Keywords: algebraic grid generation, interpolation function, finite Fourier series, finite difference method, finite element, boundary value problem.

Numerical grid generation is a useful tool for use in the numerical solution of partial differential equations. The procedures for the grid generation are divided into two categories: the numerical solution procedure of partial differential equations and the algebraic procedure by interpolation function [1].

Lagrange polynomials are widely used as interpolation functions in engineering. However, polynomials of high degree exhibit occasionally considerable oscillation, hence calculated values by polynomials are often inaccurate.

On the other hand, interpolation functions derived by calculating the finite Fourier series of relative vectors between two adjacent boundary nodes may vary in higher order including linear variation, and fit the curvilinear boundary shapes and unknown quantities accurately [2]. Furthermore, the finite Fourier series also has features that the shapes of equations are almost the same without relating for the order of differentiation, since the trigonometric functions are used.

This paper shows an algebraic grid generation by interpolation functions using finite Fourier series. The region defined in physical domain is mapped into computational domain of rectangular grid, and partial differential equation in Cartesian coordinates is transformed and calculated numerically on the regularly-interval orthogonal grid in the transformed computational domain.

In order to evaluate the efficiency of the proposed method, finite difference analysis and finite element analysis are performed on the mapped plane. First, as a problem in which exact solution of model with curvilinear boundary shape is known, a torsion problem of a bar with elliptical cross section is analysed by the finite difference method. Several calculations are performed by varying the number of divisions between data points on the boundary. The calculated value of the torsional constant approaches to the exact value, as number of divisions between data points on the boundary increases, and the relative error for the exact value becomes less than 1% when the number of divisions is two. We next consider a transverse vibration of a circular membrane fixed at the edge, where the deflected surface of the membrane is symmetric with respect to the center of the circle [3]. For this problem, two different discretized models are used in the finite difference analysis, and calculations are carried out by changing the number of divisions of one side of the rectangle on the mapped plane. Results calculated are satisfactory for both models, for example, relative errors are less than 1% in the case that the number of divisions of one side of the rectangle on the mapped plane is 16. Grids generated by the proposed method are arranged in regularly and have rectangles of the equal size on the mapped plane. Then, it is possible to calculate without requiring numerical integration and make the re-division of elements only by changing the number of divisions of the boundary. Utilising these features, finite element analysis on the mapped plane is carried out, where each rectangle is used as a isoparametric finite element with 4 nodes. A potential flow problem of incompressible and non-viscous fluid is analysed. Using the proposed interpolation function, the calculated model consists of only ten nodes on the boundary as input data. The finite element analysis is performed with eight divisions between adjacent data nodes on the boundary. The calculated results are compared with the usual finite element analysis with 4-nodes isoparametric elements discretized in the physical plane. In the usual finite element analysis, 425 elements and 384 nodes are used, and numerical integrals for each element are evaluated using the Gaussian quadrature. Calculated results by both method agree well and the maximum value of the difference between results by the usual finite element analysis and results by the proposed method is 0.8%. Input data required are 51,877 bytes for the usual finite element analysis in the physical plane and 198 bytes for the proposed method, respectively.

As a result of numerical calculations, the method is found to be sufficiently accurate and effective for analysing boundary value problems with curvilinear boundary shape.

Interpolation functions used in the present work are ones for the case of general open curved elements [2,3]. According to problems to be analysed, however, it is possible to apply interpolation functions of which first-order derivatives are equal to be zero at the beginning and/or ending points of open curved elements. In such case, it may be able to calculate with less input data.

1

J.F. Thompson et al., "Numerical Grid Generation", North-Holland, 1985.

2

T. Kusama, T. Ohkami and Y. Mitsui, "Application of The Finite Fourier Series to The Boundary Element Method", Computers and Structures, 32(6), 1267-1273, 1989. doi:10.1016/0045-7949(89)90304-0

3

T. Kusama and T. Ohkami, "Improvement of Interpolation Function Using Finite Fourier Series", Proc.JSCE, 446/I-19, 167-175, 1992.(in Japanese)

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