An iterative solver enhanced with extrapolation for steady-state high-frequency Maxwell problems

Transcription

1 An iterative solver enhanced with extrapolation for steady-state high-frequency Maxwell problems K. Hertel 1,2, S. Yan 1,2, C. Pflaum 1,2, R. Mittra 3 kai.hertel@fau.de 1 Lehrstuhl für Systemsimulation Department Informatik Friedrich-Alexander-Universität Erlangen-Nürnberg 2 School for Advanced Optical Technologies Friedrich-Alexander-Universität Erlangen-Nürnberg 3 Electromagnetic Communication Laboratory Electrical Engineering Department Pennsylvania State University Copper Mountain March 22 nd / 24

2 Basic Outline 1 Model and Discretization 2 Iterative Solver 3 Extrapolation 4 Numerical Results 2 / 24

3 Overview of the model problem High-frequency Maxwell problem (optical regime, Hz < f < Hz, 0.3µm < λ < 1.1µm resp.) Simple box-shaped domain geometries Both media of positive and negative permittivities Rough interfaces between media Periodic and absorbing boundaries (PML) Figure : Schematic of a multi-junction thin-film silicon solar cell 3 / 24

4 Overview of the numerical scheme Finite Difference Frequency Domain discretization (FDFD) Spatial discretization based on Yee cells Finite Integration Technique (FIT) Explicit leap-frog time stepping scheme Algebraically speaking a Richardson method Prony s extrapolation method Figure : EM field in a solar cell structure 4 / 24

5 Under model assumptions valid for optical applications (linearity, isotropy and non-dispersivity), Maxwell s equations can be stated as follows: Faraday s Law: Ampère s Law: µ t H = E σ H (1) ɛ t E = H σe (2) Gauß s Law for Electric Fields: (ɛe) = 0 (3) Gauß s Law for Magnetic Fields: H = 0 (4) 5 / 24

6 In a time-harmonic setting of (1),(2), we seek a solution: H(ξ, t) = Ĥ(ξ) exp(iωt) E(ξ, t) = Ê(ξ) exp(iωt) Therefore, the problem with source terms SÊ, SĤ of frequency ω can be stated as: Find (Ĥ, Ê) that satisfies: iωµ(ξ)ĥ(ξ) = Ê(ξ) σ (ξ)ĥ(ξ) + µ(ξ)s Ĥ (ξ) iωε(ξ)ê(ξ) = Ĥ(ξ) σ(ξ)ê(ξ) + ε(ξ)s Ê (ξ) with ξ Ω R 3 a box-shaped domain. 6 / 24

7 After discretization in time and space (FDFD, Yee cells, explicit leap-frog), we get the following iterative scheme: µ (α 2 Ĥ k+1 τ Ĥk) = α h Êk σ (Ĥk + α 2 Ĥ k+1) + µsĥ 2 ε ( α 2 Ê k+1 τ Êk) = α h Ĥk+1 σ (Êk + α 2 Ê k+1) + εsê 2 for time steps τ > 0, where α := α(τ) = exp (iω 12 ) τ and ( H k (ξ, kτ) = Ĥk (ξ)α 2k, E ξ, 2k + 1 ) τ = 2 Êk (ξ)α 2k+1 ( H ξ, 2k + 1 ) τ 1 ( ) 2 2 α2k Ĥ k (ξ) + α 2 Ĥ k+1 (ξ) 7 / 24

8 Introduce a local modification to the iteration scheme (THIIM 1 ) to deal with instabilities in media of negative permittivity ε < 0: ε ( α 2 Ê k τ Êk+1) = α h Ĥk+1 σ ( α 2 Ê k + 2 Êk+1) + εsê This yields a sparse system matrix consisting of blocks of the following kind: M := 2µ τσ α 2 (2µ+τσ ) 2τ α(2ε+τσ) h 2µ τσ α 2 (2µ+τσ ) 2τα 2 ε +τσ h 2µ τσ α 2 (2µ+τσ ) 2τ α(2µ+τσ ) h 2ε τσ α 2 (2ε+τσ) 2τ α(2ε+τσ) h 2τ α(2µ+τσ ) h α 2 (2 ε τσ) 2 ε +τσ 2τα 2 ε +τσ h 2τ α(2µ+τσ ) h with eigenpairs (λ m, e m ), e m = 1, λ m < 1 m for 0 < τ sufficiently small and M(τ) = m e m λ m e T m τ 0 Id. 1 Pflaum, Rahimi: An iterative solver for the FDFD method for the simulation of materials with negative permittivity, Num.Lin.Alg.Appl., / 24

9 Define algebraic state variables and load vectors: x k (Ĥk :=, Êk) T (Ĥk ) T =, ε>0 Êk, ( ) T ε<0 Êk, b := τ SĤ, SÊ and restate the discretized time-harmonic Maxwell problem: Find a solution x = (Ĥ, Ê)T R d such that: or equivalently: Solve the system x = Mx + b (Id M)x = b 9 / 24

