Efficient symmetric boundary condition for Gralerkin finite volume solution of 3D temperature field on tetrahedral meshes

In some of the engineering problems, it is necessary to analyze the three-dimensional temperature profiles. In order to solve a typical problem numerically, the three-dimensional temperature diffusion equation is chosen as the mathematical model. The finite volume formulation is derived using Galerkin approach for the mesh of tetrahedral elements, which facilitates solving temperature problems with complicated geometries. In this approach, the Poisson equation is multiplied by the piece wise linear shape function of tetrahedral element and integrated over the control volumes which are formed by gathering all the elements meeting every computational node. The linear shape functions of the elements vanish by some mathematical manipulations and the resulted formulation can be solved explicitly for each computational node. The algorithm not only is able to handle the essential boundary conditions but also the natural boundary conditions using a novel technique. Accuracy and efficiency of the algorithm is assessed by comparison of the numerical results for a bench mark problem of heat generation and transfer in a block with its analytical solution. Then, introduced technique for imposing natural boundary conditions on unstructured tetrahedral mesh is examined for cases with inclined symmetric boundaries.