Exponential Recursion for Multi-Scale Problems in Electromagnetics

Transcription

1 Exponential Recursion for Multi-Scale Problems in Electromagnetics Matthew F. Causley Department of Mathematics Kettering University Computational Aspects of Time Dependent Electromagnetic Wave Problems in Complex Materials ICERM, Brown University June 28, 2018 Exponential Recursion 1 / 49

2 Outline 1 Introduction and Motivation 2 Havriliak-Negami Model 3 Wave Solver 4 Conclusions Exponential Recursion 2 / 49

3 Table of Contents 1 Introduction and Motivation 2 Havriliak-Negami Model 3 Wave Solver 4 Conclusions Exponential Recursion 3 / 49

4 Exponential recursion is a simple concept, but a powerful tool. We focus on EM wave propagation in complex media. (1) Anomalous dielectric relaxation (fractional relaxation models) 1 Complex hetergoeneous materials (soil, biological tissues) 2 Empirical dispersion models involve (iω) α. 3 Power-law decay, requires time history of fields. 4 Useful for similar problems in acoustics, solid mechanics, etc. (2) Plasmas 1 Plasma phenomena occur at vastly disparate time scales. 2 Geometry, fine spatial scales must be resolved. 3 Experiments often have non-local effects. 4 Pros and Cons of kinetic vs. fluid models of plasma. Exponential Recursion 4 / 49

5 In time, exponential recursion truncates time history. Consider φ + yφ = f (t), 0 < t < T Multiply by integrating factor, and integrate over [t δ, t] t t δ ( e yt φ ) dt = t e yt φ(t) e y(t δ) φ(t δ) = t δ t t δ Rearranging, we find the exponential recursion e yt f (t) dt e yτ f (τ) dτ δ φ(t) = e yδ φ(t δ) + e yu f (t u) du. 0 Discretize using any exponential integrator. Exponential Recursion 5 / 49

6 In space, exponential recursion localizes the solution. Consider w 1 α 2 w = u = w p (x) = α 2 b a e α x y u(y) dy Split the integral at y = x, and let w p = w L + w R. Then w L (x) = α 2 x a e α(y x) u(y) dy, w R (x) = α 2 b A few steps of algebra produce the exponential recursion δl w L (x) = e αδ L w L (x δ L ) + w R (x) = e αδ R w R (x + δ R ) + 0 δr Discretize using any collocation method. 0 x e αz u(x z) dz e αz u(x + z) dz. e α(x y) u(y) dy. Exponential Recursion 6 / 49

7 Table of Contents 1 Introduction and Motivation 2 Havriliak-Negami Model 3 Wave Solver 4 Conclusions Exponential Recursion 7 / 49

8 Consider EM propagation through a region Ω D t = H J, D = ρ B = E, t B = 0 In the absence of external charges and currents, ρ = 0, J = 0. Assume the material is non-magnetic, so B = µ 0 H. Anomalous dielectric relaxation, ˆD = ɛ 0ˆɛÊ, where ˆɛ(s) = ɛ + ɛ (1 + (sτ) α ) β with ɛ 1, ɛ > 0, 0 < α, β 1. Exponential Recursion 8 / 49

9 Decompose D = ɛ 0 (ɛ E + P), where (I + τ α cd α t ) β P = ɛe fractional PDO! Solve for the polarization using Laplace transforms. P(x, t) = t 0 χ(u)e(x, t u) du, ˆχ(s) = ɛ (1 + (sτ) α ) β χ(t) { t αβ 1 t 1 t α 1 t 1 Exponential Recursion 9 / 49

10 We first re-cast the susceptibility as an integral over R χ(t) = ɛ 2πi = = 0 ζ+i ζ i e st (1 + (sτ) α ) β ds f (y)e yt/τ dy f (e z )e z ez t/τ dz, where { }) f (y) = ɛ sin (β cos 1 y α cos (πα)+1 y 2α +2 cos (πα)y α +1 πτ (y 2α + 2 cos (πα)y α + 1) β/2. Exponential Recursion 10 / 49

11 In general, consider K(t) = 0 f (y)e yt dy for non-negative f L 1 (R + ). We set y = e z. Then, for h > 0 the Poisson summation formula yields ( ) 2πk h f (e nh )e nh enht = ˆf h where ˆf (k) = n= k= [ f (e z )e z ez t ] e ikz dz, ˆf (0) = K(t) Under mild assumptions, the integrand is analytic for Im{z} θ. ˆf (k) C k e k θ. Exponential Recursion 11 / 49

12 K(t) = h f (e nh )e nh enht ( ) 2πk ˆf h n= k 0 h f (e nh )e nh enht + O (e ) π2 /h nh<z r Exponential Recursion 12 / 49

