EE 5337 Computational Electromagnetics (CEM) Introduction to CEM

Transcription

1 Instructor Dr. Raymond Rumpf (915) EE 5337 Computational Electromagnetics (CEM) Lecture #1 Introduction to CEM Lecture 1These notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited Slide 1 Outline What is CEM? CEM wisdom General concepts in CEM Classification of methods Overview of methods Lecture 1 Slide 2 1

2 What is CEM? Lecture 1 Slide 3 Computational Electromagnetics Definition Computational electromagnetics (CEM) is the procedure we must follow to model and simulate the behavior of electromagnetic fields in devices or around structures. Most often, CEM implies using numerical techniques to solve Maxwell s equations instead of obtaining analytical solutions. Why is this needed? Very often, exact analytical solutions, or even good approximate solutions, are not available. Using a numerical technique offers the ability to solve virtually any electromagnetic problem of interest. Z r 1 out c cosh 2 rin Z? c Lecture 1 Slide 4 2

3 Popular Numerical Techniques Transfer matrix method Scattering matrix method Finite difference frequency domain Finite difference time domain Transmission line modeling method Beam propagation method Method of lines Rigorous coupled wave analysis Plane wave expansion method Slice absorption method Finite element analysis Method of moments Boundary element method Spectral domain method Discontinuous Galerkin method Lecture 1 Slide 5 CEM Wisdom Lecture 1 Slide 6 3

4 The Key to Computation is Visualization Is there anything wrong? If so, what is it? i, j, k1 i, j, k 1, i, j k i, j, k E E E E y y i, j, k i, j, k H xx z z x y z, j, k i, j, k H y 1, j, k i1, j, k H y, j1, k i, j1, k H y 1, j1, k i1, j1, k H y xy xy xy xy i i i i 4 i H H H H i i i , jk, i, jk, xz z, jk, xz i, jk, z 1, jk, xz i1, jk, z 1, jk, i1, jk, xz z 4, j, k i, j, k 1 i, j, k i, j, k i1, j, k i, j, k E E E E yx i H y i1, j, 1, j,, j1,, j1, 1, j1, 1, j1, k i k k i k k i k yx i yx i H H H x x x x x x x z z z x 4 H yy i, j, k i, j, k y i k i, j,, j,, j, k i, j, k, j i, j, j, j k k i k yz H yz yz yz i i i H H H 1 1 1, k1 1, k1 1, 1, z z z z 4 i1, j, k i, j, k i, j1, k i, jk, i, jk, i, jk, i1, jk, i1, jk, i1, jk, 1 i1, jk, 1 i, jk, 1 i, j, Ey Ey Ex Ex zx H x zx H x zx H x zx H k1 x x y 4 i, j, k i, j, k i, j1, k i, j1, k i, j1, k1 i, j1, k1 i, j, k1 i, j, k1 zy H y zy H y zy H y zy H y 4 i, jk, i, jk, zz H z i, j, k i, j 1, k i, j, k i, j, k1 H z H H y H z y i, j, k i, j, k xx Ex y z i, j, k i, j, k i, j1, k i, j1, k i1, j1, k i1, j1, k i1, j, k i1, j, k xy Ey xy Ey xy Ey xy Ey 4 i, jk, i, jk, i, jk, 1 i, jk, 1 i1, jk, 1 i1, jk, 1 i1, jk, i1, jk, xz Ez xz Ez xz Ez xz Ez 4 Response, j, k i, j, k yx i E y i H H, j1,, j1, 1, j1, 1, j1, 1, j, 1, j, H H k i k k i k k i k 1 1, yx i yx i E E E i, j, k i, j, k i, j, k i j, k x x z z x x x x x z x 4 i, j, k i, j, k yy Ey i , j, k i, j, k z, j1, k i, j1, k z, j1, k i, j1, k Ez yz yz yz i i i, j, k, j, k yz i E E E z 4 H H H H i k i k i k i k k i k k i k i, j, y 1, j,, j, x, j1, x, j,, j, x 1, j, 1, j, 1, j, k1 1, j, k1, j, k1 i, j, k1 Ex zx zx zx zx i i i i E E E y x x x y 4 k i k k i k, j,, j, y, j1,, j1, y, j1, k1 i, j1, k1 Ey, j, k1 i, j, k1 Ey zy zy zy zy i i i i E E 4 zz i, j, k i, j,k Ez Lecture 1 Slide 7 Golden Rule #1 All numbers should equal 1. Why? ( ) + ( ) = Lost two digits of accuracy!! Solution: NORMALIZE EVERYTHING!!! m E H or 0 E 0 0 H 0 x kx 0 y k y 0 z k z 0 Lecture 1 Slide 8 4