10 Explicit FDFD iteration scheme: x 0 = 0 x k+1 = Mx k + b = k 1 M κ b κ=0 converges slowly 2. Approximation error: x x k = M(x x k 1 ) = M k (x x 0 ) = M k x = M κ b κ=k 2 Recall the corresponding Neumann series: k=0 M k = (Id M) 1 10 / 24

11 High iteration numbers (typically ) to convergence up to a suitable prescribed tolerance 0.2 Real Imag Number of Iterations Wave Length [nm] Figure : Transient signal and iterations to convergence up to a tolerance of / 24

12 How to speed up convergence? x k+1 x k = M k b = m e m λ k me T m Idea: Use low order model to capture the (few) dominant eigenvalue contributions of the system matrix to the iterates. M e m λ m e T m m {n N: λ n >θ} Problem: How to get an estimate of the dominant eigenpairs (or their effect on given iterates)? Approach: Use an extrapolation method based on component-wise samples of successive iterates Pros: Allows for observation of oscillatory behavior caused by resonant modes. Cons: Decouples the degrees of freedom. 12 / 24

13 Step 1: Establish dependence of iterates on previous iterates x k l s x k l 1 s... x k 1 s u 1 x k l+1 x k s l+1 s... x P s := k 2 s....., u := u 2., r x k l+2 s s :=. x k n 1 s x k n 2 s... x k n l u s l x kn s We solve for u: Equivalent to: P s u = r s l m=0 u m x k t m s = 0 t = l n, where u 0 = 1 (5) 13 / 24

14 Step 2: Exponential basis Represent x k s as linear combination of exponentials z k = exp(ifk) Complex-valued f relates to frequency (Re f) and attenuation ( Im f) Determine roots {z m : p(z m ) = 0} of: p(z) = 1 + l u m z l+1 m (6) m=1 Roots z m = exp(if m ) represent distinct frequencies of the system (as detected in the respective degree of freedom s corresponding to some spatial position ξ Ω). 14 / 24

15 Step 3: Determine amplitudes Q s := z k 1 1 z k z k 1 l z k z k 2 l....., a := z1 kn z2 kn... z kn l a 1 a 2. a l, w s := x k 1 s x k 2 s. x kn s Denote iterates x kt s as a superposition of frequencies: x kt s = l a m exp(if m t) m=1 Solve system to get corresponding amplitudes: Q s a = w s Filter amplitude-frequency pairs to get behavior for t 15 / 24

16 Remarks System generally overdetermined (n 2l) Solve in the least squares sense Leads to singular value decomposition (SVD) Minimizes: Numerically robust P s u r s 2 = inf v R l P s v r s 2 (7) Q s a w s 2 = inf v R l Q s v w s 2 (8) 16 / 24

17 Typical distribution of amplitude and frequency pairs as determined by Prony s method 0.1 Distribution of Frequencies and Amplitudes Amplitude Frequency Figure : The signal s frequency-amplitude pairs (abs) Stabilization: Attenutation a = Im f. Discard frequencies with a > ε, assume that frequencies a ε are / 24

18 Convergence of a single degree of freedom: Order 6 extrapolation error vs iterative error, filter by frequency, 0.97 frequencies used on average 10-1 extrapolation iterated signal Figure : Error: Iterated vs extrapolated signal 18 / 24

19 Convergence of the whole simulation domain: Figure : 2 Error: Iterated vs extrapolated signal 19 / 24

20 Numerical Results Figure : Experimental and simulation results 20 / 24

21 Figure : Absorption for smooth and rough interfaces Figure : Quantum efficiency under different angles of incidence 21 / 24

22 Figure : Surface plasmons Figure : Scattering behavior of rough interfaces 22 / 24

23 Summary Explicit iteration scheme for high-frequency Maxwell problems Stable for negative permittivity Extrapolation reduces error Future work How to improve the extrapolation accuracy? (e.g. by choice of better samples) Can a hierarchical basis approach reduce local inaccuracies? What about the divergence of the extrapolated signal? (a constraint in the original system of PDEs) How to make use of multigrid methods? 23 / 24

24 Thank you for your attention. References: A. Taflove, S. Hagness: Computational Electrodynamics, Artech House, 2005 C. Pflaum, Z. Rahimi: An Iterative Solver for the Finite-Difference Frequency Domain (FDFD) Method for the Simulation of Materials with Negative Permittivity, Numerical Linear Algebra with Applications, 2010, DOI: /nla.746 C. Jandl, K. Hertel, C. Pflaum, H. Stiebig: Simulation of Thin-film Silicon Solar Cells with Integrated AFM Scans for Oblique Incident Waves, Proceedings of PVSEC, 2011 R. Prony: Essai expérimental et analytique sur les lois de la dilatibilite des fluides elastiques et sur celles de la force expansive de la vapeur de l eau et de la vapeur de l alkool a differentes temperatures, J. l Ecole Polytechnique, Paris, 1795 M. van Blaricum, R. Mittra: A Technique for Extracting the Poles and Residues of a System Directly from Its Transient Response, IEEE Transactions on Antennas and Propagation, 1975 R. Kumaresan, D. Tufts: Estimating the Parameters of Exponentially Damped Sinusoids and Pole-Zero Modeling in Noise, IEEE Transactions on Acoustics, Speech, and Signal Processing, / 24