DL_PERU

1Higher Order Modelling forComputational Electromagnetics

Roberto D. GragliaDipartimento di ElettronicaPolitecnico di Torino Italy

e-mail: [email protected]

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6Higher Order Modelling forComputational Electromagnetics

Roberto D. GragliaDipartimento di Elettronica e Telecomunicazioni

Politecnico di Torino Italy

e-mail: [email protected]

Considered equations & problems

We deal with Electromagnetic problems modelled byMaxwells equations.

The techniques described have applications to manypractical problems, e.g.: EMC & EMP; Shielding radiationfrom printed circuits; Microwave hazards; Electromagneticradiation from and penetration into vehicles, aircrafts,ships, missiles; Antennas near ground; Design offrequency selective surfaces for reflector antennas andradomes; Radar scattering; Propagation in opticalcomponents & systems; etc.

7

Numerical methods (for linear problems)- in 3 slides -

9

Introduce a set of testing functions {wm}, and a suitable inner product

For a linear, inhomogeneous equation:

10

Then obtain a matrix equation

for a square, non-singular matrixwith

The solution

or approximate depending on the choice of {fn} and {wm}

if wn=fn Galerkins method

may be exact

How to choose {fn}, {wm}? The set {fn} (linearly independent functions) has to

approximate f reasonably well; The set {fn} could be chosen in order to satisfy the

BCs (FEM applications) ; The set {wm} (linearly independent functions) has to

be chosen so that depends on relatively independent properties of g;

The choice depends on the desired accuracy of the solution;

The ease of evaluation of matrix elements; The realization of a well-conditioned matrix [m,n].

11

12

Representations of Fields in Terms of Potentials (Frequency-Domain)

Solutions of Maxwells equations can be expressed in terms of potentials:

13

Representations of Fields in Terms of Potentials (Frequency-Domain)

Solutions of Maxwells equations can be expressed in terms of potentials:

To invert the differential operator is to solve the problem to do this one has to know (find) the Green function G of the given problem;

If G is known, then the solution is simply obtained by integration;

Unfortunately, for all problems of practical interest G is unknown numerical approaches to differential/integral problems are required because of the ignorance of the Green function;

The numerical problem could be well- or bad-posed; this depends on the properties of the operator;

14

In general, in integral formulations, use of the proper (unknown) G is avoided by resorting to general principles (e.g., equivalence principle), or to integral theorems (e.g., Green theorems);

Integral formulations guarantee the BCs and all the physical properties of the solutions, as a matter of fact, in many cases, the proper integral equations are obtained only at the moment of imposing the BCs (in the finite region);

This is why IE are widely used in EM (since radiation conditions / far-field conditions are automatically satisfied);

Because of the Green kernel, the matrix approximation of the problem is given by a dense matrix.

15

FEM MOMDifferential formulation Integral formulation

Non linear problems can be dealt withApplications to linear problems are well known

BCs have to be enforced BCs automatically satisfied

Infinite domains: there is some problems Infinite domains: no problem

Sparse matrices Dense matrices (problems in dealing with very large problems)

The integrals to be evaluated are simpleThe singularities of the kernel render numerical integration quite difficult

Complex geometry are easily dealt with

Higher order models have been introduced more recently

16

Higher order modelling

Required:

To better represent the geometry of the problem. To reduce the L2 (least squares norm) of the

solution error on regions of interest.

- It usually improves the convergence of the results and/or reduces the size of the numerical matrices.


18

Regarding the geometry of the problem;

need to approximate the curvature of the geometry;need to approximate small details in multiscale problems.

use subsectional bases thereby subdividing the geometry intosimple (curved) elements of small size.

Use subsectional bases thereby subdividing the geometryinto simple (curved) elements of small size.

19

Pablo Picasso: "Retrato de Ambroise Vollard (1910).

Regarding the geometry of the problem

20

Contribution from specular points (in the optical limit).Points of reflection on the body at which the angle of incidence isequal to the angle of reflection relative to the observation point.

e.g., the RCS of a metallic sphere of radius a is =a2

21

In the high-frequency range, the PEC-sphere models shown here poorly model the commonly used sphere-benchmark (see below)

NASA Communications SatelliteEcho Project (1960-1969)


22

Travelling waves (contribution from points with high curvature).

e.g., estimation of the near-nose-oncross-sections of long, thin bodies.


23

Creeping waves

of importance for the analysis in the shadow region(the importance of the creeping waves depends on the dimension in wavelength of the body).


