Orthogonal Polynomial Expansions for Solving Random Eigenvalue Problems

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.