NASA Technical Pape 3485 Finite Element Method fo Eigenvalue Poblems in Electomagnetics C. J. Reddy, Manoha D. Deshpande, C. R. Cockell, and Fed B. Beck Decembe 994
NASA Technical Pape 3485 Finite Element Method fo Eigenvalue Poblems in Electomagnetics C. J. Reddy Langley Reseach Cente Hampton, Viginia Manoha D. Deshpande ViGYAN, Inc. Hampton, Viginia C. R. Cockell and Fed B. Beck Langley Reseach Cente Hampton, Viginia National Aeonautics and Space Administation Langley Reseach Cente Hampton, Viginia 368- Decembe 994
This publication is available fom the following souces: NASA Cente fo AeoSpace Infomation National Technical Infomation Sevice (NTIS) 8 Elkidge Landing Road 585 Pot Royal Road Linthicum Heights, MD 9-934 Spingeld, VA 6-7 (3) 6-39 (73) 487-465
Contents Symbols.................................. vii Abstact.................................... Intoduction................................. Two-Dimensional Poblems........................... Homogeneous Waveguides Scala Fomulation.................. Fomulation............................... Discetization..............................3. Field Computation Fom Scala Potential............... 4..4. Numeical Examples......................... 4..5. Summay.............................. 8.. Inhomogeneous Waveguides Vecto Fomulation.............. 8... Solution of Homogeneous Waveguide Poblem With Two-Component Tansvese Vecto Fields........................... Fomulation............................ Discetization............................3. Finite element fomulation......................4. Finite element matices.......................5. Numeical examples......................... Inhomogeneous Waveguide Poblems Using Thee-Component Vecto Fields................................... Fomulation......................... 3... Discetization......................... 3...3. Finite element fomulation................... 3...4. Finite element matices.................... 4...5. Numeical examples...................... 4..3. Wave-Numbe Detemination fo Given Popagation Constant..... 5..3.. Fomulation......................... 5..3.. Discetization......................... 6..3.3. Finite element fomulation................... 6..3.4. Finite element matices.................... 7..3.5. Numeical example...................... 7..4. Dispesion Chaacteistics of Waveguides............... 7..4.. Fomulation......................... 7..4.. Discetization......................... 7..4.3. Finite element fomulation................... 8..4.4. Finite element matices.................... 8..4.5. Numeical examples...................... 9..5. Summay............................. 9 3. Thee-Dimensional Poblems........................ 9 3.. Eigenvalues of Thee-Dimensional Cavity Vecto Fomulation....... 9 3... Fomulation........................... 3... Discetization........................... 3..3. Finite Element Fomulation..................... 3..4. Finite Element Matices...................... iii
3..5. Numeical Examples........................ 3 3..6. Summay............................. 4 4. Concluding Remaks........................... 4 Appendix................................. 6 Refeences................................. 7 iv
Tables Table. Cuto Wave Numbes fo Rectangula Waveguide............. 5 Table. Cuto Wave Numbes fo Cicula Waveguide.............. 5 Table 3. Cuto Wave Numbes fo Coaxial Line With = = 4.......... 8 Table 4. Cuto Wave Numbes fo Rectangula Waveguide............ Table 5. Cuto Wave Numbes fo Cicula Waveguide............. 3 Table 6. Cuto Wave Numbes fo Rectangula Waveguide............ 4 Table 7. Cuto Wave Numbes fo Cicula Waveguide............. 4 Table 8. Wave Numbes fo LSM Modes of Squae Waveguide With =............................... 7 Table 9. Dispesion Chaacteistics of Patially Filled Rectangula Waveguide of Figue.......................... 9 Table. Dispesion Chaacteistics of Patially Filled Rectangula Waveguide of Figue 3.......................... Table. Fomation of Edges of Tetahedal Element.............. Table. Eigenvalues of Ai-Filled Rectangula Cavity.............. 3 Table 3. Eigenvalues of Half-Filled Rectangula Cavity............. 4 Table 4. Eigenvalues of Ai-Filled Cicula Cylindical Cavity.......... 4 Table 5. Eigenvalues of Spheical Cavity With Radius of cm.......... 4 v
Figues Figue. Geomety of poblem.......................... Figue. Single tiangula element........................ Figue 3. Flowchat fo FEM solution....................... 4 Figue 4. Geomety of ectangula waveguide................... 5 Figue 5. Electic eld distibution of some modes fo ectangula waveguide...... 6 Figue 6. Coss section of cicula waveguide.................... 6 Figue 7. Electic eld distibution of some modes fo cicula waveguide....... 7 Figue 8. Coss section of coaxial line....................... 8 Figue 9. Electic eld distibution of some modes fo coaxial line.......... 9 Figue. Conguation of tangential edge elements............... Figue. Patially lled squae waveguide................... 7 Figue. Patially lled ectangula waveguide with b =a =:45, d=b =:5, and " =:45............................... 9 Figue 3. Patially lled waveguide with b =a =:45 and " =:45........ 9 Figue 4. Fist-ode tetahedal element.................... Figue 5. Ai-lled ectangula cavity. Size: by.5 by.75 cm.......... 3 Figue 6. Half-lled ectangula cavity with dielectic mateial " = : and lled fom z =:5 to cm. Size: by. by cm.................. 3 Figue 7. Ai-lled cicula cylindical cavity. Dimensions ae in centimetes.... 4 vi
Symbols A, B geneic vectos used in vecto identities A aea of tiangle A m ;A xm ;A ym ;A zm coecients dened by equations (57), (64), (67), and (7), m =;;:::;6 a i ;b i ;c i coecients dened by equations (6), (7), and (8), i =;;3 a ;b dimensions of ectangula waveguide a tm ;b tm ;c tm ;d tm coecients of tm as dened in equation (6), m =;;3;4 B m ;B xm ;B ym ;B zm coecients dened by equations (58), (65), (68), and (7), m =;;:::;6 C m ;C xm ;C ym ;C zm coecients dened by equations (59), (66), (69), and (7), m =;;:::;6 D m coecient dened in equation (6) dl ds dv d E E t E x ;E y ;E z e m line integal suface integal volume integal bounday contou of waveguide electic eld, E x bx + E y by + E z bz tansvese electic eld, E x bx + E y by components of vecto E unknown coecient at edges of tetahedal element, m =;;:::;6 e tm unknown coecient at edges of tiangula element, m =;;3 e zi FEM3D, FEM3D f H H t unknown coecient of eld at nodes of tiangle as dened by equation (94), i =;;3 compute codes fo thee-dimensional eigenvalue solves geneic scala used in vecto identities magnetic eld, H x bx + H y by + H z bz tansvese magnetic eld, H x bx + H y by H x ;H y ;H z components of vecto H HELM compute code fo two-dimensional scala eigenvalue solve HELMVEC, HELMVEC, compute codes fo two-dimensional vecto eigenvalue solves HELMVEC, HELMVEC3 k c cuto wave numbe k o fee-space wave numbe, o L m length of edges fo tetahedal element, m =;;3;4 L tm length of edges fo tiangula element, m =;;3 vii
LSM bn PEC longitudinal section magnetic unit nomal pefect electic conducto adius of cicula waveguide adius of inne conducto of coaxial line adius of oute conducto of coaxial line S, T FEM global matices S el ; T el FEM element matices S el(tt) ;S el(tz), FEM element submatices as dened in equations () S el(zt) ;S el(zz) and (39) S tt ;S tz ;S zt ;S zz FEM global submatices as dened in equations (9) and (35) S t ;T t S z ;T z T global matices fo tansvese elds as dened in equation (9) global matices fo longitudinal elds as dened in equation (9) vecto testing function, T x bx + T y by + T z bz T el(tt) ;T el(tz), FEM element submatices as dened in equations () T el(zt) ;T el(zz) and (9) Tt T s two-dimensional vecto testing function, T x bx + T y by two-dimensional scala testing function T tt ;T tz ;T zt ;T zz FEM global submatices as dened in equations (9) and (35) T x ;T y ;T z components of vecto T TE TM tansvese electic tansvese magnetic b tm unit vecto along edge of tetahedal element, m =;;:::;6 b ttm unit vecto along edge of tiangula element, m =;;3 V volume of tetahedon W m edge basis function fo tetahedal elements, m =;;:::;6 W tm edge basis function fo tiangula elements, m =;;3 X; Y; Z ectangula coodinate axes x; y; z ectangula coodinates bx; by; bz x tet ; y tet ; z tet x ti ; y ti Zo TM i unit vectos along X-, Y -, and Z-axis, espectively centoid of tetahedal element centoid of tiangula element chaacteistic impedance of TE mode st-ode shape function fo tiangula element dened by equation (5), i =;;3 viii
tm st-ode shape function fo a tetahedal element, m =;;3;4 waveguide popagation constant waveguide coss-section aea integation ove suface of tiangula element integation ove volume of tetahedal element " elative pemittivity o t fee-space wavelength elative pemeability two-dimensional scala potential function, TE o TM i unknown coecients of at nodes of tiangula element, i =;;3 thee-dimensional gadient opeato in ectangula coodinates two-dimensional gadient opeato in X-Y plane ix
