Lecture 1 -- Introduction to CEM
5/11/2015 Instructor Dr. Raymond Rumpf (915) 747‐6958 [email protected] EE 4395/5390 – Special Topics Computational Electromagnetics (CEM) Lecture #1 Introduction to CEM These notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited Lecture 1 Slide 1 Outline • • • • What is CEM? Classification of methods General concepts in CEM Overview of methods Lecture 1 Slide 2 1 5/11/2015 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. Zc Lecture 1 2 r cosh 1 out rin Zc ? Slide 4 2 5/11/2015 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 Discontinuous Galerkin method Lecture 1 Slide 5 Classification of Methods Lecture 1 Slide 6 3 5/11/2015 Classification by Size Scale Low Frequency Methods High Frequency Methods 0 a 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. Fields can be accurately treated as scalar quantities. • • • • • • • • • • 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 • • • • • Ray tracing Geometric theory of diffraction Physical optics Physical theory of diffraction Shooting and bouncing rays Lecture 1 Slide 7 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 8 4 5/11/2015 Comparison of Method Types Time‐Domain Frequency‐Domain + resolves sharp resonances + handles oblique incidence + longitudinal periodicity + can be very fast ‐ scales at best NlogN ‐ can miss sharp resonances ‐ active & nonlinear devices + wideband simulations + scales near linearly + active & nonlinear devices + easily locates resonances Fully Numerical + better convergence + scales better than SA + complex device geometry ‐ memory requirements ‐ long uniform sections Real‐Space ‐ slow for low index contrast + high index contrast + metals + resolving fine details + field visualization Structured Grid + easy to implement + rectangular structures + easy for divergence free ‐ less efficient ‐ curved surfaces ‐ longitudinal periodicity ‐ sharp resonances ‐ memory requirements ‐ oblique incidence Semi‐Analytical ‐ convergence issues ‐ scales poorly ‐ complex device geometry + very fast & efficient + layered devices + less memory Fourier‐Space + moderate index contrast + periodic problems + very fast and efficient ‐ field visualization ‐ formulation difficult ‐ resolving fine details Unstructured Grid + most efficient + handles larger structures + conforms to curved surfaces ‐ difficult to implement ‐ spurious solutions Lecture 1 Slide 9 General Concepts in Computational EM Lecture 1 Slide 10 5 5/11/2015 The Key to Computation is Visualization Is there anything wrong? If so, what is it? i , j , k 1 Ezi , j 1,k Ezi , j ,k E y y E yi , j ,k z xxi , j ,k H xi , j ,k xyi , j ,k H yi , j ,k xyi 1, j ,k H yi 1, j ,k xyi , j 1,k H yi , j 1,k xyi 1, j 1,k H yi 1, j 1,k 4 xzi , j ,k H zi , j ,k xzi , j ,k 1 H zi , j ,k 1 xzi 1, j ,k 1 H zi 1, j ,k 1 xzi 1, j ,k H zi 1, j ,k 4 i 1, j , k i 1, j , k i , j ,k i , j ,k Hx yxi , j 1,k H xi , j 1,k yxi 1, j 1,k H xi 1, j 1,k Exi , j ,k 1 Exi , j ,k Ezi 1, j ,k E zi , j ,k yx H x yx 4 z x yyi , j ,k H yi , j ,k yzi , j ,k H zi , j ,k yzi , j ,k 1 H zi , j ,k 1 yzi , j 1,k 1 H zi , j 1,k 1 yzi, j 1,k H zi, j 1,k 4 E yi 1, j ,k E yi , j ,k Exi , j 1,k E xi , j ,k zxi , j ,k H xi , j ,k zxi 1, j ,k H xi 1, j ,k zxi 1, j ,k 1 H xi 1, j ,k 1 zxi , j ,k 1 H xi , j ,k 1 4 x y i , j ,k H yi , j ,k zyi , j 1,k H yi , j 1,k zyi , j 1,k 1 H yi , j 1, k 1 zyi, j ,k 1 H yi , j ,k 1 zy 4 i , j ,k i , j ,k zz H z i, j,k i , j , k 1 H zi , j ,k H zi , j 1,k H y H y xxi , j , k Exi , j ,k y z xyi , j ,k E yi , j ,k xyi , j 1,k E yi , j 1, k xyi 1, j 1,k E yi 1, j 1, k xyi 1, j , k E yi 1, j ,k 4 xzi , j ,k Ezi , j ,k xzi , j , k 1 Ezi , j ,k 1 xzi 1, j , k 1 Ezi 1, j ,k 1 xzi 1, j , k Ezi 1, j ,k 4 i , j 1, k i , j 1, k i , j ,k i , j ,k yxi 1, j 1,k Exi 1, j 1, k yxi 1, j ,k Exi 1, j ,k Ex H xi , j ,k H xi , j ,k 1 H zi , j , k H zi 1, j , k yx Ex yx z x 4 yyi , j , k E yi , j ,k yzi , j ,k Ezi , j ,k yzi , j , k 1 Ezi , j ,k 1 yzi , j 1, k 1 Ezi , j 1,k 1 yzi , j 1, k Ezi , j 1,k 4 H yi , j ,k H yi 1, j ,k H xi , j , k H xi , j 1, k zxi , j , k Exi , j ,k zxi 1, j ,k Exi 1, j , k zxi 1, j ,k 1 Exi 1, j , k 1 zxi , j ,k 1 Exi , j ,k 1 x y 4 zyi , j ,k E yi , j ,k zyi , j 1,k E yi , j 1, k zyi , j 1,k 1 E yi , j 1,k 1 zyi , j ,k 1 E yi , j ,k 1 4 zzi , j , k Ezi , j ,k Lecture 1 Slide 11 Golden Rule #1 All numbers should equal 1. Why? (1.234567…) + (0.0123456…) = Lost two digits of accuracy!! Solution: NORMALIZE EVERYTHING!!! 0 0 1 m E 0E x k0 x or y k0 y 0 0 H H 0 Lecture 1 z k0 z Slide 12 6 5/11/2015 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 x2 y 2 R x2 y 2 r2 g r exp 2 R g R exp 2 Lecture 1 Slide 13 Golden Rule #3 Write clean code. • Well organized • Well commented • Compact • No junk code Why? 1. 2. 3. 4. 