1. Numerical Methods

2. Outlines Numerical Methods (Approximate Solutions):
Extended Boundary Conditions Method
Finite Difference Method
Finite Difference Time Domain (FDTD) method
Pseudo spectral Time Domain (PSTD) method
Finite Difference Frequency Domain (FDFD) method
Variational Technique; it form a common base for both MOM and FEM.
Moment Method (Method Of Weighted Residual) as EFIE & MFIE
Finite Element Method
Finite Element Time Domain (FETD) is state-of-the-art for time domain.
Discontinuous Galerkin finite element methods (DG-FEM); DG-FEM=FETD+FVTD
Finite Volume Time Domain (FVTD)
Transmission-line Method (TLM)
Method of Lines
Monte Carlo Method
Generalized Multi-Pole Technique
Conjugate Gradient Method
Boundary Element Method
High-Frequency Asymptotic (GTD and UTD)
Multi-pole methods
Finite Integral Techniques (FIT)
Hybrid methods
Analytical Methods (Exact Solutions):
Separation of variables
Series expansion
Conformal mapping
Integral solutions, e. g., Laplace and Fourier transforms
Perturbation methods

3. Numerical Methods (Approximate Solutions):

4. Numerical Methods Finite Element Method:
Finite Element Methods (FEM) to complex problems.
Dividing configuration into a number of small homogeneous pieces or elements.
Goal of FEM is to determine field quantities at nodes.
FEM solve for unknown field quantities by minimizing an energy functional.
Moment Methods (Method of Weighted Residuals):
Moment Methods are expressed in frequency domain.
Moment Methods are in 2 way as:
First step is to expand J as a finite sum of basis (or expansion) functions:
It well suited for modeling thin-wire structures
It is better suited for analyzing metal plates

5. Numerical Methods Finite Difference Method (FDM):
For modeling a variety of physical behaviors using PDE.
FDM method involves three steps:
Meshing problem.
Approximating deferential equation.
Using boundary conditions.

6. Numerical Methods Finite Difference Time Domain (FDTD) method
Time stepping is continued until a steady state solution.
Frequency domain results can be obtained by applying a DFT.
Wide-band frequency-domain analysis using impulse source.
Basic elements are cubes, then curved surfaces are staircase.
sharp edges require a very small grid having increasing computational size.
to combat this problem, Surface Conforming techniques with non-rectangular elements have been introduced.
One of these, which permits each element in grid to have an arbitrary shape, is Finite Volume Time Domain (FVTD).

7. Numerical Methods Finite Difference Time Domain (FDTD) method
FDTD and FVTD methods are widely used for radar cross section (RCS) analysis.
Primary advantage is great flexibility and arbitrary waveforms.
Another advantage is implementation on massively parallel computers.

8. Numerical Methods Transmission Line Matrix Method (TLM) :
TLM is similar to FDTD but its approach is unique.
Instead of interleaving E & H-fields, a single grid is established & interconnected by virtual transmission Lines (TL).
Absorbing boundaries are constructed by terminating each boundary node of with its Z0
Similar FDTD:
Analysis in time and entire region of grids.
Modeleling Of Complex, Nonlinear Materials.
Impulse responses and time-domain behavior of systems
is suitable for implementation on massively parallel machines.
Disadvantages of FDTD method are also shared by this technique.
Symmetrical Nodes

9. History of FDTD Method
A Perspective on 40-Year History of FDTD Computational Electrodynamics.

10. History of FDTD Method Paper Number 1: Kane Yee
IEEE AP-S Transactions, May 1966.
2441 citations as of March 7, 2006.
IEEE MTT, Aug. 1975
IEEE MTT, Aug. 1975
IEEE EMC, Aug. 1980

11. History of FDTD Method
Timeline 1966 to 1980:
Timeline 1981 to 1990:

12. History of FDTD Method
Timeline 1991 to 1995:
Timeline 1996 to 2000:

13. Emerging Applications
Timeline 2000 to 2005:
Some Major Technical Paths Since Yee
Absorbing boundary conditions
Numerical dispersion
Numerical stability
Conforming grids
Digital signal processing
Dispersive and nonlinear materials
Multi-physics coupling to Maxwell’s equations
Some Interesting Emerging Applications
Earth / ionosphere models in geophysics.
Wireless personal communications devices.
Ultra wideband microwave detection of early-stage.
Breast Cancer.
Ultra high-speed band pass digital interconnects.
Micron / nanometer-scale photonic devices.
Bio photonics, especially optical detection of early.
Stage Epithelial Cancers.