13 Given ɛ > 0, for t [ t, T ] 1 Discretization: Choose h so that e π2 /h ɛ. 2 Truncation: Choose z r so that hf (e zr )e zr tezr ɛ. 3 Compression: Choose z l so that a minimal number J (typically 2 or 3) compressed nodes satisfies h z z l f (e nh )e nh enht = J w j e y j T + O(ɛ) j=1 Next, merge the weights and nodes. Then, we have a uniform relative error bound M max t<t<t K(t) w m e ymt K(t)ɛ m=1 Exponential Recursion 13 / 49

14 Exponential Recursion 14 / 49

15 For t [0, T ], P(x, t) = t 0 χ(u)e(x, t u) du + M w m φ m (t) + E(t) m=1 where we have the diagonalized linear system φ m + y m φ m = E. Discretize using any exponential integrator. For E L 2 ([0, T ]), M E(t) L 2 χ(t) w m e ymt L E(x, t) L 2 m=1 1 M T χ(t) w m e ymt L E(x, t) L 2 m=1 ɛt E(x, t) L 2 Exponential Recursion 15 / 49

16 Replace the electric field with a polynomial interpolant, and perform product integration to arrive at 1 P(t) = L A l E(x, t l t) + l=0 φ m (x, t) = e ym t φ m (x, t t) + M w m φ m (t) m=1 L B l,m E(x, t l t). The polarization law can now be evaluated with L levels of time history, and M = O (log N t ) terms in memory. l=0 The operation count is O(LM). Here, N t = T t. 1 JCP 2013, with S. Jiang and P. Petropoulos Exponential Recursion 16 / 49

17 Theorem. The numerical scheme, based on the second order accurate FDTD method, and is stable under the standard CFL stability condition. We solve a signaling problem (1d TEM wave) using the finite difference time domain (FDTD) technique, with square pulse E(0, t) = 1 t d (H(t) H(t t d )). Exponential Recursion 17 / 49

18 Similar asymptotics for Cole-Cole, Cole-Davidson models. 2 2 IEEE Trans. Ant with P. Petropoulos Exponential Recursion 18 / 49

19 Table of Contents 1 Introduction and Motivation 2 Havriliak-Negami Model 3 Wave Solver 4 Conclusions Exponential Recursion 19 / 49

20 Consider the Vlasov-Maxwell system for a plasma E ɛ 0 µ 0 t = B µ 0J, E = ρ ɛ 0 B = E, t B = 0 ρ = q f (x, v, t) dv, v f t + v xf + F m v f = 0 F = q (E + v B) J = q vf (x, v, t) dv v Exponential Recursion 20 / 49

21 Define EM fields using scalar potential φ and vector potential A by E = A t φ, B = A. Impose the Lorenz gauge 1 φ c 2 t + A = 0, so that 1 2 A c 2 t 2 2 A = µ 0 J 1 2 φ c 2 t 2 2 φ = 1 ρ ɛ 0 Exponential Recursion 21 / 49

22 1 Apply the method of lines transpose (MOL T ) Discretize in time first. Re-formulate as a semi-discrete boundary integral. Exponential Recursion 22 / 49

23 1 Apply the method of lines transpose (MOL T ) Discretize in time first. Re-formulate as a semi-discrete boundary integral. 2 Build a (1d) spatial solver Spatial discretization over a (perhaps nonuniform) mesh. Fast O(N) matrix-free convolution, via exponential recursion. Exponential Recursion 22 / 49

24 1 Apply the method of lines transpose (MOL T ) Discretize in time first. Re-formulate as a semi-discrete boundary integral. 2 Build a (1d) spatial solver Spatial discretization over a (perhaps nonuniform) mesh. Fast O(N) matrix-free convolution, via exponential recursion. 3 Multi-dimensional solver Spatial solver works in a line-by-line fashion. Embedded boundaries. Local interpolation for normal derivatives. Exponential Recursion 22 / 49

25 1 Apply the method of lines transpose (MOL T ) Discretize in time first. Re-formulate as a semi-discrete boundary integral. 2 Build a (1d) spatial solver Spatial discretization over a (perhaps nonuniform) mesh. Fast O(N) matrix-free convolution, via exponential recursion. 3 Multi-dimensional solver Spatial solver works in a line-by-line fashion. Embedded boundaries. Local interpolation for normal derivatives. 4 Raise order of accuracy in time Taylor expansion reformulated with convolution operators. Coefficients determined with resolvent expansion. Stability guaranteed by introducing a free parameter β > 0. Exponential Recursion 22 / 49