Regarding the expansion/testing functions used in the numerical application.

reduce the L2 (least squares norm) of the solution error on regions of interest;improve convergence of the results;reduce the size of the numerical matrices.

use vector functions of high polynomial order.

24

How to get curved (i.e., distorted) elements.

25Source: Zinkiewiczs book

How to get curved / distorted elements.

Any curved cell is obtained by mapping a parent cell into the object domain.

The parent cell for triangular patches is a rectilinear triangle. In object space, the 3 edges are the zero-coordinate lines 1, 2, 3=0.

i is the area coordinate i = Ai /AT Dependency relations: 1 + 2 + 3 = 1

How to get curved / distorted elements.

The rectilinear triangle on the right-hand side is r = 1 r1+ 2 r2+ 3 r3

Where 1, 2, 3 are 3 linear shape functions, each associated to a different corner node of the cell.

Interpolatory polynomialsfor shape functions.

Use Lagrange interpolation polynomials written interms of interpolatory polynomials of Silvester.

Interpolatory polynomials of Silvester

Use polynomials of degree i in , where is in the interval[0, 1].

The parameter p indicates the number of uniform subintervals into which the interval is divided.

The polynomial is unity at =i/p and has zeros at =0, 1/p, 2/p, , (i-1)/p

01

1)(!1

),(1

0

i

pikpipR

i

ki

Silvester polynomials: example for p=1

There are 2 polynomials: one of 0th and one of 1st degree. Please draw the polynomials on the interval [0, 1].

01

1)(!1

),(

),1(;1),1(

1

0

10

i

pikpipR

RR

i

ki


There are 2 polynomials: one of 0th and one of 1st degree. Please draw the polynomials on the interval [0, 1].

),1(;1),1( 10 RR


Three polynomials: one of 0th, one of 1st and one of 2nd degree. Please draw these polynomials on the interval [0, 1].

01

1)(!

1),(

21)12(2),2(;

12),2(;1),2(

1

0

210

i

pikpipR

RRR

i

ki


Please draw these 3 polynomials on the interval [0, 1].

21)12(2),2(;

12),2(;1),2( 210

RRR

34

Scalar Lagrangian interpolation on the canonical elements: the segment.

There are two coordinates: 1, 2 . These are dependent coordinates (1+2 =1)

ij(1, 2 )=Ri(p, 1)Rj(p, 2 ) with i+j=pis a p-th order Lagrangian polynomialinterpolating points within a segment whosenormalized coordinates (1, 2 ) are (i/p, j/p).

35

Interpolation on a segment, example for p=1

There are two interpolatory polynomials of 1st degree. ij(1, 2 )=Ri(p, 1)Rj(p, 2 ), (with i+j=1).

Please draw the polynomials on the interval 1=[0, 1].

221102101

120112110

),1(),1(),(),1(),1(),(

RRRR

36


221102101

120112110

),1(),1(),(),1(),1(),(

RRRR

37


Example for p=2: there are three interpolatrory polynomials of 2nd degree.

ij(1, 2 )=Ri(p, 1)Rj(p, 2 ), (with i+j=2). Please draw the polynomials on the interval 1=[0, 1].

2)12(2),2(),2(),(

22),2(),2(),(2

)12(2),2(),2(),(

2222102102

2121112111

1120122120

RR

RR

RR

38

Interpolation on a segment for p=2

2)12(2),2(),2(),(

22),2(),2(),(2

)12(2),2(),2(),(

2222102102

2121112111

1120122120

RR

RR

RR

39

Scalar Lagrangian interpolationon a segment.

For the p-th order Lagrangian polynomial interpolation of a segment there are (p+1) interpolating polynomials of p-th degree.

Parameterization of the points on a triangle.

Recall that the coordinates of a rectilinear triangle may beparameterized as

where ri is a vertex position vector for vertex i.

40

,332211r rrr

Parametrization of the points on a triangle.

A Lagrange parametrization of order q for a curvilineartriangle can be expressed in terms of interpolatingpolynomials of the Silvester form as:

where a triple indexing scheme is used to label the positionvector rijk interpolating the point with normalized coordinates(1, 2, 3)=(i/q, j/q, k/q).

qkjiqRqRqR kjq

kjiiijk

),,(),(),( 32

0,,1

rr

Geometry description for a triangle.

Normalized coordinates related by 1 + 2+ 3 =1

Edge vectors derived from the independent coordinates 1 and 2i=r/ i , i=1,2,

from which the edge vectors that follow are found:

Edge vectors of a triangle.

1 = - 2, 2 = 1, 3 = 2 - 1

Gradient vectors of a triangle.