14. Emerging Applications
Earth/Ionosphere Models in Geophysics:
There is a rich history of investigation of ELF and VLF electromagnetic wave propagation within the Earth ionosphere waveguide.
Applications:
Submarine communications
Remote-sensing of lightning and sprites
Global temperature change
Subsurface structures
Potential earthquake precursors
Snapshots of FDTD-Computed Global Propagation of ELF Electromagnetic Pulse Generated by Vertical Lightning Strike off South America Coast.

15. Emerging Applications
Wireless Personal Communications Devices:
Motorola T250 Cellphone:
High-resolution FDTD model:
Lattice-cell size is as fine as 0.1mm to resolve individual circuit board layers and the helical antenna.

16. Emerging Applications
Phantom Head Validation at 1.8GHz:
Final Head Model Results:
Head model has 121 slices.
1mm thick in ear region & 3mm thick elsewhere.
Having a transverse resolution of 0.2mm.
Ultra-wideband microwave detection of early-stage breast cancer.
Modeled detection of a 2mm tumor.
FDTD simulation of UWB microwave detection of a 2mm diameter malignant tumor embedded 3cm within an MRI derived numerical breast model.
The cancer’s signature is 15 to 30db stronger than the clutter due to the surrounding normal tissues.
Source: Bond et al., IEEE trans. Antennas and propagation, 2003, pp. 1690–1705.

17. Emerging Applications
Substrate Integrated Waveguides (SIW):
Pass band is 27-81GHz with negligible multimoding.
It is confirmed by measurements at Intel Corporation.
Photonic Band-gap Defect Mode Cavities:

18. Emerging Applications
Photonic Band-gap Defect Mode Laser Cavities
Laterally Coupled Photonic Disk Resonators:

19. Emerging Applications
Vertically Coupled Photonic Racetrack (Fully 3-D Model):
Pulse Propagation in the Vertically Coupled Racetrack :

20. Emerging Applications
Nanoplasmonics:
Enhanced Transmission Through a Sub-Micron Hole in a Gold Film.
Focusing Plasmonic Lens:

21. Emerging Applications
Lasing in a Random Clump of ZnO Particles:
Backscattering Spectroscopy:
FDTD modeling has shown that observing spectrum of retro-reflected light from living tissues yields much greater information regarding health of these tissues than existing diagnostic techniques.
Backscattering Detection of Nano-scale Features:

22. Emerging Applications
following practical examples demonstrate some of outstanding simulation capabilities of time- domain solvers.
For these examples, finite integration technique (FIT) method in time domain and TLM are used.
Simulations were performed with commercial software packages
Conformal UWB Antenna:
Lightning Strike on an Airplane:
TDM is the most appropriate, for two reasons:
The input signal is known in the time domain
The structure is typically very large.

23. Emerging Applications
Lightning Strike on an Airplane (cont.):
Magnitude of the surface current:
This example was set up using 300,000 mesh.
That is ensures an accuracy of wave resolution well above typical limit.
Total number of unknown field components to be calculated is 1.9 million.
Over the relatively long pulse duration required here, a total of 120,000 time steps must be executed.
The simulation time for creating the mesh is so short that it cannot be measured.
It takes 46s to calculate the matrix coefficients and 44min to perform entire time stepping on an office PC with an INTEL Core 2 Duo running at 3.16GHz.

