This 3-day course teaches the basics of CEM with application examples. Fundamental concepts in the solution of EM radiation and scattering problems are presented. Emphasis is on applying computational methods to practical applications. You will develop a working knowledge of popular methods such as the FEM, MOM, FDTD, FIT, and TLM including asymptotic and hybrid methods. Students will then be able to identify the most relevant CEM method for various applications, avoid common user pitfalls, understand model validation and correctly interpret results. Students are encouraged to bring their laptop to work examples using the provided FEKO Lite code. You will learn the importance of model development and meshing, post- processing for scientific visualization and presentation of results. Participants will receive a complete set of notes, a copy of FEKO and textbook

1. Slides From ATI Professional Development Short Course
Computational Electromagnetics
Instructor:
Dr. Keefe Coburn
http://www.ATIcourses.com/schedule.htm
ATI Course Schedule:
ATI's ComputationalElectromagnetics http://www.aticourses.com/Computational_Electromagnetics.htm

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3. Coarse Outline
• Day 1
– Review EM Theory
– Review Antennas
– Review Antenna Arrays
– Review Scattering
– Introduction to FEKO Lite
• Day 2
– CEM Introduction
– FEM Introduction
– MoM Introduction
– FDTD Introduction
– FVTD, TLM and FIT
– Examples
• Day 3
– FEM Tutorial
– MoM and FDTD Tutorial
– Summary and Advanced Topics
• High-frequency Methods
• Hybrid Methods
– Discussion and Examples

4. Electromagnetics (EM)
Maxwell’s Equations
Faraday’s Law: Ampère’s Circuital Law:
 
E   B   H  J D
t t
Gauss’ Laws: Constitutive Equations:
B  0 D   B  H DE
Actual solution for realistic problems is complex and requires
simplifying assumptions and/or numerical approximations
Solutions to Maxwell’s equations using numerical approximations is
known as the study of Computational Electromagnetics (CEM)

5. Why Computer-Aided Methods Are Used?

6. Importance
• Key to analysis, design & optimization of RF to
optical systems.
• Basis for field-theory based and process-oriented
CAD (virtual prototyping).
• Key to economical success of a product through
shortening of development time.
• Only means for dealing with complex (non-canonical)
electromagnetic structures.
• Theoretical models must be validated by
experiments.
• Theoretical and experimental work are of equal
importance.

7. Classic Electromagnetic Solution
Mathematical Maxwell’s Boundary Material
Formulation Equations Conditions Properties
Analytical Analytical
Model
Problem-Dependent
Preprocessing
Discretization Computer
Program
User Data
Computation
User Interface Examples:
• Closed-form expressions for
Results microstrip/transmission lines;
Postprocessing
• Spectral domain program for
coplanar waveguides;
• Eigenvalue solvers for guided
waves and cavities;

8. Modern Electromagnetic Solution
Maxwell’s Huygens’ Variational
Numerical Equations Principle Principle
Formulation
And
Discretization Numerical Problem-Independent
Model
Computer
Program
Boundary Conditions
Material Properties
Computation
User Interface Examples:
• Finite Element or
Results MoM Frequency
Postprocessing
Domain Solver
• FVTD or FDTD Time
Domain Numerical
Simulation

9. Conventional Microwave Design
Circuit/Antenna
Specifications
Design
Data
Initial Design
Expensive,
Rapid Laboratory
Prototyping Model
Modifications Time-Consuming
Fail
Not Automated
Loop
Measurements Compare
Pass
Final
Fabrication

10. Computer-Aided Design (CAD)
Circuit/Antenna
Specifications
Design Data
Synthesis Methods
Initial Design
Models
Inexpensive,
Virtual
Prototyping Sensitivity
Fast,
Analysis Modifications Automated
Analysis
Optimization
Loop Fail
Measurements Compare
• Physical Modeling Error Pass
• Discretization Error Prototype
• Numerical Modeling Error Fabrication
• Measurement Error

11. Electromagnetic Simulators
• An Electromagnetic Simulator is a modeling tool that:
– solves electromagnetic field problems by numerical analysis;
– extracts engineering parameters from the field solution and
visualize fields and parameters;
– allows design by means of analysis combined with
optimization (PSO, GA, parameterized models, etc.).
• The field solver engine employs one or several numerical
methods obtained through the practice of CEM:
– is the theory and practice of solving electromagnetic field
problems on digital computers;
– provides the only viable approach to solving “real world” field
problems;
– enables Computer-Aided Engineering (CAE) and Computer-
Aided Design (CAD) of EM components and systems.