The gradient vectors are determined from the edgevectors as:

i = (n x i ) /J

where n = 1 x 2/J is the unit vector normal to thetriangle while J= | 1 x 2| is the Jacobian.

Edge and gradient vectors of a triangle.

The above definitions for edge, (height) and gradient vectors apply for triangles on curved surfaces with an extended interpretation

Triangle tangent to a curvilinear triangle at a point. The curvilinear and (rectilinear) tangent triangles have the same element coordinates, Jacobian, edge vectors, and height vectors at the point of tangency.

Why to use curved (i.e., distorted) elements?

46

the same model (that is, the same database) used by structural /mechanical engineers can be used this is a big plus!!!

better approximations of boundaries are obtainedwith very small error in volume/surface/line models (no rescaling)

usually a smaller number of unknowns is needed whenever higher-order expansion functions are used shorter computation time

Why to use curved (i.e., distorted) elements?

47

mixed approaches (FEM + MoM) are facilitated because the expansion functions of the two methods can be chosen within the same set.

there is the possibility to construct singular functions able to model singular current or field behaviors (though applications of this are rather new).

But, is there any problem?

But, is there any problem?

48

In FEM applications you now have to use quadrature to evaluate the matrix entries, whereas for non-distorted cells there are often closed form results possible.

For MoM applications people are already using quadrature, but the treatment of the Green's function singularity might be different with curved cells.

The main cost of curved cells is(1) the additional data structure necessary in the model,(2) the additional computation arising from the quadrature.

Vector functions

We first consider the lowest order functions.

Then we move on and consider higher-order vector functions.

49

The family of generalized triangle basis functions (for current representation)

50

Source: Don Wiltons notes

Div-Conf functions on a triangular element

51

Divergence-conforming functions of the Nedelec typemaintain only normal continuity across elementBoundaries (they do not prescribe tangential continuity).

Div-Conf functions on a triangular element

52

They eliminates spurious solution of the EFIE operatorwhile discarding the highest order degrees of freedomassociated with the nullspace of the divergence operator inthe EFIE operator.

53

Divergence-conforming bases on triangles.

.1

,1

,1

12213

31132

23321

J

J

J

Zeroth-order bases. Three vector basis

functions of first order. They have constant

normal and linear tangential (CN/LT) components at element edges.

54

Curl-conforming bases on triangles.

.,,

:

12213

31132

23321

n Zeroth-order bases Three vector basis functions

of first order. They have constant

tangential and linear normal (CT/LN) components at element edges(prove this by use of eq. (48) of 1997 paper).

55

Completeness of zeroth order triangular bases(we consider only the curl-conforming ones).

.,,

12213

31132

23321

The basis set is incomplete to first order since 6 degrees of freedom are required to model linear variations in two independent vector components on a surface.

56


.,,

12213

31132

23321

To render the bases first-order complete one must include the curl-free combinations:

.,,

1221

22

11

57


.,,

12213

31132

23321

Completeness to zeroth-order is proved by noticing that the following linear combinations are able to represent two independent basis vectors on a 2D element (verify this by yourself and express 3).

.,

223

132

58


.,,

12213

31132

23321

The Nedelec conditions also require completeness of the curl to the same order as the bases. Completeness of the curl to zeroth-order follows from (verify this by use of (50, 53) of the 1997 paper).

3,2,1,2 nJ

Higher order vector bases

59

Higher order bases

One can construct higher order bases complete to order p by forming the product of zeroth-order bases with complete polynomial factors of order p.

The set of polynomials factors used may take one of several different forms chosen for convenience.

60

pkji:alhierarchicptsr:oryinterpolat

jipsr :ousinhomogeneptsr:shomogeneou

sj

ri

tsr

0,),)))

,0,

,

321

321

321

,,(H(p,R(p,R(p,R

ijk

tsr

Interpolatory vector functions

For interpolatory vector functions the key idea is to use shifted polynomials of Silvester to move interpolation points away from two of the edges-those along with the tangential (for curl-conf.) and the normal (for div.-conf.) components of the 0th-order basis factor vanish.

61

Interpolation nodes for curl- or divergence-conforming bases on triangular elements. Only nodes in basis subset 1i j k or 1i j k for p = 3 are shown.

62

Interpolation nodes for curl- or divergence-conforming bases on quadrilateral elements. Only nodes in basis subset 3ik; j or 3ik; j for p = 2 are shown.