24. Emerging Applications
EMC Simulation of an Airplane:
While previous example of lightning strike is a comparably small example with only 1.9 million unknowns.
Problem size becomes much larger when we investigate a plane wave at 900MHz hitting same airplane but now equipped with passenger and interior detail as shown:
In order to obtain a solution of reasonable quality, at least ten steps per wavelength are employed.
This yields a problem size of over 2500 million unknowns requiring only moderate memory usage of 52GB.
Besides ability of TDM to solve such very large problems, one also obtains fields over full frequency band from 0-950MHz.
This results in 414 million mesh cells and aforementioned over 2500 million unknown field components to be calculated at 15,000 time steps.
Calculation of matrix coefficients takes 330min, and time stepping roughly 28h of CPU time on an INTEL Workstation with two XEONs X5472 running at 3GHz in 64-bit mode.
As TDM can be parallelized, CPU time may be scaled down easily by adding more CPUs, typically in a cluster of pizza-box PCs.

25. Emerging Applications
Electromagnetic Compatibility Simulation of a PCB:
EM compatibility (EMC) simulations are an ideal terrain for transient solvers, since EMC issues are inherently broadband.
For example, a typical problem setup consists of a PCB and some shielding mechanism as shown in:
Before device is built and operated, its virtual operation is modeled and simulated, no one can reliably predict in which frequency range problems might occur.
In addition to EM properties of PCB, its radiation may depend on various structural features, such as vents, seams, cables, box dimensions, etc.
a metallic enclosure

26. Emerging Applications
Coupled Simulation of Electromagnetic Field and Nonlinear Elements:
Often, 3D components need to be connected in a larger network, which may include various linear and non-linear circuit elements.
It is well known that, for strong nonlinearities, a circuit simulation in time domain is most reliable.
This is because frequency-domain methods, such as harmonic balance, may lead to inaccuracies if an insufficient number of harmonics is considered.
A possible approach would be to first simulate 3D component alone, calculate full S-parameter matrix to obtain a behavioral model for device, and then connect this model (for example ADS format) to circuit to perform transient non- linear simulation.
For applications whose 3D model has a significant number of ports, however, transient co-simulation
approach, in which circuit elements are directly connected to the 3-D model, is more efficient one.
Moreover, this type of transient co-simulation also allows EM field resulting from nonlinear effect to be studied.
An example of such a co-simulation is step-recovery diode (SRD) pulse generator.
The pulse generator contains a 3D EM structure shown in:

27. Emerging Applications
Particle-in-Cell Simulation of Traveling Wave Tubes (TWTs):
TWTs are used to amplify signals to high power at high frequencies.
For this purpose several elements are required:
A continuous electron beam is created in a gun.
Electrons are emitted from heated cathode.
They are accelerated in a static electric field and exit gun section through anode.
Electrons then enter a slow-wave structure, where some fraction of the kinetic energy of electrons is converted into a high-frequency wave.
This wave interacts with particle beam and has to be coupled out of system.
Electron beam is securely dumped in a collector.
A TWT with a micro-machined folded waveguide structure, operating at 220GHz is:

28. Emerging Applications
TWT’s wave forms:

29. Why FDTD Method? There are seven primary reasons to use FDTD:
FDTD uses no linear algebra:
Being a fully explicit computation, FDTD avoids difficulties with linear algebra.
Linear algebra limit size of frequency-domain IE and finite-element electromagnetics models to generally fewer than 106 electromagnetic field unknowns.
FDTD models with as many as 109 field unknowns have been run
There is no intrinsic upper bound to this number.
FDTD is accurate and robust:
Sources of errors in FDTD are well understood, and can be bounded to permit accurate models for a very large variety of electromagnetic wave interaction problems.
FDTD treats impulsive behavior naturally.
Directly calculates impulse response of an electromagnetic system.
Therefore, a single simulation have UWB wave forms or sinusoidal steady-state response at any frequency with in excitation spectrum.
FDTD treats nonlinear behavior naturally.
Being a time-domain technique, FDTD directly calculates nonlinear response of an EM system.

