A COMPARISON OF TWO APPROACHES TO EXTEND NODAL INTEGRAL METHODS TO ARBITRARY GEOMETRIES—APPLIED TO THE CONVECTION–DIFFUSION EQUATION

Nodal Integral Methods (NIM) are a class of coarse-mesh numerical methods developed to solve partial differential equations (PDEs). They are more accurate and efficient than the conventional finite-difference, finite-volume, and finite-element methods because of the use of approximate analytical solutions to the governing differential equations in the scheme’s development. The transverse integration process, which reduces the PDE into a set of ODEs, restricts the application of NIM to domains discretized by rectangular elements in 2D and cuboid elements in 3D. Two approaches presented in this paper relax this restriction and extend NIM’s efficiency to arbitrary geometries in 3D. In the first approach, NIM is derived in general 3D curvilinear coordinates and applied to solve problems in domains discretized by hexahedral elements. The hexahedral elements in the Cartesian system are transformed into cubes in curvilinear coordinates, where the transverse integration procedure can be applied. The second approach is a hybrid nodal-integral/finite-element approach. In this approach, the bulk of the domain is discretized into regular cuboid elements, and the regions adjacent to curve boundaries are discretized by tetrahedral elements. The standard NIM is applied to the cuboid elements, while the finite–element is used for the tetrahedral elements. The two approaches are used to numerically solve the convection–diffusion equation (CDE) in four different computational domains. The method of manufactured solution (MMS) is used to construct exact temperature profiles for all test cases. The computational domains of the first two test cases are cylindrical annulus and solid cylinder. The first two test cases are solved using both approaches, where the accuracy and efficiency of both methods are compared. The third test case is solved in a spherical domain using approach I only, while the fourth is solved in a cuboid domain with a hemispherical cavity using approach II only.