Abstact Finite element method (FEM) has been a vey poweful tool to solve many complex poblems in electomagnetics. The goal of the cuent eseach at the Langley Reseach Cente is to develop a combined FEM/method of moments appoach to thee-dimensional scatteing/ adiation poblem fo objects with abitay shape and lled with complex mateials. As a st step towad that goal, an execise is taken to establish the powe of FEM, though closed bounday poblems. This pape demonstates the development of FEM tools fo two- and theedimensional eigenvalue poblems in electomagnetics. In section, both the scala and vecto nite elements have been used fo vaious waveguide poblems to demonstate the exibility of FEM. In section 3, vecto nite element method has been extended to thee-dimensional eigenvalue poblems.. Intoduction The nite element method (FEM) has been widely used as an analysis and design tool in many engineeing disciplines like stuctues and computational uid mechanics. Though FEM has been applied to electomagnetic poblems, it was mainly con- ned to electical machines and magnetics (ef. ). In the past yeas thee has been a geat inteest in application of this method to micowave components such as waveguides and antennas. But fo many yeas, its use has been esticted because of the so-called spuious solutions in vecto nite elements (ef. ). Vey ecently, the \edge elements" have been employed successfully fo vecto fomulations without esulting in \spuious solutions." In the ecent past, use of these edge elements in FEM has evived an inteest in applying FEM to micowave engineeing poblems (ef. 3). This in combination with the advances in compute hadwae and softwae helped to make FEM an attactive tool fo electomagnetics. Also, thee ae a vaiety of commecial geometical modelling tools to accuately model any theedimensional geomety and to geneate the equied mesh with any kind of elements such as tiangles and tetahedals (efs. 4 and 5). In this pape, the FEM tools fo analyzing eigenvalue poblems in electomagnetics have been descibed. This pape is divided into two pats: section deals with the two-dimensional poblems; section 3, with the thee-dimensional poblems. Thoughout this pape tiangula elements ae used fo modelling two-dimensional poblems and tetahedals ae used to model the thee-dimensional poblems. In section., a scala FEM fomulation is used fo two-dimensional abitaily shaped waveguides. Tiangula elements with nodal basis functions ae used to fomulate the FEM matices. The eigenvalues fo dieent types of waveguides ae obtained and the eld intensity plots ae pesented fo vaious waveguide modes. In section., a vecto FEM is intoduced with two-dimensional edge elements fo analyzing inhomogeneous waveguides. Fo the sake of claity in fomulation, section. is divided into fou sections. Section.. gives the solution of homogeneous waveguide poblem with two-component tansvese vecto elds. Section.. gives the calculation of eigenvalues fo inhomogeneous waveguides using the theecomponent vecto elds. Combination of edge and nodal basis functions have been used fo tansvese and longitudinal eld components, espectively. Sections..3 and..4 extend the fomulation in section.. to detemine eithe the wave numbe o the popagation constant fo inhomogeneously lled waveguides when one of them is specied. In section 3., fomulation fo thee-dimensional vecto FEM is descibed. Edge basis functions fo tetahedal elements ae intoduced to fomulate - nite element matices fo thee-dimensional cavities lled with inhomogeneous mateial. In sections.,., and 3., numeical examples ae pesented to show the validity of the analysis and the compute pogams developed. Fo all the examples, FEM esults in good accuacy. All the numeical examples have been checked fo numeical convegence. By vitue of FEM, the compute codes pesented in this pape can handle any abitaily shaped geometies lled with inhomogeneous mateials, unless othewise mentioned.
. Two-Dimensional Poblems.. Homogeneous Waveguides Scala Fomulation This section deals with the solution of twodimensional waveguide poblems with closed boundaies using the Galekin nite element method. The wave equation is solved fo a genealized poblem with nodal-based st-ode tiangula elements (ef. ). Finite element matices ae deived and a compute pogam to calculate the eigenvalues and electic eld distibutions is pesented.... Fomulation The scala potential function satises the Helmholtz equation with wave numbe k c t + k c = () within and on the bounday d indicated in gue. This is the \stong" fom of the scala Helmholtz equation. In the stong fom, the unknown appeas within a second-ode dieential opeato. To make the equation suitable fo a numeical solution, it can be conveted into the \weak" fom by multiplying both sides with a testing function T s and integating ove the suface ; that is, ht s t +k c T s ids = () The st tem in equation () can be witten as T s t ds = T s ( t t )ds (3) The following vecto identities can be used to modify equation (3): whee @ =@n is the nomal deivative of along the bounday d. The tem on the ight-hand side vanishes as T s vanishes on the PEC bounday fo the TM case and @ =@n vanishes at the bounday fo the TE case. Hence equation (6) can be witten as ( t T s t )ds = k c... Discetization T s ds (7) The poblem egion is discetized with the stode tiangula elements. Within the tiangula element given in gue, is adequately appoximated by the expession (ef. ) = a + bx + cy (8) The solution is piecewise plana but continuous eveywhee. At the vetices,, and 3, the potential can be expessed as Y = a + bx + cy (9) = a + bx + cy () 3 = a + bx 3 + cy 3 () Γ dγ Figue. Geomety of poblem. X and t [T s ( t )] = t T s t +T s ( t s ) (4) t t ds = Z d t bndl (5) whee bn is the unit nomal along the bounday d. Equation () can now be witten as ( t T s t )ds k c T s ds = Z d T s @ @n dl (6) Y (x,y ) (x,y ) Figue. Single tiangula element. 3 (x 3,y 3 ) X
Fom equations (9), (), and (), the coecients a, b, and c ae evaluated as 6 4 3 a 7 b 5= c 6 4 x y x y x 3 y 3 3 3 7 5 6 4 7 3 5 () Hence, equation (8) can be ewitten by substituting fo a, b, and c =[ x y] 4 a b c 3 5 =[ x y] Equation (3) can be witten as = i= whee i (x; y) is given by 4 x y x y x3 y3 3 5 3 4 5 3 (3) i i (x; y) (4) i (x; y) = A (a i + b i x + c i y) (i =;;3) (5) and a i, b i, and c i ae given by a i = x j y k x k y j (6) b i = y j y k (7) c i = x k x j (8) whee i, j, and k ae cyclical; that is (i =,j=, k = 3), (i =,j=3,k= ) and (i =3,j=, k= ), and A is given by A= x y x y x 3 y 3 (9) Using the testing function (as pe Galekin's technique (ef. )), T s = j (x; y) (j =;;3) () and the element epesentation in equation (4), the left-hand side of equation (7) can be evaluated ove a single element as ( t T s t )dxdy = i= i ( t i t j )dx dy (j =;;3) () and the ight-hand side as T s dxdy = i= i i j dx dy (j =;;3) Hence fo each element, equation (7) becomes i= i = k c i= ( i j )dx dy i () i j dx dy (j =;;3) (3) And this can be witten in a matix fom as whee [S el ]= [S el ][ ]=k c [T el][ ] (4) [T el ]= ( i j )dx dy (5) ( i j )dx dy (6) i = @ i @x bx + @ iby (7) @y Fom equation (5), i can be witten as and hence, i = A (b ibx + c i by) (8) 3 [ i j ]= 4 3 3 3 3 33 5 (9) Substituting equation (8) into equation (9) gives 3 [ i j ]= 4 b + c bb+ cc bb3+ cc3 4A bb+ cc b + c bb3+ cc3 5 b3b+ c3c b3b+ c3c b3 + c 3 (3) The matix [S el ] can be evaluated by using equation (3) to obtain [S el ]= 4 3 ( i j )dx dy5 =A[ i j ] (3) [T el ]= 4 3 i j dx dy5 (3) 3
The matix [T el ] has been evaluated by Silveste (ef. 6) and is given in a simple fom as [T el ]= A 6 4 3 7 5 (33) The matices [S el ] and [T el ] ae evaluated fo each element and ae assembled ove the entie egion accoding to the global node numbeing to obtain a global matix equation (ef. ) as follows: walls of the waveguide to satisfy the Diichlet bounday conditions fo the longitudinal electic eld component. A vey simple way of implementing this is to ignoe the nodes on PEC while foming nite element matices. This will esult in lowe ode matices fo the TM case than those fo the TE case. Once the scala potential is obtained, the tansvese electic elds fo the TM modes ae given by E x = Z TM o @ @x (4) [S][ ]=k c[t][ ] (34) E y = Z TM o @ @y (4) This esults in matices of the ode n n whee n is total numbe of nodes. With equations (3), (3), and (33), the eigenvalue equation (eq. (34)) is solved fo k by the standad eigenvalue solves fom the c EISPACK libay (efs. 7 and q 8). The cuto wave numbes ae then given by kc...3. Field Computation Fom Scala Potential Once the scala potential is calculated at evey node, the electic eld can be calculated fo both the TE and TM modes by the following fomulation. The scala potential at any point (x; y) inside a tiangula element is given by = i= i i(x; y) (35) These scala potentials can be dieentiated with espect to x and y to obtain the following expessions: @ = @x A @ = @y A i= i= i b i (36) i c i (37) Fo the TE modes, the tansvese electic eld components E x and E y inside an element ae given by E x = @ E y = @ @y @x (38) (39) When obtaining the scala potential fo the TM modes, the scala potential is set to zeo on PEC 4 whee Z TM is the chaacteistic wave impedance fo o the TM mode...4. Numeical Examples A compute code HELM was witten to implement the fomulation pesented in section.. The owchat fo the implementation of the FEM solution is given in gue 3. Numeical examples fo the ectangula waveguide, the cicula waveguide, and the coaxial line ae given as follows: Rectangula waveguide: The cuto wave numbes k c of a ectangula waveguide calculated by using HELM ae pesented in table along with analytical esults fom efeence 9. The geomety of the ectangula waveguide (a =b = ) is shown in gue 4. The numeical esults pesented ae achieved with 4 tiangula elements ove the coss section of the waveguide. The eigenvectos fo some of the modes have been calculated and the electic elds of the coesponding modes ae plotted in gue 5. Waveguide geomety COSMOS (mesh geneato) HELM (FEM code) Eigenvalues field distibution TECPLOT/AVS Figue 3. Flowchat fo FEM solution.
