Optimizing the perfectly matched layer by F. Collino, P . B. Monk - PowerPoint PPT Presentation

Optimizing the perfectly matched layer by F. Collino, P . B. Monk Norbert Stoop Optimizing the perfectly matched layer – p. 1

Overview PML constructed using a change of variables Cartesian coordinates (review) Comparison to Bérenger’s approach in cylindrical coordinates Discretization of PMLs and resulting effects Optimization of cartesian PMLs Effects of boundary conditions Framework: Planar Maxwell equations Optimizing the perfectly matched layer – p. 2

PML construction - Overview We have already seen that a PML can be understood in two ways: Split the magnetic field and introduce a damping term σ (Bérenger’s approach) Perform a complex change of variables We will see: 1. cartesian case: both are equivalent 2. cylindrical coordinates: inequivalent, efficiency differs Optimizing the perfectly matched layer – p. 3

Planar PML for cartesian coords Consider a TE wave ( E z = 0 ) in free space ( ǫ 0 = µ 0 = c = 1 ). The two-dimensional Maxwell equations then reduce to: ∂H z ∂E x ∂y − ∂E y = ∂t ∂x ∂E y = − ∂H z ∂E x = ∂H z , ∂t ∂x ∂t ∂y Suppose we’d like to construct a 2D PML for x > 0 : y E y PML H z E x x z x=0 Optimizing the perfectly matched layer – p. 4

Bérenger PML for cart. coords Bérenger: 1. Split H field: H z = H zx + H zy such that the MW equations can be written as: ∂H zx = − ∂E y ∂H zy = ∂E x , ∂t ∂x ∂t ∂y ∂E y = − ∂H z ∂E x = ∂H z , ∂t ∂x ∂t ∂y 2. Introduce damping term σ ( x ) ( σ ( x ) = 0 for x < 0 ) in all equations which contain x -derivatives: ∂H zx + σ ( x ) H zx = − ∂E y ∂H zy = ∂E x , ∂t ∂x ∂t ∂y ∂E y ∂t + σ ( x ) E y = − ∂H z ∂E x = ∂H z , ∂x ∂t ∂y Optimizing the perfectly matched layer – p. 5

Bérenger PML for cart. coords II 3. In time harmonic regime, E i ( x, y, t ) = ˆ H zi ( x, y, t ) = ˆ E i ( x, y ) exp( − iωt ) , H zi ( x, y ) exp( − iωt ) , i = x, y , the PML equations can be written as: H z = ∂ ˆ ∂ ˆ E x 1 E y − iω ˆ ∂y − ∂x , 1 + iσ/ω ∂ ˆ E x = ∂ ˆ 1 H z H z − iω ˆ − iω ˆ E y = − ∂x , 1 + iσ/ω ∂y Optimizing the perfectly matched layer – p. 6

Change of variables technique 1. Start again in time harmonic regime, but don’t split fields: E i ( x, y, t ) = ˆ H z ( x, y, t ) = ˆ E i ( x, y ) exp( − iωt ) , H z ( x, y ) exp( − iωt ) , i = x, y 2. In frequency domain, the Maxwell equations become: ∂ ˆ ∂y − ∂ ˆ E x E y − iω ˆ H z = ∂x , − ∂ ˆ E x = ∂ ˆ H z H z − iω ˆ − iω ˆ E y = ∂x , ∂y � x 3. Change of variables: x → x ′ = x + i 0 σ ( s ) ds ω Optimizing the perfectly matched layer – p. 7

Change of variables technique II If we use the chain rule to replace x ′ by x , we get: ∂ ˆ ∂ ˆ E x 1 E y − iω ˆ H z = ∂y − ∂x , 1 + iσ/ω ∂ ˆ E x = ∂ ˆ 1 H z H z − iω ˆ − iω ˆ E y = − ∂x , 1 + iσ/ω ∂y This is exactly the Bérenger PML in the frequency domain: Both approaches are equivalent! Practical computation: truncate PML. We impose Dirichlet BC: R δ ˆ R = e − 2 ik x 0 (1+ iσ ( s ) /ω ) ds E y ( x = δ, y, t ) = 0 = ⇒ Note: Pick σ large to minimize R (if k x ∈ R ). Optimizing the perfectly matched layer – p. 8

