Machine learning approach to the temperature gradient in the case of discontinuous temperature boundary conditions in a triangular cavity

Abstract

In this study, a prediction for the temperature gradient close to the sharp corners on a right triangle having singularity is proposed utilizing statistical techniques. This singularity is a result of different Dirichlet types of boundary conditions at the corners. In most of the numerical methods, refinement on corners is done, and computational difficulty in gradient calculation is softened. However, the increased refinement results in a blow-up in the temperature gradient close to the problematic corner. A natural convection flow problem is concerned in a right isosceles triangle in two cases of singularity at the left top corner. In Case 1, the left vertical hot wall and the right inclined cold wall, and in Case 2 the left vertical cold wall and the right inclined hot wall are considered. In the present approach, input data is built from the numerical solution of the problem by cutting the region at a chosen cutting point where almost blow-up starts at the temperature gradient along the hot wall. Then, model functions are derived by using machine learning techniques either multivariate adaptive regression splines (Mars) in R project or Trilayer Neural Network (TNN) in Regression Learner Toolbox in Matlab. The obtained model function for the temperature gradient is used to predict the temperature gradient of points above the cutting point. Therefore, the temperature gradient along the hot wall is calculated by combining the temperature gradients of numerical-below and predicted-above. Average Nusselt number as an integral of the temperature gradient along the concerned hot wall is compared with a reference study, and good agreement is observed. That is, modeling of the temperature gradient on singular points is a way of approach for computing accurate results of the temperature gradient.