5 Golden Rule #2 Never perform calculations. Why? 1. Golden Rule #1. 2. Finite floating point precision introduces round off errors. Solution: MINIMIZE NUMBER OF COMPUTATIONS!!! 1. Take problems as far analytically as possible. 2. Avoid unnecessary computations. r x y r grexp 2 R x y 2 2 R grexp 2 Lecture 1 Slide 9 Golden Rule #3 Why? Solution Write clean code. Well organized Well commented 1. It will run faster and more reliably. 2. Easier to catch mistakes. 3. Easier to troubleshoot. 4. Easier to pick up again at a later date. 5. Easier to modify. 1. Outline your code before writing it. 2. Delete obsolete code. 3. Comment every step. 4. Use meaningful variable names. Compact No junk code Lecture 1 Slide 10 5

6 The CEM Process There is a rhythm to computational electromagnetics and it repeats itself constantly. Starts with Maxwell s equations and derives all the necessary equations to implement the algorithm in MATLAB. Equations everywhere! Only a few are needed. Implementation does not resemble the formulation. Organizes the equations derived in the formulation and considers other numerical details. Consider all numerical best practices. Should end with a detailed block diagram. Actually implements the algorithm in computer code. Implementation should be simple and minimal. The art of simulation begins here. Practice, practice, practice! Lecture 1 Slide 11 Don t Be Lazy A little extra time making your program more efficient or simulating a device in a more intelligent manner can save you lots of time, energy, and aggravation. Lecture 1 Slide 12 6

7 Formulation Wisdom Derive equations as far and as simple as possible. Build big/complicated matrices from small/simple matrices. This usually requires converting to matrix form early in the formulation. Make your formulation documents very detailed. A good understanding of the formulation gives you the ability to modify your algorithm or to add/subtract features. Lecture 1 Slide 13 Implementation Wisdom Make a detailed block diagram! In the block diagram, include only the equations you will incorporate into your code. Add all other steps to your block diagram. Sources Building devices Extracting information Post processing data Etc. Lecture 1 Slide 14 7

8 Coding Wisdom Work hard to write clean, simple, and well commented code. Indent code inside loops, if statements, etc. Let linear algebra do the work for you. Use lots of comments. Match your code to your formulation. Try to use the same variables. Do not fix your code with incorrect equations. Changing signs arbitrarily is a common way to make things work, but you are hiding a problem and possibly creating more. Lecture 1 Slide 15 Simulation Wisdom #1 Simulate devices in multiple steps using models of increasing complexity. Avoid the temptation to jump straight to the big, bad, and ugly 3D simulation in all of its glorious complexity. Model your device with slowly increasing levels of complexity. You will get to your final answer much faster this way! R. C. Rumpf, Engineering the dispersion and anisotropy in periodic electromagnetic structures, Solid State Physics 66, Lecture 1 Slide 16 8

9 Simulation Wisdom #2 It must be standard practice to ensure your results are converged. Effective Refractive Index w 2.0 m h 0.6 m a 0.25 m n n n n sup rib core sub neff 1.750, t 1.1 sec a h w n rib n sup n core n sub neff t , sec Grid Resolution Lecture 1 Slide 17 Simulation Wisdom #3 Those who simulate the most, trust the simulations the least. Never trust your code or your results. Benchmark. Benchmark. Benchmark. Lecture 1 Slide 18 9

