Computational Electromagnetics Definitions, applications and research

Transcription

1 Computational Electromagnetics Definitions, applications and research Luis E. Tobón Pontificia Universidad Javeriana Seminario de investigación Departamento de Electrónica y Ciencias de la Computación November 30, 2012

2 Outline Definition and some areas of application Maxwell s equations and numerical methods Finite Elements Method Discretized EM The De Rham diagram Basis functions in tetrahedral element Galerkin s method: FETD based on E and H fields Cases: Heat sink and MW filters Domain Decomposition: Discontinuous Galerkin s method Implicit time integration: Block-Thomas Crank-Nicholson (BT-CN) method Cases: MW filters and On-Chip interconnection Current work: Improved BT-CN method FETD based on E and B fields Conclusions and future work 2

3 Computational Electromagnetics Definition Computational Electromagnetics is the process of modeling the interaction of electromagnetic fields with physical objects and the environment. Wiki Computational Electromagnetics deals with the art and science of solving Maxwell s equations numerically using computers. Jian-Ming Jin It is used to analyze: Antenna performance EM compatibility EM Wave Propagation EM devices (RF, MW, photonics) 3

4 Areas of Application Devices Metamaterials Active and passives Microstrip Antennas 4

5 Areas of Application Signal Integrity Real Scheme Model 5

6 Areas of Application EM compatibility Emission spectrum spreading for new generation components Continuous decrease of power supply voltages. 6

7 Areas of Application EM imaging and sensors Oil exploration Landmine detection acoustics.org 7

8 Computational electromagnetics Maxwell s equations Topological Laws Constitutive Laws Oliver Heaviside ( ) From Continuous to Discrete! 1. Which fields must be selected? E? H? D? B? 2. What is the correct discrete representation of these fields? 3. What is the definition of discrete constitutive laws? 4. What is the numerical dispersion of these schemes? James Clerck Maxwell ( ) 8

9 Computational electromagnetics Some methods Time Domain FDTD Directly PDE from Maxwell s Equation (Yee s cell) FETD Weak form of Maxwell s Equation Frequency Domain FDFD FD in Frequency domain FEM Weak form of Helmholtz Equation MoM Volume and Surface Integral Equations (Electric and Magnetic) Hybrid Techniques FDTD-FETD MoM-FEM 9

10 Finite Elements Method 10

11 Discretized Maxwell s Equations The De Rham Diagram relates function from the Hilbert-Sobolev spaces by means of differential operators 11

12 Constituve Topological Discretized Maxwell s Equations Continuous Laws Discrete Representation 1-forms Curl-Conforming 2-forms Div-Conforming 12

13 Basis Functions Curl-Conforming Tetrahedral element 6 Edge BF 13

14 Basis Functions Curl-Conforming Tetrahedral element 12 Edge BF 14

15 Basis Functions Curl-Conforming Tetrahedral element 8 Face BF 15

16 Basis Functions Div-Conforming Tetrahedral element 4 Face BF 16

17 Basis Functions Div-Conforming Tetrahedral element 12 Face BF 17

18 Basis Functions Div-Conforming Tetrahedral element 3 Volume BF 18

19 Basis Functions The DeRham diagram Tetrahedral element E B D H 19

20 Weak form of Maxwell s equations Computational electromagnetics FETD, Galerkin s method 20

21 Computational electromagnetics Discretization using tetrahedrons Edge basis functions (Ct/Ln) sop.inria.fr cst.com 21

22 Computational electromagnetics Case 1. Heat Sink, model 150 mm 150 mm PEC cavity 60 mm 48 mm 4 mm 48 mm Observer Ez: (63,63,45) mm Fmax 10 GHz 30 mm 5 mm Source: (0,0,-4) mm Ez BHW Fo = 4 GHz 22

23 Computational electromagnetics Case 1. Heat Sink, discretization and results Matlab Model using brick elements Good agreement between commercial software and our results 23

24 Computational electromagnetics Case 3. Strip line fmax=10 GHz 110 mm Source 16.5 mm Voltage in Port 2 r=4.8 W=5.7 mm H=3.18 mm T=0.32 mm Zo=50 L=80 mm 58.4 mm Good agreement S21 24

25 Computational electromagnetics Case 4. -filter, results fmax=10 GHz 5.7 mm 110 mm 17.5 mm 7.5 mm 16.5 mm 2.2 mm 7.5 mm 7.5 mm 58.4 mm r=4.8 W=5.7 mm H=3.18 mm T=0.32 mm Zo=50 L=80 mm 25

