3rd-order Spectral Representation Method: Part I -- Multi-dimensional random fields with fast Fourier transform implementation

TL;DR

This paper presents a generalized 3rd-order spectral representation method for efficiently simulating multi-dimensional stochastic fields with specified spectral properties, leveraging FFT for scalability.

Contribution

It introduces a novel 3rd-order spectral simulation approach for multi-dimensional fields that is computationally efficient and scalable using FFT, extending classical methods.

Findings

Efficient FFT-based implementation for multi-dimensional fields.

Exponential computational savings with increasing dimensions.

Successful simulation of 2D and 3D random fields with prescribed spectra.

Abstract

This paper introduces a generalised 3rd-order Spectral Representation Method for the simulation of multi-dimensional stochastic fields with asymmetric non-linearities. The simulated random fields satisfy a prescribed Power Spectrum and Bispectrum. The general d-dimensional simulation equations are presented and the method is applied to simulate 2D and 3D random fields. The differences between samples generated by the proposed methodology and the existing classical Spectral Representation Method are analysed. An important feature of this methodology is that the formula can be implemented efficiently with the Fast Fourier Transform, details of which are presented. Computational savings are shown to grow exponentially with dimensionality as a testament of the scalability of the simulation methodology.