30. Why FDTD Method? FDTD is a systematic approach.
With FDTD, specifying a new structure to be modeled is reduced to a problem of mesh generation rather than potentially complex reformulation of an IE.
For example, FDTD requires no calculation of structure-dependent Green functions.
Computer memory capacities are increasing rapidly.
While this trend positively influences all numerical techniques, it is of particular advantage to FDTD methods, which are founded on discretizing space over a volume, and therefore inherently require a larger and random access memory.
Computer visualization capabilities are increasing rapidly.
While this trend positively influences all numerical techniques, it is of particular advantage to FDTD methods.
That is, FDTD generate time-marched arrays of field quantities suitable for use in color videos to illustrate field dynamics.

31. Why FDTD Method?
Time-domain versus volumetric frequency-domain methods:
Various higher-order FDTD schemes have been proposed:
Finite-volume Time-domain (FVTD).
Pseudo Spectral Time-domain (PSTD)
Multi Resolution Time-domain (MRTD).
Frequency-domain methods (FDFD) misses:
Arbitrary time signals as excitation.
broadband frequency results in a single simulation.
Transient field effects.
Transient far fields for UWB antennas.
nonlinear effects,
FDFD, on other hand, are the ideal tool for:
Low-frequency problems.
Highly resonant structures.
Eigen-mode computations.

32. Why FDTD Method? Method has following Advantages:
It is conceptually simple.
Algorithm does not require formulation of integral equations, and relatively complex scatterers can be treated without inversion of large matrices.
It is simple to implement for complicated, inhomogeneous conducting or dielectric structures because constitutive parameters (σ,μ,ε) can be assigned to each lattice point.
Its computer memory requirement is not prohibitive for many complex structures of interest.
Algorithm makes use of memory in a simple sequential order.
It is much easier to obtain frequency domain data from time domain results than converse. Thus, it is more convenient to obtain frequency domain results via time domain when many frequencies are involved.
Method has following disadvantages:
Its implementation necessitates modeling object as well as its surroundings. Thus, required program execution time may be excessive.
Its accuracy is at least one order of magnitude worse than that of the method of moments, for example.
Since computational meshes are rectangular in shape, they do not conform to scatterers with curved surfaces, as is the case of the cylindrical or spherical boundary.
As in all finite difference algorithms, the field quantities are only known at grid nodes.

33. Finite Difference Methods (FDM)

34. Finite Difference Schemes
Analytical methods may fail if:
PDE is not linear and cannot be linearized.
Solution region is complex.
Boundary conditions are mixed types.
Boundary conditions are time-dependent.
Medium is inhomogeneous or anisotropic.
Whenever a problem with such complexity arises, numerical solutions must be employed.
Finite differences method (FDM) are:
More Easily understood.
More frequently used.
More universally applicable.
FDM was first developed by A. Thom in 1920 to solve hydrodynamic equations.
FDM basically involves three steps:
Dividing the solution region into a grid of nodes.
Approximating PDE by finite difference equivalent that relates the dependent variable at a point to its values at neighboring points.
Solving PDE subject to boundary conditions.
Most commonly used grid patterns:

35. Finite Difference Schemes
Given a function f(x), its derivative is given by “forward-difference formula”:
Yielding the backward-difference formula:
Resulting in central-difference formula:
Second derivative of f (x):
A more general approach is using Taylor’s series:

36. Finite Difference Schemes
Where O(x)4 is error introduced by truncating the series.
Assuming that these terms are negligible:
To apply FDM to find the solution, we divide region in x−t:
The value of φ(x,t)at the point of P:
Central difference approximations of the derivatives of φ at (i, j)th node are:

37. Finite Differencing of Parabolic PDEs
Where:
Using:
This explicit formula can be used to compute φ(x,t +Δt) in terms of φ(x,t) as:
The first time row, t=Δt, can be calculated in terms of boundary and initial conditions.
The second time row, t=2Δt, are calculated in terms of the first time row and so on.
A graphic way is through the computational molecule as:
Circle is unknown
Square is known

38. Finite Differencing of Parabolic PDEs
An implicit formula, proposed by Crank and Nicholson in 1974, is valid for all finite values of r.
This can be rewritten as:
Computational molecule for Crank-Nicholson method for finite values of r

39. Finite Difference Schemes
Some useful finite difference approximations for φx and φxx :
FD = Forward Difference,
BD = Backward Difference,
CD = Central Difference.