5. It will run faster and more reliably. Easier to catch mistakes. Easier to troubleshoot. Easier to pick up again at a later date. Easier to modify. Solution 1. 2. 3. 4. Lecture 1 Outline your code before writing it. Delete obsolete code. Comment every step. Use meaningful variable names. Slide 14 7 5/11/2015 Do Not Fix Your Code with Incorrect Equations Sometimes you can get your code to work by making small changes that deviate from the equations and procedures given in this course. Don’t do this!! You are masking another problem!! Implement the equations and procedures exactly as they are presented. You are hiding another problem in your code that could appear later or cause other problems that are more difficult to find. Lecture 1 Slide 15 Use 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! Lecture 1 R. C. Rumpf, “Engineering the dispersion and anisotropy in periodic electromagnetic structures,” Solid State Physics 66, 2015. Slide 16 8 5/11/2015 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 17 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 18 9 5/11/2015 Full Vs. Sparse Matrices Full Matrices A Sparse Matrices A Full matrices have all non‐zero elements. Sparse matrices have most of their elements equal to zero. They are often more than 99% sparse. They tend to look banded with the largest numbers running down the main diagonal. It is most memory efficient to store only the non‐ zero elements in memory. They tend to “banded” matrices with the largest numbers running down the main diagonal. Lecture 1 Slide 19 Integral Vs. Differential Equations (1 of 2) Integral Equations f x, x 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 f x g x dx 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 20 10 5/11/2015 Algorithm Development Formulation Starts with Maxwell’s equations and derives all the necessary equations to implement the algorithm in MATLAB. • Formulation most often does not resemble the implementation at all. • Do not think your code needs to follow the formulation or do all of the work in the formulation. Implementation Organizes the equations derived in the formulation and considers other details for how to implement the algorithm. • Consider all numerical best practices. • Should end with a detailed block diagram. Coding Actually implements the algorithm in computer code. • Implementation should be simple and minimal. Lecture 1 Slide 21 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 22 11 5/11/2015 Example of Convergence w Effective Refractive Index w 2.0 m Lecture 1 h 0.6 m a 0.25 m nsup 1.0 nrib a nrib 1.9 h nsup ncore nsub ncore 1.9 nsub 1.52 neff 1.750, t 1.1 sec neff 1.736, t 6.1 sec Grid Resolution Slide 23 Tips About Convergence • Make checking for convergence a habit that you always perform. • When checking a parameter for convergence, ensure that 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 24 12 5/11/2015 How Do You Know if Your Model Works? In many cases, you may not know. 1. BENCHMARK, BENCHMARK, BENCHMARK 2. CONVERGENCE, CONVERGENCE, CONVERGENCE Common Sense – Check your model for simple things like conservation of energy, magnitude of the numbers, 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 25 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 26 13 5/11/2015 Trust and Confidence in Your Simulations Those who simulate the most, trust the simulations the least. Never trust your code or your results. Benchmark. Benchmark. Benchmark. Lecture 1 Slide 27 Overview of the Methods Lecture 1 Slide 28 14 5/11/2015 Transfer Matrix Method (1 of 2) Transfer matrices are derived that relate the fields present at the interfaces between the layers. E x ,trn E x ,2 E T3 E x ,2 y ,trn T3 Ex ,trn E y ,trn Ex ,2 Ex ,1 E T2 E x ,2 x ,1 T2 E x ,ref Ex ,1 E T1 E x ,1 y ,ref Ex ,2 E x ,2 T1 Tglobal T3T2 T1 Ex ,1 E x ,1 Ex ,ref E y ,ref Lecture 1 Ex ,trn Ex ,ref E Tglobal E y ,trn y ,ref Transmission through all the layers is described by multiplying all the individual transfer matrices. Slide 29 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 Lecture 1 Drawbacks • Limited number of geometries it can model. • Only handles linear, homogeneous and infinite slabs. • Cannot account for diffraction effects • Inefficient for transient analysis Slide 30 15 5/11/2015 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 y Ez E z x, y 2 , z Ez x, y 2 , z y y This large set of equations is written in matrix form and solved to calculate the fields. Ez E y j H x y z Ex Ez j H y z x E y Ex j H z x y H y H z j Ex y z H x H z j E y z x H y H x j Ez x y D Ey e z D Ez e y jμ xx h x D Ez e x D Ex e z jμ yy h y D Ex e y D Ey e x jμ zz h z source Ax b x A 1b D Hy h z D Hz h y jε xx e x jε e D Hz h x D Hx H z yy y D Hx h y D Hy h x jε zz e z e x x e y e z Lecture 1 Slide 31 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 Lecture 1 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 