26 Computational electromagnetics Case 4. -filter, model Low Pass Band Pass 26

27 Computational electromagnetics Multi-scale problems Challenges: Spatial discretization FDTD: too many unknowns FETD: inversion or factorization of large system matrices Time integration explicit scheme: very small time steps implicit scheme: inversion or factorization of large matrices 27

28 /Smallest Largest Introduction Domain decomposition for multiscale structures fine subdomains coarse subdomains 10 5 Implicit CN-BT CN-GS Low Tetra Hybrid IMEX LocalTS Low to High Prism Hexa Explicit RK High Brick 10 Multiscale Factor = Largest /Smallest 28

29 Domain Decomposition Method Discontinuous Galerkin FETD Maxwell s equations Galerkin s weak form perform integration by parts surface integration around subdomain 29

30 Domain Decomposition Method Riemann solver Galerkin s weak form with integration by parts surface integration Riemann solver for interface between adjacent subdomains 30

31 Domain Decomposition Method Discretized system of equations Large system matrices are divided into several middle sized matrices by the hybrid SETD/FETD method 5 X 5 X 4 subdomains Interfaces between subdomains 31

32 Domain Decomposition Method Time integration, Crank-Nicholson for sequential domains Sequential order of subdomains: Reflections (i-1)-th subdomain i-th subdomain (i+1)-th subdomain Transmissions Crank-Nicholson implicit method: Block diagonal!! 32

33 Domain Decomposition Method Time integration, Block-Thomas Crank-Nicholson method 1. Block LU decomposition 2. Solve for L (forward) 3. Solve for U (Backward) 33

34 Domain Decomposition Method Case 4. Microwave filter High Pass PEC Cavity 0.2mm 0.1mm Microstrip Z0=50 W=0.065mm T=0.67 m Dielectric: Duroid r= mm Port 2. Passive Chip Inside: mm Port 1. Active Fmax 30 GHz 4.9 mm mm Capacitor 0.065mm x 0.06 mm x 0.08 m multiscale factor =

35 Domain Decomposition Method Case 4. Microwave filter High Pass FDTD FDTD 10 times less unknowns than FDTD 4 times faster than HFSS 8 times faster than CST 9 times faster than FDTD 35

36 Domain Decomposition Method Case 5. High Q Band Pass Microwave filter Thickness of plates in layers 1, 2 and 3 is 6 m. Multiscale Factor 36

37 Domain Decomposition Method Case 5. High Q Band Pass Microwave filter In Port 1 In Port 2 37

38 Domain Decomposition Method Case 5. High Q Band Pass Microwave filter Resonance Tunning li Resonant frequency li fr 0.45mm 1.34GHz 0.65mm 1.22GHz 0.85mm 1.14GHz This analysis takes less than 1.5 hours, 5.9 hours for one simulation using FDTD 38

39 Domain Decomposition Method Case 6. Interconnect Layered structure Real model Simplified model multiscale factor =

40 Domain Decomposition Method Case 6. Interconnect Layered structure FDTD grid PPW=40 cells: 511 X 323 X 60 total DoF: > 50 million SETD / FETD mesh PPW=40 44 subdomains total DoF: 152,356 40

41 Domain Decomposition Method Case 6. Interconnect Layered structure Relatively big difference for S 31 and S 41 S 31 and S 41 are very small quantities (< -50 db) Interfaces bring artificial dissipation and dispersion 41

42 Domain Decomposition Method Case 7. Packaging-to-Chip interconnect GND 11 mm 6 mm Connectors Port 4 Port 5 Port 6 Port 1 Port 2 Port 3 IC Active port: Port 1 (50 Ohms) Passive port: Port 4 (50 Ohms) V s : BHW fc=2.6 GHz 42

43 Domain Decomposition Method Case 7. Packaging-to-Chip interconnect 7 Layer-Domains SPrism DG-FETD Total DoF: CPU time: 9 min Mem. Cost: 192 MB FDTD Total DoF: 1.4 M CPU Time: 36 min HFSS (30 freq.) CPU time: 11:26 min Mem. Cost: 66 M 43

44 Optimization BT-CN 1 T 12 2 M 1 S 1 S 2 M 2 T 32 S 3 M 3 3 T 21 T 23 LU decomposition and Block-Thomas L U 44

45 New LDU Decomposition 1 T 12 2 M 1 S 1 S 2 M 2 T 32 S 3 M 3 3 T 21 T 23 LDU decomposition Volumes Surface to volume Volume No Transpose Volume to surface Interfaces Connection between interfaces in same domain. Usually are zeros Interface 45

46 New LDU Decomposition 1 T 12 2 M 1 S 1 S 2 M 2 T 32 S 3 M 3 3 T 21 T 23 LDU-Block decomposition Volumes 3 Interfaces source to volume 2 Volume source to interfaces 1 Interfaces BT Advantages: 1. Highly parallelizable 2. Smaller matrices 3. Memory cost 4. CPU time 5. General formulation? 46

