FDTD for 1D wave equation Equation: 2 H = t 2 c2 2 H x 2 Notations: o t = nδδ, x = iδx o n H nδδ, iδx = H i o n E nδδ, iδx = E i discretization H 2H + H H 2H + H n+ 1 n n 1 n n n i i i 2 i+ 1 i i 1 = c 2 2 ( t) ( x) t n +1 n +1 n 1 i + 1 i i 1 x 2 n+ 1 c t n n 2 n n 1 i i i i i 2 ( + 1 1) 21 ( γ ) H = H + H + H H x γ 26
FDTD for 1D wave equation (cont d) Properties o Fully explicit second-order accurate scheme o Two previous time steps are required to be stored o The analysis is recursive in time o Each field is found at every time step at every node t n +1 n +1 n 1 i + 1 i i 1 x 27
FDTD for 1D Maxwell s equations Equation: Notations: o Redefine: 2ΔΔ ΔΔ o Central difference: H 1 E E 1 H = ; = t µ x t ε x H Ht ( + t2) Ht ( t2) t t o Involved time steps: t = nδδ, t = nδδ + ΔΔ/2 o Involved spatial steps: x = iδx, x = iδx + Δx/2 o Involved fields: Hn ( ti, x+ x2) = H En ( t+ t 2, i x) = E n i+ 12 n+ 12 i t n +1 n +1 2 n i 12 i i + 12 x H E H 28
FDTD for 1D Maxwell s equations (cont d) Update equations: H = H t 1 + E x µ E E = E t 1 + H x ε H ( ) n+ 1 n n+ 12 n+ 12 i+ 12 i+ 12 i+ 1 i ( + 12 12) n+ 12 n 12 n n i i i i n +1 n +1 2 n Yee algorithm: o The fields are given in different space and time locations o Interleaved E & H grids o Leapfrog in time o Explicit stepping no matrix inversions, no iterations o The solution is more robust and straightforward o E & H boundary conditions are enforced automatically t i 12 i i + 12 x H E H 29
Numerical dispersion of FDTD Consider a time harmonic wave H = Continuous dispersion relation is linear: n j( n t ki x) Numerical wave: Hi = e ω Use the FDTD update equation γ ( ) 21 ( γ ) e = γ e + e + e 2 ω t 2 2 k x sin = γ sin 2 2 jω t 2 jk x jk x 2 jω t j( t kx) e ω ω = ck Properties: o In general the phase velocity depends on the frequency numerical dispersion! o ΔΔ, Δx 0 ω 2 = cc 2 - weak dispersion o Δx = cδt, γ = 1 magic time step with no dispersion (for the 1D case) ( + 1 1) 21 ( γ ) H = H + H + H n 1 2 n n 2 n 1 i i i i 30
Stability of FDTD Consider H = e j(ωω kk) and H i n = e j(ωωωω kkkk) Allow ω to be complex, i.e. ω = ω + jω The propagation is stable if ω 0 (wave~e ω t ) For ω < 0 (wave~e ω t ) the propagation is unstable Define jω Ω= ω t 2, K = k x 2, g = e and use sin Ω γ sin K = 0 2 2 2 g + 2 jγg sin K 1 = 0 g = jk sin K ± 1 γ sin K Stable solution: 2 2 g 1 1 γ sin K 0 γ 1 Courant stability criterion for 1D case: t x c 31
Stability of FDTD (cont d) Physical meaning: the wave should not pass through more than one cell in a single time step Under the magic time step the scheme is exact How to choose ΔΔ and Δt If no fine geometrical features are present: ΔΔ < λ min 10 to satisfy Nyquist sampling criterion If fine geometrical features are present: ΔΔ is determined by the features Choose Δt < ΔΔ/c (note that for fine geometrical features, ΔΔ is small and the time step is small as well!) 32
Source modeling Initial conditions: Define E i 0, H i 1 for all i. Hard source o Specify fields at certain locations independent of surrounding cells, i.e. replace the Yee equations, e.g. n = F(t n E z,is o Easy to implement but lead to reflections Soft source o Sources that allow field updates, e.g. current sources o Equations are updated and no reflections but harder to implement Plane wave injection: Scattered field / total field approach 33
Time dependence of sources Time harmonic sources o Simple sin functions leads to discontinuities and a relatively broadband spectrum o Use sin with a smooth transition function (g(t)) Ft () = Fsin( ω n tgt ) () 0 0 Pulsed sources o Gaussian: Contains DC a problem o Time derivative of a Gaussian no DC o Modulated Gaussian 34
Domain truncation: Perfectly matched layer (PML) Add an additional domain where a truncation is needed Fill the domain with a matching lossy material with σ & σ e satisfying σ m σ µ 1 µ 2 = & = µ 1 ε1 ε1 ε2 No reflections are obtained because computational domain PML domain µ µ 1 + σ ( jωµ ) η = = η = 1 ( ) 1 2 m 2 1 2 ε1 ε 2 + σ jωε 2 This idea can be extended to 2D and 3D but the lossy materials should be anisotropic to match all directions Other absorbing boundary conditions can be used as well 35
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Outline Introduction Maxwell s equations Finite Difference Time Domain method Finite element method Integral equation methods 42
Integral equation procedure Formulate an integral equation Expand the unknowns in terms of basis functions Define a discrete matrix representation of the continuous integral equation Solve the matrix equation Post-process to derive the required parameters 43
Advantages Only the structure of interest is discretized Outgoing conditions are satisfied automatically No numerical dispersion Can be formulated in time and frequency Can be defined for surfaces and volumes Can be formulated for complex backgrounds Can be higher order accurate 44
Challenges Computational cost of simple methods is ~O(N 2 ) Need for fast methods (~O(N) or O(NlogN)) for N degrees of freedom Different formulations are required for different problem types, e.g. volumes vs. surfaces 45
