Three-Dimensional Streamline Tracing Method over Tetrahedral Domains

Abstract

Getting a clear understanding of the fluid velocity field in underground porous media is critical to various engineering applications, such as oil/gas reservoirs, CO2 sequestration, groundwater, etc. As an effective visualization tool and efficient transport behaviors solution algorithm, the streamline-based method was improved significantly by numerous studies conducted in the last couple of decades. However, the implementation of streamline simulation is still challenging while working with Finite Element Method (FEM) over 3D tetrahedral domains, where the mass conservation is not guaranteed. Considering the increased computational cost to enforce mass conservation in FEM and additional complexity, a new three-dimensional streamline tracing algorithm is presented that only relies on the velocity vector of a flow field on each vertex of a tetrahedron in a 3D unstructured mesh system. Owning to the shape functions and transformation equations between the master element and actual element, the exit coordinate leaving a tetrahedral element can be determined effectively. As a result, Time of Flight (TOF), the coordinate variable along each streamline, can be calculated accurately and efficiently because that the analytical solution depicting the trajectory in Master Element is deduced. The presented streamline-based method is tested under FEniCS, a programming framework for FEM, which eases the implementation and further development of the presented method.

Predicting the airborne microbial transmission via human breath particles using a gated recurrent units neural network

“…Because we use a tetrahedral mesh, the calculation of trajectory requires tracing a particle through a tetrahedron. Here, it is necessary to calculate the leaving point in the tetrahedron surface for a given entering point alongside the propagation time through the cell (Luo et al , 2020). For this, we use the linear shape function ξ i associated to the nodes of the tetrahedron.…”

“…A trajectory in the linearly varying velocity field can be expressed analytically through the exponential function, but to find the leaving point, we have to solve a numerically algebraic equation (Luo et al , 2020). Instead, we propose here a fast and accurate predictor-corrector type method.…”

“…Despite aerosol transport within indoor environments has been extensively studied in the past decades (Luo et al, 2020), there is a pressing need for the establishment of efficient computational tools for the prediction of transmission and infectivity levels of airborne viruses (such as COVID-19 virus) throughout enclosed spaces. To the authors' knowledge, ML methods and specifically gated recurrent units (GRU) neural network (GRU-NN) (Cho et al, 2014;Chung et al, 2014) have not been used before for this purpose.…”