CEM Workshop Lectures (6/11): FVTD Method in CEM

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FVTD Method in CEM
Presentation on FVTD CEM solver
V3
Prof GR Shevare

2
©ZeusNumerixPvtLtd:ConfidentialDocument
Overview
16 June 2020 FVTD method for CEM: Zeus Numerix
 Introduction
 Governing equations
 Integral formulation
 Finite volume implementation
 Boundary conditions
 Mesh quality
 Test cases
 Solver features
 Conclusions
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3
Introduction
 Methods used for Computational field simulation in Electro Magnetics are FDTD,
FEM and FVTD
 All methods have their respective advantages and disadvantages
 FDTD :
 Advantage: Explicit time marching , less complexity, easy mesh generation
 Disadvantage: Huge number of grid point requirement, staircase effect of non-conformal boundary
 FEM :
 Advantage: Boundary conforming mesh with local refinement, implicit iteration - faster convergence
 Disadvantages: Matrix inversion at every step need large computing resource, applicable to frequency
domain only
3

4
Introduction contd…
 FVTD
 Advantage
 Supports body fitted Structured, Unstructured and Hybrid meshes. Accurately resolves geometric
details.
 Can handle complex geometries and allows multi scaling for crucial locations
 Like FDTD, uses explicit time marching; hence less complexity in modeling
 Conservation is preserved at cell level
 Easily parallelizable
 Disadvantages
 More approximation errors
 Sensitive to mesh quality
4

5
Governing Equations
 Maxwell’s curl Equation with losses
 Constitutive Relations
E = Electric field intensity H = Magnetic Field Intensity
D = Electric Flux B = Magnetic Flux
μ = Magnetic Permeability ε = permittivity
5
𝜕𝐷
𝜕𝑡
− ∇ × 𝐻 + 𝐽 + 𝜎𝐸 = 0
𝜕𝐵
𝜕𝑡
+ ∇ × 𝐸 + 𝜌𝐻 = 0
…(1)
…(2)
…(3)
…(4)
𝐷 = 𝜀𝐸𝐵 = 𝜇𝐻
𝜎 = 𝜔𝜀″
𝜌 = 𝜔𝜇″

6
Governing Equations
 Maxwell’s curl Equation in homogeneous form without losses
Equation (1) & (2) becomes
6

7
Governing Equations
 Expanding Equation (1)
 Similarly from Equation (4)
7
𝜕𝐷
𝜕𝑡
−
𝜕𝐻𝑧
𝜕𝑦
−
𝜕𝐻 𝑦
𝜕𝑧
𝑖 +
𝜕𝐻 𝑥
𝜕𝑧
−
𝜕𝐻𝑧
𝜕𝑥
𝑗 +
𝜕𝐻 𝑦
𝜕𝑥
−
𝜕𝐻 𝑥
𝜕𝑦
𝑘 = − 𝐽 − 𝜎𝐸 …(6)
𝜕
𝜕𝑡
𝐷 𝑥
𝐷 𝑦
𝐷𝑧
+ ∇.
0
𝐻𝑧
−𝐻 𝑦
𝑖 +
−𝐻𝑧
0
𝐻 𝑥
𝑗 +
𝐻 𝑦
−𝐻 𝑥
0
𝑘 =
−𝐽 𝑥 − 𝜎𝐸 𝑥
−𝐽 𝑦 − 𝜎𝐸 𝑦
−𝐽𝑧 − 𝜎𝐸𝑧
…(7)
𝜕
𝜕𝑡
𝐵𝑥
𝐵𝑦
𝐵𝑧
+ ∇.
0
−𝐸𝑧
𝐸 𝑦
𝑖 +
𝐸𝑧
0
−𝐸 𝑥
𝑗 +
−𝐸 𝑦
𝐸 𝑥
0
𝑘 =
−𝜌𝐻 𝑥
−𝜌𝐻 𝑦
−𝜌𝐻𝑧
…(8)