Interpolatory vector functions

The polynomials with interpolating nodes as shown in figure are of global order p=3, and have the form:

63

Interpolation nodes for curl- or divergence-conforming bases on triangular elements. Only nodes in basis subset 1i j k or 1i j k for p = 3 are shown.

21,,2,1,;,,1,0

),2(),2(),2( 321

pkjipkjpipRpRpR kji

with

64

Volumetric Elements

65

Volumetric Elements

66

Volumetric Elements

67

Volumetric Elements

68

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

k0*a

k

z

/

k

0

m=1

2

3

4

5

6 7

3

8

2

1

1

0

2

3

4

5

6

0

Example of results for surface elements: INHOMOGENEOUSLY FILLED WAVEGUIDES

a=2b, h=0.1br =10

a

bhr

0

analytical

+, * FEM

69

102 103 10410-5

10-4

10-3

10-2

10-1

100

MATRIX DIMENSIONS

R

E

L

A

T

I

V

E

E

R

R

O

R

INHOM. FILLED WG - RELATIVE ERROR

r0

P=1

P=2P=3

70

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

Normal field component at the air-dielectric interface.

P=0~ 1800 UNKNOWNS

E

y

a

/

E

y

d

x

P=1

P=2 P=3a

bhr

0

P N0 16411 18732 18973 1681

a =2b, h =0.2 br =10, k0a =7

Fundamental mode

71

Image waveguide

0 2 4 6 820

25

30

35

40

F

R

E

Q

U

E

N

Z

A

G

H

z

Kz mm-1

a

ba`

b`

a =1.3 mm, b=1.6 mm, a` =0.55 mm, b`=0.82 mm

rx =170, ry = ry 85

[1] J.I.Askne, E.L. Kolberg, L. Pettersson,``Propagation in a waveguide partially filled with anisotropic dielectric material``, IEEE Trans. MTT, vol.30, n5, pp.795-799, Maggio 1982.

24 triangoli19 nodi

291 triangoli166 nodi

- - p=0, (37 incognite), mesh densa- p=3, (325 incognite), mesh lasca

espansione modale [1]

modo 1

modo 2

3

4

72

Hierarchical Vector Basis Functions for Meshes withHexahedra, Tetrahedra, and Triangular Prism Cells

Roberto D. Graglia*, and Andrew F. Peterson

* Politecnico di Torino ITALY Georgia Institute of Technology USA

e-mail: [email protected] ; [email protected]

73

NEW VECTOR BASESHierarchical curl / divergence-conforming vector bases ofthe Nedelec kind for 2D and 3D cells that:

A.Allow one to mesh a structure with differently shaped cells2D Triangles & Quadrilaterals3D Tetrahedrons & Bricks & Triangular Prisms

B.Yield well-conditioned system matrices, even in case ofhigh order bases

Why Nedelec spaces?

The Nedelec curl-conforming spaces eliminate the degrees of freedom associated with the gradient of a higher degree polynomial.

They do this locally, without introducing global constraints that complicate the definition of basis functions, the sparseness of the FEM system, boundary conditions, etc.

These are the easy degrees of freedom to discard (eliminating all the gradient DoF requires global operations, matrix partitioning, etc.)

75

Other existing vector bases

In the open literature, only one group has developed (curl-conforming) bases with similar features; however those basesdo not satisfy Nedelecs constraints:S.Zaglmayr, High order Finite Element Methods forelectromagnetic field computation, Ph. D. Thesis, JohannesKepler Universitt, Linz, Austria, July 2006.

76

OUR PAPERS ON THIS HIERARCHICAL SUBJECT FOR

TRIANGULAR & TETRAHEDRAL cells, QUADRILATERAL & BRICK & PRISM cells;

1. A. F. Peterson, R. D. Graglia, Scale factors and matrix conditioning associated withtriangular-cell hierarchical vector basis functions IEEE Antennas and WirelessPropagation Letters, vol. 9, pages 40-43, 2010.2. R. D. Graglia, A. F. Peterson, and F. P. Andriulli, Curl-conforming hierarchical vectorbases for triangles and tetrahedra, IEEE TAP, vol. 59, no. 3, pp. 950-959, Mar. 2011.3. A. F. Peterson, R. D. Graglia, Evaluation of hierarchical vector basis functions forquadrilateral cells, IEEE Trans. Magn., due to appear May 2011.4. R. D. Graglia, and A. F. Peterson, Hierarchical curl-conforming Nedelec elements forquadrilateral and brick cells, IEEE TAP, accepted for publication, Dec. 2010.5. R. D. Graglia, and A. F. Peterson, Hierarchical curl-conforming Nedelec elements fortriangular-prism cells,'' in preparation, Nov. 2010.6. R. D. Graglia, and A. F. Peterson, Hierarchical divergence-conforming Nedelec elementsfor volumetric cells,'' Mar. 2011.