26 1 c 2 u tt 2 u = S 2 Math Comp with A. Christlieb, B. Ong, and L. Van Groningen Exponential Recursion 23 / 49

27 Discretize u tt in time 1 c 2 u tt 2 u = S u tt (x, t n ) un+1 2u n + u n 1 t 2. 2 Math Comp with A. Christlieb, B. Ong, and L. Van Groningen Exponential Recursion 23 / 49

28 Discretize u tt in time 1 c 2 u tt 2 u = S u tt (x, t n ) un+1 2u n + u n 1 t 2. The Laplacian is treated semi-implicitly, so that u n+1 2u n + u n 1 ( (c t 2 2 u n + un+1 2u n + u n 1 ) ) β 2 with 0 < β 2. S n, 2 Math Comp with A. Christlieb, B. Ong, and L. Van Groningen Exponential Recursion 23 / 49

29 L [u n + un+1 2u n + u n 1 ] β 2 = u n + S n α 2, where the modified Helmholtz operator is L[u](x) := (1 1α ) 2 2 u(x), α = β c t. Update equation u n+1 = 2u n u n 1 β 2 D[u n ](x) + β 2 L 1 [ S n α 2 ] (x), where D[u](x) := u(x) L 1 [u](x) 1 α 2 2 u. Exponential Recursion 24 / 49

30 Rather than invert the 3d Helmholtz operator, we utilize dimensional splitting L L x L y L z In one spatial dimension, where u n+1 = 2u n u n 1 β 2 D[u n ](x) + β 2 L 1 [ S n b α 2 ] (x), L 1 [u](x) := α u(y)e α x y dy + Ae α(x a) + Be α(b x). 2 a }{{}}{{} Particular Solution Homogeneous Solution Theorem: This semi-discrete scheme is second order accurate in time, and A-stable for β (0, 2]. Exponential Recursion 25 / 49

31 w p (x) = I [u](x) = α 2 b Split the particular solution at y = x, x a e α x y u(y)dy. b I [u](x) = α e α(x y) u(y)dy + α e α(y x) u(y)dy. 2 a 2 x }{{}}{{} I L (x) I R (x) Each characteristic is updated using exponential recursion I L (x + δ) =e αδ I L (x) + α 2 I R (x δ) =e αδ I R (x) + α 2 x+δ x x x δ e α(x j y) u(y)dy e α(y x j ) u(y)dy. Exponential Recursion 26 / 49

32 Partition [a, b] into N subintervals [x j 1, x j ], h j = x j x j 1. Replace u with a local Lagrange interpolant, of order 2M. Then M I L (x j ) = e αh j I L (x j 1 ) + wk L u(x j+k), j = 1,... N, k= M M I R (x j ) = e αh j+1 I R (x j+1 ) + wk R u(x j+k), j = N 1, k= M Convolution computed in O(MN) operations. Exponential Recursion 27 / 49

33 Transmission conditions can be formulated, for domain decomposition and (1d) outflow BCs. b L 1 [u](x) := α u(y)e α x y dy + Ae α(x a) + Be α(b x). 2 a }{{}}{{} Particular Solution Homogeneous Solution Compare with the free space solution. Then, for x [a, b], we see A(t) = α 0 u(a y, t)e αy dy. Using exponential recursion, storing time history at the boundary A n = α a a ct n e α(a y) u(y, t n )dy = e β A n 1 + α c t 0 ( e αy u a, t n y ) dy. c Exponential Recursion 28 / 49

34 The exponential recursion can also be employed hierarchically. 3 Ω J R m J L m+1 Ω A m B m Ω 0 Ω m Ω M 1 Ω 0 Ω m Ω M 1 (a) Fine-Coarse Pass (b) Coarse-Fine Pass I L (X j ) = e α(x j X j 1 ) I L (X j 1 ) + J L (X j ), J L (X j ) = α Xj X j 1 e α(x j y) u(y)dy 1 Compute each local particular solution on Ω j. 2 Find particular solution using global exponential recursion. 3 Pass transmission conditions, boundary conditions to each Ω j. 3 Arxiv, with A. Christlieb, Y. Guclu and E. Wolf Exponential Recursion 29 / 49

35 Undergraduate thesis with C. Seipp Exponential Recursion 30 / 49

36 In order to approximate higher order powers of the Laplacian operator using dimensional splitting 4, we first define L γ := 1 2 γ α 2, D γ := 1 L 1 γ, γ = x, y, z, and C xyz := L 1 y L 1 z D x + L 1 z L 1 x D y + L 1 x L 1 y D z. so that ( ) m 2 α 2 = Cxyz m p=m ( ) p 1 Dxyz p m. m 1 4 SINUM 2013, with A. Christlieb Exponential Recursion 31 / 49

