A nodal approach to arbitrary geometries, and adaptive mesh refinement for the nodal method

The nodal integral method (NIM) has an advantage over many conventional numerical methods in that the nodal integral method requires less CPU time to achieve a given accuracy. However, there are three limitations of the NIM that either hinder its application to a wider class of problems or restrict full exploitation of the efficiency associated with coarse meshes. First, the transverse integration step limits the method to problems with domains of regular geometries—those formed by the union of non-overlapping rectangles. Second, the coarse mesh efficiency is not fully realized in problems that are characterized by a wide range of spatial scales. Third, the coarse mesh efficiency can be hampered by a poor choice of the solver—the routine that solves the resulting set of algebraic equations obtained in the NIM.; To remove the first limitation, two hybrid numerical methods based on the NIM are developed. In these hybrid methods, the computational domain is divided into rectangular and triangular nodes (cells). The conventional NIM is applied to the interfaces between rectangular nodes (cells) while a finite element approach (for the hybrid nodal-integral/finite-element method) or a finite analytic approach (for the hybrid nodal-integral/finite-analytic method) is applied to the triangular-rectangular node (cell) interfaces. The application of these hybrid methods to several convection-diffusion and fluid flow problems clearly shows that these hybrid methods relax the restriction on the geometric shape of the computational domain without sacrificing the nodal efficiency.; To eliminate the second limitation, adaptive mesh refinement (AMR) capability is developed for the NIM. The results of several convection-diffusion and fluid flow problems demonstrate that the combination of the AMR with the NIM can recover the coarse mesh efficiency by allowing high degrees of resolution in specific localized areas where it is needed and using a lower resolution everywhere else. Moreover, the NIM has several inherent properties that can be exploited by the components of the AMR process.; The third limitation associated with the nodal method is removed by modifying an existing implementation of the NIM to incorporate fast and accurate preconditioner and solver routines. Application of this modified implementation to convection-diffusion problems proves that the coarse mesh efficiency of the nodal method can be maintained by taking advantage of “off the shelf” advanced solver technology.