10 Final Word of Wisdom Do not EVER share your codes! Only bad things will happen. The best thing that can happen is that you become useless. Instead, offer to simulate devices for them and make yourself a collaborator. Lecture 1 Slide 19 General Concepts in Computational EM Lecture 1 Slide 20 10

11 Physical Vs. Numerical Boundary Conditions Physical Boundary Conditions Physical boundary conditions refer to the conditions that must be satisfied at the boundary between two materials. These are derived from the integral form of Maxwell s equations. Tangential components are continuous Numerical Boundary Conditions Numerical boundary conditions refer to the what is done at the edge of a grid or mesh and how fields outside the grid are estimated. Lecture 1 Slide 21 Full Vs. Sparse Matrices A Full Matrices Full matrices have all non zero elements. They tend to look banded with the largest numbers running down the main diagonal. Sparse Matrices A Sparse matrices have most of their elements equal to zero. They are often more than 99% sparse. It is most memory efficient to store only the nonzero elements in memory. They tend to banded matrices with the largest numbers running down the main diagonal. Lecture 1 Slide 22 11

12 Integral Vs. Differential Equations (1 of 2) Integral Equations, f xx dx g x Integral equations calculate a quantity at a specific point using information from the entire domain. They are usually written around boundaries and lead to formulations with full matrices. They do not require boundary conditions. Differential Equations df x dx f x g x Differential equations calculate a quantity at a specific point using only information from the local vicinity. They are usually written for points distributed throughout a volume and lead to formulations with sparse matrices. They require boundary conditions. Lecture 1 Slide 23 Frequency Domain Vs. Time Domain This is what a frequency domain code calculates. This is what a timedomain code calculates. Frequency domain solutions are at a single frequency. Time domain solutions look different because there are inherently a broad range of frequencies involved. Lecture 1 Slide 24 12

13 Definition of Convergence Virtually all numerical methods have some sort of resolution parameter that when taken to infinity solves Maxwell s equations exactly. In practice, we cannot this arbitrarily far because a computer will run out of memory and simulations will take prohibitively long to run. There are no equations to calculate what resolution is needed to obtain accurate results. Instead, the user must look for convergence. There are, however, some good rules of thumb to make an initial guess at resolution. Convergence is the tendency of a calculated parameter to asymptotically approach some fixed value as the resolution of the model is increased. A converged solution does not imply an accurate solution!!! Lecture 1 Slide 25 Tips About Convergence Make checking for convergence a habit that you always perform. When checking a parameter for convergence, ensure that it is the only thing about the simulation that is changing. Simulations do not get more accurate as resolution is increased. They only get more converged. Lecture 1 Slide 26 13

14 Conservation of Power When electromagnetic wave is applied to a device, it can be absorbed (i.e. converted to another form of energy), reflected and/or transmitted. Without a nuclear reaction, nothing else can happen. A RT 1 Reflectance, R Fraction of power from the applied wave that is reflected from the device. Transmittance, T Fraction of power from the applied wave that is transmitted through the device. Absorptance, A Fraction of power from the applied wave that is absorbed by the device. Applied Wave Reflected Wave Absorbed Wave Transmitted Wave Lecture 1 Slide 27 How Do You Know if Your Model Works? In many cases, you may not know. 1. BENCHMARK, BENCHMARK, BENCHMARK 2. Check for power conservation 3. CONVERGENCE, CONVERGENCE, CONVERGENCE Common Sense Check your model for simple things like conservation of power, magnitude of the numbers, consistency with the physics, etc. Benchmark You can verify your code is working by modeling a device with a known response. Does your model predict that response? Convergence Your models will have certain parameters that you can adjust to improve accuracy usually at the cost of computer memory and run time. Keep increasing accuracy until your answer does not change much any more. When modeling a new device, benchmark your model using as similar of a device as you can find which has a known response. Compare your experimental results to the model. Do they agree? Reconcile any differences. Lecture 1 Slide 28 14

