Helmholtz equation solution using 2-D FEM technique

3. A Hybrid MM/FE Technique to Analyze Horn having Discontinuities

This chapter presents the solution of Helmholtz equation using 2-D finite element method (FEM) to calculate the fields and their cutoff wave-numbers for non-regular surfaces. Delaunay triangulations are used to develop the FEM solution. The related theory for accurate prediction of eigenvalue and eigenvector for the solution of Helmholtz equation is discussed. The issues related to the calculation of reaction matrix for non-regular junction employing MM technique are elaborately discussed. The performance of this hybrid technique which is a combination of MM and 2-D FEM is evaluated in case of stepped cylindrical horn containing inner posts.

3.2 Helmholtz equation solution using 2-D FEM technique

as circular, rectangular, elliptical etc. When we need to model waveguide having non-regular geometry or having discontinuity, numerical techniques become more useful. Several numerical techniques such as FDM, FEM, FDTD are available for analysis of such geometry. From literature [53,54], we ﬁnd that FEM provides very good accuracy. FEM technique for solving the wave equation in guided structure are reported in this chapter.

The wave equation in a homogenous medium as mentioned in Section 2.1 is given by∇²φ+k_c²φ= 0.

Here,φis the scaler potential of longitudinal magnetic field for TE, longitudinal electric field for TM mode and electric field for TEM mode, respectively. Solution using variation method for φ can be obtained by optimizing the integral equation as given below:

F(φ) = 1

2 |∇φ|²−k²_cφ²

dS (3.1)

The natural boundary conditions i.e. Dirichlet or Neumann boundary conditions must be satisﬁed.

Now, the total surface is split into small sized meshes like triangular, rectangular etc. This small sized mesh is called an element and every element has some nodal points. The scalar potential and nodal point potential for n^th element can be written as φⁿ_e(x, y) and φⁿ_ei, where iis the local node number of the n^th mesh element. Let the scalar potential in an element vary as:

φⁿ_e(x, y) = [

2^ndorder

$ %& '

1^storder

$ %& '

1 x y x² y² xy x³ y³ x²y xy²

& '$ %

3^rdorder

...]

(

a_n b_n c_n d_n...

)_T

& '$ %

Constant

(3.2)

The choice of number of node points (n_p) depends upon the order p of the scalar potential function.

The relation between number of node points and order of polynomial isn_p = ⁽^p⁺¹⁾⁽₂^p⁺²⁾. The constant values of scalar potential function can be evaluated using node points, as speciﬁed using cartesian

3. A Hybrid MM/FE Technique to Analyze Horn having Discontinuities

coordinate values for equation (3.2). The constant values using equation (3.2) can be written as:

⎡

⎢⎢

⎣ a_n b_n c_n ... ...

⎤

⎥⎥

⎦

⎡

⎢⎢

⎣

1 x₁ y₁ x²₁ y₁² x₁y₁ · · · · 1 x2 y2 x²₂ y₂² x2y2 · · · · 1 x3 y3 x²₃ y₃² x3y3 · · · ·

· · · · · · · · · · ·

⎤

⎥⎥

⎦

−1⎡

⎢⎢

⎣ φⁿ_e₁ φⁿ_e₂ φⁿ_e₃ ... ...

⎤

⎥⎥

⎦

(3.3)

Combining equations (3.2) and (3.3), the scalar potential function can be written as

φⁿ_e(x, y) = [ 1 x y x² y² xy · · · · ]

⎡

⎢⎢

⎣

1 x₁ y₁ x²₁ y²₁ x₁y₁ · · · · 1 x₂ y₂ x²₂ y²₂ x₂y₂ · · · · 1 x₃ y₃ x²₃ y²₃ x₃y₃ · · · ·

· · · · · · · · · · ·

⎤

⎥⎥

⎦

−1⎡

⎢⎢

⎣ φⁿ_e₁ φⁿ_e₂ φⁿ_e₃ ... ...

⎤

⎥⎥

⎦

(3.4)

Equation (3.4) can also be written as

φⁿ_e(x, y) = (

α₁ α₂ α₃ · · · · ) (

φⁿ_e₁ φⁿ_e₂ φⁿ_e₃ · · · · )_T

(3.5)

where α is called the shape function. Substituting the equation (3.5) into equation (3.1), we obtain

F(φⁿ_e) = 1 2

(

φⁿ_e₁ φⁿ_e₂ · · · · )

⎡

⎢⎢

⎢⎣

∇α₁· ∇α₁ds

∇α₁· ∇α₂ds · · · · ∇α₂· ∇α₁ds

∇α₂· ∇α₂ds · · · ·

· · · · · · · · · ·

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣ φⁿ_e₁ φⁿ_e₂ ... ...

⎤

⎥⎥

⎥⎦

−k²_c 2

(

φⁿ_e₁ φⁿ_e₂ · · · · )

⎡

⎢⎢

⎢⎣

α1α1ds

α1α2ds · · · · α2α1ds

α2α2ds · · · ·

· · · · · · · · · ·

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣ φⁿ_e₁ φⁿ_e₂ ... ...