77

HIERARCHICAL BASES: The basis of order m is a subset of the basis of order (m+1). In FEM and MoM applications, these bases enable different expansion orders on different elements in the same mesh (p-adaptation)

Example 2D-polynomial hierarchical bases:

0__ 1

1__ x ______ y

2__ x2 ___ xy ____ y2

3__ x3 __ x2 y ____ x y2 __ y3

4__ x4 __ x3 y __ x2 y2 ___ x y3 __ y4

Our ideaWe define and work only with hierarchical

polynomial bases;Then define a redundant complete vector set by

multiplying the zeroth-order vector functions withthe hierarchical polynomials of the base;

and then eliminate redundancy to define the(unisolvent) hierarchical vector bases.

Same scheme used to define interpolatoryvector bases on elements of different shape.

78

Our ideaor what we actually do

We linearly combine the terms of the existingHIGH-ORDER INTERPOLATORY VECTORbases to formHIGH ORDER HIERARCHICAL VECTORbasesR. D. Graglia, D. R. Wilton, and A. F. Peterson, Higher order interpolatory vectorbases for computational electromagnetics, special issue on Advanced NumericalTechniques in Electromagnetics, IEEE Trans. Antennas Propagat., vol. 45, no. 3,pp. 329342, Mar. 1997.

79

For the curl-conforming bases, the Generating Polynomials are subdivided from the outset into three different groups of edge (E), face (F), and volume-based (V) functions.

For each family, the number of the generating polynomials and their maximum polynomial order is the same as in the interpolatory family.

80

81

The orthogonalization along edges leads toLegendre polynomials

(Pro) are hierarchical and could define edge-based polynomials;(Pro) are either symmetric or antisymmetric;(Con) are orthogonal on the cell edge but not on the faces attached to that edge.

Linear combinations of the face-based polynomials are addedto the edge-based ones to make them orthogonal on the facealso.

82

Linear combinations of the volume-based polynomialsare added to the face-based polynomials to make themorthogonal on the volume also.

Curl-conforming basesThus, the definition process for the curl-conformingcase is:

1) First define the volume-based polynomials;2) Then the face-based ones;3) Finally the edge-based ones.

83

Div-conforming bases The generating polynomials are subdivided from the

outset into two different groups of face (F), and volume-based (V) functions.

For each family, the number of the generating polynomials and their maximum polynomial order is the same as in the interpolatory family.

The definition process for the divergence-conforming case is slightly simpler than the curl-conforming:

1. First define the volume-based polynomials;2. Then the face-based ones.

84

Div-conforming bases In this case, the zeroth-order functions are not

associated to the edges but to each cell-face. Thus, there is the need to choose two reference

parent variables on each cell-face to write the generating orthogonal polynomials in a way to easily ensure the continuity of the normal component of the vector functions across adjacent elements (i.e., by sign adjustment).

The other face variables are obtained from the dependency relations.

85

Div-conforming bases The reference-variables are easily individuated by the

pivoting edges of the face. The pivoting edges depart from the face corner-node with

the lowest global node-number, and each reference-variable vanishes only on one of the two pivoting edges.

86

To obtain and write down these polynomialssymmetry considerations are extensively used.

This is done to make the tangent (or normal) vector component continuous across adjacent cells by sign adjustment only.

The zeroth-order vector functions ALSO show some symmetry properties.

87

To obtain and write down these polynomialssymmetry considerations are extensively used.

The polynomials at issue are functions of the element parent variables; keep in mind that:

Line element 2 dependent parent variables; Triangular 3 dependent parent variables; Quad. & Tetra. 4 dependent parent variables; Prism 5 dependent parent variables; Brick 6 dependent parent variables.

88

There are two important issuesone has to consider in the process

of defining a base

1. Possibly, the orthogonality of the elements inthe base [this requires us to introduceappropriate inner products!!], and

2. The scale-factor of each element in the base.

89

Example: 3 different bases for vectors on-a-plane.

Which is the best? How do we get it?

90

We normalize the polynomialsin some clever manner

(inner-product definition):

the integral over the edge for edge-based polynomials Ep

the integral over the face for face-based polynomials Fp

the integral over the volume for volume-based polynomials Vp

91

ALL the edge-based polynomials are madeorthogonal over their associated edge, face, andvolume;

ALL the face-based polynomials are madeorthogonal over their associated face andvolume;

ALL the volume-based polynomials are madeorthogonal over the associated volume.