b Y a Figue 4. Geomety of ectangula waveguide. X Cicula waveguide: The cuto wave numbes fo a cicula waveguide of unit adius wee calculated with HELM and compaed with analytically available data fom efeence 9. (See table.) A coss section of the cicula waveguide is shown in gue 6. Two hunded tiangula elements have been used to model the geomety. The eigenvectos of selected modes have been calculated and the electic elds of these modes ae plotted in gue 7. Coaxial line: The coss section of the coaxial line is shown in gue 8. The HELM pogam is used to calculate the cuto wave numbes and coesponding electic eld stength of highe ode TE and TM modes. A tiangula mesh with 34 elements has been used to model the geomety. Table 3 pesents the cuto wave numbes computed fo = =4by HELM and the analytically available cuto wave numbes in the liteatue (ef. ). Fo TM modes the potential on the inne and oute conductos is set to zeo. The tansvese electic eld components ae calculated and plotted in gue 9. Table. Cuto Wave Numbes fo Rectangula Waveguide kc a TE TM Analytical (ef. 9) HELM 3.4 3.44 6.85 6.38 6.85 6.38 7.7 7.7.958 3. 8.889 8.993 Table. Cuto Wave Numbes fo Cicula Waveguide kc Mode Analytical (ef. 9) HELM TE 3.83 3.858 TE.84.843 TE 5.33 5.379 TE 3.54 3.68 TE 6.76 6.688 TM.45.43 TM 3.83 3.854 TM 7.6 7.36 TM 5.35 5. TM 8.47 8.668 5
(a) TE mode. (b) TE mode. (c) TE mode. (d) TM mode. Figue 5. Electic eld distibution of some modes fo ectangula waveguide. Y X Figue 6. Coss section of cicula waveguide. 6
(a) TE mode. (b) TE mode. (c) TE mode. (d) TM mode. (e) TM mode. (f) TM mode. Figue 7. Electic eld distibution of some modes fo cicula waveguide. 7
Table 3. Cuto Wave Numbes fo Coaxial Line With ==4 kc Mode Analytical (ef. ) HELM TE.4.4 TE.75.754 TE3.48.55 TM.4.3 TM.. Y X inteested in the dispesion chaacteistics of the waveguide. Fo this a vecto nite element appoach has to be followed. Unfotunately, the nodal-based nite element method when applied to the vecto wave equation esults in nonphysical o spuious solutions, which ae geneally attibuted to the lack of enfocement of divegence condition. (See efs. and.) Many attempts have been made to avoid these spuious modes by dieent vaiations of the nodal-based nite element method. Fotunately, a evolutionay appoach has been discoveed ecently. This appoach uses the so-called \vecto basis" o \vecto elements," which assign degees of feedom to the edges athe than to the nodes of the element. Fo this eason they ae populaly efeed to as \edge elements." Figue 8. Coss section of coaxial line...5. Summay In section, a nodal-based two-dimensional nite element method has been descibed fo homogeneous waveguides using Galekin's technique. The pocedue outlined hee is valid fo any abitay coss section of the waveguide lled with homogeneous mateials. The HELM compute pogam gives the cuto wave numbes and the tansvese electic eld components fo any mode of popagation in such a waveguide. Examples of a ectangula waveguide, a cicula waveguide, and a coaxial line ae pesented to validate the compute code. Accuacy of the numeical esults depends on the numbe of elements used to epesent the geomety... Inhomogeneous Waveguides Vecto Fomulation Fo inhomogeneous waveguide poblems, the scala potential appoach is not applicable if one is Although the edge elements wee st descibed by Whitney (ef. ) 35 yeas ago, thei impotance in electomagnetics was not ealized until the late 98's (ef. 3). In section., we descibe the fomulation of the nite element method fo vecto wave equation using edge elements. This fomulation, in geneal, follows the one given by Lee, Sun, and Cendes (ef. 3). But instead of attempting to solve the dispesion poblem ight away, we use the following steps to build the edge element fomulation:. Solve fo the homogeneous waveguide poblem by using two-component tansvese vecto elds. Solve fo the genealized waveguide poblem by using thee-component vecto elds 3. Solve fo the wave numbe ko when the popagation constant is specied 4. Solve fo the popagation constant of any genealized waveguide fo a given fequency of opeation 8
(a) TE mode. (b) TE mode. (c) TE3 mode. (d) TM mode. (e) TM mode. (f) TM mode. Figue 9. Electic eld distibution of some modes fo coaxial line. 9
... Solution of Homogeneous Waveguide Poblem With Two-Component Tansvese Vecto Fields Fo a homogeneous waveguide, the popagating modes can be divided into tansvese electic (TE) o tansvese magnetic (TM) modes, which can be solved sepaately. Fo the TE mode E z =, the tansvese electic eld vecto E t satises the vecto wave equation t t E t k " c E t = (4) and fo the TM mode H z =, the tansvese electic eld vecto H t satises the vecto wave equation t t H t k " c H t = (43) whee and " ae the pemeability and pemittivity, espectively, of the mateial in the waveguide. In this section, we illustate the pocedue fo the TE mode and the same pocedue can be applied fo the TM mode, except the Neumann bounday conditions ae applied at PEC boundaies instead of the Diichlet bounday conditions.... Fomulation. Fo the TE mode, econside the wave equation (4), which is t t E t k " c E t = Dot multiplying equation (4) by a vecto testing function T t and integating ove the coss section of the waveguide, we get T t t E t Fom the vecto identities, k c " T t E t ds = (44) T t ( t A) =( t T t )A t (T t A) (45) (T t A) bn = T t (bn A) (46) and the divegence theoem, t (T t A)ds = Z d (T t A) bndl (47) equation (44) can be witten in weak fom as ( t T t )( t E t )ds = k " c T t E t ds Z T t bn t E t dl (48) d On a pefect electic conducting (PEC) bounday, the contou integal vanishes as T t is set to zeo to satisfy the Diichlet bounday conditions. Hence, fo all the poblems closed by PEC walls, the last tem on the ight-hand side of equation (48) is set to zeo. Thus, the nal equation can be witten as ( t T t )( t E t )ds = k " c T t E t ds (49)... Discetization. The nite elements used in section. ae scala and have unknown paametes, the values of the scala eld at the nodes of the element. Nodal-based nite elements ae not suitable to epesent vecto elds in electomagnetics, as the bounday conditions often take the fom of a specication of only the pat of the vecto eld that is tangent to the bounday. With nodal-based elements, the physical constaint must be tansfomed into linea elationships between the Catesian components, and at nodes whee the bounday changes diection, an aveage tangential diection must be detemined st. These ae vey dicult (if not impossible) to implement in nodal-based nite elements. The failue to implement pope conditions esults in spuious modes which ae nonphysical. The most elegant and simple appoach to eliminate the disadvantages of the nodal-based elements is to use edge elements. Edge elements ae ecently developed nite element bases fo vecto elds. With edge elements, only the tangential continuity of the vecto elds is imposed acoss the element boundaies. The advantages of edge elements ae as follows:. The edge elements impose the continuity of only the tangential components of the electic and magnetic elds, which is consistent with the physical constaints on these elds.. The inteelement bounday conditions ae automatically obtained though the natual bounday conditions. 3. The Diichlet bounday condition can be easily imposed along the element edges.
