USE OF WEB-SPLINES OF ARBITRARY DOMAIN FOR WAVEGUIDES G. Apaydin, N. Ari University of Technology Zürich, Applied Research and Development, Technoparkstrasse 1, 8005 Zürich, Switzerland, email:
[email protected],
[email protected] Abstract – This paper illustrates the weighted extended b-splines as basis functions of finite element method of arbitrary domain for waveguides. This method does not need mesh generation for applications. The results are compared with the finite element method with triangulation method using Lagrange basis functions. This method can be used in more electromagnetic applications for future studies. I. INTRODUCTION The advances of mathematics have supported the development of electromagnetic theory. This study uses web-spline method of numerical solution for solving waveguides of arbitrary domain which the analytical solutions do not exist for the first time. The finite element method (FEM) is one of the numerical solution methods for partial differential equations. Some engineering applications use partial differential equations which are difficult to solve. This method is powerful to find solutions for irregular domains or inhomogeneous media. The wave equations can be solved by using FEM. The idea is to use piecewise continuous functions and meshes for approximation. Mesh generation causes time consumption [1-2]. The basis splines (b-splines), which are used in data fitting, numerical approximation, or computer aided design etc., were developed by Carl de Boor, de Casteljau, Bezier [3-4]. The b-splines should be considered as basis functions for FEM instead of Lagrange functions. Hollig developed extended b-splines and weighted extended b-splines (web-splines) as basis functions for different boundary conditions [5-6]. Therefore, accurate results have been obtained by using fewer nodes. The finite element and finite difference methods have been applied successfully to electromagnetic field calculations in two dimensions for many years. However, the applications in three dimensions have been faced with severe practical difficulties. The method of this paper uses FEM with web-splines, and thus overcomes the restrictions of meshes. The applications of one dimensional electromagnetic problems using FEM with b-splines are given in [7]. According to [8-9], the cutoff wave numbers for the coaxial waveguides (regular domain) were studied and the numerical results were compared with the analytical results given in [10]. Now, this paper presents web-splines and their implementations for waveguides of arbitrary domain. The cutoff wave number analysis for arbitrary domain has been studied and compared with the standard FEM method. II. WEB-SPLINES The b-splines can be used as basis functions for their flexibility and continuity between points. However, boundary conditions can cause problems considering stability. Homogeneous essential boundary conditions should be modeled by using weight functions. Well conditioned basis functions can be obtained by using extended b-splines. Combining these two ideas gives rise to the definition of web-splines, which is a new type of meshless finite elements. The application of two dimensional electromagnetic problems using FEM with b-splines is given in [8-9]. The tensor product b-splines help to construct b-splines in two dimensions. It is defined by multiplying b-splines of each direction [5]. The tensor product b-splines bkn (x) for x ∈ ℜ 2 is bkn ( x) = h −1b n ( x / h − k ) ,
(1)
where h is the grid width, n is the degree, k = [k1 , k 2 ]∈ Z 2 is the grid index. The support of each b-spline consists of (n + 1) 2 grid cells ( kh + [0, h]2 ). The b-splines are positive on their supports, and n − 1 times continuously differentiable.
The relevant b-splines ( bk , k ∈ K ) supporting in the domain are classified as inner and outer b-splines. The inner b-splines ( bi , i ∈ I ) have at least one complete grid cell of their support in the domain. The others are outer b-splines ( b j , j ∈ J ). For the outer b-splines, the grid cells of their supports are not entirely contained in the domain. Although the outer b-splines have small effect, they must be taken into consideration for stability. Therefore, they are adjoined to the closest inner b-splines in order to form the extended b-splines as
Bi = bi +
∑e
i , k bk
for i ∈ I , k ∈ J (i ) ,
(2)
k
where ei , k are the extension coefficients using Lagrange polynomials [9]. Equation (2) can be used to solve boundary value problem with the Neumann boundary conditions. If the Dirichlet boundary conditions are taken into consideration, the extended b-splines are multiplied by the weight function w(x ) , which is a continuous positive function in the domain, and zero on the boundary. It can be constructed by using distance function which is shown in Fig. 1a [5]. As a result, the web-splines are obtained as
Bi =
w( x) bi + w( xi )
∑e
i , k bk
k
,
(3)
where xi is in the center of a grid cell which intersects the support of b-spline and the domain completely for normalization, w(x) is the weight function for x ∈ ℜ 2 , and w( xi ) is the value of weight function at the center of grid cell. According to the center of their supports, Fig. 1b shows the standard (●), extended inner (▲) and outer (○) quadratic b-splines respectively.