40. Finite Differencing of Parabolic PDEs
Other schemes are the Leapfrog method and the Dufort-Frankel method.
further schemes Smith [5] and Ferziger [6].

41. Finite Differencing of Parabolic PDEs
Example:
Solve the diffusion equation of:
Solution:
This is the temperature distribution in a rod of L=1m with its end in contacts with ice blocks at 0oC and the rod initially at 100oC.
Our problem is finding internal temperature as a function of position and time.
Using explicit methods:
From symmetric respect to x=0.5, solution is performed for 0≤x≤0.5.
First, boundary values are shown in:
Notice that the values of φ(0,0) and φ(1,0) are taken as average of 0 and 100.
Other values can be calculated at free nodes using:
Using:

42. Finite Differencing of Parabolic PDEs
Solution (cont.):
Analytic solution is:
Comparison of explicit solution with analytic solution at x=0.4 is shown in:
LN02T01: Due: 12.1
Do by MATLAB.

43. Finite Differencing of Parabolic PDEs
Solution (cont.):
Using implicit method, we have x=0.2, r=1 so that Δt =0.04:
Values at fixed nodes are calculated as in previous part.
For free nodes, using (3-16) as:
By denoting φ(i,j +1) by φi(i=1,2,3,4) and using of this figure,
Values φ for first time step (t=0.04) can be obtained by solving following simultaneous equations:
Another set of simultaneous equations for t=0.08:
Using programming, accuracy can be increased by choosing more points for each time step.

44. Finite Differencing of Hyperbolic PDEs
A Hyperbolic PDE is wave equation:
Can be written as:
To be stable, aspect ratio r≤1:
If we choose r=1:
Example:
Solve wave equation:
Solution:
Analytical solution is easily obtained as:
Where:
Computational Molecule for arbitrary r≤1
Computational Molecule for r=1

45. Finite Differencing of Hyperbolic PDEs
Solution (cont.):
Assuming:

46. Finite Differencing of Elliptic PDEs
A typical Elliptic PDE is Poisson’s equation:
Assuming:
PDE can be summarized as:
If g(x,y)=0, this leads to Laplace’s equation as:
Each point is the average of those at the four surrounding points.
An alternative fourth order difference is:
Where
Computational molecules for Laplace’s equation based on second order approximation
value of coefficient
ℎ is called the mesh size
Computational molecules for Laplace’s equation based on second order approximation

47. Finite Differencing of Elliptic PDEs
Application of FDM to elliptic PDEs leads to a large equations.
Two commonly used methods of solving these equations are:
Band Matrix Method.
Elimination Methods.
Gauss’s Method.
Cholesky’s Method.
Iterative Methods.
Jacobi’s Method.
Gauss-Seidel Method.
Successive Over-Relaxation (SOR) Method.
Gradient Methods. (the convergence is faster)
Matrix Inversion.
Band Matrix Method:
[A] is a sparse matrix and it has many zero elements.
[X] is a column matrix consisting of unknown values of φ at free nodes.
[B] is a column matrix containing known values of at fixed nodes.
Appendix D
An example of Sparse Matrix:

48. Finite Differencing of Elliptic PDEs
What is the drawback of band matrix method?
The answer is rich, [A] is a sparse matrix and it has many zero elements.
For small domains, band matrix method works perfectly well.
Size of [A] grows directly with square of size of x.
For example, given a rectangular simulation domain of 4 x 4 = 16 voltage samples:
X=[V1, V2, … , V16]
We need [A] = 16 x 16 =256 elements.
A given rectangular simulation domain of 100x100=10000 voltage have [A] = 10,000 x 10,000 =100,000,000 elements.
V13
V16 V1 V4 V2 V3
V14
V15
Example:
Solve Laplace’s equation
Solution:
Applying: V1 V2 V3 V4