Y e t (x,y ) (x,y ) e t e t3 3 (x 3,y 3 ) Figue. Conguation of tangential edge elements. 4. As the edge elements ae chosen to be divegence fee, the spuious nonphysical solutions ae completely eliminated. Fo a single tiangula element shown in gue, the tansvese electic eld can be expessed as a supeposition of edge elements. The edge elements pemit a constant tangential component of the basis function along one tiangula edge while simultaneously allowing a zeo tangential component along the othe two edges. Thee such basis functions ovelapping each tiangula cell, povide the complete expansion (ef. ): E t = m= e tm W tm (5) whee W tm = L tm ( i t j j t i ) (5) i is the st-ode shape function associated with nodes,, and 3 dened by equation (5); and L tm is the length of edge m connecting nodes i and j, that is, explicitly, the basis functions epesenting edges,, and 3 with coecients e t, e t, and e t3 witten as W t = L t ( t t ) (5) W t = L t ( t 3 3 t ) (53) W t3 = L t3 ( 3 t t 3 ) (54) The thee unknown paametes ae e tm. It can be shown that b ttm E t = e tm (on edge m) (55) whee b ttm is a unit vecto along edge m in the diection of the edge; fo example, b tt = (x x )bx +(y y )by L t X fo edge connecting nodes and. In othe wods, e tm contols the tangential eld on edge m. Late t W tm = is veied and hence the electic eld obtained though equation (5) satises t E t =, the divegence equation, within the element. Theefoe, the nite element solution is fee of spuious solutions. With the use of the simplex coodinates as de- ned by equation (5), the basis function given by equation (5) can be witten as W tm = L tm 4A [(A m + B m y)bx +(C m +D m x)by] (56) whee A m = a i b j a j b i (57) B m = c i b j c j b i (58) C m = a i c j a j c i (59) D m = b i c j b j c i = B m (6) Fom equation (56), t W tm =, which esults in a divegenceless electic eld; that is, t E t =....3. Finite element fomulation. Substituting equation (5) into equation (49) fo a single tiangula element gives the following equation: = k c " ( t W tn ) ( t W tn )e tm ds m= (W tm W tn )e tm ds m= (n =;;3) (6) whee indicates the integation ove the tiangula element. By intechanging the integation and summations, equation (6) can be witten in a matix fom as [S el ][e t ]=kc[t el ][e t ] (6) Hence the nite element matices fo a single element ae given by [S el ]= ( t W tm ) ( t W tn ) ds (63) and [T el ]=" (W tm W tn ) ds (64)
These element matices can be assembled ove all the tiangles in the coss section of the waveguide to obtain a global eigenmatix equation. [S][e t ]=k c [T][e t] (65)...4. Finite element matices. Closed-fom expessions fo integals in equations (63) and (64) can be witten as and whee ( t W tm ) ( t W tn ) ds = L tml tn 4A 3 D m D n (66) (Wtm Wtn) ds = L tmltn 6A 3 (I t+it+it3+it4+it5) (67) I t = A (A ma n + C m C n ) I t = A (C md n + C n D m ) I t3 = A (A mb n + A n B m ) I t4 = A (B mb n ) I t5 = A (D md n ) dx dy (68) xdxdy (69) ydxdy (7) y dx dy (7) x dx dy (7) These equations can be educed futhe by the following integation fomulas fo integating ove a tiangle given in efeence 4: I t =(A m A n +C m C n ) (73) I t =(C m D n +C n D m )x ti (74) I t3 =(A m B n +A n B m )y ti (75) I t4 = B mb n I t5 = D md n i= i= y i +9y ti x i +9x ti!! (76) (77)...5. Numeical examples. A compute pogam HELMVEC was witten to solve the eigenvalue poblem pesented in section... Numeical data computed with this pogam fo a ectangula waveguide and a cicula waveguide ae given as follows. Rectangula waveguide: The cuto wave numbes of an ai-lled ectangula waveguide (shown in g. 4, a =b = ) wee calculated with the pocedue in section.. (HELMVEC) and ae pesented in table 4 fo both the TE and TM modes along with analytical data fom efeence 9. Fou hunded tiangula elements have been used to epesent the geomety. Table 4. Cuto Wave Numbes fo Rectangula Waveguide kca Mode Analytical (ef. 9) HELMVEC TE mode TE 3.4 3.4 TE 6.85 6.74 TE 3 9.48 9.396 TE 6.85 6.74 TE 7.7 7.4 TE 8.889 8.897 TE 3.33.339 TE.57.497 TM mode TM 7.7 7.4 TM 8.889 8.889 TM 3.33.34 TM.958.9 TM 4.55 4.44 TM 3 5.7 5.757 TM 3 9.7 8.96 TM 3 9.877 9.78 Cicula waveguide: The cuto wave numbes fo an ai-lled cicula waveguide (Radius = ) wee also computed by HELMVEC and compaed with analytically calculated values fom efeence 9. These esults ae pesented in table 5 fo both the TE and TM modes. The waveguide geomety is epesented by tiangula elements.... Inhomogeneous Waveguide Poblems Using Thee-Component Vecto Fields In this section, a genealized appoach fo nding the cuto fequencies of inhomogeneously lled waveguides is descibed.
Table 5. Cuto Wave Numbes fo Cicula Waveguide Mode Analytical (ef. 9) HELMVEC TE mode TE 3.83 3.834 TE.84.846 TE 3.54 3.7 TE 3 4. 4.9 TE 7.6 7. TE 5.33 5.33 TE 6.76 6.6 TE 3 8.5 8.78 TM mode TM.45.46 TM 3.83 3.83 TM 5.36 5.55 TM 3 6.38 6.358 TM 5.5 5.58 TM 7.6 7.54 TM 8.47 8.43 TM 3 9.76 9.644... Fomulation. This genealized appoach can be followed by using eithe the E o H eld. We kc will illustate the case fo the E eld. wave equation fo E is given by t The vecto t E k c " E = (78) The electic eld can be witten as E = E t + bze z (79) Hence equation (78) can be divided into two pats one consisting of the tansvese electic elds and the othe, the z-component of the electic eld that is, t t t E t k c " E t = (8) t E z + kc " E z = (8) Equation (8) can be witten in its weak fom as ( t Tt)( t E t )ds = k c " TtE t ds (8) and equation (8) can be witten in its weak fom by following equations () though (7) to obtain ( t T z t E z )ds = kc " whee the testing function is T = Tt + bzt z. T z E z ds (83)... Discetization. As the vecto Helmholtz equation is divided into two pats, vecto-based tangential edge elements can be used to appoximate the tansvese elds, and nodal-based st-ode Lagangian intepolation functions can be used to appoximate the z-component. Fom equation (5), which is E t = e tm W tm m= whee m indicates the mth edge of the tiangle and W tm is the edge element fo edge m. The testing function Tt is chosen to be the same as the basis function in equation (5) (shown above); that is, Tt = W tm. The z-component of the electic eld can be witten as E z = i= e zi i (84) Hee i indicates ith node and i is the simplex coodinate of node i as given in equation (5). Also the testing function Tz is chosen to be the same as the basis function in equation (84); that is, Tz = j....3. Finite element fomulation. Substituting equations (5) and (84) into equations (8) and (83), espectively, we can get the following equations: = k c " m= ( t W tm ) ( t W tn )e tm ds X m= (W tm W tn )e tm ds (n =;;3) (85) e zi ( t i t j )dx dy i= = k c " i= e zi i j dxdy (j =;;3) (86) 3
