Stochastic Harmonic Function Representation of Random Fields for Material Properties of Structures

Taking account of the spatial variation of material properties is of paramount importance to the safety and reliability evaluation of engineering structures. This necessitates the highly efficient and accurate representation of random fields of material properties. Although theoretically it is only an extension of stochastic process representation, in practice this is nontrivial because the dimension of space increases. To reduce the number of basic random variables and improve the accuracy, a stochastic harmonic function (SHF) representation method for homogenous random fields is proposed. Compared to the classical spectral representation method, besides the phase angles, the wave numbers of each harmonic component are also random variables, of which the supports could be specified by the Voronoi partition of the bounded wave-number domain. It is rigorously proved that the proposed SHF representation could reproduce the target wave-number spectral density exactly rather than approximately with a finite number of random variables. In practice, for the same number of random variables and samples, it is numerically proved that the SHF is accurate to the target wave-number spectral density, while spectral representation is only accurate on specific points. Further, it is demonstrated that the SHF fields are homogeneous and asymptotically Gaussian, with the convergence rate being higher than the spectral representation method. In contrast to the Karhunen-Loève expansion, on the other hand, the solution of the integral equation is avoided. The response analysis of a shear wall with random-field material parameters is taken as an example to illustrate the application of the proposed method. It is shown that the scale of fluctuation will affect the variation of dissipated energy greatly, and thereby will affect the reliability. Problems to be further studied are also discussed.