Neural networks for determining the vector normal to the surface in CFD, LBM and CA applications

Abstract

Purpose The well-known discrete methods of computational fluid dynamics (CFD), lattice Boltzmann method (LBM), cellular automata (CA), volume-of-fluid (VoF) and others rely on several parameters describing the boundary or the surface. Some of them are vector normal to the surface, coordinates of the point on the surface and the curvature. They are necessary for the reconstruction of the real surface (boundary) based on the values of the volume fractions of several cells. However, the simple methods commonly used for calculations of the vector normal to the surface are of unsatisfactory accuracy. In light of this, the purpose of this paper is to demonstrate a more accurate method for determining the vector normal to the surface. Design/methodology/approach Based on the thesis that information about the volume fractions of the 3 × 3 cell block should be enough for normal vector determination, a neural network (NN) was proposed for use in the paper. The normal vector and the volume fractions of the cells themselves can be defined on the basis of such variables as the location of the center and the radius of the circumference. Therefore, the NN is proposed to solve the inverse problem – to determine the normal vector based on known values of volume fractions. Volume fractions are inputs of NNs, while the normal vector is their output. Over a thousand variants of the surface location, orientations of the normal vector and curvatures were prepared for volume fraction calculations; their results were used for training, validating and testing the NNs. Findings The simplest NN with one neuron in the hidden layer shows better results than other commonly used methods, and an NN with four neurons produces results with errors below 1° relative to the orientation of the normal vector; for several cases, it proven to be more accurate by an order of magnitude. Practical implications The method can be used in the CFD, LBM, CA, VoF and other discrete computational methods. The more precise normal vector allows for a more accurate determination of the points on the surface and curvature in further calculations via the surface or interface tracking method. The paper contains the data for the practical application of developed NNs. The method is limited to regular square or cuboid lattices. Originality value The paper presents an original implementation of NNs for normal vector calculation connected with CFD, LBM and other application for fluid flow with free surface or phase transformation.