15 Tips for Matching Simulations to Experiments 1. Model the geometry of the device as accurately as possible. 2. Incorporate accurate material properties, including dispersion. 3. Ensure the source replicates the experimental source as closely as possible. 4. Ensure you detect power in your simulation in a way that is consistent with laboratory experiments. Lecture 1 Slide 29 Classification of Methods Lecture 1 Slide 30 15

16 Classification by Size Scale Low Frequency Methods 0 a High Frequency Methods 0 a Structural dimensions are on the order of the wavelength or smaller. Structural dimensions much larger than the wavelength. Polarization and the vector nature of the field is important. Finite difference time domain Finite difference frequency domain Finite element analysis Method of moments Rigorous coupled wave analysis Method of lines Beam propagation method Boundary element method Spectral domain method Plane wave expansion method Fields can be accurately treated as scalar quantities. Ray tracing Geometric theory of diffraction Physical optics Physical theory of diffraction Shooting and bouncing rays Lecture 1 Slide 31 Classification by Approximations Rigorous Methods A method is rigorous if there exists a resolution parameter that when taken to infinity, finds an exact solution to Maxwell s equations. Finite difference time domain Finite difference frequency domain Finite element method Rigorous coupled wave analysis Method of lines Full Wave Methods A method is full wave if it accounts for the vector nature of the electromagnetic field. A full wave method is not necessarily rigorous. Method of moments Boundary element method Beam propagation method Scalar Methods A method is scalar if the vector nature of the field is not accounted for. Ray tracing Lecture 1 Slide 32 16

17 Comparison of Method Types + resolves sharp resonances + handles oblique incidence + longitudinal periodicity + can be very fast + better convergence + scales better than SA + complex device geometry Frequency Domain scales at best NlogN can miss sharp resonances active & nonlinear devices Fully Numerical memory requirements long uniform sections + wideband simulations + scales near linearly + active & nonlinear devices + easily locates resonances + very fast & efficient + layered devices + less memory Time Domain Semi Analytical longitudinal periodicity sharp resonances memory requirements oblique incidence convergence issues scales poorly complex device geometry + high index contrast + metals + resolving fine details + field visualization + easy to implement + rectangular structures + easy for divergence free Real Space Structured Grid slow for low index contrast + moderate index contrast + periodic problems + very fast and efficient less efficient curved surfaces + most efficient + handles larger structures + conforms to curved surfaces Fourier Space Unstructured Grid field visualization formulation difficult resolving fine details difficult to implement spurious solutions + sparse matrices + easier to formulate + easier to implement Differential Based volume mesh spurious solutions Integral Based + surface mesh + Very efficient for many structures full matrices more difficult to formulate more difficult to implement Lecture 1 Slide 33 Multiphysics Simulations A multiphysics simulation is one that accounts for multiple simultaneous physical mechanisms at the same time. Electromagnetic Thermal Fluids Motion Chemical Acoustic Optical Lecture 1 Slide 34 17

18 Any Method Can Do Anything Any method can be made to do anything. The real questions are: What devices and information is a particular method best suited for? How much of a force fit is it for that method? Lecture 1 Slide 35 Overview of the Methods Lecture 1 Slide 36 18

19 Transfer Matrix Method (1 of 2) Transfer matrices are derived that relate the fields present at the interfaces between the layers. Ex,trn Ex,2 3 E T y,trn E x,2 T 3 Ex,2 Ex,1 2 E T x,2 E x,1 E E x,trn y,trn T 2 E E x,1 x,ref 1 E T x,1 E y,ref T 1 Ex,1 E x,1 Ex,ref E y,ref individual transfer matrices. Lecture 1 Slide 37 E E x,2 x,2 Tglobal TTT Ex,trn Ex,ref global E T y,trn E y,ref Transmission through all the layers is described by multiplying all the Transfer Matrix Method (2 of 2) This method is good for 1. Modeling transmission and reflection from layered devices. 2. Modeling layers of anisotropic materials. Benefits Very fast and efficient Rigorous Near 100% accuracy Unconditionally stable Robust Simple to implement Thickness of layers can be anything Able to exploit longitudinal periodicity Easily incorporates material dispersion Easily accounts for polarization and angle of incidence Excellent for anisotropic layered materials Drawbacks Limited number of geometries it can model. Only handles linear, homogeneous and infinite slabs. Cannot account for diffraction effects Inefficient for transient analysis Lecture 1 Slide 38 19