PML for curvilinear coordinates Do Bérenger’s and the complex change of variables approach also result in equivalent PMLs for non-Cartesian coordinate system? Maxwell’s equations in polar coordinates ( ρ , θ ): ∂H z = 1 � ∂E ρ ∂θ − ∂ � ∂ρ ( ρE θ ) ∂t ρ ∂E ρ = 1 ∂H z ∂E θ ∂t = − ∂H z ∂θ , ∂t ρ ∂ρ Assume the layer starts at ρ = a , so σ ( ρ ) > 0 for ρ > a and 0 otherwise. PML e θ ρ =a z e ρ Optimizing the perfectly matched layer – p. 9

Change of variables for polar coords � ρ Start in the frequency domain. Let ρ ′ = ρ + i a σ ( s ) ds and introduce ω � ρ d ( ρ ) = 1 + iσ ( ρ ) d ( ρ ) = 1 + i 1 ¯ and σ ( s ) ds ω ρω a such that ρ ′ = ρ ¯ dρ ′ d and dρ = d . We thus have in freq. domain: − iωH z = 1 � ∂E ρ ∂θ − 1 ∂ � ∂ρ ( ¯ dρE θ ) ¯ dρ d − iωE ρ = 1 ∂H z − iωE θ = − 1 ∂H z ∂θ , ¯ d ∂ρ dρ Note: 1 ∂ ∂ ∂ ∂ρ = ∂ρ ′ = ∂ ( ¯ d dρ ) Optimizing the perfectly matched layer – p. 10

Change of variables for polar coords II E θ = ¯ Using ˜ E ρ = dE ρ and ˜ dE θ we get the traditional Helmholtz equations: � � ∂ ˜ dH z = 1 ∂θ − ∂ E ρ − iωd ¯ ∂ρ ( ρ ˜ E θ ) ρ ¯ d E ρ = 1 ∂H z − iω d E θ = − ∂H z ˜ ˜ − iω ∂θ , ¯ d ρ ∂ρ d ρ = 1 /d ˜ θ = 1 / ¯ d ˜ We can return to time domain by introducing E ∗ E ρ , E ∗ E θ , � ρ z = ¯ σ ( ρ ) = 1 H ∗ dH z and ¯ a σ ( s ) ds : ρ ! ∂ ˜ ∂E ∗ ∂E ∗ ∂H ∗ z = 1 E ρ − ∂ ρ = 1 ∂H z θ = − ∂H z ρ z ∂ρ ( ρ ˜ θ ∂t + σH ∗ σE ∗ ∂t + σE ∗ E θ ) , ∂t +¯ ∂θ , ρ ∂θ ρ ∂ρ ∂ ˜ ∂ ˜ ∂E ∗ = ∂E ∗ E ρ E θ ∂H z σH z = ∂H ∗ ρ z θ + σE ∗ σE ∗ = ρ , + ¯ θ , + ¯ ∂t ∂t ∂t ∂t ∂t ∂t Optimizing the perfectly matched layer – p. 11

Comparison to Bérenger’s PML In order to compare the two constructions, assume that ∂H z ∂θ = 0 and choose E ρ = 0 . ⇒ ∂H ∗ ∂E ∗ z = − 1 ∂ θ = − ∂H z ∂ρ ( ρ ˜ z + σH ∗ ∂t + σE ∗ θ = E θ ) , ∂t ρ ∂ρ ∂ ˜ = ∂E ∗ σH z = ∂H ∗ E θ ∂H z z θ σE ∗ ∂t + ¯ θ , + ¯ ∂t ∂t ∂t Bérenger’s construction would yield: ∂H z + σH z = − 1 ∂ ∂E θ ∂t + σE θ = − ∂H z ∂ρ ( ρE θ ) , ∂t ρ ∂ρ They are clearly different! Question: How do they perform qualitatively? Optimizing the perfectly matched layer – p. 12

