Limitations of the Leap-Frog Scheme

Transcription

1 Limitations of the Leap-Frog Scheme 1. Structures with high quality factor (resonators, filter,...): ω Ws frequency stored energy The quality factor Q is defined as Q= = Pv losses long decay time of resonant oscillation High Q long rise time of the fields 4 Example: resonator with copper walls typically has Q 7 10 Q 4 only after 10 periods the stored energy decayed to about 2π 37% of it's initial value For a typical discretization and 20 time steps per period this means that 5 more than 10 time integrations would be necessary Long computation time in time domain usually frequency domain is chosen in such cases

2 Limitations of the Leap-Frog Scheme 2. Modelling of small geometrical details: Functions of the grid: i) sampling the wave in space and time ii) map the material distribution to discrete space Difficulty if ratio of wavelength to geometrical detail is large: over-sampling of the wave in space and time Example : Plane wave of 30 MHz (wavelength 10 m) incident on perfect conductor with surface details in mm-range x= y = z t 12 For 1 mm we get 2 10 s from CFL T 8 For a period of 3,3 10 s we would need more than 17,000 time steps per period! This leads to an enormous computational effort why subgrids or a conformal approach are used then.

3 Limitations of the Leap-Frog Scheme 3. Skin depth modelling: Skin depth of good conductors wave length in free space To sample the exponential field decay inside the conductors sufficiently fine a fine grid is needed which in turn leads to extremely small time steps and thus to a strong loss in efficiency Solution: special layer models for skin depth without discretization

4 Limitations of the Leap-Frog Scheme 4. Slowly varying fields ("low-frequency fields", Quasistatics): wavelength dimension (e.g. motors or generators at 50 Hz) i incident field in conductor has to be modelled by discretization i matei r als often non-linear Example: Copper cube of 10 cm side length inside of cubic grid of d = 1m side length and uniform step size of 5 mm; 7 conductivity S/m, applied alternating current of 50 Hz skin depth of δ = 2 / ( ωµ κ) 1cm while vacuum wave length 6 is λ = c/ f 6 10 m 1 m (grid size). For the time integration of one period T = 0.02 s the maximal stable t 12 9 time step is max s affording 2 10 tim Solution: Implicite time integration scheme e steps!

5 Initial Value Problem of Time Domain For stability studies we regard a single system of differential equations: d y = Ay + q, y( 0 ) = y t= t dt representing the discrete Faraday's and Ampere's law. ( 0) initial value problem of the time domain formulation of FIT with: h 0 M 1C µ solution vector: y=, system matrix: A= 1 1 e Mε C Mε Mσ source vector: 0 q=, initial value: y 1 ε S M j ( t= t ) 0 = y ( 0)

6 Properties of the System Matrix ( σ = M = ) In the loss-free case 0 0 similarity transformation A' 1/2 1/2 1 1 µ µ = to transform A to the real skew-symmetric matrix A' A ' = M M 0 M 0 A 1/2 1/2 0 Mε 0 Mε 0 M CM 1/2 T 1/2 ε CM 0 1 µ with identical spectrum. σ 1/2 1/2 1 µ ε we may apply the

7 Properties of the System Matrix Properties of the system matrix: i All eigenvalues of A lie on the imaginary axis. λ A,i They are either 0 or complex-conjugate in pairs. i A complete basis of eigenvectors exists. Eigenvectors of different eigenvalues are orthogonal. A simple estimate for the absolute largest eigenvalue ω yields ω max A ' for any arbitrary matrix norm. = max λ max A,i Example: µ µ, ε ε, u = v= w= yields : C 4c A' = A' = A' = 1 µ ε C 0 0

8 Properties of the System Matrix Generally, the following estimate may be used: Cell i : u v w with ε, µ ω max i ì i i i ωi = + + ε µ u v w max i ( ω ) i i i i i In the lossy case similar estimates are possible: { i} { } M M Re 0 1 σ ε λ ω Im λ ω max i max

9 Recursive Time Integration Method Formal solution of the initial value problem: () ( 0) t 0 ( ( ) ( )) y t = y + Ay t' + q t' dt' t Exact solution of these large systems not possible in practical applications (complete decomposition in eigenvector basis or exponential matrix of A). Therefore: recursive solution. We distinguish: i explicite and implicite recursion, depending on the formula for a new value: explicite or affording to solve an equation or a system of equations i single and multiple step methods, depending on the number of old values taken into account. Linear single step methods may be written as ( m+ 1) ( m) ( m) y = Gy + q with the recursion matrix G.