By intechanging the summation and integation in equation (85), these two equations can be combined to be witten in a matix fom as " Sel(t) # et " Tel(t) # = k et c (87) S el(z) e z T el(z) e z Hence the element matices ae given by S el(t) = S el(z) = T el(t) = " ( t W tm ) ( t W tn ) ds (88) T el(z) = " ( i j )dx dy (89) (W tm W tn ) ds (9) i j dx dy (9) These element matices can be assembled ove all the tiangles in the coss section of the waveguide to obtain a global eigenmatix equation: St e t T = k t e t (9) S z e z T z e z Cicula waveguide: The cuto wave numbes fo an ai-lled cicula waveguide (Radius = ) ae also computed and compaed with analytically calculated values. These ae given in table 7. Table 6. Cuto Wave Numbes fo Rectangula Waveguide kc a HELMVEC fo Mode Analytical (ef. 9) E H TE mode TE 3.4 3.4 3.44 TE 6.85 6.74 6.38 TE3 9.48 9.396 9.5 TE 6.85 6.74 6.38 TE 7.7 7.4 7.4 TE 8.889 8.897 8.897 TE3.33.339.34 TE.57.497.77 TM mode TM 7.7 7.7 7.7 TM 8.889 8.995 8.993 TM3.33.54.537 TM.57.498.77 Table 7. Cuto Wave Numbes fo Cicula Waveguide...4. Finite element matices. The nite element matices shown in equations (88) though (9) ae aleady deived in the pevious sections. Equation (88) is given by equation (66), equation (89) by equation (3), equation (9) by equation (67), and equation (9) by equation (33)....5. Numeical examples. A compute pogam HELMVEC was witten fo the theecomponent vecto fomulation to calculate cuto wave numbes of waveguides with abitay coss sections. Numeical data computed with HELMVEC fo a ectangula waveguide and a cicula waveguide ae given as follows. Rectangula waveguide: The cuto wave numbes of an ai-lled ectangula waveguide (shown in g. 4, a =b = ) ae calculated with the pocedue in section.. (HELMVEC) and ae pesented in table 6 along with analytical data fom efeence 9. In the pesent fomulation, it is not equied to calculate eigenvalues sepaately fo TE and TM modes. Both E and H fomulations esult in almost identical numeical esults. 4 HELMVEC fo Mode Analytical (ef. 9) E H TE mode TE 3.83 3.798 3.89 TE.84.869.875 TE 3.54 3.9 3.7 TE3 4. 4.58 4.84 TE 7.6 6.899 6.9 TE 5.33 5.46 5.6 TE 6.76 6.653 6.746 TE3 8.5 7.93 8.6 TM mode TM.45.48.439 TM 3.83 3.798 3.89 TM 5.36 5.8 5.83 TM3 6.38 6.57 6.335 TM 5.5 5.493 5.55 TM 7.6 7.49 7.4 TM 8.47 8.59 8.449 TM3 9.76 9.769 9.84 kc
..3. Wave-Numbe Detemination fo Given Popagation Constant A genealized appoach fo nding the wave numbe k o is descibed fo a given popagation constant...3.. Fomulation. Again the same appoach as used ealie can be followed by using eithe the E o H eld. We illustate the case fo the E eld, which is the same fo the H eld. The vecto wave equation fo the E eld is given by E k o " E= (93) whee E = E xbx + E yby + E zbz exp(jz). By doing the cul-cul opeation and sepaating the tansvese fom the longitudinal components, equation (93) can be divided into two equations and ewitten as t j t E z E t t E t = k " o E t (94) [ t ( t E z + je t )] = k " o E z (95) and E t = E xbx + E yby. Sepaate fom k o and to have eal-valued matices intoduce the scaling E t = E t E z = E z j Then equations (94) and (95) can be witten as (96) (97) t t E t + t E z + E t = k o " E t (98) [ t ( t E z + E t )] = k o " E z (99) To apply Galekin's method to equations (98) and (99), multiply equation (98) with the testing function T t and equation (99) with the testing function T z and integate both the equations ove the coss section of the waveguide ; that is, T t t t E t + (T t t E z +T t E t ) ds = k o " T t E t ds () T z [ t ( t E z +E t )] ds = k o " With the vecto identities T z E z ds () A ( t B) =( t A)B t (AB) () t (AB)ds = Z d (A B) bndl = Z d A(bnB)dl (3) t f A = A t f+f t A (4) t Ads = Z d A bndl (5) equations () and () can be witten in thei weak fom as = k o " (t T t )( t E t )+ T t E z + T t E t 3 ds T t E t ds Z = k o " d T t (bne t )ds (6) ( t T z t E z + t T z E t )ds T z E z ds + @E z T z @n + T zbn E t Z d ds (7) If the waveguide bounday d is assumed to be pefectly conducting, then T t = and T z =on d. Hence, the line integals on the ight-hand side of equations (6) and (7) can be neglected. Multiplying equation (7) with fo the sake of symmety, equations (6) and (7) can be ewitten as h ( t T t )( t E t ) + T t E z + T t E t = k o " i ds T t E t ds (8) 5
( t T z t E z + t T z E t )ds = k o " T z E z ds (9)..3.. Discetization. As the vecto Helmholtz equation is divided into two pats, vecto-based tangential edge elements can be used to appoximate the tansvese elds, and nodal-based st-ode Lagangian intepolation functions can be used to appoximate the z-component. Fom equation (5), which is E t = e tm W tm m= whee m indicates the mth edge of the tiangle and W tm is the edge element fo edge m. The testing function Tt is chosen to be the same as the basis function in equation (5); that is, Tt = W tm. The z-component can be witten as E z = i= e zi i (This equation is eq. (84).) Hee i indicates ith node and i is the simplex coodinate of node i as given in equation (5). Also the testing function Tz is chosen to be the same as the basis function in equation (85) (given above); that is, Tz = i...3.3. Finite element fomulation. Substituting equations (5) and (84) into equations (8) and (9), espectively, integating ove a single tiangula element, and intechanging the integation and summation give 6 m= + + ( t W tm ) ( t W tn )e tm ds (W tm j )e zj ds m= m= = ko m= " (W tm W tn )e tm ds (W tm W tn )e tm ds (n =;;3; j =;;3) () ( i j )e zi ds i= + i= = ko i= " ( i W tn )e tn ds i j e zi ds (j =;;3; n =;;3) () Subscipts fo indicate node numbes and subscipts fo W t indicate edge numbes. Equations () and () can be witten in matix fom as " Sel(tt) S el(tz) S el(zt) S el(zz) # et e z = k o The element matices ae given by S el(tt) = + S el(tz) = " Tel(tt) T el(zz) # et e z ( t W tm ) ( t W tn ) ds S el(zt) = () (W tm W tn ) ds (3) (W tm j )ds (4) S el(zz) = T el(tt) = " T el(zz) = " ( i W tn ) ds (5) ( i j )ds (6) (W tm W tn ) ds (7) i j ds (8) These element matices can be assembled ove all the tiangles in the coss section of the waveguide to obtain a global eigenvalue equation as follows: Stt S zt S tz S zz e t e z = k o T tt T zz et e z (9)
..3.4. Finite element matices. Closed-fom expessions ae deived fo the nite element matices in equations (3) though (7). Fom equations (66) and (67), S el(tt) = " L tm L tn 6A 3 D m D n +!# 5X I tk k= () With equations (73) though (77) and the pocedues in sections.. and.., the emaining element matices ae deived as follows: S el(tz) = Ltm 8A bj (Am+ Bmy ti )+cj(cm+dmx ti ) 3 () S el(zt) = L tn 8A [b i(a n + B n y ti )+ c i (C n + D n x ti )] () S el(zz) = b i b j + c i c j (3) 4A T el(tt) = " L tm L tn 6A 3 T el(zz) = A " 6 = A " 5X k= (m = n) (m 6= n) I tk (4) 9 >= >; (5)..3.5. Numeical example. A compute pogam HELMVEC was witten to calculate the wave numbe of any genealized waveguide stuctue fo a given popagation constant. To check the validity of the pocedue, the st wave numbes with = fo LSM modes of a patially lled squae waveguide (g. ) have been obtained by HELMVEC and compaed with the available esults in the liteatue (ef. 5). (See table 8.) The geomety is modelled by 7 tiangula elements. L Table 8. Wave Numbes fo LSM Modes of Squae Waveguide With = kol Mode HELMVEC Hayata et al. (ef. 5) 8.85 8.893 9.443 9.3896 3.35.75 4.4.3 5.89.677 6.446.45 7.46.988 8.5894.6686 9.837.89.9987.9575..4. Dispesion Chaacteistics of Waveguides This section is simply an extension of section..3. The nite element equations ae eaanged to obtain when the opeating fequency (o the wave numbe) is specied...4.. Fomulation. The weak fom deived in section..3. can be eaanged and equations (8) and (9) can be witten as ( t Tt)( t E t )ds k o " = @ = ko " Tt t E z ds + t T z t E z ds + TtE t ds TtE t ds A (6) t T z E t ds T z E z ds (7) Y L/ ε =. ε =.5 L/ Figue. Patially lled squae waveguide. X..4.. Discetization. As the vecto Helmholtz equation is divided into two pats, vecto-based tangential edge elements can be used to appoximate the tansvese elds, and nodal-based st-ode Lagangian intepolation functions can be used to appoximate the z-component. Fom equation (5), which is E t = m= e tm W tm 7
