On the Application of BEM to Boundary-fitted Conformal Grid Generation and Associated Singularity Problems
Date Issued
2005
Date
2005
Author(s)
Wang, Jennhann
DOI
en-US
Abstract
In this thesis, the process of a conformal
grid generation by the boundary element method and the complex mapping technique is presented in details.
To elaborate the grid generation method, the associated singularity
problems of mathematical singularity and geometrical singularity, which are related to the BEM and the complex mapping,
are studied comprehensively. It is found that the improvement in eliminating the numerical boundary
layer leads to the ability of present grid generation method producing grids near the boundary in $10^{-9}$ scale
when the boundary element method is applied. The problem
of the grids' overlapping at a sharp corner in a conformal grid system is dealt with a complex mapping
technique, which can manage such a geometrical singularity adequately.
However, there are still some limitations on the application of the complex mapping. In this dissertation, discussions are made to
illustrate the practical way in generating a conformal grid
system that was once thought difficult to construct.
In the end of this thesis, the boundary-fitted curvilinear grid
system is applied to calculate a case of computational
fluid dynamics in connection with storm surge.
In the present study, the fundamental researches on Boundary Element
Method (BEM) are performed at first. Through the studies on the mathematical singularity and the geometrical singularity, the
resolution of the conformal grids near the boundary and the overlapping of grids at a sharp corner can be improved.
In Chapter
ef{chap2}, detailed comparisons of the contour method
and the direct method to evaluate the accuracy of singular
boundary integrals are made. In contrast to conventional numerical
integration methods which suffer from the numerical boundary
layer, the so-called boundary layer can be shown to vanish when
the contour and the direct methods are applied. The singularity
problems, including mathematical singularities and geometrical
singularities, are discussed respectively in Chapter
ef{chap2}
and Chapter
ef{chap3}. Furthermore, in Chapter
ef{chap3}, the
geometrical singularities are overcome by applying a complex
mapping technique. Several benchmarks are proposed to offer
detailed discussions. In Chapter
ef{chap4}, based on the BEM and
the complex mapping method, a powerful numerical grid generation
method is presented. By using the correct evaluation technique in
the integrals of the BEM, the new grid generation method can avoid
the effect of mathematical singularity mentioned in Chapter
ef{chap2}. This enables us to produce conformal grids near the
boundary. Moreover, by using a complex mapping technique in
Chapter
ef{chap3}, the methods serve to avert the overlapping of
the grids at a sharp corner, which occurred in the past studies.
The new grid generation can produce a boundary-fitted orthogonal
grids. In addition, it maintains conformal properties, either
angles or length ratios, so as to map a domain of Laplace equation
to another domain of still the Laplace equation. This makes the
governing equation most concise after the conformal
transformation. Nevertheless, some defects of the grid generation
arising from the mathematical singularity and the geometrical
singularity are found. The grid generation method appears to be
destined to have multi-value and the branch-cut lines' problems,
thus limiting its application in natural geometry.
In this study, angle constraints are presented to serve as a
criterion in checking the overlapping of grids. It helps in the
trial and error operation during the step-by-step complex
mappings. In addition, various numerical examples are employed to
examine the validity of this grid generation system.
Finally, in Chapter
ef{chap5}, further applications of the
orthogonal grid system are made on the storm surge around Taiwan;
the trend of the results is the same as the observed data.
Possible ways for improvement are discussed.
Subjects
物理奇異性
數學奇異性
複變轉換
邊界元素法
邊界符合座標
保角網格
Conformal Grids
Boundary Element Method, Mathematical Singularity
Complex Mapping
Boundary-fitted Coordinate System
Physical Singularity
Type
thesis