Exponential Recursion for Multi-Scale Problems in Electromagnetics - PowerPoint PPT Presentation

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions 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 M. Causley Kettering University Exponential Recursion 1 / 49

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions Outline 1 Introduction and Motivation 2 Havriliak-Negami Model 3 Wave Solver 4 Conclusions M. Causley Kettering University Exponential Recursion 2 / 49

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

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions 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 ( i ω ) 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. M. Causley Kettering University Exponential Recursion 4 / 49

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions In time, exponential recursion truncates time history. Consider ˙ φ + y φ = f ( t ) , 0 < t < T Multiply by integrating factor, and integrate over [ t − δ, t ] � t � t � ′ dt = � e yt φ e yt f ( t ) dt t − δ t − δ � t e yt φ ( t ) − e y ( t − δ ) φ ( t − δ ) = e y τ f ( τ ) d τ t − δ Rearranging, we find the exponential recursion � δ φ ( t ) = e − y δ φ ( t − δ ) + e − yu f ( t − u ) du . 0 Discretize using any exponential integrator. M. Causley Kettering University Exponential Recursion 5 / 49

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions In space, exponential recursion localizes the solution. Consider � b w − 1 w p ( x ) = α α 2 w ′′ = u e − α | x − y | u ( y ) dy = ⇒ 2 a Split the integral at y = x , and let w p = w L + w R . Then � x � b w L ( x ) = α w R ( x ) = α e − α ( y − x ) u ( y ) dy , e − α ( x − y ) u ( y ) dy . 2 2 a x A few steps of algebra produce the exponential recursion � δ L w L ( x ) = e − αδ L w L ( x − δ L ) + e − α z u ( x − z ) dz 0 � δ R w R ( x ) = e − αδ R w R ( x + δ R ) + e − α z u ( x + z ) dz . 0 Discretize using any collocation method. M. Causley Kettering University Exponential Recursion 6 / 49

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

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions Consider EM propagation through a region Ω ∂ D ∂ t = ∇ × H − J , ∇ · D = ρ ∂ B ∂ t = −∇ × E , ∇ · 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, ˆ ǫ ˆ D = ǫ 0 ˆ E , where ∆ ǫ ˆ ǫ ( s ) = ǫ ∞ + (1 + ( s τ ) α ) β with ǫ ∞ ≥ 1 , ∆ ǫ > 0 , 0 < α, β ≤ 1. M. Causley Kettering University Exponential Recursion 8 / 49

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions Decompose D = ǫ 0 ( ǫ ∞ E + P ), where t ) β P = ∆ ǫ E ( I + τ α c D α fractional PDO! Solve for the polarization using Laplace transforms. � t ∆ ǫ P ( x , t ) = χ ( u ) E ( x , t − u ) du , χ ( s ) = ˆ (1 + ( s τ ) α ) β 0 � t αβ − 1 t ≪ 1 χ ( t ) ∼ t − α − 1 t ≫ 1 M. Causley Kettering University Exponential Recursion 9 / 49

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions We first re-cast the susceptibility as an integral over R � ζ + i ∞ e st χ ( t ) = ∆ ǫ (1 + ( s τ ) α ) β ds 2 π i ζ − i ∞ � ∞ f ( y ) e − yt /τ dy = 0 � ∞ f ( e z ) e z − e z t /τ dz , = −∞ where � � �� y α cos ( πα )+1 β cos − 1 √ sin f ( y ) = ∆ ǫ y 2 α +2 cos ( πα ) y α +1 . ( y 2 α + 2 cos ( πα ) y α + 1) β/ 2 πτ M. Causley Kettering University Exponential Recursion 10 / 49

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions In general, consider � ∞ f ( y ) e − yt dy K ( t ) = 0 for non-negative f ∈ L 1 ( R + ). We set y = e z . Then, for h > 0 the Poisson summation formula yields ∞ ∞ � 2 π k � � � f ( e nh ) e nh − e nh t = ˆ h f h n = −∞ k = −∞ where � ∞ � f ( e z ) e z − e z t � e ikz dz , ˆ ˆ f ( k ) = f (0) = K ( t ) −∞ Under mild assumptions, the integrand is analytic for Im { z } ≤ θ . | ˆ f ( k ) | ≤ C k e −| k | θ . M. Causley Kettering University Exponential Recursion 11 / 49

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions ∞ � 2 π k � � � f ( e nh ) e nh − e nh t − ˆ K ( t ) = h f h n = −∞ k � =0 � e − π 2 / h � � f ( e nh ) e nh − e nh t + O ≈ h nh < z r M. Causley Kettering University Exponential Recursion 12 / 49

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions Given ǫ > 0, for t ∈ [∆ t , T ] 1 Discretization: Choose h so that e − π 2 / h ≈ ǫ . 2 Truncation: Choose z r so that hf ( e z r ) e z r − ∆ te zr ≈ ǫ . 3 Compression: Choose z ℓ so that a minimal number J (typically 2 or 3) compressed nodes satisfies J � � f ( e nh ) e nh − e nh T = w j e − y j T + O ( ǫ ) h z ≤ z ℓ j =1 Next, merge the weights and nodes. Then, we have a uniform relative error bound � � M � � � � w m e − y m t � max � K ( t ) − � ≤ K ( t ) ǫ � � ∆ t < t < T m =1 M. Causley Kettering University Exponential Recursion 13 / 49

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions M. Causley Kettering University Exponential Recursion 14 / 49

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions For t ∈ [0 , T ], � ∆ t M � P ( x , t ) = χ ( u ) E ( x , t − u ) du + w m φ m ( t ) + E ( t ) 0 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 � � � w m e − y m t � � ||E ( t ) || L 2 ≤ � χ ( t ) − || E ( x , t ) || L 2 � � � m =1 L 1 � � M � � � w m e − y m t � � ≤ T � χ ( t ) − || E ( x , t ) || L 2 � � � m =1 L ∞ ≤ ǫ T || E ( x , t ) || L 2 M. Causley Kettering University Exponential Recursion 15 / 49

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions Replace the electric field with a polynomial interpolant, and perform product integration to arrive at 1 L M � � P ( t ) = A ℓ E ( x , t − ℓ ∆ t ) + w m φ m ( t ) ℓ =0 m =1 L � φ m ( x , t ) = e − y m ∆ t φ m ( x , t − ∆ t ) + B ℓ, m E ( x , t − ℓ ∆ t ) . ℓ =0 The polarization law can now be evaluated with L levels of time history, and M = O (log N t ) terms in memory. The operation count is O ( LM ). Here, N t = ⌈ T ∆ t ⌉ . 1 JCP 2013, with S. Jiang and P. Petropoulos M. Causley Kettering University Exponential Recursion 16 / 49

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions 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 ( H ( t ) − H ( t − t d )) . t d M. Causley Kettering University Exponential Recursion 17 / 49

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions Similar asymptotics for Cole-Cole, Cole-Davidson models. 2 2 IEEE Trans. Ant. 2011 with P. Petropoulos M. Causley Kettering University Exponential Recursion 18 / 49

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

Introduction and Motivation Havriliak-Negami Model Wave Solver Conclusions Consider the Vlasov-Maxwell system for a plasma ∂ E ∇ · E = ρ ǫ 0 µ 0 ∂ t = ∇ × B − µ 0 J , ǫ 0 ∂ B ∂ t = −∇ × E , ∇ · B = 0 � � ρ = q f ( x , v , t ) dv , J = q vf ( x , v , t ) dv v v ∂ f ∂ t + v · ∇ x f + F m · ∇ v f = 0 F = q ( E + v × B ) M. Causley Kettering University Exponential Recursion 20 / 49