12. Solving EM Field Problems
• Find electromagnetic field and/or source functions
such that they
– obey Maxwell’s equations,
– satisfy all boundary conditions,
– satisfy all interface and material conditions,
– satisfy all excitation conditions.
(In both time and space, or at one frequency in space)
• Field solutions are then unique when tangential
fields on conductors and initial conditions are known
• But numerical solution depends on
• Physical Modeling Error
• Discretization Error
• Numerical Modeling Error
• Measurement Error

13. Field-Solving Methods
Methods for solving Maxwell’s Equations:
• Analytical Methods
– Exact explicit solutions (only a few ideal cases)
• Semi-Analytical Methods
– Explicit solutions requiring final numerical evaluation
– Numerical solutions with analytical “preprocessing”
• Approximate analytical models
– Approximate analytical solutions for simplified structures
(provides physical insight)
– Only practical way to handle very large electrical structures
• Numerical Methods
– Differential or integral equations are transformed into matrix
equations by numerical approximations (sampling) and
solved iteratively or by matrix inversion

14. Overview of CEM
• Central to all CEM techniques is the idea of discretizing some
unknown EM property, for example:
– In MoM the surface current is typically used
– In FE, the Electric Field
– In FDTD, the Electric and Magnetic Field
• Meshing is used to subdivide a large geometry into a number
of nonoverlapping subregions or elements, for example:
– In two dimensional regions triangles maybe used
– In three dimensional geometries a tetrahedral shape may be used
• Within each element, a simple functional dependence (basis
functions) is assumed for the spatial variation of the unknown
• CEM is a modeling process and therefore a study in
acceptable approximation and numerical solution
In other words, CEM replaces a real field problem with an
approximate one which causes physical (geometric) and
numerical limitations that one must keep in mind

15. Overview of CEM
• During the creation of the approximate “model”, assumptions and
simplifications are generally introduced so limitations on the solution
accuracy, for example:
– Assuming an infinite ground plane or substrate in an antenna
structure
– Simplifying a thin wire by a current filament (dipole impedance)
• When analyzing solutions generated from a CEM techniques keep in
mind limitations of the solution introduced by manufacturing tolerances:
– Small changes in dimensions may affect the performance
– Frequency dependent (or unknown) material properties
• Limitations are also introduced by the finite discretization
– The mesh must be fine enough so that the basis functions can
adequately represent the electromagnetic fields
– Fine mesh required for critical variations such as source region
• Numerical approximations and finite precision will limit the analysis
– Limited computational resources (i.e., mesh size)
– Double precision accuracy will not help if the problem is ill conditioned
– Finer mesh is often the only choice (multi-scale codes)

16. Fundamentals of Simulations

17. Fundamentals of Simulations

18. Fundamentals of Simulations
Accuracy Checklist
To use verified CEM tools
successfully requires:
• Knowledge and understanding of
electromagnetics and RF engineering
• Validation of simulation results
(analytical models, code-to-code,
experiments, prototyping)

19. Computational Hierarchy
computational
electromagnetics
numerical methods High frequency
IE DE
current
field based
based
TD FD TD FD
FDTD
TWTD MoM FDFD GO/GTD PO/PTD
TLM

20. Classification of Methods
• Maxwell’s Equations-based methods
Method Frequency Time Domain
Domain
Boundary Method of Moments
Element (MoM)
Finite Element FEM
Finite Difference FDTD
Finite Volume FVTD
• Optics-based methods
Method
Physical Optics
GTD/UTD
19

21. Classification of Methods
Frequency Domain Methods Time Domain Methods
(Time-Harmonic) (Transient)
Fourier
Transform
• This distinction is based more on human experience
than on physical or mathematical considerations.
• The time dimension can be treated as a fourth dimension
in Minkowski space in the form jct, where c is the speed
of light.
• The user requires knowledge of different methods to be
able to choose the most suitable design tool and setup
the calculation correctly.
• In the most general sense, solution methods can thus be
classified according to the number of dimensions upon
which the field and source functions depend.

