Numerical Modeling of Electromagnetic Problems Using EM.Cube

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An Overview of Computational Electromagnetics

Mathematically speaking, all electromagnetic modeling problems require solving some form of Maxwell's equations subject to certain initial and boundary conditions. Radiation and scattering problems are defined over an unbounded domain. Circuit and device problems are often formulated as shielded structures within finite domains. Aside from a few well-known canonical problems, there are no closed-form solutions available for most electromagnetic problems due to the complexity of their domains and boundaries. Numerical analysis, therefore, is the only way to solve such problems.

The numerical techniques used in computational electromagnetics (CEM) are generally divided into three categories:

• Full-Wave Techniques: These techniques solve the dynamic form of Maxwell's equations either in the time domain or in the frequency domain. They typically require a very fine discretization of the physical structure with a frequency-dependent mesh resolution. Some of the most well-know methods in this category include the finite difference time domain (FDTD) method, the method of moments (MoM) and the finite element method (FEM). Full-wave techniques provide the highest level of modeling accuracy among CEM methods.
• Quasi-Static Techniques: These techniques assume static DC or low-frequency conditions, which ignore wave retardation effects. Under these conditions, Maxwell's equations reduce to the electrostatic or magnetostatic forms of Laplace/Poisson equations. These methods are effective in solving lumped devices or structures with small electrical dimensions.
• Asymptotic Techniques: These techniques assume quasi-optical or high-frequency conditions, and solve the asymptotic forms of Maxwell'e equations. These methods are effective in solving structures or scenes with very large electrical dimensions. All the ray tracing techniques like the shoot-and Bounce-Rays (SBR) method fall into this category. Another example is the physical optics (PO) method.

Click here for A Review of Maxwell's Equations & Computational Electromagnetics (CEM).

The modified radiation pattern of a patch antenna installed on the front hood of a Volkswagen Golf automobile computed using the full-wave FDTD technique.

Typical Steps of Computer Simulation of an Electromagnetic Problem

Using a numerical method to solve a certain electromagnetic modeling problem typically involves a recurring sequence of steps:

1. Geometrical construction of the physical structure
2. Material assignment to geometric objects
3. Definition of the computational domain and boundary conditions
4. Definition of excitation sources
5. Definition of simulation observables
6. Geometrical reduction and mesh generation
7. Running the numerical solver
8. Post-processing and visualization of the output data

The above steps first transform your original physical modeling problem into a computational problem, which must be solved using an appropriate numerical solver.

A question often asked in conjunction with electromagnetic modeling is: "Does one really need more than one simulation engine?" Different numerical techniques have different strengths and weaknesses with respect to modeling versatility and breadth of scope, modeling accuracy and computational efficiency. There is no single numerical technique that can solve all the electromagnetic problems at all frequencies and involving all length scales from microns to miles. A true challenge of electromagnetic modeling is the right choice of numerical technique for any given problem. Depending on the electrical length scales and the physical nature of your problem, some modeling techniques may provide a more accurate or computationally more efficient solution than the others. Full-wave techniques provide the most accurate solution of Maxwell's equations in general. In the case of very large-scale problems, asymptotic methods sometimes offer the only practical solution. On the other hand, static or quasi-static methods may provide more stable solutions for extremely small-scale problems. Having access to multiple simulation engines in a unified modeling environment provides many advantages beyond getting the best solver for a particular problem. Some complex problems involve dissimilar length scales which cannot be compromised in favor of one or another. In such cases, a hybrid simulation using different techniques for different parts of the larger problem can lead to a reasonable solution. In addition, verifying and benchmarking different solvers in the same simulation environment helps you better strategize, formulate and validate a definitive solution.

A Functional Comparison of EM.Cube's Numerical Solvers

EM.Cube uses a number of computational electromagnetic (CEM) techniques to solve your modeling problems. All of these techniques are based on a fine discretization of your physical structure into a set of elementary cells or elements. A discretized form of Maxwell's equations or some variations of them are then solved numerically over these smaller cells. From the resulting numerical solution, the quantities of interest are derived and computed.