20 Finite Difference Frequency Domain (1 of 2) Space is converted to a grid and Maxwell s equations are written for each point using the finite difference method., y, 2, y E E, z xy zez xy 2 z z y y This large set of equations is written in matrix form and solved to calculate the fields. E E z y jh y z Ex Ez jh z x Ey Ex jh x y H H z y j Ex y z H x H z j Ey z x H y H x j Ez x y x y z De De jμ h E E y z z y xx x De De jμ h E E z x x z yy y De De jμ h E E x y y x zz z D h D h jε e H H y z z y xx x D h D H jε e H H z x x z yy y D h D h jε e H H x y y x zz z source Ax b x 1 A b ex x ey e z Lecture 1 Slide 39 Finite Difference Frequency Domain (2 of 2) This method is good for 1. Modeling 2D devices with high volumetric complexity. 2. Visualizing the fields. 3. Fast and easily formulation of new numerical techniques. Benefits Accurate and robust Highly versatile Simple to implement Easily incorporates dispersion Excellent for field visualization Error mechanisms are well understood Good method for metal devices Excellent for volumetrically complex devices Good scaling compared to other frequency domain methods Drawbacks Does not scale well to 3D Difficult to incorporate nonlinear materials Structured grid is inefficient Difficult to resolve curved surfaces Slow and memory innefficient Lecture 1 Slide 40 20

21 Finite Difference Time Domain (1 of 2) Fields are evolved by iterating Maxwell s equations in small time steps. Reflection Plane TF/SF Planes Maxwell s equations are enforced at each point at each time step. Spacer Region Unit cell of real device Spacer Region Transmission Plane Lecture 1 Slide 41 Finite Difference Time Domain (2 of 2) This method is good for 1. Modeling big, bad and ugly problems. 2. Modeling devices with nonlinear material properties. 3. Simulating the transient response of devices. Benefits Excellent for large scale simulations. Easily parallelized. Excellent for transient analysis. Accurate, robust, rigorous, and mature Highly versatile Intuitive to implement Easily incorporates nonlinear behavior Excellent for field visualization and learning electromagnetics Error mechanisms are well understood Good method for metal devices Excellent for volumetrically complex devices Scales near linearly Able to simulate broad frequency response in one simulation Great for resonance hunting Drawbacks Tedious to incorporate dispersion Typically has a structured grid which is less efficient and doesn t conform well to curved surfaces Difficult to resolve curved surfaces Slow for small devices Very inefficient for highly resonant devices Lecture 1 Slide 42 21

22 Transmission Line Modeling Method (1 of 2) Space is interpreted as a giant 3D circuit. Waves propagating through space are represented as current and voltage in extended circuits. Also called transmission line matrix method (TLM). Lecture 1 Slide 43 Transmission Line Modeling Method (2 of 2) This method is good for 1. Modeling big, bad and ugly problems. 2. Hybridizing models with microwave devices. 3. Representing digital waveforms. Benefits Essentially the same benefits at FDTD and FDFD. Excellent for large scale simulations. Easily parallelized. Excellent for transient analysis. No convergence criteria. Inherently stable. Time and frequency domain implementations exist. Excellent fit with network theory in microwave engineering. Drawbacks Essentially the same drawbacks as FDTD and FDFD. Lecture 1 Slide 44 22