92

Some of the polynomial bases follow

93

Brick div-conforming bases

Few polynomial bases follow

94

Quadrilateral & brick curl-conforming bases

The polynomial bases are normalized

95

Quadrilateral & brick curl-conforming bases

Some numerical results

96

97

Obtained by use of the Triangular family

98

Obtained by use of the Triangular family

99

Obtained by use of the Quad/Brick family

100

The Table shows the matrix condition numbers arising from an 18-cell model of a 2:1 rectangular cavity, constructed from a 6 by 3model with identical, uniform rectangular cells (3 rows of 6columns each). These results are also obtained from the set of 24quadratic-tangential/cubic-normal (QT/CuN) bases from eachfamily.

Comparison for quadrilateral bases

101

The Table compares the matrix condition numbers arising from theregular 18-cell model of a 2:1 rectangular cavity, constructed from a 6 by3 model with identical, uniform rectangular cells (3 rows of 6 columnseach).


The Table shows the matrix condition numbers arising from a different 18-cellmodel of a 2:1 rectangular cavity, constructed from a 6 by 3 model withinterior nodes irregularly located to produce skewed quadrilateral cells. Thisexample also used the set of 24 quadratic-tangential/cubic-normal (QT/CuN)bases from each family.

102


Since the Jorgensen and Graglia functions of order p=2 outperformed theother families, their performance for order p=3 was also investigated. Forthis comparison, the set of 40 unscaled p=3 bases (using just the originalscale factors) was employed, producing a system of order 612. The Tablepresents the condition numbers, for the two 18-cell models consideredabove.

103


In summary, the Graglia and Jorgensen bases perform in a very similarmanner as indicated by their matrix condition numbers. The other basisfamilies produce more ill-conditioned matrices, suggesting that theirlinear independence is not as good.

For completeness, this Table reports the eigenvalues obtained for theregular 18-cell mesh, for orders p = 0, 1, 2, and 3, compared to the exactresults.

104

With our construction scheme we have obtained the bases fortriangular & quadrilateral cells.

With our construction scheme we have obtained the curl-conformingbases for tetrahedral, brick and prism cells.

With our construction scheme we have obtained the divergence-conforming bases for 3D elements: tetrahedron, brick and prism.

Our curl-conforming basis family produce well-conditioned matrices;the other basis families produce more ill-conditioned matrices,suggesting that their linear independence is not as good.

The transitioning strategy for p-refinement is reported elsewhere. Our bases can be used to mesh a structure with differently shaped

cells.

105

CONCLUSION

Singular vector bases

106

107

Circular vaned waveguide FEM application

=1/2=0

elements type and conformity

108

Circular vaned waveguide: eigenmodes

Singular behavior

Numerical precision

Regular mode

Singular mode

=1/2

109

Modeling capability: very small thicknessDouble-vaned Circular homogeneous waveguide

=2/3

110

Modeling capability: Multiple singular verteces and curvilinear singular elements

=2/3=/2

=1/2=

=3/4=2/3

111

The square PEC-plate problem at normal incidence MoM Application

The results at left (a, c) were obtained by using the zeroth-order regular base (p = 0) on the dense mesh A. The results at right (b, d) were obtained by using the coarse mesh B and the singular base of order [p = 2, s = 0].

112

The square PEC-plate problem at normal incidence

113

(10 1) PEC-strip Normal incidence Ex Singualar bases p = 2, s = 0

114

The circular PEC-plate at normal incidence

115

The circular PEC-plate at normal incidence (d = /100)

116

Normal incidence on a (1 1) square PEC-plate with a hole of radius r = /10 centered at (x = 0.15, y = +0.15);the incident magnetic field is polarized in the y-direction.

117

Spherical PEC-shell of radius a = /(2 and aperture angle = 120illuminated by a planewave propagating in the positivez direction

118

FEM Analysis of DielectricLoaded Waveguides with Additive

Hierarchical Singular Vector Elements

Roberto D. Graglia*, Andrew F. PetersonLadislau Matekovits*, and Paolo Petrini*

* Politecnico di Torino ITALY Georgia Institute of Technology USA

e-mail: [email protected] ; [email protected]

APS/URSI 2014, Memphis, TN, USA Thursday, July 10, 2014.

