Gaussian (normal) distribution is a basic continuous probability distribution in statistics, it plays a substantial role in scientific and engineering problems that related to stochastic phenomena. This paper aims to review state of the art of Gaussian random field generation methods, their applications in scientific and engineering issues of interest, and open-source software/packages for Gaussian random field generation. To this end, first we briefly introduce basic mathematical concepts and theories in Gaussian random field, then seven commonly-used Gaussian random field generation methods are systematically presented. The basic idea, mathematical framework of each generation method are introduced in detail and comparisons of these methods are summarized. Then representative applications of Gaussian random field in various areas, especially of engineering interest in recent two decades, are reviewed. For readers' convenience, four representative example codes are provided, and several relevant up-to-date open-source software and packages that freely available from the Internet are introduced.

The double-layer composite electrode has attracted increasing attention in the field of intermediate-temperature solid oxide electrolysis cells (IT-SOEC). To investigate the effects of the cathode diffusion layer (CDL) and cathode functional layer (CFL) structure on performance, a three-dimensional multi-scale IT-SOEC unit model is developed. The model comprehensively considers the detailed mass transfer, electrochemical reaction and heat transfer processes. Meanwhile, percolation theory is adopted to preserve the structural characteristics and material properties of the composite electrode. The mesostructure model and the macroscopic model are coupled in the solution. The effects of the porosity of the CDL, the electrode particle size and the composition of the composite electrode in the CFL on the mass transport process and electrolysis performance of the IT-SOEC unit are analyzed. The results show that the appropriate mass flux and energy consumption in the electrode are obtained with a CDL porosity in the range of 0.3–0.5. The decrease in the electrode particle size is conducive to the improvement of the electrolysis reaction rate. The maximum reaction rate in the CFL increases by 32.64% when the radius of the electrode particle is reduced from 0.5 μm to 0.3 μm. The excellent performance can be obtained when the volume fractions of the electrode phase and electrolyte phase in the CFL tend to be uniform. This study will provide guidance for the performance optimization of IT-SOEC and further promote the development of IT-SOEC hydrogen production technology in engineering applications.

The composite electrode structure plays an important role in the optimization of performance of the intermediate-temperature solid oxide electrolysis cell (IT-SOEC). However, the structural influence of the composite electrode on the performance of IT-SOEC is not clear. In this study, we developed a three-dimensional macroscale model coupled with the mesoscale model based on percolation theory. We describe the electrode structure on a mesoscopic scale, looking at the electrochemical reactions, flow, and mass transport inside an IT-SOEC unit with a composite electrode. The accuracy of this multi-scale model was verified by two groups of experimental data. We investigated the effects of operating pressure, volume fraction of the electrode phase, and particle diameter in the composite electrode on electrolysis reaction rate, overpotential, convection/diffusion flux, and hydrogen mole fraction. The results showed that the variation in the volume fraction of the electrode phase had opposite effects on the electrochemical reaction rate and multi-component diffusion inside the composite electrode. Meanwhile, an optimal range of 0.8–1 for the particle diameter ratio was favorable for hydrogen production. The analysis of IT-SOEC with composite electrodes using this multi-scale model enables the subsequent optimization of cell performance and composite electrode structure.

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