23 Beam Propagation Method (1 of 2) The beam propagation method (BPM) is a simple method to simulate forward propagation through a device. It calculates the field one plane at a time so it does not need to solve the entire solution space at once. A μ D μ D μ ε n I H 1 E 2 i xx, i x zz, i x xx, i yy, i eff 1 jz jz e I A I A e i1 i y i1 i y 4neff 4neff Lecture 1 Slide 45 Beam Propagation Method (2 of 2) This method is good for 1. Nonlinear optical devices. 2. Devices where reflections and abrupt changes in the field are negligible (i.e. forward only devices) Benefits Simple to formulate and implement (FFT BPM is easiest) Numerically efficient for faster simulations Well established for nonlinear materials (unique for frequencydomain method). Easily incorporates dispersion Excellent for field visualization Error mechanisms are well understood Well suited for waveguide circuit simulation Drawbacks Not a rigorous method Limited in the physics it can handle Typically uses paraxial approximation Typically neglects backward reflections FFT BPM is slower, less stable, and less versatile than FDM BPM Lecture 1 Slide 46 23

24 Method of Lines (1 of 2) x y source reflected BCs The method of lines is a semi analytical method. BCs BCs Modes are computed in the transverse plane for each layer and propagated analytically in the z-direction. Boundary conditions are used to matched the fields at the interfaces between layers. Transmission through the entire stack of layers is then known and transmitted and reflected fields can be computed. z transmitted BCs Lecture 1 Slide 47 Method of Lines (2 of 2) This method is good for 1. Long devices. 2. Long devices with metals. Benefits Excellent for longitudinally periodic devices Rigorous method Excellent for devices with high index contrast and metals Good for resonant structures Less numerical dispersion than fully numerical methods Easier field visualization than RCWA Drawbacks Scales very poorly in the transverse direction Cumbersome method for field visualization Less efficient than RCWA for dielectric structures. Rarely used in 3D analysis, but this may change with more modern computers Lecture 1 Slide 48 24

25 Rigorous Coupled Wave Analysis (1 of 2) Field in each layer is represented as a set of plane waves at different angles. Plane waves describe propagation through each layer. Layers are connected by the boundary conditions. Lecture 1 Slide 49 Rigorous Coupled Wave Analysis (2 of 2) This method is good for 1. Modeling diffraction from periodic dielectric structures 2. Periodic devices with longitudinal periodicity Benefits Excellent for modeling diffraction from periodic dielectric structures. Extremely fast and efficient for all dielectric structures with low to moderate index contrast Accurate and robust Unconditionally stable Thickness of layers can be anything without numerical cost Excellent for longitudinally periodic structures. Excellent for structures large in the longitudinal direction. Easily incorporates polarization and angle of incidence. Drawbacks Scales poorly in transverse dimensions. Less efficient for high dielectric contrast and metals due to Gibb s phenomenon. Poor method for finite structures. Slow convergence if fast Fourier factorization is not used. Lecture 1 Slide 50 25

26 Plane Wave Expansion Method (1 of 2) The plane wave expansion method (PWEM) calculates modes that exist in an infinitely periodic lattice. It represents the field in Fourier space as the sum of a large set of plane waves at different angles. Lecture 1 Slide 51 Plane Wave Expansion Method (2 of 2) This method is good for 1. Analyzing unit cells 2. Calculating photonic band diagrams and effective material properties. Benefits Excellent for all dielectric unit cells Fast even for 3D Accurate and robust Rigorous method Drawbacks Scales poorly. Weak method for high dielectric contrast and metals. Limited to modal analysis. Cannot model scattering. Cannot incorporate dispersion. Lecture 1 Slide 52 26

27 Slice Absorption Method (1 of 2) Virtually any method that converts Maxwell s equations to a matrix equation can order the matrix to give it the following block tridiagonal form. This allows the problem to be solved one slice at a time. Lecture 1 Slide 53 Slice Absorption Method (2 of 2) This method is good for 1. Modeling structures with high volumetric complexity 2. Modeling finite size structures (i.e. not infinitely periodic) Benefits Excellent for modeling devices with high volumetric complexity Easily incorporates dispersion Easily incorporate polarization and oblique incidence Potential for transverse devices Excellent for finite size devices Excellent framework to hybridize different methods. Transverse sources Stacking in three dimensions. Drawbacks New method and not well understood. Lecture 1 Slide 54 27