whee m indicates the mth edge of the tiangle and W tm is the edge element fo edge m. The testing function Tt is chosen to be the same as the basis function in equation (7); that is, Tt = W tm. The z-component is witten as E z = i= e zi i (which is eq. (84)). Hee i indicates ith node and i is the simplex coodinate of node i as given in equation (5). Also the testing function Tz is chosen to be the same as the basis function in equation (5) (given above); that is, Tz = i...4.3. Finite element fomulation. Substituting equations (7) and (5) into equations (5) and (6), espectively, integating ove a single tiangula element, and intechanging the integation and summation give m= ds ko m= = 4 + m= ( t W tm ) ( t W tn )e tm " m= (W tm W tn )e tm ds (W tm j )e zj ds (W tm W tn )e tm ds 3 5 (n =;;3; j =;;3) (8) i= + ( i j )e zi ds i= = i= k o " ( i W tn )e tn ds i j e zi ds (j =;;3; n =;;3) (9) Subscipts fo indicate node numbes and subscipts fow t indicate edge numbes. Equations (84) 8 and (8) can be witten in matix fom as Sel(tt) et e z The element matices ae given by S el(tt) = k o " " Tel(tt) # = T el(tz) et T el(zt) T el(zz) e z (3) ( t W tm ) ( t W tn ) ds T el(tt) = " T el(tz) = T el(zz) = T el(zt) = (W tm W tn ) ds (3) (W tm W tn ) ds (3) (W tm j )ds (33) ( i W tn ) ds (34) ( i j )ds k o " i j ds (35) These element matices can be assembled ove all the tiangles in the coss section of the waveguide to obtain a global eigenvalue equation. Stt e t Ttt =( T tz et ) (36) e z T zz e z T zt..4.4. Finite element matices. Closed-fom expessions ae deived fo the nite element matices in equations (3) though (34). Fom equations (66) and (67),!# 5X I tk " S el(tt) = L tm L tn 6A 3 D m D n ko " k= (37) With equations (73) though (77) and the pocedues in sections.. and.., the emaining element matices ae deived as follows: T el(tz) = L tm bj 8A (A m + B m y ti ) 3 + c j (C m + D m x ti ) (38) T el(zt) = L tn 8A [b i(a n + B n y ti ) + c i (C n + D n x ti )] (39)
T el(zz) = b i + c i 4A S el(zz) = b i b j + c i c j 4A T el(tt) = " L tm L tn 6A 3 = b i b j + c i c j 4A + k o " A 6 5X k= + ko " A (4) I tk (4) (i = j) (i 6= j) 9>= >; (4)..4.5. Numeical examples. A compute pogam HELMVEC3 was witten to calculate the dispesion chaacteistics of inhomogeneously lled waveguides. Fo the st example fo a patially lled ectangula waveguide with b =a =:45, d=b =:5, and " =:45 shown in gue, the values of =k o ae calculated and compaed with analytical esults given by Haington (ef. 9, p. 6). The numeical esults ae pesented in table 9 as a function of a = o. The ectangula waveguide fo this st example is modelled by tiangula elements. b Table 9. Dispesion Chaacteistics of Patially Filled Rectangula Waveguide of Figue =k o b = o Analytical (ef. 9) HELMVEC3..48.47.3...4.8.7.5.6.8.6.3.35 ε =. ε =.45 a Figue. Patially lled ectangula waveguide with b =a =:45, d=b =:5, and " =:45. d Y X Fo the second example fo a patially lled ectangula waveguide with b =a =:45 and " =:45 shown in gue 3, the dispesion chaacteistics wee also calculated and compaed with those given by Haington (ef. 9, p. 6). The numeical esults ae pesented in table. The ectangula waveguide fo this second example is also modelled by tiangula elements. d ε a Figue 3. Patially lled waveguide with b =a =:45 and " =:45...5. Summay In section., a detailed fomulation fo vecto nite elements fo two-dimensional eigenvalue poblems in electomagnetics has been pesented. Edge elements ae used to impose divegence fee condition on elds and hence the nonphysical modes (ef. ), which plagued vecto nite elements fo many yeas ae avoided. When assembling the global matices fom element matices with edge basis functions, a unique global edge diection is dened to ensue eld continuity acoss all edges. The basis function is multiplied by, if the local edge vecto does not have the same diection as the global edge diection. The numeical examples pesented show the validity and accuacy of the analyses and compute codes, espectively. Both the fomulation and the compute codes ae valid fo any abitaily shaped waveguides lled with inhomogeneous mateials. 3. Thee-Dimensional Poblems 3.. Eigenvalues of Thee-Dimensional Cavity Vecto Fomulation The poblem of calculating esonant fequencies of thee-dimensional cavities has been plagued by spuious modes fo many yeas. As mentioned in section., this poblem has been ecently ovecome by using vecto-based tangential edge elements (efs. 5 and 6). In section 3., we fomulate the Galekin nite element method fo thee-dimensional cavities and pesent esults fo vaious shapes of cavities. b Y X 9
Table. Dispesion Chaacteistics of Patially Filled Rectangula Waveguide of Figue 3 =ko a=o Analytical (ef. 9) HELMVEC3 d=a=.4.5.3.4.6.5.56.7.7.7.8.79.78.9.83.83..88.87 d=a=:67.4.5..8.6.6.59.7.7.74.8.8.8.9.88.87..9.9 d=a=:86.4.5.5.44.6.78.74.7.9.88.8.99.5.9.3.3...9 d=a=:375.4.5.68.66.6.9.9.7.5.3.8.3..9..8..5.3 d=a=:5.4.4.4.5.9.89.6..9.7..9.8.5.4.9.3.3..35.35 d=a=:6.4.7.67.5..3.6.8.9.7.3.7.8.3.33.9.38.37..4.4 d=a=:8.4.9.9.5.8.8.6.9.3.7.38.37.8.4.4.9.43.44..44.47 3... Fomulation This fomulation can be followed by using eithe the E o H eld. We illustate the E eld fomulation. Conside the vecto wave equation, whee E E = E xbx + E yby + E zbz k o " E= (43) Intoduce a testing function T = T xbx + T yby + T zbz. To apply Galekin's method, multiply equation (43) with T and integate ove the volume of the cavity v: Z T E k o " T E dv = v With the vecto identity, (44) A ( B)=(A)B(AB) (45) equation (44) can be witten as Z v + (T) E Z v dv = k o " T E Invoking the divegence theoem and Z v Adv = S Z v TEds dv (46) Abnds (47) (A B) bn = A (bn B) (48) whee v indicates the integation ove the volume of the cavity, S indicates the integation ove the oute suface of the cavity, and bn is the outwad unit vecto nomal to the suface, equation (46) becomes Z v S (T) E T bn E dv = k o " Z v TEdv ds (49) Fo a cavity bounded by pefectly conducting electic conducto, the eld as well as the testing function T has to be zeo on the oute suface; hence,
the last tem on the ight-hand side of equation (49) vanishes. Equation (49) can be witten in its nal fom as Z v (T) E 3... Discetization dv = k o" Z v TEdv (5) The volume of the cavity is discetized by using st-ode tetahedal elements such as the one shown in gue 4. The st-ode tetahedon has fou nodes and six edges. The six edges ae fomed as shown in table. e e 4 e 4 e 5 6 e e 3 Figue 4. Fist-ode tetahedal element. Table. Fomation of Edges of Tetahedal Element Node Edge, m i j 3 3 3 4 4 5 4 6 3 4 The electic eld in a single tetahedal element is epesented as E = 6X m= e m W m (5) The six unknown paametes associated with each edge ae e, e, :::, e6. The total eld is obtained by evaluating equation (5). 3 The vecto tangential edge elements W m ae given by (ef. 3) W m = L m ( ti tj tj ti ) (5) In equation (5), m stands fo edge numbe, i and j stand fo the nodes connecting edge i, L m is the length of the edge m, ti and tj ae the simplex coodinates associated with nodes i and j. The simplex coodinates fo the nodes of a tetahedon element ae given by Silveste and Feai in efeence t = V V t = V V t3 = V 3 V t4 = V 4 (53) (54) (55) (56) V whee V is the volume of the tetahedon given by V = 6 V = 6 V = 6 V3 = 6 V4 = 6 x y z x y z x3 y3 z3 x4 y4 z4 x y z x y z x3 y3 z3 x4 y4 z4 x y z x y z x3 y3 z3 x4 y4 z4 x y z x y z x y z x4 y4 z4 x y z x y z x3 y3 z3 x y z Fo any node (i =;;3;4), ti is given by ti = a ti + b ti x + c ti y + d ti z 6V (57) (58) (59) (6) (6) (6)