⎤

⎥⎥

⎥⎦

= 1 2[φⁿ_e]^T

C_e⁽ⁿ⁾

[φⁿ_e]−k²_c 2 [φⁿ_e]^T

T_e⁽ⁿ⁾

[φⁿ_e]

(3.6)

3.2 Helmholtz equation solution using 2-D FEM technique

Equation 3.6 is derived for a single element. Solution for the entire region is given by

F(φ) =

*N n=1

F(φⁿ_e) (3.7)

From the equations (3.6) and (3.7), F(φ) can be expressed in matrix from as

F(φ) = 1

2[φ]^T [C] [φ]−k_c²

2 [φ]^T[T] [φ] (3.8)

whereCandTare the global matrices consisting of local C⁽_eⁿ⁾ andT⁽_eⁿ⁾ matrices coeﬃcients, respectively and [φ] represents the scalar value of all the node points over the global surface. N presents the total number of elements of entire surface.

3.2.1 Solution of TEM and TE modes

TEM and TE modes must satisfy the Neumann boundary condition. The solution of TEM and TE mode will be optimization of equation 3.8 with respect to [φ]. So, ^∂F_∂_[Φ]⁽^φ⁾ = 0.

[C−k²_cT][φ] = 0 or

[T⁻¹C−k_c²I][φ] = 0 (3.9) where I is the unit matrix. k²_c is the eigenvalues of T⁻¹C matrix and it should be real quantity as C and T are symmetric matrices. In case of TEM mode, k_c is equal to zero and the solution of the eigenvector will be the scalar potential.

3.2.2 Solution of TM modes

In case of TM mode, the boundary condition should support the Dirichlet boundary condition.

So, the scalar potential at the boundary nodes (φ_p) should be zero. From the equation (3.8), we can write for the prescribed nodes [φ_p] and the free nodes [φ_f] as:

F(φ) = 1 2

(

φ_f φ_p )⎡

⎢⎣ C_{f f} C_{f p} C_pf C_pp

⎤

⎥⎦

⎡

⎢⎣ φ_f φ_p

⎤

⎥⎦−k_c² 2

(

φ_f φ_p )⎡

⎢⎣ T_{f f} T_{f p} T_pf T_pp

⎤

⎥⎦

⎡

⎢⎣ φ_f φ_p

⎤

⎥⎦ (3.10)

Here ^∂F_∂_[Φ⁽^φ⁾

f] = 0. So, TM mode solution should be [T_{f f}⁻¹C_{f f} −k²_cI][φ_f] = 0.

3. A Hybrid MM/FE Technique to Analyze Horn having Discontinuities

3.2.3 Surface Meshing

Surface mesh plays an important role in performance of the numerical electromagnetic calculations.

Several types of mesh such as triangular, rectangular, hexagonal or their mixed types are generally used for surface meshing. Mainly, triangular mesh having the ability to handle any arbitrary surface are widely used. One of the triangular mesh is Delaunay triangle. It is used for numerical electromagnetic calculation due to the availability of fast algorithm for Delaunay triangulation and in such schemes the absence of similar type of triangle improves the error performance in electromagnetic calculations done numerically [55]. To develop the FEM code, PDE Toolbox in MATLAB for Delaunay triangulation has been used. Some codes for graphics interface are developed to draw the desired surface in PDE Toolbox and collect the information as required. Basically, the information of triangulation are presented as position vector of node points (it specifies the node number and their position in cartesian coordinate) and a vector which contains nodes’ number for every triangle and its sub-area information. To close the PDE tool box after it is used, a code has been developed. Any type of planer objects in PDE Toolbox can be drawn using circle, rectangular, elliptical, polygon and using the operations of substraction, intersection and union etc. Also, a MATLAB code is written to identify the boundary nodes and interior nodes of the surface. Specifically, interior nodes are used for TM mode solution and sometimes boundary nodes are also used for triangulation of 3D objects. The boundary edge size can be controlled using step size for triangulation. In Figure 3.1-(a), we represent the property of Delaunay triangular mesh in which the nodes of a triangle cannot be located inside the circum-circle. Also, different objects (rectangle R₁,R₂,R₃ and circle C₁) for triangulation using the operation of intersection, union and substraction, respectively are drawn in Figure 3.1-(b). The sub-area of its surfaces (1, 2, 3 and 4) are also presented in same Figure.

3.2.4 Evaluation of local coeﬃcient matrix in closed form

2D-FEM code has been implemented based on scalar potential as a 1^st order polynomial using the Delaunay triangular mesh. Closed form matrix of local C_e⁽ⁿ⁾ and T_e⁽ⁿ⁾ are formulated on the basis of Appendix 8.2 as given below:

C⁽_eⁿ⁾= 1

4A_n[p_ip_j+q_iq_j]₃_×₃ where i= 1,2,3 &j= 1,2,3 (3.11)

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