Analysis of Quadruple Corner-Cut Ridged Elliptical Waveguide by NURBS Enhanced Scaled Boundary Finite Element Method

The scaled boundary finite element method(SBFEM) is a semi-analysis method, combing the advantages of boundary element method and finite element method. However, in solving the quadruple corner-cut ridged elliptical(QCRE) waveguide, the traditional SBFEM employ the Lagrange polynomials as the basis functions which leads to the curved boundaries cannot be exactly represented and the continuity order across element is low. In this paper, a non-uniform rational B-spline (NURBS) enhanced SBFEM is firstly extended to solve the QCRE waveguide, which can exactly describe the curved boundaries, reduce the spatial dimensions by one and obtain analytically results in the radial direction. According to its symmetry, only a quarter of the ridged elliptical waveguide needs to be simulated and subdivided into several subdomains. The curved boundaries and straight boundaries of the subdomains are described and discretized by the NURBS and Lagrange basis functions, respectively. The side-face boundaries do not need to be discretized. Then, the NURBS enhanced SBFEM governing equation of the waveguide eigenvalue problem is derived based on the vibrational principle and scaled boundary coordinate transforming. Finally, a generalized eigenvalue equation respecting to the cut-off wave numbers is established by introducing the boundary dynamic stiffness and employing a continued fraction solution. Numerical results verify the high computational efficiency and accuracy of the NURBS enhanced SBFEM with exactly describing the curved boundaries. The influence of the corner-cut on the cut-off wave numbers of several modes and single-mode bandwidth are investigated in details.


I. INTRODUCTION
In order to meet the performance requirements of micro-wave and millimeter-wave communication systems, it is necessary to continuously explore and research new waveguides with special cross-section shapes. Due to the characteristics of low cut-off frequency, low impedance and extended bandwidth, various kinds of ridged waveguide have been applied in millimeter-wave and micro-wave communication systems.
The analysis of the waveguide eigenvalue problem is of great significance in designing ridged waveguide devices.
The associate editor coordinating the review of this manuscript and approving it for publication was Wen-Sheng Zhao .
Thus, a variety of numerical methods has been used to analyze the characteristics of a ridged waveguide. Such as the magnetic field integral equation (MFIE) method [1], the finite difference method(FDM) [2], meshless method [3], the mode-matching method (MMM) [4], [5], boundary element method (BEM) [6], [7], multipole theory (MT) [8], the finite element method (FEM) [9]- [11], the isogeometric analysis (IGA) [12] and the scaled boundary finite element method(SBFEM) [13], [14]. Among those above mentioned methods, the FEM as the widely used numerical technique has a remarkable capability to solve the ridged waveguide problems. However, the traditional FEM cannot exactly represent the curved boundaries of the computational domain with low-order basis functions. Besides, the whole computational domain need to be discretized in FEM, which leads to considerable workload in mesh refinement and generation for the complex waveguide problems. Comparing with the traditional FEM, only the boundaries of the unknown domain need to be discretized in BEM, which can reduce the spatial dimension by 1. Hence, the consuming time in mesh generation and CPU time are spared. However, the fundamental solutions of complex waveguide problems are complicated to express.
The SBFEM [15], [16] as a semi-analysis approach has been widely used to analyze different problems, such as fracture mechanics [17]- [19], electromagnetic field [20]- [22], hydraulic structures [23]- [27], heat conduction [28], and so on. The SBFEM inherits the advantages of the BEM and FEM with some unique properties of its own. Such as the fundamental solution is not required, only the boundaries need to be discretized, the governing equations are weakened in the circumferential direction and solved analytically in the radial direction. However, the traditional SBFEM employ the Lagrange polynomials as the basis functions which leads to the curved boundaries cannot be exactly represented and the continuity order across element is low.
Isogeometric analysis (IGA) as an alternative to the conventional FEM was developed by Hughes et al. [29], in which NURBS is used to replace conventional Lagrange polynomials. Because the NURBS is employed as a standard approach in CAD systems, the IGA can seamlessly link the CAE model with the CAD model. In the framework of IGA, the geometry can be exactly described by NURBS, and the accuracy of results is superior. The IGA has been extended to solve heat conduction [30], shape optimizing [31], contact problems [32], electromagnetics [33], [34] to name a few.
According to the concept of IGA, the NURBS basis function is used to replace the Lagrange basis function in the traditional SBFEM [35]. The NURBS-based SBFEM has been successfully used in different problems, such as elastostatics problems [36], [37], electrostatic problems [38], heat conduction problems [39] and fracture problems [40]. In this paper, the NURBS enhanced SBFEM is firstly extended to analyze the QCRE waveguide. The NURBS enhanced SBFEM inherits the advantages of IGA and SBFEM, which can exactly describe the curved boundaries, reduce the spatial dimensions by one and obtain analytically results in the radial direction. It worthy note that, the NURBS enhanced SBFEM only employ the NURBS basis function to represent and discretize the curved boundaries of the elliptical waveguide. The side-face boundaries do not need to be discretized. As shown in Fig.1, the QCRE waveguide contains re-entrant corners and curved boundaries. Thus, the NURBS enhanced SBFEM is an attractive approach to solve the QCRE waveguide.
This paper is organized as follows. The concepts of B-splines and NURBS are briefly reviewed in Section II. In Section III, the governing equation of the waveguide eigenvalue problem is derived by the NURBS enhanced SBFEM. A generalized eigenvalue equation is obtained by introducing the boundary dynamic stiffness and employing a continued fraction solution in Section IV. With the NURBS enhanced SBFEM, the QCRE waveguide is solved and discussed in Section V. The conclusions are stated in Section VI.