whee a ti, b ti, c ti,and d ti ae appopiate cofactos picked fom the deteminants in V, V, V 3, and V 4 fo m = ;;3, and 4, espectively. Fom equations (5) and (6), the edge elements ae given as W m = L m 36V [(A xm + B xm y + C xm z)bx +(A ym + B ym x + C ym z)by whee +(A zm + B zm x + C zm y)bz] (63) A xm = a ti b tj a tj b ti (64) B xm = c ti b tj c tj b ti (65) C xm = d ti b tj d tj b ti (66) A ym = a ti c tj a tj c ti (67) B ym = b ti c tj b tj c ti = B xm (68) C ym = d ti c tj d tj c ti (69) A zm = a ti d tj a tj d ti (7) B zm = b ti d tj b tj d ti = C xm (7) C zm = c ti d tj c tj d ti = C ym (7) Also W m can be shown to satisfy the condition, b tm W (edge m) m = (othe edges) (73) whee b tm is the unit vecto along the diection of the edge. 3..3. Finite Element Fomulation Substituting equation (5) into equation (5), integating ove the volume of one tetahedal element, and intechanging the summation and integation give m= 6X Z = k o (W m )(W n )e m dv 6X Z " m= (W m W n )e m dv (n =;;:::;6) (74) whee indicates integation ove the volume of tetahedon. This can be witten in matix fom as [S el ][e] =k o [T el][e] (75) whee the element matices ae given by [S el ]= Z [T el ]=" Z (W m )(W n )dv (76) (W m W n )dv (77) These element matices can be assembled ove all the tetahedal elements in the cavity volume to obtain a global eigenmatix equation, [S][e] =ko [T][e] (78) To ensue eld continuity acoss all edges, a unique global edge diection is dened (i.e., always pointing fom the smalle node numbe to the lage node numbe) so that equation (5) has to be multiplied by, if the local edge vecto does not have the same diection as the global edge diection. The electic eld is zeo on the PEC boundaies. It is imposed by taking the coecients of equation (5) as zeos. In othe wods, the edges on the bounday ae simply ignoed when foming the nite element matices and hence educing the ode of the matices to be solved. 3..4. Finite Element Matices The aim of this section is to obtain closed-fom expessions fo equations (76) and (77). Fom equation (63), W m = and hence L m 8V Czm bx + C xm by + B ym bz (79) ( W m )(W n )= L ml n (8V ) (C zm C zn + C xm C xn + B ym B yn ) (8) Fom equations (76), (77), and (8), and making use of the integation fomulas given in efeence 8, the closed-fom expessions fo element matices ae given as S el = L ml n 34V 3 (C zm C zn + C xm C xn + B ym B yn ) (8) T el = " L m L n 96V 3 X k= I k (8)
whee I = AxmAxn + AymAyn + AzmAzn I =(AymByn + AynBym + AzmBzn + AznBzm)x tet Z Y X I 3 =(AxmBxn + AxnBxm + AzmCzn + AznCzm)y tet I 4 =(AxmCxn + AxnCxm + AymCyn + AynCym)z tet I 5 4X! = (B zmczn + BznCzm) xiyi +6x tet y tet i= I 6 4X! = (B xmcxn + BxnCxm) yizi +6y tet z tet i= I 7 4X! = (B ymcyn + BynCym) xizi +6x tet z tet I 8 = (B ymbyn + BzmBzn) I 9 = (B xmbxn + CzmCzn) I = (C xmcxn + CymCyn) 3..5. Numeical Examples 4X i= 4X i= 4X i= i= x i +6x tet y i +6y tet z i +6z tet A compute pogam FEM3D was witten to calculate the eigenvalues of a thee-dimensional cavity. Fo a thee-dimensional poblem, the numbe of vaiables inceases dastically compaed with those fo a two-dimensional poblem. Hence it is not economical to use a genealized eigenvalue solve. The pogam FEM3D was witten to take advantage of the spase natue of the nite element matices. This pogam exploits the symmety of the matices and stoes only the nonzeo enties in the lowe tiangle of the matices; hence, consideable savings in memoy esult. It also makes use of the spase eigenvalue solves available in VECLIB (ef. 9) and esults in faste computation. The following numeical examples have been taken fom Chattejee, Jin, and Volakis (ef. 7), in which the esonant wave numbes fo vaious theedimensional cavities have been calculated by using a dieent kind of edge elements. The analytical esults mentioned hee ae also taken fom efeence 7. Ai-lled ectangula cavity: The eigenvalues of an ai-lled ectangula cavity (shown in g. 5)!!! Figue 5. Ai-lled ectangula cavity..75 cm. Z Y X Size: by.5 by Figue 6. Half-lled ectangula cavity with dielectic mateial " =: and lled fom z =:5 to cm. Size: by. by cm. wee calculated by FEM3D and ae pesented in table. The cavity geomety is epesented by 343 tetahedal elements. Table. Eigenvalues of Ai-Filled Rectangula Cavity k o,cm Analytical FEM3D Mode (ef. 7) (343 elements) Refeence 7 TE 5.36 5.4 5.3 TM 7.5 6.94 6.977 TE 7.53 7.37 7.474 TE 7.53 7.56 7.573 TE 8.79 8.64 7.99 TM 8.79 8.64 8. TM 8.886 8.75 8.57 TE 8.947 8.87 8.795 Half-lled ectangula cavity: A half-lled (lled fom z = :5 to cm) ectangula cavity with dielectic mateial " = : is shown in gue 6. 3
The eigenvalues wee calculated by FEM3D and ae pesented in table 3. The cavity geomety was modelled with 65 tetahedal elements. Table 3. Eigenvalues of Half-Filled Rectangula Cavity ko,cm FEM3D Mode Analytical (65 elements) Refeence 7 TE z 3.538 3.54 3.534 TE z 5.445 5.4 5.44 TE z 5.935 5.93 5.96 TE z3 7.53 7.38 7.5 TE z 7.633 7.56 7.56 TE z3 8.96 8.3 8.56 Cicula cylindical cavity: The eigenvalues of an ai-lled cicula cylindical cavity (shown in g. 7) wee calculated by FEM3D and ae pesented in table 4. The geomety of this cavity was epesented by 633 tetahedal elements. Z Y X Figue 7. Ai-lled cicula cylindical cavity. Dimensions ae in centimetes. Table 4. Eigenvalues of Ai-Filled Cicula Cylindical Cavity ko, cm FEM3D Mode Analytical (633 elements) Refeence 7 TM 4.8 4.78 4.89 TE 7.83 7. 7. TE 7.83 7.9 7.88 TM 7.65 7.575 7.633 TM 7.65 7.6 7.74 TM 7.84 7.9 7.94 TE 8.658 8.676 8.697 TE 8.658 8.865 Spheical cavity: The spheical cavity geomety was epesented by 473 tetahedal elements. The.5 eigenvalues, calculated by FEM3D, ae pesented in table 5. The spheical cavity had a adius of cm. Table 5. Eigenvalues of Spheical Cavity With Radius of cm ko, cm FEM3D Mode Analytical (473 elements) Refeence 6 TM.744.799.799 TM.744.8.8 TM.744.87.8 TM 3.87 3.96 3.948 TM, even 3.87 3.976 3.986 TM, odd 3.87 3.986 3.994 TM, even 3.87 3.994 4.38 TM, odd 3.87 3.998 4.48 TE 4.493 4.4 4.433 TE, even 4.493 4.478 4.47 TE, odd 4.493 4.5 4.549 3..6. Summay In section 3., vecto nite elements intoduced in section. ae extended to solve thee-dimensional eigenvalue poblems by using tangential edge basis functions fo tetahedal elements. Spuious solutions ae completely avoided because of the divegence fee natue of the edge elements. Spase matix eigenvalue solves ae used to take advantage of the spasity and symmety of the nite element matices; this esults in consideable savings in compute memoy and computational time. The numeical esults pesented fo cavities with dieent shapes pove the validity of the analysis and accuacy of the compute codes pesented in this section. 4. Concluding Remaks A thoough fomulation of FEM fo vaious eigenvalue poblems in electomagnetism has been pesented. The use of ecently developed edge basis functions fo vecto nite elements has been demonstated fo two-dimensional and thee-dimensional poblems. Tiangula elements fo two-dimensional poblems and tetahedal elements fo theedimensional poblems ae used to model complex geometical shapes because of thei ability to epesent such shapes accuately. Implementation of conventional nodal-based scala fomulation fo two-dimensional homogeneous poblems is demonstated in section.. The compute code developed can be used to calculate the 4
eigenvalues and eld intensity pattens of any abitaily shaped waveguide lled with homogeneous mateial. The numeical esults fo vaious waveguides and the eld intensity pattens fo vaious modes have been pesented in section. which show the validity of the analysis and compute code. Simple waveguide shapes have been chosen fo demonstation puposes because thei eld distibutions ae well known. In section., the two-dimensional edge basis functions fo tiangula elements ae intoduced to model tansvese vecto elds in waveguides. Fo waveguides o micowave cicuit design poblems in geneal, knowledge of a popagation constant at a given fequency is desiable. A step-by-step fomulation is done to detemine eithe the wave numbe o the popagation constant, if one of them is specied. Numeical examples ae pesented to validate the analysis and the compute codes. Since eal poblems involve thee-dimensional geometies, a fomulation fo the calculation of eigenvalues fo such geometies is pesented in section 3. The thee-dimensional edge basis functions ae intoduced fo tetahedal elements. Numeical examples fo vaious geometies ae pesented. Compaison of the numeical data with the available data in the liteatue shows the validity and accuacy of the analysis. Fo the thee-dimensional poblems, the numbe of vaiables inceases dastically compaed with those fo the two-dimensional poblems. In the theedimensional compute codes, spasity of FEM matices have been exploited by stoing only the nonzeo elements and symmety is utilized by stoing only eithe the uppe o lowe pat of the matices. The spase matix eigenvalue solves ae used to eciently solve the FEM equations. NASA Langley Reseach Cente Hampton, VA 368- Octobe 3, 994 5