The numerical techniques used by EM.Cube are:

• Finite Different Time Domain (FDTD) method
• Shooting-and-Bouncing-Rays (SBR) method
• Finite Difference (FD) method for electrostatic, magnetostatic and steady-state thermal Laplace/Poisson equations
• Mixed Potential Integral Equation (MPIE) method for multilayer planar structures also known as Planar method of Moments (PMOM)
• Wire Method of Moments (WMOM) based on Pocklington integral equation
• Surface Method of Moments (SMOM) with Adaptive Integration Equation (AIM) accelerator
• Physical Optics (PO) method: Geometrical Optics - Physical Optics (GO-PO) method and Iterative Physical Optics (IPO) method

Of EM.Cube's computational modules, EM.Tempo serves as a general-purpose electromagnetic simulator that can handle most types of modeling problems involving arbitrary geometries and complex material variations in both time and frequency domains. The table below compares EM.Cube's computational modules and its simulation engines with regards to modeling accuracy, frequency limitations and the type of numerical solution they offer:

	Module Name	Simulation Engine(s)	Modeling Accuracy	Solver Type	Frequency Range	Fundamental Solution	Applications
	EM.Tempo	FDTD	Full-wave	3D volumetric solver	Ultra-wideband time-domain	Electric and magnetic fields in the entire domain	General-purpose field simulator capable of handling complex geometrical and material variations
	EM.Terrano	SBR	Asymptotic	3D ray tracer	High-frequency harmonic	Electric fields and ray tubes received at receiver locations	Radio wave propagation in very large scenes
	EM.Ferma	FD	Static or quasi-static	3D volumetric solver	DC or low-frequency	Electric or magnetic fields or temperature in the entire domain	Small-scale devices and structures
	EM.Picasso	MPIE (PMOM)	Full-wave	2.5D planar solver	Arbitrary harmonic	Electric and magnetic currents on traces	Multilayer planar circuits, antennas & arrays, FSS, homogeneous substrates
	EM.Libera	WMOM & SMOM	Full-wave	3D wire & surface solvers	Arbitrary harmonic	Electric and magnetic currents on surfaces or wires	Radiation and scattering problems involving metals and homogeneous dielectric materials
	EM.Illumina	GO-PO & IPO	Asymptotic	3D surface solver	High-frequency harmonic	Electric and magnetic currents on surfaces	Scattering from very large surface structures & antenna-platform combinations

Geometrical Construction of the Physical Structure

A physical structure in EM.Cube is typically made up of one or more geometric objects, which you either draw in its project workspace or import from an external CAD model file.

When you start the EM.Cube application, you land in its CubeCAD module by default. CubeCAD is a comprehensive, parametric, 3D CAD modeler along with integrated mesh generation, data processing and visualization capabilities and a powerful Python scripting environment. With the click of a few buttons, you can build complex 3D models and structures in seconds using a large variety of intuitive mouse-based object creation and transformation tools. Import of standard CAD formats allows you to fly in external CAD models with utmost ease. Imported structures can easily be augmented with your own geometrical constructions created using the native standard geometric objects.

All of EM.Cube's computational modules use CubeCAD together with individually customized navigation trees as their graphical user interface and geometry definition utility. Once you have mastered the basics of CubeCAD, using the other modules will be very straightforward.

Click here to learn more about CubeCAD.

An Apache helicopter.

The imported model of the Apache helicopter.

The table below compares EM.Cube's computational modules with regards to their geometrical variety:

	Module Name	Geometry Types
	EM.Tempo	General solid, surface & curve objects
	EM.Terrano	General solid & surface objects - no curve objects
	EM.Ferma	General solid, surface & curve objects
	EM.Picasso	Planar surface objects only - no solid or curve objects (vias are automatically constructed from cross-sectional planar objects as vertical prisms)
	EM.Libera	General solid & surface objects for Surface MoM solver, general curve objects and wireframe structures for Wire MoM solver
	EM.Illumina	General solid & surface objects - no curve objects

Material Composition of the Physical Structure

In CubeCAD, geometric objects are simply grouped together by their color. They do not have any physical properties. In all of EM.Cube's computational modules, however, you need to assign physical properties to each geometric object. From an electromagnetic modeling point of view, the difference between a material block and a free-space region is the constitutive relations that govern the electric and magnetic fields in these media and/or their boundary conditions. In EM.Cube's computational modules, geometric objects are grouped together by their common physical properties as well as their color. The types of physical properties may differ in different computational modules, but they are typically related to the material properties or boundary conditions.