Slide 32 16 5/11/2015 Finite‐Difference Time‐Domain (1 of 2) Fields are evolved by iterating Maxwell’s equations in small time steps. Maxwell’s equations are enforced at each point at each time step. Reflection Plane TF/SF Planes Spacer Region Unit cell of real device Spacer Region Transmission Plane Lecture 1 Slide 33 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” Lecture 1 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 Slide 34 17 5/11/2015 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 35 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 Drawbacks • 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. • Essentially the same drawbacks as FDTD and FDFD. Lecture 1 Slide 36 18 5/11/2015 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. 2 A i μ xx ,i D Hx μ zz1,i D Ex μ xx ,i ε yy ,i neff I 1 j z j z i eiy1 I A i 1 I Ai e y 4 4 n neff eff Lecture 1 Slide 37 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 frequency‐ domain method). • Easily incorporates dispersion • Excellent for field visualization • Error mechanisms are well understood • Well suited for waveguide circuit simulation Lecture 1 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 Slide 38 19 5/11/2015 Method of Lines (1 of 2) source reflected x y BCs The method of lines is a semi‐analytical method. 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. BCs Transmission through the entire stack of layers is then known and transmitted and reflected fields can be computed. Lecture 1 transmitted z BCs Slide 39 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 Lecture 1 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 Slide 40 20 5/11/2015 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 41 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. Lecture 1 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. Slide 42 21 5/11/2015 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 43 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 Lecture 1 Drawbacks • Scales poorly. • Weak method for high dielectric contrast and metals. • Limited to modal analysis. • Cannot model scattering. • Cannot incorporate dispersion. Slide 44 22 5/11/2015 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 45 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. Lecture 1 Drawbacks • New method and not well understood. Slide 46 23 5/11/2015 Finite Element Method (1 of 2) Step 1: Describe Structure Step 2: Mesh Structure This is a VERY important and involved step. 1 1.0 r 1.50 2 2.5 Step 3: Build Global Matrix Step 4: Solve Matrix Equation Incorporate a source. Iterate through each element to populate the global matrix. Ax b Calculate field. x A 1b Ax 0 Lecture 1 Slide 47 Finite Element Method (2 of 2) This method is good for… 1. Modeling volumetrically complex structures in the frequency‐ domain. Benefits Drawbacks • Very mature method • Tedious to implement • Excellent representation of • Requires a meshing step curved surfaces • Unstructured grid is highly efficient • Unconditionally stable • Scaling improved with domain decomposition Lecture 1 Slide 48 24 5/11/2015 Method of Moments (1 of 2) f an v n Lf g a n a Lv n n n n v1 , Lv1 v 2 , Lv1 Galerkin Method Integral Equation • Converts a linear equation to a matrix equation •Usually uses PEC approximation •Usually based on current g n v m , Lv n v m , g v1 , Lv 2 v 2 , Lv 2 a v , Lg 1 1 a1 v 2 , Lg aN v N , Lg Ezinc j L2 L 2 2 e jkr I z z k 2 2 dz z 4 r The Method of Moments i1 v1 i2 v2 i3 v3 i4 v4 i5 i6 i7 v5 v6 v7 z11 z 21 z31 z41 z51 z61 z 71 z12 z22 z13 z23 z14 z24 z15 z25 z16 z26 z32 z42 z33 z43 z34 z44 z35 z45 z36 z46 z52 z62 z72 z53 z63 z73 z54 z64 z74 z55 z65 z75 z56 z66 z76 z17 i1 v1 z27 i2 v2 z37 i3 v3 z47 i4 v4 z57 i5 v5 z67 i6 v6 z77 i7 v7 Lecture 1 Slide 49 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 Lecture 1 Drawbacks • Not a rigorous method • Poor method for incorporating dispersion and dielectrics • Long a tedious formulation • Inefficient for volumetrically complex structures Slide 50 25 5/11/2015 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. 409‐418, 1982. Governing equation exists only at the boundary of a device so many fewer elements are needed. 5000 elements 400 elements Lecture 1 Slide 51 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 Lecture 1 Drawbacks • • • • Tedious to implement Requires a meshing step Not usually a rigorous method Inefficient for volumetrically complex geometries Slide 52 26 5/11/2015 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 53 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) Lecture 1 Drawbacks • • • • Tedious to implement Requires a meshing step Not usually a rigorous method Inefficient for volumetrically complex geometries Slide 54 27
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