8
Governing Equations
Since and equation (7) and (8) together can be written as
Equation (9) has the form of,
8
𝐷 = 𝜀𝐸 𝐵 = 𝜇𝐻
𝜕
𝜕𝑡
𝐸 𝑥
𝐸 𝑦
𝐸𝑧
𝐻 𝑥
𝐻 𝑦
𝐻𝑧
+ ∇.
0
𝐻𝑧/𝜀
−𝐻 𝑦/𝜀
0
−𝐸𝑧/𝜇
𝐸 𝑦/𝜇
𝑖 +
−𝐻𝑧/𝜀
0
𝐻 𝑥/𝜀
𝐸𝑧/𝜇
0
−𝐸 𝑥/𝜇
𝑗 +
𝐻 𝑦/𝜀
−𝐻 𝑥/𝜀
0
−𝐸 𝑦/𝜇
𝐸 𝑥/𝜇
0
𝑘 =
(−𝐽 𝑥 − 𝜎𝐸 𝑥)/𝜀
(−𝐽 𝑦 − 𝜎𝐸 𝑦)/𝜀
(−𝐽𝑧 − 𝜎𝐸𝑧)/𝜀
−𝜌𝐻 𝑥/𝜇
−𝜌𝐻 𝑦/𝜇
−𝜌𝐻𝑧/𝜇
…(9)
𝜕(𝑈)
𝜕𝑡
+ ∇. 𝐹 = 𝑄
…(10)

9
Governing Equations
 Equation (10) in Integral form over a control volume V
 Using Divergence theorem
 Equation (12) over a small control volume(ΔV) can be represented as
 Where R(U) is residue
 Where ‘ΔS’ is the bounding surface of the control volume and ‘n’ is the surface
outward unit normal
9
𝜕
𝜕𝑡
𝑉
𝑈𝑑𝑣 +
𝑉
∇. 𝐹(𝑈)𝑑𝑣 =
𝑉
𝑄𝑑𝑣
𝜕
𝜕𝑡
𝑉
𝑈𝑑𝑣 +
𝑆
𝐹(𝑈). 𝑛𝑑𝑠 =
𝑉
𝑄𝑑𝑣
𝑑𝑈
𝑑𝑡
+ 𝑅(𝑈) = 0
𝑅(𝑈) =
1
Δ𝑉
∑
𝑖
𝐹(𝑈). 𝑛Δ𝑆𝑖 − 𝑄
n
ΔS
ΔV

10
Governing Equations
 Equation (13) in discretized form can be represented
 Equation is solved using multistage Runge-Kutta time marching method
10
𝑈 𝑛+1 − 𝑈 𝑛
Δ𝑡
= −𝑅(𝑈)

11
Steps of Finite Volume Method
 Chose a suitable outer domain surrounding the object (generally 5λ away from the
reflecting surface)
 Discretize the object and Domain suitably for a given range of frequencies
 Apply Boundary conditions on the Conductor surface and Far-field faces
 Calculate Fluxes at each interfaces of a control volume taking information from its
neighbor
 Solve Governing PDEs within each of these control volume and store properties at
the center of the CV
11
Conduct
or
surface
Far-field
Boundary

12
Numerical Technique
 Numerical simulation of FVTD method is
represented in the flow chart
 Characteristic based flux calculation method
at the interface
 Higher order time integration using
multistage R-K time stepping
12
Read Input Data
structure building
Start
Iteration?
Apply BC
Incident field
Flux calculation
End
Memory allocation
and initialization
Field calculation
RK loop?
yes
no
yes
completed
Field Update n+1 th

13
Flux Calculation
 It is important to calculate cell Interface Fluxes correctly so that the numerical
method become stable. This is done using Upwind flux splitting methods; which
assures that any Right moving characteristic must come from left cell and Left
moving characteristic from Right cell, at the interface
 Flux term on the control volume face are then split into left and right going
characteristics
13
𝐹(𝑈). 𝑛
𝐹(𝑈). 𝑛 =
𝜕( 𝐹(𝑈). 𝑛)
𝜕𝑈
𝑑𝑈 = 𝐴(𝑛)𝑑𝑈
𝐴(𝑛)𝑑𝑈 = 𝐴(𝑛)+ 𝑈𝐿 + 𝐴(𝑛)− 𝑈 𝑅
nˆ