10 Stability Stability is one of the most important properties of a numerical integration scheme. ( t) The linear recursion operator G depends on t. Several slightly different definitions can be found for the term "stability", the following plays an important role in convergence analysis, too. Definition by Lax-Richtmeyer: 0 ( m+ 1) ( ) ( m t ) ( t) ( m) A recursion y = G y is stable if a constant C and a value t > 0 exist for each (fixed) value T such that ( ) ( m) G C m t T and t < t holds. i.e. G t remains bounded for t 0 and constant T ( m ). 0

11 A sufficient condition for G A sufficient stability condition is G ( t) 1 ( t) since this also involves G m ( t) G( t) 1. Then, for y y ( m) we get G Stability ( m) ( m 1) m ( 0) m ( 0) ( 0) i.e. the solution norm stays bounded (note: no excitation assumed!) λ 1 for all eigenvalues of. m = Gy = = G y G y ( t) 1 ( t) G ( t) It may be used to study different recursion methods w.r.t. their stability. is y,

12 Stability of Some Recursion Methods The three most simple time integration schemes are based on approximating the time derivative by: 1. forward difference quotient: 2. backward difference quotin e t: 3. central difference quotient: (constant time step t) ( ) f t 0 ( ) f t f 0 ( t ) 0 ( + t) f ( t ) f t f 0 0 t ( t ) f ( t t) 0 0 t ( + t/2) f ( t t /2) f t 0 0 t

13 y ( m+ 1) ( m) y t Forward Difference Quotient ( m) ( m) = Ay + q ( m+ 1) ( m) ( m) y = ta+ I y + t q ( m+ 1) explicie t recursion formula for y : G (no inversion, just matrix-vector multiplications) In loss-free case ( M =0) A has purely imaginary eigenvalues σ λ Ai, ωi for it's eigenvectors y eigenvalues of G: ( t ) ( t jω ) Gyi = A+ I yi = i + 1 yi. Each eigenvalue λ i λ Gi, 0 leads to an eigenvalue λ A, i Gi, > 1 = j independently of t, the forward difference quotient is never stable!

14 Forward Difference Quotient Im ( λ G ) ( ω t) max 1 Eigenvalues always lay outside of the (shaded) stability area with λ 1. G ( ω t) = 0 1 Re ( λ ) G ( ω t) max

15 y ( m+ 1) ( m) Resolving for y ( ) y t Backward-Difference Quotient = A y ( m+ 1) ( t A I) ( m+ 1) ( m+ 1) + q ( ) ( ) ( + t ) m+ 1 1 m m+ 1 y = + y q G yields an implicit scheme, i.e. either we need to compute an inverse or solve a linear system (more efficient if A is sparse). 1 1 For the eigenvalues of G we get Gy = ( t A+ I) y = y t λai, jωi + 1 i i i no eigenvalues of G are larger than 1, i.e. the backward difference quotient is stabl e for any arbitrary t. λ Gi,

16 Backward-Difference Quotient ( ) Eigenvalues Im λ always G ( ω tlay inside of ) max the (shaded) stability area with λg 1. 1 Yet, accuracy depends on t ( ω t) = 0 why a time step control is necessary. Im ( λ ) G 1 ( ω t) max ( ω t) = 0 1 Re ( λ ) G ( ω t) max Re( λg) 1 ( ω t ) max

17 Central Difference Quotient In the loss-free case ( = Mσ 0) and allocating h in t + 0 m t, 1 e in t0 + m+ t the leap-frog scheme results from the 2 central difference quotient which we will study now as single step scheme: d 0 A12 y = y+ q dt A with A12 = M 1 C, A µ 21 = Mε C h 0 and the vectors y =, q=. 1 e Mε js