49. Finite Differencing of Elliptic PDEs
Iterative Methods:
Because direct matrix inversion is an intense operation, simulations can quickly require excessive computational resources.
To reduce computational cost, because that A is a sparse matrix, a solution through use of sparse matrix solvers take advantage of this property.
There are many available methods such as iterative methods to solve these problems.
We will focus on a very simple algorithm called successive over-relaxation (SOR).
It is used to solve a large system of simultaneous equations ( h —› 0 ).
In this method, a first approximation is used to second approximation, which in turn is used to third approximation, and so on.
Three iterative methods are (Appendix D):
Jacobi
Gauss-Seidel
Successive Over Relaxation (SOR)
SOR is special case of Gauss-Seidel.

50. Finite Differencing of Elliptic PDEs
To apply SOR to equation of :
First step is to define residual R(i,j) as:
Rk(i, j) at kth iteration is a correction which must be added to φ(i, j) to make it nearer to correct value.
For converging to correct value, Rk(i, j) tends to zero.
To improve rate of convergence, we multiply residual by a number ω and add that to φ(i, j) at kth iteration to:
Or:
1<ω<2 is relaxation factor.
ωopt must be found by trial and error.
In order to start:

51. Finite Differencing of Elliptic PDEs
Example: (by using SOR)
Solve Poisson’s equation:
Solution:
Assuming:
For exact solution, we use superposition theorem and let:
V1 from Laplace’s equation subject to inhomogeneous Boundary conditions
V2 from Poisson’s equation subject to homogeneous boundary conditions
Solution (cont.):
From example Analytical Methods:
By using FDM-SOR, ωopt is given by smaller root of quadratic equation [10]
Successive Over-relaxation (SOR) Solution
LN2T2:
Apply SOR to solve this example and compare results with exact solution.

52. Accuracy & Stability
Stability: increase magnitude of solution in time.
Accuracy: closeness of approximate solution to exact.
sources of errors:
 Modeling Errors  Truncation (Or Discretization) Errors  Round Off Errors
Modeling errors: assumptions at mathematical model such as:
a nonlinear system may be represented by a linear PDE.
Truncation errors selecting a finite terms from infinite series.
Example: higher-order terms in Taylor series expansion were neglected.
Can be reduced by using a large number of terms in series.
Can be reduced by using finer meshes, that is, by reducing mesh Size.
instability may result if we apply an high order PDE.
higher-order terms may introduce “spurious solutions.”
Round off errors occur when that computations can be done only with a finite precision on a computer.
This error is due to limited size of registers in arithmetic unit of computer.
Round off errors can be minimized by using of double-precision arithmetic.
only way to avoid round off errors completely it to code all operations using integer arithmetic.

53. Accuracy & Stability
Although reducing mesh size h will increase accuracy, it is not possible to indefinitely reduce h.
Decreasing truncation error, by using a finer mesh, may result in increasing round off error due to the increased number of arithmetic operations.
A point is reached where minimum total error occurs for any particular algorithm as illustrated:
A numerical algorithm is said to be stable if a small error at any stage produces a smaller cumulative error. It is unstable otherwise.
To determine stability, error εn , at time step n, assuming that there is one independent variable.
We define amplification of this error at time step n+1 as:
In more complex situations, there are two or more independent variables, and then:
For stability, it is required that:
Error as a function of the mesh size
where g is known as amplification factor
where [G] is the amplification matrix

54. Accuracy & Stability Von Neumann’s Method:
One useful method of finding a stability criterion is to construct a Fourier analysis of PDE and thereby derive g.
By Appling Von Neumann’s Method for explicit scheme of:
Or:
Let the solution be:
By substituting:
For example:
or:
or:

55. Practical Applications
FDM has been applied successfully to solve many EM problems:
Transmission-line problems,
Waveguides,
Microwave circuit,
EM penetration and scattering problems,
EM pulse (EMP) problems,
EM exploration of minerals,
EM energy deposition in human bodies.
Example:
Transmission Lines
Assuming TEM mode & biaxial symmetry about two axes only one quarter of cross section:
Using FDM approximation of Laplace’s equation:
let us denote:
Shielded double strip line
Simplified by making full use of symmetry