28 Finite Element Method (1 of 2) Step 1: Describe Structure Step 2: Mesh Structure This is a VERY important and involved step. r Step 3: Build Global Matrix Iterate through each element to populate the global matrix. Ax 0 Step 4: Solve Matrix Equation Incorporate a source. Ax b Calculate field. x 1 A b Lecture 1 Slide 55 Finite Element Method (2 of 2) This method is good for 1. Modeling volumetrically complex structures in the frequencydomain. Benefits Very mature method Excellent representation of curved surfaces Unstructured grid is highly efficient Unconditionally stable Scaling improved with domain decomposition Drawbacks Tedious to implement Requires a meshing step Lecture 1 Slide 56 28

29 Method of Moments (1 of 2) Lf g an n anlvn g n n an vm, Lvn vm, g n a1 v1, Lg v1, Lv1 v1, Lv2 a 1 2, 2, 1 2, v Lg v Lv v Lv2 a, N vn Lg f v Galerkin Method Integral Equation Converts a linear equation to a matrix equation Usually uses PEC approximation Usually based on current L 2 2 jkr inc j 2 e Ez Izz k dz 2 L 2 z 4 r The Method of Moments i 1 v 1 i 2 v 2 i 3 v 3 i 4 v 4 i 5 v 5 i 6 v 6 i 7 v 7 z11 z12 z13 z14 z15 z16 z17 i1 v1 z21 z22 z23 z24 z25 z26 z 27 i 2 v 2 z31 z32 z33 z34 z35 z36 z 37 i 3 v 3 z41 z42 z43 z44 z45 z46 z47 i4 v4 z51 z52 z53 z54 z55 z56 z 57 i 5 v 5 z61 z62 z63 z64 z65 z66 z67 i6 v6 z71 z72 z73 z74 z75 z76 z 77 i 7 v 7 Lecture 1 Slide 57 Method of Moments (2 of 2) This method is good for 1. Modeling metallic devices at radio frequencies 2. Modeling large scale metallic structures at radio frequencies Benefits Extremely efficient analysis of metallic devices Full wave Very fast Excellent scaling using the fast multipole method No boundary conditions Simple implementation Mature method with lots of literature Can by hybridized with FEM Drawbacks Not a rigorous method Poor method for incorporating dispersion and dielectrics Long a tedious formulation Inefficient for volumetrically complex structures Lecture 1 Slide 58 29

30 Boundary Element Method (1 of 2) The boundary element method (BEM) is also called the Method of Moments, but is applied to 2D elements. The most famous element is the Rao Wilton Glisson (RWG) edge element. S. M. Rao, D. R. Wilton, A. W. Glisson, Electromagnetic Scattering by Surfaces of Arbitrary Shape, IEEE Trans. Antennas and Propagation, vol. AP 30, no. 3, pp , Governing equation exists only at the boundary of a device so many fewer elements are needed elements 400 elements Lecture 1 Slide 59 Boundary Element Method (2 of 2) This method is good for 1. Modeling large devices with simple geometries. 2. Modeling scattering from homogeneous blobs. Benefits Highly efficient when surface to volume ratio is low Excellent representation of curved surfaces Unstructured grid is highly efficient Unconditionally stable Can be hybridized with FEM Domain can extend to infinity Simpler meshing than FEM Drawbacks Tedious to implement Requires a meshing step Not usually a rigorous method Inefficient for volumetrically complex geometries Lecture 1 Slide 60 30

31 Discontinuous Galerkin Method (1 of 2) The discontinuous Galerkin method (DGM) combines features of the finite element and finite volume framework to solve differential equations. Lecture 1 Slide 61 Discontinuous Galerkin Method (2 of 2) This method is good for 1. Solving very complex equations. 2. Modeling very electrically large structures. 3. Time domain finite element method. Benefits Mesh elements can have any arbitrary shape. Fields may be collocated instead of staggered. Inherently a parallel method. Easily extended to higher order of accuracy. Allows explicit time stepping Low memory consumption (no large matrices) Drawbacks Tedious to implement Requires a meshing step Not usually a rigorous method Inefficient for volumetrically complex geometries Lecture 1 Slide 62 31