37 u n+1 2u n + u n 1 = = = = m=1 m=1 2β 2m (2m)! ( ) 2 m u n α 2 ( 1) m 2β2m p p=1 m=1 P p=1 m=1 ( ) p 1 (2m)! Cm D p m [u n ] m 1 p=m ( ) p 1 C m D p m [u n ] (2m)! m 1 ( 1) m 2β2m p A p,m (β)c m D p m [u n ] + O( t 2P+2 ). Exponential Recursion 32 / 49

38 P = 1 P = 2 P = 3 t Error Rate Time (s) Error Rate Time (s) Error Rate Time (s) E-1 * E-1 * E-1 * E E E E E E E E E E E E Table: Refinement and computational efficiency for a 2d rectangular mode u(x, y, 0) = sin(πx) sin(πy). The mesh is held fixed at x = y = The algorithm scales linearly with the number of spatial points. Exponential Recursion 33 / 49

39 The characteristic polynomial satisfied by the amplification factor is P ρ 2 2ρ + 1 = ρ A p (β) ˆD p, β > 0, 0 ˆD 1. p=1 Def. The scheme will be A-stable provided that ρ 1. Theorem. For each finite P, there exists β max such that the semi-discrete scheme will be A-stable for 0 < β β max, and where P A p (β max ) = 4. p=1 P Order β max Exponential Recursion 34 / 49

40 (a) 2nd order (b) 4th order Figure: Propagation due to a point source in 2d, on a mesh,with CFL number 2. Exponential Recursion 35 / 49

41 1 Add a ghost point, exterior to the domain. 5 2 Interpolate u I and u II from local interior grid points (bilinear). 3 Interpolant through u I and u II, such that u n (ξ B) = 0. 4 Extrapolate to find u G, and update only local grid points. Iterate to convergence. 5 J. Sci. Comp. 2017, with A. Christlieb and E. Wolf Exponential Recursion 36 / 49

42 { ( ) x + y 2 C = (x, y) : + (x y) 2 = 1} y 0 y x (a) x-sweep x (b) y-sweep Figure: The boundary points (red) close each line of the x and y sweeps. Exponential Recursion 37 / 49

43 Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions Solution is 6th order in time and space. No stability restriction. M. Causley Exponential Recursion Kettering University 38 / 49

44 (a) t = 0 (b) t = 0.1 (c) t = 0.2 (d) t = 0.25 (e) t = 0.45 (f) t = 0.7 (g) t = 0.8 (h) t = 1.0 Exponential Recursion 39 / 49

45 Exponential Recursion 40 / 49

46 Exponential Recursion 41 / 49

47 Implicit PIC, quasi-electrostatics. 6 Initial Prediction 1 Charge: {ξi n} ρn. 2 Potential (MOL T ): φ n 1, φ n, ρ n φ. 3 Positions: ξi = ξi n + tvi n. 4 Fields: φ E E i = E (ξi ). Correction Iteration 5 Velocities: vi = vi n + t(αe i + (1 α) E i n ). 6 Positions: ξi = ξi n + t(αv + (1 α)v n 7 Charge: {ξ i } ρ. 8 Potential/Fields: φ n 1, φ n, ρ φ E i. 9 Repeat steps 5-8 to convergence. i i ). 6 JCP 2016, with M. Bettencourt A. Christlieb and E. Wolf Exponential Recursion 42 / 49

48 5 x Total Energy energy (J) α = 0.5 α = 0.6 α = time (s) x periods of an electron-ion oscillatory system. The CFL is 10, with 100 cells in the domain. Standard spatial smoothing operators are applied to grid quantities. The relaxation parameter is varied, no grid heating is observed when α = 0.5. Exponential Recursion 43 / 49

49 energy (J) 6 x Kinetic, Field and Total Energies, α = Field Energy Kinetic Energy Total Energy time (s) x Total energy is conserved, as the underlying field solver is non-dissipative. Exponential Recursion 44 / 49

50 Ph.D. thesis of M. Thavappiragasam Exponential Recursion 45 / 49

51 Ph.D. thesis of M. Thavappiragasam Exponential Recursion 46 / 49

52 Table of Contents 1 Introduction and Motivation 2 Havriliak-Negami Model 3 Wave Solver 4 Conclusions Exponential Recursion 47 / 49

53 Future Work: 1 Analysis of outflow boundary conditions 2 Domain decomposition in multiple dimensions 3 Consistent treatment of particles 4 Fully parallel 3d Maxwell solver Exponential Recursion 48 / 49

54 Thank you! References The Christlieb Group Exponential Recursion 49 / 49