Appendix Compute Pogams Compute pogams wee witten to implement the analysis pesented in this epot. All the pogams ae witten in FORTRAN language. These pogams take *.MOD le containing the meshing infomation fom COSMOS/M (ef. 4). Also these compute pogams make use of the optimized libay outines of EISPACK (efs. 7 and 8) and VECLIB (ef. 8) on the CONVEX compute. HELM is a two-dimensional nite element pogam to calculate the eigenvalues of homogeneously lled waveguides. This pogam implements the analysis descibed in section.. HELMVEC is a two-dimensional nite element pogam to calculate the eigenvalues of homogeneously lled waveguides. Unlike HELM, this pogam uses the vecto basis functions and implements the analysis pesented in section... HELMVEC is a two-dimensional nite element pogam to calculate the eigenvalues of inhomogeneously lled waveguides. This pogam implements the thee-component vecto eld fomulation discussed in section.. and makes use of edge basis functions fo tansvese elds and scala basis functions associated with nodes fo longitudinal elds. HELMVEC is a two-dimensional nite element pogam fo nding out the wave numbe of an inhomogeneously lled waveguide when the popagation constant is specied. This pogam implements the fomulation pesented in section..3. HELMVEC3 is a two-dimensional nite element pogam fo detemination of popagation constant of an inhomogeneously lled waveguide at any fequency. This pogam implements the fomulation given in section..4. FEM3D and FEM3D ae thee-dimensional - nite element pogams to calculate the eigenvalues of inhomogeneously lled cavities. These pogams implement the fomulation pesented in section 3.. FEM3D uses the EISPACK (efs. 7 and 8) outines, wheeas FEM3D exploits the spasity and symmety of FEM matices and uses VECLIB (ef. 9) outines. These compute pogams ae available on equest fom Infomation and Electomagnetic Technology Division Electomagnetic Reseach Banch M.S. 49 NASA Langley Reseach Cente Hampton VA 368-6
Refeences. Silveste, P. P.; and Feai, R. L.: Finite Elements fo Electical Enginees. Second ed., Cambidge Univ. Pess, 99.. Rahman, B. M. Azizu; and Davies, J. Bian: Penalty Function Impovement of Waveguide Solution by Finite Elements. IEEE Tans. Micow. Theoy & Techni., vol. MTT-3, no. 8, Aug. 984, pp. 9{98. 3. Bossavit, Alain: Simplicial Finite Elements fo Scatteing Poblems in Electomagnetism. Comput. Methods Appl. Mech. & Eng., vol. 76, 989, pp. 99{36. 4. COSMOS/M Use Guide. Volume Pepocessing, Analysis and Postpocessing Inteface, Vesion.7. Stuctual Reseach and Analysis Cop., May 993. 5. I-DEAS Relational Data Base. I-DEAS V Pod. Bul., Stuctual Dynamics Reseach Cop., Jan. 99. 6. Silveste, P.: Constuction of Tiangula Finite Element Univesal Matices. Int. J. Nume. Methods Eng. vol., no., 978, pp. 37{44. 7. Smith, B. T.; Boyle, J. M.; Dongaa, J. J.; Gabow, B. S.; Ikebe, Y.; Klema, V. C.; and Mole, C. B.: Matix Eigensystem Routines EISPACK Guide. Second Ed., Spinge-Velag, 976. 8. Gabow, B. S.; Boyle, J. M.; Dongaa, J. J.; and Mole, C. B.: Matix Eigensystem Routines EISPACK Guide Extension. Spinge-Velag, 977. 9. Haington, Roge F.: Time-Hamonic Electomagnetic Fields. McGaw-Hill Book Co., Inc., 96.. Macuvitz, N., ed.: Waveguide Handbook. McGaw-Hill Book Co., Inc., 95.. Koshiba, Masanoi; Hayata, Kazuya; and Suzuki, Michio: Impoved Finite-Element Fomulation in Tems of the Magnetic Field Vecto fo Dielectic Waveguides. IEEE Tans. Micow. Theoy & Techni., vol. MTT-33, no. 3, Ma. 985, pp. 7{33.. Whitney, Hassle: Geometic Integation Theoy. Pinceton Univ. Pess, 957. 3. Lee, J.-F.; Sun, D. K.; and Cendes, Z. J.: Full-Wave Analysis ofdielectic Waveguides Using Tangential Vecto Finite Elements. IEEE Tans. Micow. Theoy & Techni., vol. 39, no. 8, Aug. 99, pp. 6{7. 4. Reddy, J. N.: An Intoduction to the Finite Element Method. McGaw-Hill Book Co., Inc., 984. 5. Hayata, Kazuya; Koshiba, Masanoi; Eguchi, Masashi; and Suzuki, Michio: Vectoial Finite-Element Method Without Any Spuious Solutions fo Dielectic Waveguiding Poblems Using Tansvese Magnetic-Field Component. IEEE Tans. Micow. Theoy & Techni., vol. MTT-34, no., Nov. 986, pp. {4. 6. Lee, Jin-Fa; and Mitta, Raj: A Note on the Application of Edge-Elements fo Modeling Thee-Dimensional Inhomogeneously-Filled Cavities. IEEE Tans. Micow. Theoy & Techni., vol. 4, no. 9, Sept. 99, pp. 767{773. 7. Chattejee, A.; Jin, J. M.; and Volakis, J. L.: Computation of Cavity Resonances Using Edge-Based Finite Elements. IEEE Tans. Micow. Theoy & Techni., vol. 4, no., Nov. 99, pp. 6{8. 8. Zienkiewicz, O. C.: The Finite Element Method in Engineeing Science. McGaw-Hill Book Co., Inc., 97. 9. CONVEX VECLIB Use's Guide, Seventh ed., CONVEX Compute Cop., 993. 7
REPORT DOCUMENTATION PAGE Fom Appoved OMB No. 74-88 Public epoting buden fo this collection of infomation is estimated to aveage hou pe esponse, including the time fo eviewing instuctions, seaching existing data souces, gatheing and maintaining the data needed, and completing and eviewing the collection of infomation. Send comments egading this buden estimate o any othe aspect of this collection of infomation, including suggestions fo educing this buden, to Washington Headquates Sevices, Diectoate fo Infomation Opeations and Repots, 5 Jeeson Davis Highway, Suite 4, Alington, VA -43, and to the Oce of Management and Budget, Papewok Reduction Poject (74-88), Washington, DC 53.. AGENCY USE ONLY(Leave blank). REPORT DATE 3. REPORT TYPE AND DATES COVERED Decembe 994 Technical Pape 4. TITLE AND SUBTITLE Finite Element Method fo Eigenvalue Poblems in Electomagnetics 5. FUNDING NUMBERS WU 55-64-5-4 6. AUTHOR(S) C. J. Reddy, Manoha D. Deshpande, C. R. Cockell, and Fed B. Beck 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) NASA Langley Reseach Cente Hampton, VA 368-8. PERFORMING ORGANIZATION REPORT NUMBER L-739 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) National Aeonautics and Space Administation Washington, DC 546-. SPONSORING/MONITORING AGENCY REPORT NUMBER NASA TP-3485. SUPPLEMENTARY NOTES Reddy: NRC-NASA Resident Reseach Associate, Langley Reseach Cente, Hampton, VA; Deshpande: ViGYAN, Inc., Hampton, VA; Cockell and Beck: Langley Reseach Cente, Hampton, VA. a. DISTRIBUTION/AVAILABILITY STATEMENT b. DISTRIBUTION CODE Unclassied{Unlimited Subject Categoy 3 Availability: NASA CASI (3) 6-39 3. ABSTRACT (Maximum wods) Finite element method (FEM) has been a vey poweful tool to solve many complex poblems in electomagnetics. The goal of the cuent eseach at the Langley Reseach Cente is to develop a combined FEM/method of moments appoach to thee-dimensional scatteing/adiation poblem fo objects with abitay shape and lled with complex mateials. As a st step towad that goal, an execise is taken to establish the powe of FEM, though closed bounday poblems. This pape demonstates the development of FEM tools fo twoand thee-dimensional eigenvalue poblems in electomagnetics. In section, both the scala and vecto nite elements have been used fo vaious waveguide poblems to demonstate the exibility of FEM. In section 3, vecto nite element method has been extended to thee-dimensional eigenvalue poblems. 4. SUBJECT TERMS 5. NUMBER OF PAGES Finite element; Waveguide; Eigenvalues; Computational method; Electomagnetic 36 6. PRICE CODE A3 7. SECURITY CLASSIFICATION 8. SECURITY CLASSIFICATION 9. SECURITY CLASSIFICATION. LIMITATION OF REPORT OF THIS PAGE OF ABSTRACT OF ABSTRACT Unclassied Unclassied Unclassied NSN 754--8-55 Standad Fom 98(Rev. -89) Pescibed by ANSI Std. Z39-8 98-