119

Scope of Presentation:

Treatment of vertex singularities in a 2D triangular-cell mesh wedge angle singularity with known exponents

High order hierarchical representations scalar (already done and published) vector curl-conf (done and in publication) vector div-conf (future work) substitutive vs. additive

Examples from cavity/waveguide FEM problems

120

ReferencesR.D. Graglia, A.F. Peterson, L. Matekovits, Singular, hierarchical scalarbasis functions for triangular cells, IEEE Trans. AP, vol. 61, no. 7, pp. 3674-3692, July 2013.

R.D. Graglia, A.F. Peterson, L. Matekovits, and P. Petrini, Hierarchicaladditive basis functions for the finite-element treatment of cornersingularities, Special Issue on Finite Elements for Microwave Engineering,Electromagnetics, vol. 34, pp. 171-198, March 2014.

R.D. Graglia, A.F. Peterson, L. Matekovits, and P. Petrini, Singularhierarchical curl-conforming vector bases for triangular cells, IEEE Trans.AP, due to appear July 2014.

R.D. Graglia, P. Petrini, A.F. Peterson, and L. Matekovits, Full-waveanalysis of inhomogeneous waveguiding structures containing corners withsingular hierarchical curl-conforming vector bases, IEEE AWPL, April 2014.

121

Conclusions from previous works(PEC edges or corners)

Can achieve true high order behavior, even with edges

However, representation in edge cells requires an additive expansion with multiple integer exponents and multiple fractional powers

cells often a quarter wavelength or more in dimension

1 2 3 4 5 6108

106

104

102

limit of accuracy

purely polynomial

p,1,0

p,1,1

p,2,2

p,p,p

R

e

l

a

t

i

v

e

e

r

r

o

r

f

o

r

k

c

1

s

t

T

E

m

o

d

e

122

PEC Wedge Singularity

Use the infinite wedge solution to identify the series of fractional exponents needed

n n(2 )

123

Dielectric Wedge Singularity

Use the infinite wedge solution to identify the series of fractional exponents needed

Unbounded fields region

Wedge aperture angle (degrees)

n

c

o

e

f

f

s

.

f

o

r

d

i

e

l

e

c

t

r

i

c

w

e

d

g

e

s

(

r

e

l

.

=

1

0

)

0 45 90 135 180 225 270 315 3600.5

1

1.5

2

2.5

3

3.5

4

4.5

1~

~n

n

t

z

EE

124

Substitutive vs Additive:

Substitutive: replace one regular basis function with a singular function Not recommended for high order bases

Additive: keep original basis functions and add singular functions to that set necessary for high order/accuracy maximum flexibility difficulty: matrix condition numbers

125

Conclusions from previous works

As expected, conditioning is a problem with additive functions some orthogonalization essential more complete orthogonality leads to lower condition

numbers (in MoM applications the CN can often be lowered by

using a non-Galerkin approach)

126

Proposed singular bases:

Singular bases to be added to regular, hierarchical bases without changing the regular bases

Hierarchical Accommodate a general set of exponents

may have any number of exponents (may also have singularities at more than one vertex) may be orthogonal to any number of regular

polynomial bases

127

The key idea:

Field component approximation:

nqn

znpolyzz aFF

1

ddaaFF

nq

nnnpolytt

1

kn

bb nn1

0

128

Notation: Modified coordinates

Singularity incorporatedthrough radial functions

n: index of exponent k: number of polynomials used in R to enforce

orthogonality to regular basis functions : exponent (for the singular functions we use only

the non integer ones)

i1 i11i

1 i

Rn (k,,)

129

Orthogonality

and to Ri , for all i

130

Normalization

1),,(:

1),,(:

1),,(:

1

0

2

1

0

2'

1

0

2

dkRnC

dkRnB

dkRnA

n

n

n

(3 possibilities)

131

Singular scalar basis functions:

Combine radial dependence with Jacobi polynomials in to obtain

where

fm ( ) (2m 5)(m 3)(m 4)32(m1)(m 2) Pm(2,2) ( )

j0i1 Rj ()1 4 j1i1 Rj ()1 4 jm Rj () fm2 ( ) 1 2

132

Singular vector basis functions: hierarchical polynomial basis subsets are those used in

Graglia, Peterson, Andriulli, Curl conforming hierarchical vector bases for triangles and tetrahedra, IEEE Trans. AP, March 2011

Idea is to construct vector bases from gradients of the scalar bases; these form gradient bases functions with zero curl

Then we define also (irrational) non-zero curl vector functions

Linear combinations with regular bases used to minimize matrix condition numbers

133

Singular vector basis functions:The (irrational) zero-curl subspace is modelled by:

81),,(;