47 New LDU Decomposition, algorithm 0. Pre-Processing: Solve and store: Parallel 1. Algorithm Volume to interface: Parallel 47

48 New LDU Decomposition, algorithm No needed 2. Algorithm Interface solution: 1. Solved as a whole 2. Apply BT 48

49 New LDU Decomposition, algorithm 1 T 12 2 M 1 S 1 S 2 M 2 T 32 S 3 M 3 3 T 21 T Algorithm Interface to Volume: LUPQR decomposition Sparse Sparse 49

50 Models Total SD1: SD2: SD3: Total SD1: SD2: SD3: SD4: Total Total Total 385k Total SD1: SD2:

51 Cases of study 51

52 Field accuracy case 1 Fmax = 670 MHz 20 ppw Perfect agreement 52

53 Field accuracy case 2 Fmax = 670 MHz 20 ppw Perfect agreement Accuracy is not an issue 53

54 Computational cost, Memory DoF per SD fixed, Number of SD changed Out of memory It is not the limit 4 times less memory 54

55 Computational cost, time DoF per SD fixed, Number of SD changed The new method is always faster 55

56 Computational cost, memory Number SD fixed, DoF per SD changed No a general solution!! Out of memory < 3k Interface linear system is solved as a whole > 4k DoF on interface Block-Thomas algorithm for interface linear system 56

57 Computational cost, memory Block-Thomas for Interface Linear System > 6k DoF on interface No limit of memory, yet 590 MB 726 s More study is required 320 MB 376 s 57

58 Constituve Topological Maxwell s Equations Continuous Laws Discrete Representation 1-forms Curl-Conforming 2-forms Div-Conforming 58

59 Discrete Maxwell s Equations Wave Equation Formulation 59

60 Discrete Maxwell s Equations EH Formulation Sparse square matrices length(h) >> length(e) 60

61 Discrete Maxwell s Equations EB Formulation Sparse square matrices size(m ee ) size(m bb ) 61

62 Discrete Maxwell s Equations EB-Hodge Formulation dense size(m ee ) size(m bb ) 62

63 Validation Eigenvalues 0.5 cm 0.75 cm =2.5mm 1.0 cm Mode Analytical result (GHz) E1H2 (GHz) Error (%) E1B1 (GHz) Error (%) E2B2 (GHz) Error (%) TE , ,3452 2, ,7779 0, ,9871-0,0164 TM , ,9491 4, ,9020 1, ,5275-0,0251 TE , ,1833 4, ,3638 1, ,0425-0,3036 TE , ,3456 4, ,5245 1, ,0542-0,3362 TM , ,5466 6, ,0133 2, ,0603-0,0899 TE , ,3066 4, ,7656 0, ,0767-0,1320 TM , ,5894 6, ,5081 2, ,4103-0,0276 TE , ,6803 4, ,9204-0, ,6889 0,0019 Large Error E DoF: 309 H DoF: 2770 Same as Wave E DoF: 309 B DoF: 818 Same as Wave E DoF: 1910 B DoF:

64 Eigenvalues Maxwell s equations First Mode Second Mode 64

65 Transient solutions Maxwell s equations, Explicit 65

66 Transient solutions Maxwell s equations, Explicit 66

67 Transient solutions Maxwell s equations, Implicit 67

68 Conclusions FETD for Mawxell s equations was defined. Correct basis functions to approximate E, B, H, and D fields according with the De Rham diagram were presented. Efficient and accurate locally implicit DG-FETD schemes have been discussed: The spatial discretization is based on discontinous Galerkin s method The time stepping consists of the Crank-Nicolson method with free-iterative Block- Thomas algorithm. It was showed the DG-FETD s capacity of solving large systems and layered structures for multiscale simulations. Implicit time integration for sequential domains is improved performing a new memory efficient and highly paralellizable LDU decomposition. A new implementation of FEM based on E-B fields shows improvements in accuracy and computational costs, for both frequency and time responses. 68

69 Nest work 1. Numerical dispersion analysis of FETD based on EB fields 2. Implementation of EB-scheme in hexahedral and prismatic elements 3. Realistic cases of application 1. On-chip 2. Oil exploration 3. Photonic device (photonic crystal or metamaterial) 4. Writing 69

70 Acknowledgments Prof. Qing Liu s group Dr. Jiefu Chen Pratt School of Engineering Duke University Pontificia Universidad Javeriana, Cali Universidad del Quindío Fulbright Colciencias Intel Co. Family and friends!!! 70