A structure made up of a human head (lossy dielectric) and a handheld radio unit with plastic and metallic parts.

In general, an isotropic material medium is macroscopically characterized by four constitutive parameters:

• Permittivity (ε) having units of F/m
• Permeability (μ) having units of H/m
• Electric conductivity (σ) having units of S/m
• Magnetic conductivity (σ_m) having units of Ω/m

EM.Cube offers a large variety of material types listed in the table below:

Material Type	Supporting Module(s)
Perfect Electric Conductor (PEC)	EM.Tempo, EM.Ferma, EM.Picasso, EM.Libera, EM.Illumina
Thin Wire	EM.Tempo, EM.Libera
Perfect Magnetic Conductor (PMC)	EM.Tempo, EM.Picasso, EM.Illumina
Perfect Thermal Conductor (PTC)	EM.Ferma
Dielectric (or Magnetic) Material	EM.Tempo, EM.Ferma, EM.Picasso, EM.Libera, EM.Terrano
Insulator Material	EM.Ferma
Impedance Surface	EM.Illumina
Conductive Sheet	EM.Picasso
Anisotropic Material	EM.Tempo
Dispersive Material (Debye, Drude, Lorentz, Generalized Metamaterial)	EM.Tempo
Gyrotropic Material (Ferrite, Magnetoplasma)	EM.Tempo

The table below compares EM.Cube's computational modules with regards to their material variety:

	Module Name	Material Types
	EM.Tempo	PEC, thin wire, PMC, dielectric, anisotropic, dispersive, gyrotropic
	EM.Terrano	Material surfaces, thin walls and material volumes
	EM.Ferma	PEC, PTC, dielectric or magnetic or insulator materials
	EM.Picasso	PEC and slot traces, short vias, infinite substrate layers
	EM.Libera	PEC, thin wire, homogeneous dielectric
	EM.Illumina	PEC, PMC, impedance surfaces

Click here for a more detailed discussion of Assigning Material Properties to the Physical Structure.

Click here to access Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types.

Defining the Computational Domain & Boundary Conditions in EM.Cube

The table below compares EM.Cube's computational modules with regards to their computational domain and boundary condition types:

	Module Name	Domain Type	Domain Boundary Conditions
	EM.Tempo	Finite box	PEC, PMC, PML, Periodic
	EM.Terrano	Open-boundary free space with optional half-space ground	Radiation BC
	EM.Ferma	Finite box	Dirichlet, Neumann, Adiabatic and Convective (for thermal simulation)
	EM.Picasso	Open-boundary with multilayer background medium	Radiation BC
	EM.Libera	Open-boundary free space	Radiation BC
	EM.Illumina	Open-boundary free space	Radiation BC

For more information, refer to the manuals of individual computational modules.

Exciting a Physical Structure Using Sources or Devices in EM.Cube

In order to perform an electromagnetic simulation in any of EM.Cube's computational modules, you need to excite your physical structure using some kind of source. In most cases, you can define more than one source if necessary.

• In EM.Tempo, a source pumps energy into your FDTD computational domain in the form of a temporal waveform varying as a function time.
• In the MoM-based modules, EM.picasso and EM.Libera, a source provides the "right-hand-side (RHS)" vector of the MoM linear system resulting from the integral equation formulation of your boundary value problem.
• In EM.Illumina, a source is used to illuminate your surfaces, and the scattrering of the incident fields from those surfaces are then computed.
• In EM.Terrano, a source acts as a transmitter that launches the broadcast signal into the free space. The transmitter rays travel in the free space until they reach a receiver or are intercepted by an obstructing surface.
• In EM.Ferma, sources like volume charges and volume currents provide the source term for the Poisson equation, while fixed-potential PEC objects set the boundary conditions for the Laplace equation.