II. B-SPLINES AND NURBS
The concepts of B-Splines and NURBS are briefly outlined in this section. More detailed description can be referred to the literature [29], [41].
A B-Spline basis is defined based on a series of nondecreasing real numbers named knot vector. The knot vector can be written as: where, B i,p denotes the i th B-spline basis function with p degree. Univariate NURBS can be defined by the rational and weighted form of B-spline: where N i,p is the i th NURBS basis function with p degree. ω i denotes the i th weights. The NURBS basis function inherits the advantage of the B-spline basis function, such as nonnegativity partition of unity, flexible continuity, local support VOLUME 9, 2021 and linear independence. NURBS can exactly represent conic sections comparing with the B-splines.

FIGURE 2. Quadratic NURBS functions with open knot vector.
A NURBS curve can be represented as a linear combination of the NURBS basis function: where P i denotes the i th control point.

III. NURBS ENHANCED SBFEM FOR WAVEGUIDE PROBLEMS
For the ridged waveguide with homogeneous permeability and permittivity distributions, the propagation characteristics are determined by the longitudinal component of the magnetic field H z or electric field E z . The basic equations of the magnetic field H z or electric field E z satisfy the Helmholtz equation: with essential boundary conditions for TM mode and natural boundary conditions for TE mode where TE mode: φ = H z . TM mode: φ = E z . k c denotes the cut-off wave number which is abbreviated as CWN in the following passage. As shown in Fig.3, in order to analyze the waveguide eigenvalue problems by using the NURBS enhanced SBFEM, a scaled boundary coordinate system (ξ , η) is establishing. ξ denotes the radial coordinate which ranges from 0 at the scaling center to 1 on the boundary b , η denotes the circumferential coordinate and a scaling center O x 0 ,ŷ 0 is chosen at a point which satisfies the scaling requirement. The scaling requirement is defined as the whole boundary must be visible from the scaling center. The relationship of the global Cartesian and the scaled boundary coordinate systems can be expressed as: where, x,ŷ denotes any point in the domain. (x(η), y(η)) denotes an arbitrary point on the boundary b . The gradient operators between the global Cartesian and the scaled boundary coordinate system are related as: where, J is independence of ξ named boundary Jacobian matrix: The derivative of J can be written as: Then, the Laplace operator is expressed as: As shown in Fig.3, the whole boundary is formed by the side-face boundary s and the outer boundary b . Only the outer boundary b needs to be discretized in the SBFEM, which reduce the spatial dimension by one. In the traditional SBFEM, the whole outer boundary b is discretized by the Lagrange basis function. Comparing with the traditional SBFEM, the curved segment of the boundary b is described and discretized by the NURBS basis function and the straight segment of the outer boundary b is discretized by the Lagrange basis function in the NURBS enhanced SBFEM. As the open knot vector is adopted, then the first and the last basis function have interpolation property at the both ends(A and B in Fig3). Thus, the control points A and B are shared with the straight segment of the outer boundary.
By applying the NURBS enhanced SBFEM to discretize the outer boundary, any point (x(η), y(η)) on the boundary b is expressed as: (13) where the coordinate η ranges from s i to s i +1 in each knot span or ranges from −1 to 1 in each Lagrange element. (x, y) represents the coordinates of control points or nodes. N(η) denotes the NURBS or the Lagrange basis functions. By substituting equation (13) into equation (8), any point in the domain is expressed as: Based on the isoparametric concept, φ (ξ, η) at any point is expressed as: where φ (ξ ) represents the unknown functions in the radial direction.