18 Leap-Frog Update Equations ( m) ( m+ 1) ( m) ( m+ 1/2) ( m) h h = h + t A12e y = ( m+ 1/2) e ( m+ 3/2) ( m+ 1/2) ( m+ 1) 1 ( m+ 1) e = e + t A21h t Mε js ( ) ( ) ( ) = e + A h + A e M j = ( ) ( ) ε S m+ 1/2 m m+ 1/2 m+ 1 t 21 t 12 t ( 2 ) ( m+ 1/2 ) ( m ) 1 ( m+ 1 I t A ) 21A12 e t A21h t Mε js which yields ( m ) 0 ( m+ 1) ( m) ( m) ( m) h ( m) y = G y + q with y = = q ( m+ 1/2) 1 ( m+ 1) t e Mε js I ta G = 12 and the recursion matrix. 2 ta21 I+ t A21A12

19 Central Difference Quotient (Leap Frog) To determine the eigenvalues of G we first study the eigenvalue equation h i of the system matrix A: Ayi = λa, i yi with yi = ei A e = λ h and A h = λ e 12 i A, i i h i Let y' i = be an eigenvalue of G. αei Multiply inserting the last 2 equations yields i( 1 α t λa, i) h + hi G = 1 2. α i α i( 1+ α t λa, i + ( t λ e e A, i) ) 21 i A, i i

20 Central Difference Quotient y ' is an eigenvalue of G with Gy ' = λ y ' i i G, i i ( ) 1+ α t λ = 1 + α t λ + t λ holds for α. 1 Ai, Ai, Ai, λ G, i 1 Inserting λ G, i= 1+ α t λa, i α = t λ yields ( t ) λ λ 2+ λ + 1= 0 or G, i G, i A, i ( t λ, ) Ai ( t λ Ai, ) λ G, i= ± describing the relation between the eigenvalues of A and G. 2 2 Ai,

21 Central Difference Quotient (Leap Frog) ( for ω t = 2) t ω max λg, i 1 ωi t 2 t t = with ω = max λ max max A, i ωmax Im ( λ ) G 1 " + " ω t = 0 ( ω t) max ± " " ( ω t) max ± " + " 1 Re ( λ ) G ω t = 2 " " stability limit ωi t = 2

22 Central Difference Quotient (Leap Frog) Inserting the estimate ωmax max = max + + εiµ u i i vi wi for the absolute largest eigenvalue yields again 1 t t max min εiµ i, i ui vi wi i.e. the Courant-Friedrichs-Levy condition. ( ωi ) ω max can also be determined numerically, yielding an exact stability limit for the leap-frog scheme.

23 Central Difference Quotient (Leap Frog) The stability limit has to be kept strictly: exceeding the limit by 1% only, the instable portion per interation step increases by the factor of f = /100 = 1.28 (this is f after 100 steps)

24 Example b v e 1 b 4 1 b 3 e t = 410 t = w b 2 u 2 0 t = t -2-4 t = stability limit for time step: (experimentally found) t t max As example we regard a computational domain built out of four cells only, with equidistant Cartesian grid of N u x N v x N ω = 3 x 3 x 2 ( u = v = ω= ) lines. We assume ideal electric boundaries such that only the field components displayed in the sketch are non-vanishing (indices are chosen different from usual). Further, we assume a homogeneous material distribution with ε = ε 0 and µ = µ 0. The simple expressions resulting for the material matrices are given above.

25 Central Difference Quotient - Example system matrix µ µ µ µ ε ε ε ε = A b 1 4 b b 3 b 2 e 1

26 Central Difference Quotient - Example b e 1 4 b 1 b 2 b 3 (absolute) largest eigenvalue (dimensionless): λ ± 2 j 0 Amax, = = =± µ ε 0 0 Amax, ± 2 jc maximal time step : 2 tmax = = = λ c 0 j (Identical to experimentally found value) 9 Yet, CFL condition yields only the estimate t max 3 c which, in this case, is a limit too small by a factor of

27 Properties of Explicite Leap Frog Scheme In realistic applications, the limit determined by the CFL condition is not too small but close to the stability limit. i The leap frog recursion scheme is suitable to compute loss-free systems. i The time step is limited for stability reasons. i The numerical effort per step is one matrix-vector-multiplication. i The integration with time steps fulfilling the CFL condition is sufficiently accurate. (Generally, spatial and time discretization error are equal.)