In each module, you should choose the right source type depending on the purpose of your simulation and based on the observables you define for your project. For example, for computing the radar cross section (RCS) of a target, you need a plane wave source. If you are interested in computing the S/Z/Y parameters of your structure, then you have to choose a source type like a gap or lumped source that supports a "Port Definition" observable.

Current distribution on a metallic plate excited by a plane wave source.

Current distribution on a metallic plate excited by a short horizontal dipole source above it.

EM.Cube provides a large variety of source types listed in the table below:

Source Type	Applications	Restrictions	Supporting Module(s)
Lumped Source	General-purpose point voltage source	Associated with a PEC or thin wire line	EM.Tempo
Distributed Source	General-purpose distributed voltage source	Associated with a virtual rectangle strip	EM.Tempo
Microstrip Port Source	Used for S-parameter computations	Associated with a PEC rectangle strip	EM.Tempo
CPW Port Source	Used for S-parameter computations	Associated with a PEC rectangle strip	EM.Tempo
Coaxial Port Source	Used for S-parameter computations	Associated with a PEC Cylinder	EM.Tempo
Waveguide Port Source	Used for S-parameter computations	Associated with a hollow PEC box	EM.Tempo
Wire Current Source	General-purpose current source	Stand-alone source	EM.Tempo
Wire Gap Circuit Source	General-purpose point voltage source	Associated with a PEC or thin wire line	EM.Libera
Strip Gap Circuit Source	General-purpose point voltage source	Associated with a PEC rectangle strip	EM.Picasso, EM.Libera
Probe Gap Circuit Source	Used for modeling short coaxial probes	Associated with a PEC embedded via	EM.Picasso
Scattering Wave Port Source	Used for S-parameter computations	Associated with a PEC rectangle strip	EM.Picasso
Hertzian Dipole Source	General-purpose short filament current source	Stand-alone source	EM.Tempo, EM.Terrano, EM.Picasso, EM.Libera, EM.Illumina
Plane Wave Source	Used for scattering computations	Stand-alone source	EM.Tempo, EM.Picasso, EM.Libera, EM.Illumina
Gaussian Beam Source	Used for modeling focused beams	Stand-alone source	EM.Tempo
Huygens Source	Used as a distributed equivalent	Imported from a Huygens data file	EM.Tempo, EM.Terrano, EM.Picasso, EM.Libera, EM.Illumina
Point Transmitter Set	Used as a point source with an imported radiation pattern	Associated with a point or point array	EM.Terrano
Fixed-Potential PEC	Used as a distributed DC voltage source	Requires drawing geometric objects	EM.Ferma
Volume Charge	Used as a distributed electric charge source	Requires drawing geometric objects	EM.Ferma
Wire Current	Used as a linear filament current source	Requires drawing geometric objects	EM.Ferma
Volume Current	Used as a distributed elctric current source	Requires drawing a line or polyline	EM.Ferma
Permanent Magnet	Used as a distributed magnetization source	Requires drawing geometric objects	EM.Ferma
Fixed-Temperature PTC	Used as an isothermal surface heat source for thermal simulation	Requires drawing geometric objects	EM.Ferma
Volume Heat Source	Used as a distributed volume heat source for thermal simulation	Requires drawing geometric objects	EM.Ferma

The table below compares EM.Cube's computational modules with regards to their excitation source and lumped device types:

	Module Name	Excitation/Sources	Lumped Devices
	EM.Tempo	Lumped source, distributed source, microstrip, CPW, coaxial and waveguide port sources, Hertzian dipole, wire current source, plane wave, Gaussian beam, Huygens source, arbitrary waveform	Resistor, capacitor, inductor, series RL, parallel RC, nonlinear diode, active lumped one-port and two-port devices, active distributed one-port and two-port devices
	EM.Terrano	Point transmitter set, Hertzian dipole, Huygens source	N/A
	EM.Ferma	Volume charge, volume current, wire current, permanent magnet, volume heat source	N/A
	EM.Picasso	Strip and probe gap circuit sources, scattering wave port, Hertzian dipole, plane wave, Huygens source	Simple passive RLC lumped elements
	EM.Libera	Strip and wire gap circuit sources, Hertzian dipole, plane wave, Huygens source	Simple passive RLC lumped elements
	EM.Illumina	Hertzian dipole, plane wave, Huygens source	N/A

Click here for a more detailed discussion of Defining an Excitation Source.