22. Classification of Methods
1D Methods: Fields and voltage/current vary in one space
dimension (Transmission Line Problems)
(…Touchstone, Supercompact, SPICE…)
2D Methods: Fields and currents vary in two space
dimensions (Cross-section problems,
TEn0 waveguide problems)
(…FEM-2D, MEFiSTo-2D…)
2 1/2 D Methods: Fields vary in three space dimensions,
currents vary in two space dimensions
(Planar multilayer circuits)
(…Sonnet, Momentum, Ensemble...) frequency domain
3D Methods: Fields and currents vary in three space
dimensions (General propagation, scattering
and radiation problems)
(…HFSS, FEKO, CST, XFDTD, GEMS, GEMACS…)

23. What Have All Methods In Common?
1. In all methods, the unknown solution is expressed as a sum
of known functions (expansion functions or basis functions).
2. The weight (coefficient) of each expansion function is
determined for best fit.
What distinguishes them?
• the electromagnetic quantity approximated,
• the expansion functions used,
• the strategy employed for determining the coefficients of the
expansion functions, and
• the numerical solution method.

24. Other Classifications
Quasi-TEM or full-wave?
Quasi-TEM use notions of effective dielectric constant, single
mode impedance, axial current, dispersionless propagation.
Circuit or antenna?
Static and quasi-static techniques are applicable to analysis of
circuits, except for spurious effects: radiation & surface waves.
Open or closed problem space?
Circuits are at some point usually enclosed within a metal box,
similarly, antennas are designed to operate in free space. One
would expect that techniques for boxed in structures would be
used to analyse circuits and those techniques for open structures
for antennas.
This is not what happens in practice!
Common assumptions are: infinite dielectrics and ground planes,
zero thickness of strips, etc.

25. Frequency and Time Domain Concepts
• Complex Frequency • Time dependence
• Phase angle • Delay
• Complex Dielectric • Real permittivity and
Constant conductivity
• Complex Reflection/ • Reflection/Transmission
Transmission Coeff. time response
• Complex Impedance • Impulse response
• Q-Factor • Decay time
• Complex multiplication • Time domain convolution

26. Why Model In The Frequency Domain?
• Most microwave engineers are more familiar with
FD concepts than with TD concepts
• Frequency domain simulations are steady-state
• Complex notation is elegant and efficient
• Specifications are traditionally formulated in the
FD (S-Parameters, loss tangent, dispersion)
• Time domain information can be obtained by
inverse Fourier Transform (Causality issues!)
• Dispersive materials and boundaries are easily
described by frequency-dependent parameters

27. Finite Element Method (FEM)

28. Finite Element Method (FEM)

29. Method of Moments (MoM)

30. Method of Moments (MoM)

31. Why Model In The Time Domain?
• Time domain simulations are “life-like” and allow
visualization of signal propagation
• Virtual experiments are set up as in the lab
(Source, reference planes, output probes)
• Cause and effect can be distinguished
• One simulation can cover a wide bandwidth
• Transient phenomena can be simulated
• Nonlinear behavior is modeled naturally
• Dispersive materials and boundaries are modeled
in a more physical manner
• Frequency domain information can be obtained via
Fourier transform

32. Finite Difference Time Domain (FDTD)

33. Finite Difference Time Domain (FDTD)

34. Comparison

35. Comparison

36. Summary and Conclusions
• Numerical Methods allow us to solve real life EM problems
(within certain limits). They form the engine(s) of
electromagnetic simulators.
• Electromagnetic simulators are not merely Maxwell equation
solvers, but powerful simulation and design tools with
visualization capabilities.
• Understanding EM phenomena and knowledge of radio
engineering are necessary for successful use of codes.
• Understanding the underlying numerical methods is
essential in assessing the accuracy, performance and
limitations of a particular simulation tool.
• Electromagnetic simulators are the heart of modern CAD
tools for analog microwave, digital high-speed and mixed
signal design, EMC and signal integrity engineering and
other applications of electromagnetic fields and waves.

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