41),,(

;8

1),,(;4

1),,(

2

2

2

2

fkSkS

fkRkR

nn

nn

kn

j

jnjn

nn bac

kR n1

1),,(

kn

j

jnjn

nn bac

kS n1

21),,(

with:

134

Singular vector basis functions:The (irrational) non-zero-curl subspace is modelled by:

kn

j

jnjn

nn bac

kS n1

11),,(

ddSSkT nnn 2),,(

inn PkSV 12),,( PkTJ

nV nn 12),,(

with:

135

Singular vector basis functions:What is new with respect to what we did for the scalarbases:

The radial functions are obtained numerically by usinga recursive algorithm that involves the solution of asquare linear system. (The normalizing coefficients are thesquare root of quadratic forms.)

The list of the singularity coefficients does not considerthe integer singularity coefficients BUT can be changedaccording to the aperture angle of the wedge.

137

Waveguide problem formulations

z

t

TLz

z

t

TL

tTctT

ScS

ee

DCCB

keeA

nFormulatioFieldTL

eBkeAnFormulatioFieldT

BkAnFormulatioScalar

2

2

2

000

The mass-matrices are the B-matrices

For homogeneous waveguides 222 cz kkk

138

1 2 3 4 5108

106

104

102

p order of the polynomial subset

R

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e

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r

o

r

f

o

r

k

c

1

s

t

T

E

m

o

d

e

purely polynomial, all formulations

singular bases

limit of accuracy

Scalar formulationTfield formulationTLfield formulation

1 2 3 4 5

102

104

106

108

p order of the polynomial set

C

N

T

E

p

r

o

b

l

e

m

w

i

t

h

p

u

r

e

l

y

p

o

l

y

n

o

m

i

a

l

b

a

s

e

s


0 1 20

0.5

1

1.5

2

1 2 3 4 5

102

104

106

108


C

N

T

E

p

r

o

b

l

e

m

w

i

t

h

s

i

n

g

u

l

a

r

b

a

s

e

s


Homogeneous waveguides canbe studied with all formulations

139

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6150

100

50

0

50

100

150

200

250

300

Frequency GHz

z

N

p

/

m

z

r

a

d

/

m

Propagating modesEvanescent modesComplex ModesBackward Wave

0 5 10 15 20 25 30

0

5

10

15

20

25

y=6mm

24 X 24mm metal box;12 X 12mm rod with rel=37.13;

0 5 10 15 20 25

0

5

10

15

20

25

First (regular) mode 0 5 10 15 20 250

5

10

15

20

25

Backward mode

140

0 6 12 18 240.7

1.0

1.5

2.0

2.5First mode withpolynomial base

Mesh 22Mesh 38

0 6 12 18 240.7

1.0

1.5

2.0

2.5

x (along the y=6mm line)

First mode withsingular base

Mesh 22Mesh 38

0 5 10 15 20 25 30

0

5

10

15

20

25

y=6mm

0 5 10 15 20 25 30

0

5

10

15

20

25

y=6mm

Continuity of Dnormal along the red line

1410 5 10 15 20 25 300

5

10

15

20

25

y=6mm

0 5 10 15 20 25 30

0

5

10

15

20

25

y=6mm

Continuity of Dnormal along the red line

0 6 12 18 240.7

1.0

1.5

2.0

2.5Backward mode with polynomial base

0 6 12 18 24

1

1.5

2

2.5Backward mode with singular base


1420 5 10 15 20 25 300

5

10

15

20

25

y=6mm

0 5 10 15 20 25 30

0

5

10

15

20

25

y=6mm

The field is unboundedat the corners

0 6 12 18 240

0.2

0.4

0.6

0.8

1

1.2 Backward mode with polynomial base

0 6 12 18 240

0.2

0.4

0.6

0.8

1

1.2


Backward modesingular base

Mesh 22Mesh 38

143

1 2 3 4 50

500

1000

1500

2000

2500


D

e

g

r

e

e

s

o

f

f

r

e

e

d

o

m

Polynomial basesSingular bases

1 2 3 4 5108

106

104

102


R

e

l

a

t

i

v

e

e

r

r

o

r

limit of accuracy

Singular bases

Polynomial bases

First modeBackward mode

1 2 3 4 5

102

104

106

108


C

o

n

d

i

t

i

o

n

n

u

m

b

e

r

Polynomial basesSingular bases

144

Conclusion:

Additive basis sets offer flexibility and possibility of true high order accuracy when singularities are present

Matrix conditioning is a major issue more orthogonalization = lower condition numbers

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