Click here to access Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types.

Defining Simulation Observables in EM.Cube

In most of EM.Cube's computational modules, you have to define one or more observables to generate any output data at the end of a simulation. In other words, no simulation data is generated by itself. EM.Cube provides a large variety of simulation data and observable types as listed in the table below:

Simulation Data Type	Required Observable	Supporting Module(s)
Electric and Magnetic Field Distributions	Field Sensor	EM.Tempo, Em.Terrano, EM.Ferma, EM.Picasso, EM.Libera, EM.Illumina
Electric and Magnetic Current Distributions	Current Distribution	EM.Picasso, EM.Libera, EM.Illumina
Temporal Fields	Field Probe	EM.Tempo
Far-Field Radiation Patterns	Far-Field Radiation Pattern	EM.Tempo, EM.Picasso, EM.Libera, EM.Illumina, EM.Terrano
Radiation Characteristics (D0, HPBW, SLL, AR, etc.)	Far-Field Radiation Pattern	EM.Tempo, EM.Picasso, EM.Libera, EM.Illumina, EM.Terrano
Radar Cross Section (RCS)	RCS	EM.Tempo, EM.Picasso, EM.Libera, EM.Illumina
Huygens Surface Data	Huygens Surface	EM.Tempo, EM.Picasso, EM.Libera, EM.Illumina, EM.Terrano
Port Characteristics (S/Z/Y Parameters)	Port Definition	EM.Tempo, EM.Picasso, EM.Libera
Port Voltags, Currents & Powers	Port Definition	EM.Tempo
Periodic Reflection and Transmission Coefficients	No Observables Required	EM.Tempo, EM.Picasso
Temporal Electric and Magnetic Energies and Dissipated Power	Energy-Power	EM.Tempo
Electric and Magnetic Field Densities and Dissipated Power Density	Field Sensor & Energy-Power	EM.Tempo, EM.Ferma
Specific Absorption Rate (SAR) Density	Field Sensor & Energy-Power	EM.Tempo
Complex Poynting Vector	Field Sensor & Energy-Power	EM.Tempo
Static Electric and Magnetic Energy & Ohmic Losses	Field Integral	EM.Ferma
Voltage and Current	Field Integrals	EM.Ferma
Electric and Magnetic Flux	Field Integrals	EM.Ferma
Resistance, Capacitance, Self-Inductance and Mutual Inductance	Field Integrals	EM.Ferma
Temperature and heat flux distributions	Field Sensor	EM.Ferma
Thermal energy density	Field Sensor	EM.Ferma
Thermal Flux and Thermal Energy	Field Integrals	EM.Ferma
Received Power	Receiver Set	EM.Terrano
Channel Path Loss	Receiver Set	EM.Terrano
Singal-to-Noise Ratio (SNR)	Receiver Set	EM.Terrano
Eb/N0	Receiver Set	EM.Terrano
Bit Error Rate (BER)	Receiver Set	EM.Terrano
Power Delay Profile	Receiver Set	EM.Terrano
Angles of Arrival and Departure	Receiver Set	EM.Terrano

The radar cross section of a metallic plate illuminated by a plane wave source.

The radiation pattern of a short horizontal dipole above a metallic plate.

The table below compares EM.Cube's computational modules with regards to their observable types:

	Module Name	Observables
	EM.Tempo	Near-field distributions, far-field radiation patterns and characteristics, RCS, periodic R/T coefficients, temporal waveforms, S/Z/Y parameters, port current/voltage/power, electric and magnetic field densities, dissipated power density, SAR density, complex Poynting vector, temporal domain electric and magnetic energies and dissipated power
	EM.Terrano	Received power coverage, field distributions, SNR, Eb/N0, BER, ray information (delay and angles of arrival and departure, ray fields, ray power)
	EM.Ferma	Electric or magnetic fields & potentials, electric and magnetic energy densities, dissipated power density, voltage, current, electric and magnetic energy, Ohmic power loss, electric and magnetic flux, capacitance, inductance, resistance, temperature, heat flux density, thermal energy density, thermal flux and energy
	EM.Picasso	Current distributions, exterior near-field distributions, far-field radiation patterns and characteristics, RCS, periodic R/T, S/Z/Y parameters
	EM.Libera	Current distributions, exterior near-field distributions, far-field radiation patterns and characteristics, RCS, S/Z/Y parameters
	EM.Illumina	Current distributions, exterior near-field distributions, far-field radiation patterns and characteristics, RCS

Click here to access Glossary of EM.Cube's Simulation Observables & Graph Types.