28 CST Microwave Studio

29 Example: Rectangular Waveguide

30 Example: Rectangular Waveguide

31 Example: Rectangular Waveguide

32 Example: Rectangular Waveguide

33 Example: Rectangular Waveguide - Junction

34 Example: Rectangular Waveguide - Junction

35 Example: Rectangular Waveguide - Junction

36 Example: Rectangular Waveguide

37 Example: Rectangular Waveguide

38 Example: Rectangular Waveguide

39 Example: Rectangular Waveguide

40 Example: Rectangular Waveguide

41 Example: Rectangular Waveguide

42 Example: Rectangular Waveguide

43 Example: Rectangular Waveguide

44 Example: Circular Waveguide

45 Example: Circular Waveguide TE11, both polarizations

46 Example: Circular Waveguide TM01

47 Example: Circular Waveguide TE21, both polarizations

48 Waveguide Junction CST-Demo

49 Waveguide Junction CST-Demo Port 1 Port 3

50 The Human Body Model HUGO in MWS TM HUGO: based on Visible Human Data Set of the National Library of Medicine, Maryland 40 tissues 7 detail level: min/max-voxel size 1x1x1 mm (361 MB) 8x8x8 mm ( 0,7 MB) C/C++ programming interface convertable in STL format

51 HUGO and RF-Waves of a WLAN-Card Magnetic field Magnetic field Magnetic energy density Magnetic energy density V.Motrescu, G.Pöplau, UvR

52 Examples for Parameter Extraction PCB RJ45 connection CST-Demo isolator ε = 2,1; µ = 1 r r transmission lines ε = 2, 4 ; µ = 1 r r connector PEC PCB, socket & connector ε = 2, 2 ; µ = 1 r r

53 RJ45 Connector Extraction of some SPICE-compatible network model based on the computation of S-parameters : Definition of excitation sources: all wires are terminated with discrete ports the ports are excited sequentially with a Gaussian pulse yields broadband results for all wires mesh points CST-Demo

54 RJ45 Connector Ports 1 and 6 CST-Demo

55 IC Package ε r = Box ports, mesh points Substrat Silicon ε = 12,3 PEC wires and traces r ε = 9, 2 r CST-Demo

56 IC Package CST-Demo

57 Parasitic Effects in IC s Current density of an IC at 5 GHz T. Wittig, I. Munteanu, T. Weiland, TU Darmstadt

58 Patch-Antenna Patch-Antenna Constante energy flux density Coaxial line Planes of constant field amplitude T.Weiland et al. CST

59 Patch-Antenna Folded Patch-Antenna Electric field of the excited modes Two- and three-dimensional farfield representation T.Weiland et al. CST

60 Horn Antenna Horn antenna with dielectric cone, coaxially loaded E y Antenna with farfield T.Weiland et al. CST

61 Horn Antenna Rillenhornstrahler Versorgung by Dielectrically filled waveguide Electric field strength (nearfield) T.Weiland et al. CST

62 Mobile Communication CAD-Data (STL-Format) modelled Diskretization Electric field strebgth T.Weiland et al. CST

63 Magneto-Quasistatics (MQS) J t D R r R r max max 3 3 << 0 div div rot rot = = = = B D J H t B E ρ

64 Magneto-Quasistatics (MQS) Harmonic Time-Dependence: rot E rot H div D div B = = = iωb J ρ = 0 rot div H = J B= 0

65 MQS: Position Sensor Electrical position sensor (BOSCH) - f = 10 khz - Position S of shielding ring by measurement of coil inductivity L H-field L Shielding ring (Copper) s J Excitation coil Iron Simulation: M. Clemens, TU Darmstadt

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67 MQS: Magnet Head in Hard Disc SRC hard disc magnet head Simulation: M. Clemens, TU Darmstadt (Benchmark problem of Storage Research Consortiums (SRC)) 148 µm Air gap (0.4µm) Copper coils Permalloy Transient problem (Magneto-Quasistatics) 1.2 Mio. Unknowns

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