14
Higher Order Spatial Discretization Method
 UL and UR values are interpolated from cell center to face center for higher order
spatial discretization of
14
𝐹(𝑈). 𝑛
Ui
UR
UL
Ui+1
UL = Ui+GradU*rL
UR = Ui+1+GradU*rR
rR
rL
Ui Ui+1
UL
UR
UL = Ui UR = Ui+1
First Order interpolation Second Order interpolation

15
Boundary Condition
 On Perfect Electric Conductor(PEC)
 Total tangential electric field is 0
 Total normal magnetic field is 0
 Far-Field boundaries
 Characteristic boundary condition (scattering from the boundary=0). Commonly used
methods are
 Absorbing boundary Condition
 Perfectly Matched Layer
15
𝑛 × 𝐸 = 0
𝑛. 𝐻 = 0

16
𝑛 × 𝐸 = 0
𝑛. 𝐻 = 0
𝐸𝑆
tan 𝑔𝑒𝑛𝑡
= −𝐸𝑖𝑛𝑐
𝐻𝑆
𝑛𝑜𝑟𝑚𝑎𝑙
= −𝐻𝑖𝑛𝑐
𝐻𝑠
𝐻𝑆
𝑡
𝐻𝑆
𝑛
𝐻𝑖
𝐻𝑖
𝑡
𝐻𝑖
𝑛
𝐸𝑠
𝐸𝑆
𝑡
𝐸𝑆
𝑛
𝐸𝑖
𝐸𝑖
𝑡
𝐸𝑖
𝑛
𝐸𝑖
𝑡
𝐸𝑆
𝑡
𝐸𝑖𝑛𝑐
= 𝐸𝑖𝑛𝑐 − 𝑛(𝐸𝑖𝑛𝑐. 𝑛)
𝐻𝑖𝑛𝑐
= 𝑛(𝐻𝑖𝑛𝑐. 𝑛)
16
PEC Boundary Condition

17
Scattered Field Estimation
 Total Electric and Magnetic Field is calculated at the surface of the object
 Electric and Magnetic fields are then transformed to their respective frequency
domain fields using Discrete Fourier Transform
 These frequency domain Electric and Magnetic Field are then used to calculate
equivalent Currents
 Far-field Electric scatter then calculated as
r = unit vector along R;
17
EnˆMs 
HnˆJs  (equivalent electric current)
(equivalent magnetic current)
R’
R
R-R’
Obs
𝐸𝑠 =
𝑗𝑘𝑒 𝑗𝑘𝑅
4𝜋𝑅
𝑆′
𝜇
𝜀
1/2
𝑛 × 𝐻 − 𝑛 × 𝐸 × 𝑟 − 𝑛. 𝐸 𝑟 𝑒−𝑗𝑘𝑅′
𝑑𝑆′

18
RCS Calculation
 Far-field RCS (σ) is calculated using
18
σ = 4π R2
|𝐸𝑠|2
|𝐸𝑖𝑛𝑐|2

19
Example problems and Validation
19

20
3D Cube
 Edge length = 1
 Incident Frequency = 600 MHz
 Electric Size = 12.566
 Cube Surface = PEC
 Incident wave angle = 0°to 90°
 Incident wave plane = XZ
 Polarization = Vertical
 Illumination = Sinusoidal
20

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Grid and Nomenclature
21
Figure shows the axes and the wave direction in yellow

22
Results
Surface Current color plot
22

23
Results
Comparison of Monostatic RCS with analytical result
23

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Sphere
 Diameter = 2m
 Incident Frequency = 100 MHz
 Incident wave angle = 0
24

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Grid
25

26
Sphere : Results
Surface Current
26

27
Comparison of Bi-static RCS (E-plane) with analytical results
27

28
Comparison of Bi-static RCS (H-Plane) with analytical result
28

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Ogive
 Length = 0.254m
 Incident Frequency = 1.18 GHz
29

30
Metallic Ogive : Grid
Figure shows the axes
30

31
Ogive : Results
Surface current
31
0°Incidence 90°Incidence

32
Ogive : Results
Comparison of Mono-static RCS with analytical result
32

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NASA Almond
 Edge length = 0.254m
 Incident Frequency = 1.19 GHz
33

34
34
NASA Almond : Geometry and Mesh

35
35
NASA Almond : Results
Surface Currents
Angle of incidence = 0 deg. Angle of incidence = 90 deg.

