PML for computational electromagnetics: PML for computational electromagnetics.pdf

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1. The Requirements for the Simulation of Free Space and a Review of Existing
Absorbing Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 The Maxwell Equations and the Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 The Actual Problems to be Solved with Numerical Methods . . . . . . . . . . . . . . . . . .7
1.3 The Requirements to be Satisfied by the Absorbing Boundary Conditions . . . . . 8
1.4 The Existing ABCs before the Introduction of the PML ABC . . . . . . . . . . . . . . . 9
2. The Two-Dimensional PerfectlyMatched Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 A Medium without Reflection at Normal and Grazing Incidences . . . . . . . . . . . .13
2.2 The PML Medium in the 2D TE Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Reflection ofWaves from a Vacuum–PML Interface and from a
PML–PML Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
2.4 The Perfectly Matched Layer Absorbing Boundary Condition . . . . . . . . . . . . . . . 21
2.5 EvanescentWaves in PML Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3. Generalizations and Interpretations of the PerfectlyMatched Layer . . . . . . . . . . . . . 29
3.1 The Three-Dimensional PML Matched to a Vacuum . . . . . . . . . . . . . . . . . . . . . . 29
3.2 The Three-Dimensional PML Absorbing Boundary Condition. . . . . . . . . . . . . .35
3.3 Interpretation of the PML Medium in Terms of Stretched Coordinates . . . . . . 36
3.4 Interpretation in Terms of Dependent Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5 The PML Matched to General Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6 The PML Matched to Nonhomogeneous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.7 The Uniaxial PML Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.8 The Complex Frequency Shifted PML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4. Time Domain Equations for the PMLMedium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Time Domain PML Matched to a Vacuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.1 The Split PML. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.2 The Convolutional PML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.3 The Near PML. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52
4.1.4 The Uniaxial PML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
vi PERFECTLYMATCHED LAYER (PML) FOR COMPUTATIONAL ELECTROMAGNETICS
4.2 Time Domain PML for Lossy Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.1 Split PML for Lossy Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54
4.2.2 CPML for Lossy Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.3 NPML for Lossy Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.4 Uniaxial PML for Lossy Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Time Domain PML for Anisotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.1 Split PML for Anisotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.2 CPML for Anisotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.3 NPML for Anisotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Time Domain PML for Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4.1 Time Domain CPML and NPML for Isotropic or
Anisotropic Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4.2 Time Domain Uniaxial PML for Isotropic Dispersive Media. . . . . . . . .61
5. The PML ABC for the FDTDMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1 FDTD Schemes for the PML Matched to a Vacuum . . . . . . . . . . . . . . . . . . . . . . . 63
5.1.1 FDTD Scheme for the Split PML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1.2 FDTD Scheme for the Convolutional PML. . . . . . . . . . . . . . . . . . . . . . . . 65
5.1.3 FDTD Scheme for the NPML. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66
5.1.4 FDTD Scheme for the Uniaxial PML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1.5 A Comparison of the Requirements of the Different Versions
of the PML ABC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 FDTD Schemes for PMLs Matched to Lossy Isotropic Media . . . . . . . . . . . . . . 68
5.3 FDTD Schemes for PMLs Matched to Anisotropic Media . . . . . . . . . . . . . . . . . .69
5.4 FDTD Schemes for PMLs Matched to Dispersive Media . . . . . . . . . . . . . . . . . . . 70
5.5 Profiles of Conductivity in the PML ABC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.6 The PML ABC in the Discretized FDTD Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.6.1 Propagation of PlaneWaves in the Split FDTD-PML . . . . . . . . . . . . . . 75
5.6.2 Reflection from a PML–PML Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . .77
5.6.3 Reflection from a N-Cell-Thick PML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.6.4 Reflection from the CPML, the NPML, and the Uniaxial PML . . . . . 85
5.6.5 Reflection from the CFS-PML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6. Optmization of the PML ABC inWave-Structure Interaction and
Waveguide Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1 Wave-Structure Interaction Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1.1 The General Shape of the Results Computed with a PML Placed
Close to a Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
CONTENTS vii
6.1.2 Interpretation of the Numerical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.1.3 Design of the PML Using a Regular Stretching Factor . . . . . . . . . . . . . . 93
6.1.4 Design of the PML Using the CFS Stretching Factor . . . . . . . . . . . . . . . 95
6.2 Waveguide Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2.1 Improvement of the Absorption by Means of a Real
Stretch of Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2.2 Improvement of the Absorption by Using a CFS
Stretching Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3 Concluding Remarks to the Application of the PML ABC to
FDTD Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
7. Some Extensions of the PML ABC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.1 The Perfectly Matched Layer in Other Systems of Coordinates . . . . . . . . . . . . . 107
7.2 The Perfectly Matched Layer with Other Numerical Techniques . . . . . . . . . . . 107
7.3 Use of the Perfectly Matched Layer with Other Equations of Physics . . . . . . . 109
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Author Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

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PML worthes the push

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