Equation of electrostatics Electrostatics t = ω = 0 two uncoupled sets of (static) equations are obtained for E& D and H& B Equation of electrostatics E = 0 D = ρ Boundary conditions nˆ ( E E ) = 0 12 2 1 or nˆ 12 ( D2 D1) = ρs Constitutive relation D S C = εe E dl = D ds= 0 Q 46
Electric scalar potential Potential due to various charge distributions Point charge Many discrete charges Various continuous charge distributions in free space 47
Capacitance Consider a capacitor consisting of two conductors with a voltage Opposite charges appear on the conductors Capacitance Alternative expression Relation to resistance 48
Numerical capacitance extraction (1) Integral equation method Given V = V 0 2 Two conductors Voltage on the conductors is Find the capacitance Formulation C V 0 2and V 0 2 Potential on the surfaces is given by V 0 2 and V 0 2 1 Potential everywhere is calculated as V () r = ρ s ( r ) ds 4 πε 0 r r S surface charge where ρ s is an unknown surface charge distribution Green's function Equate the potentials on the surfaces S (1) (2) + S 1 V0 2; r S1 ρ ( ) 4 s r ds = πε 0 r r V 2; S surface charge r Green's function distribution (unknown) 0 2 V = V 0 2 (1) surface S (2) surface S integral equation 49
Numerical capacitance extraction (2) Matrix equation V = V 0 2 Discretize the problem Divide the surfaces into N = N + N S small patches, N (1), N (2) are numbers S S (1) (2) of patches on surfaces S, S Replace the integration by summation Enforce the IE at the location of charges N ρ ( ) 0 2; s r ds V rm S1 = 4 πε r r V 2; r S (1) (2) n= 1 S 0 m 0 m 2 n S V = V 0 2 patch S n (1) surface S of area s (2) surface S n Obtain a matrix equation N 0 2; N ρs ( r ) ds V r 0 m S1 = ZmnQn = Vm ZQ = V 4 πε r r V 2; r S n= 1 S 0 m 0 m 2 n= 1 n matrix equation 50
Numerical capacitance extraction (3) Matrix equation and solution Matrix equation Z Z : N N matrix; Z = Q: N vector; Q ( ) s V : N vector; V Solution ZQ mn m n 1 ds 4 πε 0 rm rn mn s 4 1 n πε 0 r Sn m r Zmn ; rm = rn 2ε0 π s n = ρs rn n n m V0 2; rm S1 = V 2; r S Q 1 Z V Extracted capacitance = = V 0 m 2 total charge on surface 1 C = Q (1) S V C = 1 N S (1) n= 1 V 0 0 Q n ; r r 51
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Outline Introduction Maxwell s equations Finite Difference Time Domain method Finite element method Integral equation methods 54
Basic FEM Steps 1. Discretization/subdivision of solution domain 1-D: 2-D: 3-D: 2. Selection of interpolation schemes Linear or higher-order polynomials 3. Formulation of the system of equations Using either the Ritz or Galerkin method: Formulate elemental equations and assemble 4. Solution of the system of equations Using either a direct or iterative method 55
Advantages Works for general structures Can be formulated in time and frequency Can be higher order accurate Low cost (~O(N)) 56
Boundary-Value Problem Ω ˆn Γ 57
Equivalent Variational Problem 58
Equivalent Variational Problem 59
Equivalent Variational Problem 60
Equivalent Variational Problem 61
Basic FEM Steps 1. Discretization/subdivision of solution domain 1-D: 2-D: 3-D: 2. Selection of interpolation schemes Linear or higher-order polynomials 3. Formulation of the system of equations Using either the Ritz or Galerkin method: Formulate elemental equations and assemble 4. Solution of the system of equations Using either a direct or iterative method 62
FEM Analysis Domain subdivision Step 1: Domain Discretization 63
FEM Analysis Domain subdivision Step 1: Domain Discretization 64
FEM Analysis Domain subdivision Step 1: Domain Discretization 65
FEM Analysis Domain subdivision Step 1: Domain Discretization 66
FEM Analysis Element interpolation Step 2: Element Interpolation 3 1 e 2 67
FEM Analysis Element interpolation 68
FEM Analysis Element formulation Step 3: Formulation of the System of Equations A. Elemental equations 69
FEM Analysis Element formulation Elemental functional: 70
FEM Analysis Element formulation Integration formula: 71
FEM Analysis Element formulation Use matrix notation: 72
FEM Analysis Assembly B. Assembly 73
FEM Analysis Assembly Carry out the summation: Apply the stationarity condition: 74
FEM Analysis Assembly How to carry out the summation? 75
FEM Analysis Assembly Example: 1 3 5 1 2 3 4 2 4 6 76
FEM Analysis 1. Start from a null matrix and add in the first element: 77
FEM Analysis 2. Add in the second element: 78
FEM Analysis 3. Add in the third element: 79
FEM Analysis 4. Add in the fourth element: 80
FEM Analysis 5. Follow a similar procedure: 81
FEM Analysis Apply BC C. Impose the Dirichlet Boundary Condition: Approach #1: To impose, simply set: To maintain symmetry, set: 82
FEM Analysis Apply BC After imposing,, : Remains Symmetric! 83
FEM Analysis Apply BC Can be made smaller: Worthwhile when there are many prescribed nodes. 84
FEM Analysis Apply BC Approach #2 (Simple one): To impose, simply set: After imposing,, : 85