56. Practical Applications
Transmission Lines (cont.)
Gauss’s law for electric field:
Rearranging the terms:
This equation is used to apply FDM in boundary.
When ε1= ε2 :
Using band matrix or iterative methods, [[[ by setting potential at fixed nodes equal to their prescribed values ]]] V can be determined.
Eab .
Ebn = Enc .
boundary a b c d m n
reduces

57. Practical Applications
Example: Waveguides
Solution of waveguide problems is well suited for FDM because region is closed:
Where φ=Ez for TM modes & φ=Hz for TE modes.
To apply FDM, we discretize cross section of wave-guide by a suitable square mesh:
At boundary, Dirichlet condition (φ=0) for TM and Neumann condition (∂φ/∂n=0) for TE modes.
for TM modes (φ=Ez):
As well as, at point A, ∂φ/∂n=0 implies that φD=φE and then:
By applying this equation to all mesh points in waveguide, m simultaneous equations (φ1, φ2,... , φm) is generated.
These simultaneous equations may be conveniently cast into matrix equation.
There are several ways to determine λ and the corresponding φ :
Direct method Appendix D.4
Iterative method as:
boundary
is eigenvalue
which is introduced previously

58. FDM for Nonrectangular Systems
Cylindrical Coordinates:
We can always replace a nonrectangular solution region by an approximate rectangular one.
Laplace’s equation in cylindrical coordinates can be written as:
A special case of for an axisymmetric system, here is no dependence on φ so that V=V(ρ,z).
FDM molecule
(1)

59. FDM for Nonrectangular Systems
Using we have:
To solve Poisson’s equation:
As previously mentioned, boundary condition D1n=D2n must be imposed at interface.
By apply Taylor series expansion to point 1, 2, 5 in medium 1: =
medium 1

60. FDM for Nonrectangular Systems
Combining recent two equations:
or:
Similarly, applying Taylor series to points 1, 2, and 6 in medium 2:
Using: D1n=D2n
Substituting this to two previous equations:
This equation is only applicable to interface points.
Notice that eq. (2) becomes eq. (1) if εr1= εr2
(2)

61. FDM for Nonrectangular Systems
Typical examples of FDM approximations for boundary points:

62. FDM for Nonrectangular Systems
Spherical Coordinates:
In spherical coordinates, Laplace’s equation can be written as:
FDM approximation:
FDM molecule

63. FDM for Nonrectangular Systems
Example: (cylindrical case)
Consider an earthed metal cylindrical tank partly filled with a charge liquid, such as hydrocarbons, as illustrated in:
Determine V in entire domain using FDM.
Assume:
Plot V along ρ=0.5, 0<z<2m.
Plot V on surface of liquid.

64. FDM for Nonrectangular Systems
Exact analytic solution was given in section 2.7 (Sadiku) as:
Because symmetry about z-axis, only necessary region to investigate solution is shown:
Impose condition that z-axis is a flux line: ∂V/∂n=∂V/∂ρ=0.
Using FDM grid as shown:
With Δρ=Δz=h=0.05m, 0≤i≤Imax=20, 0≤j≤Jmax=40.
Along z-axis (i=0), we impose Neumann condition:
Because gas has dielectric constant εr1 and liquid has εr2, we impose boundary condition in eq. (2) on liquid-gas interface.
see eq (Sadiku)

65. FDM for Nonrectangular Systems
Example (cont.):
values of potential along ρ=0.5, 0<z<2 and along gas-liquid interface are plotted in:
It is evident from figure that FDM compares well with exact solution in section 2.7.
It is simplicity in concept and ease of programming finite difference schemes that make them very attractive for solving such problems.
along ρ=0.5m,0≤z≤2m
along gas-liquid interface
LN2T3: Due:
The basic code of this program is included in the book.
Apply MATLAB to solve this example and compare results with exact solution.