IV. THE GENERALIZED EIGENVALUE EQUATION
In order to solve the homogeneous second-order differential equation (19), the concept of ''boundary dynamic stiffness'' S(k c , ξ ) is introduced and defined as: where Q(ξ ) represents the external nodal flux.
According to the principle of virtual work, the relationship between Q(ξ ) and φ (ξ ) is as follows which is similar to equation (20): Based on equations (19) and (23), the governing equation in boundary dynamic stiffness can be rewritten as: Equation (24) belongs to the Riccati differential equation which can be solved by a continued fraction solution [42]. The continued fraction solution has a higher convergence rate and a larger convergence radius than the corresponding power series. Similarly, the continued fraction solution of equation (24) can be expressed as [13], [14]: where x = −k 2 c , M cf denotes the order of the continued fraction. K, M, S (i) 0 and S (i) 1 (i = 1, 2, . . . , M cf ) denote the coefficient which can be determined by a recursive procedure [42].
Appling the continued fraction solution to equation(25) which is respect to the boundary dynamic stiffness, then the relationship between the flux Q and the field function φ can be expressed as: where K h and M h are symmetric matrix. ydenotes the field function and the auxiliary variables φ (i) . F denotes the external nodal flux on the outer boundary.
Considering the boundary conditions as expressed in equations(6) and (7) of the homogeneous waveguide eigenvalue problem, F equals to 0 on the right-hand of equation (26). Thus, equation (26) changes to a generalized eigenvalue equation as follows:

A. THE EXACTNESS OF GEOMETRIC REPRESENTATION
In order to compare the exactness of geometric representation by different basis function, a quarter of ellipse is discretized into four elements by quadratic Lagrange and NURBS basis functions, respectively. A variable L is employed to evaluate the exactness of geometric representation which is defined as where M denotes the point on the elliptical boundary, F 1 and F 2 means the elliptical focal point, a denotes the lengths of major semi-axes of the ellipse. According to the definition of an ellipse, the exact value of L is one. Fig.4 shows that, the elliptical boundary can be exactly represented by the NURBS basis function, and except the nodal location the elliptical boundary is approximately represented by the Lagrange basis function.

B. THE ACCURACY AND EFFICIENCY OF THE NURBS ENHANCED SBFEM
In order to verify the accuracy and efficiency of the NURBS enhanced SBFEM, the normalized cut-off wave number(CWN) k c a of different modes which consist of the first 4 TM modes and the first 21 TE modes are solved by the traditional SBFEM and the NURBS enhanced SBFEM, respectively. The geometry of the QCRE waveguide assume a = 10mm, c = 0.5a, b = (a 2 -c 2 ) 0.5 , W a /a = W b /b = 0.4 W ac /W a = W bc /W b = 0.5 and L a /a = L b /b = 0.7. According to its symmetry, as shown in Fig.5, only a quarter of the QCRE waveguide needs to be simulated and divided to nine subdomains. The curved boundaries and straight boundaries of each subdomain are described and discretized by quadratic NURBS and Lagrange basis functions, respectively. In the analysis, four meshes are employed to validate the convergence of the NURBS enhanced SBFEM. The coarse mesh as shown in Fig.5   SBFEM are given in Tab.1. From Tab.1, the normalized cutoff numbers obtained by the NURBS enhanced SBFEM are in very good agreement with those obtained by the FEM, the accuracy and efficiency of the NURBS enhanced SBFEM are better than that of the traditional SBFEM.