36
36
NASA Almond : Results
Almond Monostatic RCS Plot (1.19 GHz, VV Polarization)

37
Almond With Duct
 Edge length = 1.25 m
 Incident Frequency = 2 GHz
37

38
38
x
z
y

39
Scattered Electric Field
 Field on the surface and the axi-symmetric section
 Figure showing “Electric Field Magnitude (Scattered)”
39

40
Almond: Results
Bi-static RCS - VV
40

41
Almond: Results
Bi-static RCS - VH
41

42
42
B2-Bomber
Image of the B2 in flightRendered Image of B2 Surface Grid

43
43
B2-Bomber : Surface Mesh

44
44
B2-Bomber : Results
Surface Currents on B2 – Nose-on incidence @ 300 MHz,
VV polarization

45
45
B2-Bomber : Results
Surface Currents on B2 – Broadside incidence @ 300 MHz,
VV polarization

46
46
B2-Bomber : Results
Surface Currents on B2 – Nose-on incidence @ 300 MHz,
HH polarization

47
47
B2-Bomber : Results
Surface Currents on B2 – Broadside incidence @ 300 MHz,
HH polarization

48
48
B2-Bomber : Results
Bi-static RCS Plot @ 50 MHz (VV)

49
49
B2-Bomber : Results
Bi-static RCS Plot @ 300 MHz (VV)

50
50
B2-Bomber : Results
Bi-static RCS Plot @ 300 MHz (HH)

51
51
B2-Bomber : Results
Monostatic RCS Plot @ 300 MHz

52
Generic Missile
 Length = 4 m
 Incident Frequency = 10 GHz
52

53
Geometry
53

54
Generic Missile : Results
Bi-static RCS E-plane and H-plane
54

55
Generic Missile : Complexity of the problem
 Mesh Details
 Length of the missile = 4.5 m
 Frequency = 10 GHz
 PPW = 6
 Number of grid points = 1.0778 Billion
 Number of blocks of multi-block mesh = 5504
 Computational Details
 Number of Nodes = 128 (16 core per Node)
 Number of processors = 2048
 Time to simulation = 4.1 hrs
55

56
Full Scale Unmanned Aerial Vehicle
 Vehicle Length = 5.93 m
 Simulated at Frequency = 10 GHz
56

57
Unmanned Vehicle : Multi-Block
Mesh
57

58
Unmanned Vehicle: Complexity of the problem
 Mesh Details
 Length of the missile = 5.93 m
 Frequency = 10 GHz
 PPW = 6
 Number of grid points = 2.17 Billion
 Number of blocks of multi-block mesh = 8336
 Computational Details
 Number of Nodes = 128 (16 core per Node)
 Number of processors = 2048
 Time to simulation = 39 hrs
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Results: Surface Current

60
General comments
 FVTD in CEM is capable of solving complex real-life problems
 Accuracy and convergence of FVTD method largely dependent on quality and resolution of
the mesh. So a good mesh is a prerequisite
 Modeling of Dielectric and Radar Absorbing Materials are possible
 High frequency EM simulation for a practical problem demands large computing resources.
e.g. Domain with a Billion of mesh points require thousands of computing processors.
 Problems of these scale, possible to solve at large clusters. Which are available only at a very
few places in the country, like CDAC, TATA-CRL etc.
 With the development of high performance and intense computing hardware and facilities,
FVTD will become more useful tool, applicable to many Computational Electromagnetic
Problems
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www.zeusnumerix.com
+91 72760 31511
Abhishek Jain
contact@zeusnumerix.com
Thank You !

CEM Workshop Lectures (6/11): FVTD Method in CEM

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