(a)
(b)
(c)
Fig. 1. a) The weight function, b) The standard (●), extended inner (▲) and outer (○) b-splines, c) The triangulation for arbitrary domain.
The cutoff wave number analyses of waveguides are determined by solving the Helmholtz equation for the function ψ with wave number β ∇ 2ψ + β 2ψ = 0 .
(4)
The solution is approximated as
ψ ap =
∑c B j
j
(5)
j
of basis functions B j supporting the domain for the coefficients c j . Replacing ψ in (4) by the approximated solution, and multiplying the equation by a smooth test function Bi
∑ c ∫ j
j
D
∇B j ⋅ (∇Bi )t − β 2 B j Bi = 0 ,
(6)
the finite element matrices are obtained and then the eigenvalues for different types of waveguides are calculated. The significance of web-splines is that the contribution of basis functions, which are near the boundary, is added to the inner basis functions, so the number of nodes and computing time is reduced. Secondly, instability problem is solved by using web-splines. III. SIMULATION RESULTS The web-spline method can be used to obtain the numerical solution of cutoff wave numbers for various Transverse Magnetic (TM) and Transverse Electric (TE) modes and compared with the analytical results for regular domain. According to the previous study, the web-spline method has been implemented easily and provides accurate results [8-9]. Compared to the weighted splines, web-splines use 20-25% less nodes and they are more stable. The analysis has been studied to find the cutoff wave numbers of waveguides for an arbitrary domain. The eigenvalue analysis is used to compare web-spline method with the standard FEM. Figure 1c shows triangulation for an arbitrary domain using 8863 nodes and 15928 triangles. For web-spline FEM analysis, Fig. 1b shows the relevant cubic b-splines of the same domain using 62 outer, 172 extended inner, and 26 standard inner web-splines. The distance weight function is used for TM mode analysis as shown in Fig. 1a. The first three cutoff wave numbers of TM and TE modes are obtained as {1.41, 1.42, 1.61} and {0.40, 0.47, 0.86} respectively. Table 1 shows the number of nodes used for triangulation and web-splines in order to get same results. The web-spline method for an arbitrary domain provides accurate results with less computation time. Table 1. The results of FEM analysis for arbitrary domain Number of nodes
FEM method
Mode
First three cutoff wave numbers
Time (sec)
8663
standard
TE
0.40
0.47
0.86
2.9
125
linear web-spline
TE
0.41
0.47
0.87
0.1
163
quadratic web-spline
TE
0.40
0.47
0.86
0.2
198
cubic web-spline
TE
0.40
0.47
0.86
0.3
10061
standard
TM
1.41
1.42
1.62
4.9
125
linear web-spline
TM
1.43
1.45
1.66
0.1
163
quadratic web-spline
TM
1.41
1.42
1.62
0.2
198
cubic web-spline
TM
1.41
1.42
1.61
0.3
IV. CONCLUSIONS The analysis presented in this study shows the suitability of the proposed method to complex electromagnetic problems. The finite element method which uses web-splines is applied to waveguides of arbitrary domain. According to [8-9], the numerical approximations were compared with the analytical results. With different ratio of radius of coaxial waveguides, accurate results are obtained by using web-splines. Web-splines and extended b-splines are used to find the wave number of TM and TE modes respectively. According to the error analysis, it
was found that the relative error for the TM mode is more accurate than the relative error for TE mode. FEM with web-spline is suitable to obtain electromagnetic solution for different frequencies. The theoretical and simulation results have been compared with the literature and the model agrees very well with published results. These comparisons show that the method is valid when kh