Discretizing a Physical Structure Using a Mesh Generator in EM.Cube

In order to transform a physical modeling problem into a computational problem that can be solved using a numerical technique, your physical structure must first be discretized into simple canonical elements or mesh cells. EM.Cube's computational modules use a number of different mesh generation schemes to discretize physical structures. Even CubeCAD provides several tools for object discretization. In general, all of EM.Cube's mesh generation schemes can be grouped into three categories based on dimensionality:

1. Linear Mesh
2. Surface Mesh
3. Volume Mesh

The linear mesh, also known as the wireframe mesh, is used by EM.Libera to discretize the physical structure for Wire MoM simulation. EM.Cube offers two types of surface mesh: triangular surface mesh and hybrid surface mesh. As its name implies, a triangular surface mesh is made up of interconnected triangular cells. EM.Terrano, EM.Illumina, EM.Libera and EM.Picasso all use triangular surface mesh generators to discretize surface geometric objects as well as the surface of solid geometric objects. The hybrid surface mesh is EM.Picasso's default mesh. It combines rectangular and triangular cells to discretize planar structures. The hybrid surface mesh generator tries to produce as many identical rectangular cells as possible in rectangular regions of your planar structure.

EM.Cube provides two types of brick mesh, also known as voxel mesh, to discretize the volume of your computational domain. Brick meshes are entire-domain volume meshes and are made up of cubic cells. They are generated by a three-dimensional arrangement of grid lines along the X, Y and Z axes. EM.Tempo offers an "Adaptive" brick mesh as well as a "Fixed-Cell" brick mesh for the FDTD simulation of your physical structure. EM.Ferma offers only a fixed-mesh brick mesh for the solution of electrostatic and magnetostatic Laplace/Poisson equations.

The geometry of a metallic torus.

The brick volume mesh of the metallic torus.

The triangular surface mesh of the metallic torus.

The objects of your physical structure are discretized based on a specified mesh density. The default mesh densities of EM.Tempo, EM.Picasso, EM.Libera and EM.Illumina are expressed as the number of cells per effective wavelength. Therefore, the resolution of the default mesh in these modules is frequency-dependent. You can also define the mesh resolution using a fixed cell size or fixed edge length specified in project units. The mesh density of EM.Terrano is always expressed in terms of cell edge length. The mesh resolution of EM.Ferma is always specified as the fixed cell size. All of EM.Cube's computational modules have default mesh settings that usually work well for most simulations.

The accuracy of the numerical solution of an electromagnet problem depends very much on the quality and resolution of the generated mesh. As a rule of thumb, a mesh density of about 10-25 cells per effective wavelength usually yields satisfactory results. Yet, for structures with lots of fine geometrical details or for highly resonant structures, higher mesh densities may be required. The particular simulation data you seek in a project also influences your choice of mesh resolution. For example, far-field characteristics like radiation patterns are less sensitive to the mesh density than the near-field distributions on a structure with a highly irregular shape and a rugged boundary.

The table below compares EM.Cube's computational modules with regards to their mesh generator types:

	Module Name	Mesh Type
	EM.Tempo	Adaptive and fixed-cell volumetric brick (voxel) mesh
	EM.Terrano	Triangular facet mesh
	EM.Ferma	Fixed-cell volumetric brick mesh
	EM.Picasso	Hybrid rectangular-triangular surface mesh
	EM.Libera	Wireframe and triangular surface mesh
	EM.Illumina	Triangular surface mesh

Click here to access Glossary of EM.Cube's Simulation-Related Operations.

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