Comparison to Bérenger’s PML II Think of the following setup: PML 1 a δ We take the source to be on the unit disc, At ρ = 1 : E θ = sin(2 πt ) for 0 ≤ t ≤ 1 and 0 otherwise , and choose a quadratic ρ -dependance for σ : σ ( ρ ) = σ 0 ( ρ − a ) 2 /δ 2 , for ρ ≥ a Optimizing the perfectly matched layer – p. 13

Comparison to Bérenger’s PML III 1.0 1.0 Bérenger Bérenger Change of variables Change of variables 0.5 0.5 σ = 50 σ = 100 0 0 a = 2.0 Magnetic field a = 2.0 Magnetic field 0.0 0.0 -0.5 -0.5 -1.0 -1.0 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 Time Time 1.0 P arameters : σ = 100 0 a = 1.02 N = 100 ( n u mber of p oints ) 0.5 n = 10 ( n u mber of p oints in P M L) l Magnetic field h = ( a-1 )/N ( s p acing ) δ = n h ( la y er thic k ness ) l 0.0 2 2 σ (ρ) = σ (ρ− a ) / δ 0 -0.5 A ll p lots sho w H at x = h / 2 ( ie. close to the z scatterer ) -1.0 0.0 1.0 2.0 3.0 4.0 Time Optimizing the perfectly matched layer – p. 14

Conclusions The change of variables PML gives a much more accurate (discrete) absorbing layer than Bérenger’s construction in polar coordinates. Unlike Bérenger’s PML, the change of variables technique allows tuning of PMLs situated very close to the scatterer, yet producing very good absorption. The quality of our PML still depends on a number of parameters (including discretization params) which need to be chosen wisely. ⇒ Is there a way to quantify the effects of discretization? = Furtermore, can we derive optimal PML parameters from there? Optimizing the perfectly matched layer – p. 15

Effects of discretization For simplicity, we restrict ourselves to the planar, two-dimensional case. Starting from Bérenger’s construction, we avoid the split fields by defining: E x = (1 + iσ/ω ) ˆ ˜ E y = ˆ ˜ H z = ˆ ˜ E x , E y , H z Now we have again a traditional curl-curl structure: H z = ∂ ˜ ∂y − ∂ ˜ E x E y − iω (1 − iσ/ω ) ˜ ∂x E y = − ∂ ˜ E x = ∂ ˜ H z iω H z − iω (1 − iσ/ω ) ˜ ˜ ∂x , − 1 + iσ/ω ∂y Optimizing the perfectly matched layer – p. 16

Discretization of planar PML We use ~ E l+1/2, j+1 a standard Yee scheme and let σ ( x ) be j+1 ~ ~ piecewise constant with jumps at x = lh , H H l-1/2, j+1/2 l+1/2, j+1/2 ~ ~ ~ l = 0 , 1 , 2 , . . . . We denote by σ l +1 / 2 E E E l-1, j+1/2 l, j+1/2 l+1, j+1/2 ~ ~ E E the value of σ in the interval ( lh, ( l + 1) h ) . l-1/2, j l+1/2, j j ~ H l+1/2, j-1/2 We then arrive ~ E l+1, j-1/2 ~ at the following discretized equations: E l+1/2, j-1 j-1 l=0 l-1 l l+1 l=n l H l +1 / 2 ,j +1 / 2 − ˜ ˜ H l +1 / 2 ,j − 1 / 2 ω ˜ − i E l +1 / 2 ,j = γ l +1 / 2 h H l +1 / 2 ,j +1 / 2 − ˜ ˜ − iω γ l +1 / 2 + γ l − 1 / 2 H l − 1 / 2 ,j +1 / 2 ˜ E l,j +1 / 2 = − 2 h E l +1 / 2 ,j +1 − ˜ ˜ E l +1 ,j +1 / 2 − ˜ ˜ E l +1 / 2 ,j E l,j +1 / 2 − iωγ l +1 / 2 ˜ H l +1 / 2 ,j +1 / 2 = − h h with γ l +1 / 2 = 1 + iσ l +1 / 2 /ω . Optimizing the perfectly matched layer – p. 17