 2008A polynomial dimensional decomposition for stochastic computing 2006Random matrix eigenvalue problems in structural dynamics 2005Analysis of Eigenvalues and Modal Interaction of Stochastic Systems

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