A nodal integral method for quadrilateral elements

Abstract

SUMMARYNodal integral methods (NIMs) have been developed and successfully used to numerically solve several problems in science and engineering. The fact that accurate solutions can be obtained on relatively coarse mesh sizes, makes NIMs a powerful numerical scheme to solve partial differential equations. However, transverse integration procedure, a step required in the NIMs, limits its applications to brick-like cells, and thus hinders its application to complex geometries. To fully exploit the potential of this powerful approach, abovementioned limitation is relaxed in this work by first using algebraic transformation to map the arbitrarily shaped quadrilaterals, used to mesh the arbitrarily shaped domain, into rectangles. The governing equations are also transformed. The transformed equations are then solved using the standard NIM. The scheme is developed for the Poisson equation as well as for the time-dependent convectiondiffusion equation. The approach developed here is validated by solving several benchmark problems. Results show that the NIM coupled with an algebraic transformation retains the coarse mesh properties of the original NIM.

Isoparametric Element Analysis of Two‐Dimensional Unsaturated Transient Seepage

“…For any element in the capacitance matrix, after the expression of formula (12) and the determinant of formula (21) of Jacobi matrix are introduced, the integrand is written as…”

“…Under the condition of shape functions sharing the same power, approximate functions constructed by values on the nodes and shape functions are defective, while the quadrilateral element has no similar problem. In this regard, some scholars begin to use the rectangular element to calculate some complex engineering problems [11,12], which demonstrates an excellent trend; however, the commercial software constituted of opaque principles is used for calculation, no determination method of stiffness and capacitance matrices is provided, and literature studies related to stiffness and capacitance matrices of the rectangular element are still scarce.…”

Pressure Correction–Based Iterative Scheme for Navier-Stokes Equations using Nodal Integral Method

“…Another approach to overcome this difficulty uses algebraic transformations to map arbitrarily shaped quadrilateral elements to rectangular elements. The convectiondiffusion equation is solved using quadrilateral elements for irregularly shaped geometry by Nezami et al [22].…”

Parallel Modified Nodal Integral Method for three-dimensional incompressible Navier-Stokes and energy equations

“…Recently, methods have been developed for fluid flow Rizwan-uddin, 2003a, 2003b) as well as for neutron diffusion equations (Gu and Rizwan-uddin, 2005), combining NIM with other numerical methods (for example finite element method and finite analytic method) to overcome this restriction. Another approach is used to transform quadrilateral cells to square cells (Nezami et al, 2009). Owing to several advantages discussed earlier, it is natural to use MNIM to simulate turbulent flows.…”