C. THE EFFECT OF THE CORNER-CUT RIDGES ON THE CWNs
Then, the effect of the corner-cut ridges on the CWNs of several lowest modes is analyzed with the NURBS enhanced SBFEM.
With the fixed aspect ratio W a /(2a) = W b /(2b), L a /a = L b /b and W ac /W a = W bc /W b = 0.5, the relationship of the normalized CWNs k c a of TEe 11 , TEe 21 , TEo 11, TEo 21 , TMe 01 and TEe 31 modes with different ratio of eccentricity e, W a /(2a) and (2a-d a )/(2a) is shown in Fig.6.
As shown in Fig.6, with the increasing of (2a-d a )/(2a), the normalized CWNs of TEe 11 , TEe 21 and TEo 11 modes remain unchanged at first, then decrease and eventually tend to the same value. The effect of the ridge thickness W a on the normalized CWNs of the TEe 11 , TEe 21 and TEo 11 modes is little.
When eccentricity e = 0.1, the normalized CWNs of the TEo 11 mode is very close to that of the TEe 11 mode. With the increasing of eccentricity e, the normalized CWNs When eccentricity e = 0.9, the normalized CWNs of the TEe 31 mode decrease at first and then tend to constant with the increasing of (2a-d a )/(2a). With the fixed eccentricity e = 0.5, W a /a = W b /b = 0.12, W ac /W a = W bc /W b and L a /a = L b /b, the relationship of the normalized CWNs k c a of TEe 11 , TEe 21 , TEo 11, TEo 21 and TMe 01 modes with different ratio of L a /a and W ac /W a is shown in Fig.7.
As shown in Fig.7(a), when the edge width W ac is fixed, the normalized CWNs of TEe 11 , TEe 21 and TEo 11 modes are determined by the ridge length L a . The normalized CWNs of TEe 11 , TEe 21 and TEo 11 modes increases with the edge width W ac increasing.
As shown in Fig.7(b), the normalized CWNs of TEo 21 and TMe 01 modes keeps almost stable when tuning the edge width W ac and the ridge length L a . Thus, the single bandwidth of the quadruple corner-cut ridged elliptical waveguide can be broadened with sufficiently suppressing or not exciting the TEe 21 and TEo 11 modes.
With the fixed eccentricity e = 0.5, W ac /(2a) = W bc /(2b) = 0.02 and d a /a = d b /b, the relationship of the normalized CWNs k c a of TEe 11 , TEe 21, TEo 11 , TEo 21 and TMe 01 modes with different ratio of d a /a and W b /W bc (W a = W -W bc + W ac ) is shown in Fig.8.
As shown in Fig.8(a), when the ridge thickness W b is fixed, the normalized CWNs of TEe 11 , TEe 21 and TEo 11 modes are mainly determined by the ridge gap d a . The normalized CWNs of TEe 11 , TEe 21 and TEo 11 modes decrease at first and then increase with the ridge thickness W b increasing. For the ratio of the ridge gap d a /a = 0.10, the normalized CWNs of TEe 11 , TEe 21 and TEo 11 modes are almost the same.
As shown in Fig.8(b), for each ridge gap d a , the normalized CWNs of TEo 21 mode increase at first and then decrease with the ridge thickness W b increasing. When the ridge thickness W b is fixed, the normalized CWNs of TEo 21 mode are mainly determined by the ridge gap d a .
As shown in Fig.8(c), for the ratio of the ridge gap with the d a /a = 0.10, 0.20 and 0.40, the normalized CWNs of TMe 01 mode increase with the ridge thickness W b increasing. For the ratio of ridge gap d a /a = 0.60, the normalized CWNs of TMe 01 mode increase at first and then tend to constant with the ridge thickness W b increasing.
Because the CWNs of TEe 21 and TEo 11 modes are close to that of the dominant TEe 11 mode. Thus, when efficiently  Fig.9 shows that, the relative single-mode bandwidths are mainly determined by the ridge thickness W b and the gap distance d a of the QCRE waveguide. As the ratio W b /W bc increasing, the relative single-mode bandwidths increase first and then decrease. Consequently, an optimal relative single-mode bandwidth can be achieved by changing the ridge thickness, ridge gap and corner-cut edge width of the QCRE waveguide.

VI. CONCLUSION
The NURBS enhanced SBFEM has been successfully extended to analyze the QCRE waveguide. In the NURBS enhanced SBFEM, the curved boundaries and straight boundaries of the subdomains are described and discretized by NURBS and Lagrange basis function, respectively, and the spatial dimensions is reduced by one. It has been shown that, the elliptical boundary can be exactly represented by the NURBS basis function, and the accuracy and efficiency of the NURBS enhanced SBFEM are better than that of the traditional SBFEM. With the NURBS enhanced SBFEM, the corner-cut influence on the CWNs and the singlemode bandwidth have been studied in details. Moreover, the NURBS enhanced SBFEM could be extended to analyze the waveguide structure with complicated geometry in engineering problems.
BINGHAN XUE was born in Henan, China, in 1991. He received the Ph.D. degree in hydraulic structure engineering from the Dalian University of Technology, in 2018. He is currently a Lecturer with the School of Water Conservancy Engineering, Zhengzhou University, Zhengzhou, China.
His research interests include scaled boundary finite element method, isogeometric analysis, and their applications in electromagnetic field and civil engineering. JIANWEI LEI was born in Inner Mongolia, China, in 1993. He received the B.S. degree in road bridge and river-crossing engineering from Zhengzhou University, Zhengzhou, China, in 2016, where he is currently pursuing the Ph.D. degree in safety and protection engineering.
His research interests include microwave components, GPR, and numerical simulation methods, such as finite-difference time-domain method and sympletic methods. SHAOKUN MA was born in Hunan, China, in 1972. He received the Ph.D. degree in geotechnical engineering from Tongji University, in 2008. He is currently a Professor and the Head of the Department of Geotechnical Engineering, Guangxi University, Nanning, China.
His research interests include waveguide antennas, underground engineering, and centrifuge modeling. His research interests include molecular dynamics simulation theory, finite element method, and their application in physic field.