Application of Modified Finite Point Method to Boundary-fitted Orthogonal Grid Generation
Date Issued
2012
Date
2012
Author(s)
Kuo, Ying-Chang
Abstract
In this thesis, the process of a 2-D orthogonal grid generation by the meshless method called Modified Finite Point Method(MFPM) and complex mapping technique is presented. Applying a complex mapping and orthogonal mapping theorem, a physical domain which is irregular geometrical shape into a hyper-rectangular shape can be transformed, the intermediate transformed domain, which is composed of four right angles and four smooth curved lines, and then MFPM is used to solve the governing equation, Laplace equations, with appropriate boundary conditions to map onto the final transformed rectangular domain. The boundary-fitted coordinates are then mapped backward onto the physical domain.
In previously research, MFPM can efficiently calculate the solutions and the partial derivatives by approximating the exact values at the nodes with polynomial collocation. A complex mapping technique is applied to transform an irregular geometrical shape to a hyper-rectangle. It is shown that orthogonality has increased since the hyper-rectangle mapped to rectangular domain is performed on the basis of conformal mapping theorem.
This paper takes the case of “an area bounded by two triangles” as a testing example and analyses comparisons between grid generations with and without complex transformation. Due to the distance between any two nodes will be stretched or shrunk in exponential ratio after complex mapping transformation, so it has a large influence on collocation. Therefore, different transforming order will result in different mapped shapes. It is shown that there exists some particular rules of transforming order to obtain more accurate approximation by MFPM. Present MFPM provides accurate solutions of scaling factors in coordinate transformations. It is also flexible to adjust nodal distribution and boundary conditions to satisfy computational needs in physical domain.
Subjects
grid generation
complex mapping theorem
conformal mapping theorem
orthogonal
Cauchy-Riemann condition
Laplace equation
meshless
modified finite point method
local polynomial approximation
Type
thesis
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