Numerical methods for non-standard fractional operators in the - PowerPoint PPT Presentation

Numerical methods for non-standard fractional operators in the simulation of dielectric materials Roberto Garrappa Department of Mathematics - University of Bari - Italy roberto.garrappa@uniba.it Fractional PDEs: Theory, Algorithms and Applications ICERM - Providence, June 18–22, 2018 R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 1 / 25

Outline Numerical methods for non-standard fractional operators in the simulation of dielectric materials The problem 1 The operator 2 The numerical method 3 Financial support: EU Cost Action 15225 - Fractional Systems Istituto Nazionale di Alta Matematica (INdAM-GNCS) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 2 / 25

Maxwell’s equations: Standard Maxwell’s equations:  ∂  ∇ × H = ǫ 0 Ampere’s law ∂ t E  ∂ H  ∇ × E = − µ 0 Faraday’s law  ∂ t E : electric field H : magnetic field R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 3 / 25

Maxwell’s equations: Standard Maxwell’s equations:  ∂  ∇ × H = ǫ 0 Ampere’s law ∂ t E  ∂ H  ∇ × E = − µ 0 Faraday’s law  ∂ t E : electric field H : magnetic field Real world applications design of data and energy storage devices design of antennas medical diagnostic (MRI), cancer therapy, ... R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 3 / 25

Maxwell’s equations: Standard Maxwell’s equations with polarization:  ∂ t E + ∂ ∂  ∇ × H = ǫ 0 Ampere’s law ∂ t P  ∂ H  ∇ × E = − µ 0 Faraday’s law  ∂ t E : electric field H : magnetic field P : polarization R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 3 / 25

Maxwell’s equations: Standard Maxwell’s equations with polarization:  ∂ t E + ∂ ∂  ∇ × H = ǫ 0 Ampere’s law ∂ t P  ∂ H  ∇ × E = − µ 0 Faraday’s law  ∂ t E : electric field H : magnetic field P : polarization The complex susceptibility The polarization P depends on the electric field E (constitutive law) ˆ χ ( ω ) ˆ P = ε 0 ˆ E χ ( ω ) is a specific feature of the matter (or system) ˆ Simplified notation (just for easy of presentation) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 3 / 25

Determining the complex susceptibility ˆ χ ( ω ) χ ′ ( ω ) − i ˆ χ ′′ ( ω ) ? How to derive ˆ χ ( ω ) = ˆ R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 4 / 25

Determining the complex susceptibility ˆ χ ( ω ) χ ′ ( ω ) − i ˆ χ ′′ ( ω ) ? How to derive ˆ χ ( ω ) = ˆ Experimental data (in the frequency domain): ˆ χ ( ω ) ˆ P = ˆ E 0 Exper. data 10 −1 10 −2 χ ′′ ( ω ) 10 ˆ −3 10 −4 10 0 2 4 6 10 10 10 10 frequency ω Match experimental data into a mathematical model R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 4 / 25

Determining the complex susceptibility ˆ χ ( ω ) 0 10 Exper. data Debye model −1 10 −2 χ ′′ ( ω ) 10 ˆ −3 10 −4 10 0 2 4 6 10 10 10 10 frequency ω 1 The Debye model: χ ( ω ) = ˆ (standard materials) 1 + i ωτ τ d Ordinary differential equation: dt P ( t ) + P ( t ) = E ( t ) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 5 / 25

Determining the complex susceptibility ˆ χ ( ω ) 0 10 Exper. data −1 10 −2 χ ′′ ( ω ) 10 ˆ −3 10 −4 10 0 2 4 6 10 10 10 10 frequency ω Materials with anomalous dielectric properties: amorphous polymers complex systems (biological tissues) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 5 / 25

Determining the complex susceptibility ˆ χ ( ω ) 0 10 Exper. data Debye model −1 10 −2 χ ′′ ( ω ) 10 ˆ −3 10 −4 10 0 2 4 6 10 10 10 10 frequency ω Debye model not satisfactory R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 5 / 25

Determining the complex susceptibility ˆ χ ( ω ) 0 10 Exper. data Debye model C−C model −1 10 −2 χ ′′ ( ω ) 10 ˆ −3 10 −4 10 0 2 4 6 10 10 10 10 frequency ω 1 The Cole-Cole model: χ ( ω ) = ˆ 0 < α < 1 1 + ( i ωτ ) α τ α d α Fractional differential equation: dt α P ( t ) + P ( t ) = E ( t ) Only partially satisfactory R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 5 / 25

Determining the complex susceptibility ˆ χ ( ω ) 0 10 Exper. data Debye model C−C model −1 H−N model 10 −2 χ ′′ ( ω ) 10 ˆ −3 10 −4 10 0 2 4 6 10 10 10 10 frequency ω 1 The Havriliak-Negami model: χ ( ω ) = ˆ 0 < α, αγ ≤ 1 � 1 + ( i ωτ ) α � γ Better matching thanks to three parameters α , γ and τ S. Havriliak and S. Negami “A complex plane representation of dielectric and mechanical relaxation processes in some polymers”. In: Polymer (1967) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 5 / 25

Determining the complex susceptibility ˆ χ ( ω ) 0 10 Exper. data Debye model C−C model −1 H−N model 10 −2 χ ′′ ( ω ) 10 ˆ −3 10 −4 10 0 2 4 6 10 10 10 10 frequency ω 1 The Havriliak-Negami model: χ ( ω ) = ˆ 0 < α, αγ < 1 � 1 + ( i ωτ ) α � γ � � γ 1 + τ α d α Fractional pseudo-differential equation: P ( t ) = E ( t ) ? dt α R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 5 / 25

Other models for complex susceptibility ˆ χ ( ω ) Modified Havriliak-Negami or JWS (Jurlewicz, Weron and Stanislavsky) 1 � − α � γ = 1 − ( i τ ⋆ ω ) αγ χ HN ( i ω ) χ JWS ( i ω ) = 1 − ˆ � � 1 + i τ ⋆ ω EW: Excess wing (Hilfer, Nigmatullin and others) 1 + ( τ 2 i ω ) α χ EW ( i ω ) = ˆ 1 + ( τ 2 i ω ) α + τ 1 i ω . Multichannel excess wing (Hilfer) 1 ˆ ξ ( s ) = � n � − 1 � ( i ωτ k ) − α k 1 + k =1 This talk focuses on the Havriliak-Negami model Garrappa R., Mainardi F. and Maione G., “Models of dielectric relaxation based on completely monotone functions”. In: Frac. Calc. Appl. Anal. 19(5) (2016) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 6 / 25

Dealing with the Havriliak-Negami model 1 ˆ (( i ωτ ) α + 1) γ ˆ P ( ω ) = E ( ω ) Few contributions on simulation of this constitutive law: C.S.Antonopoulos, N.V.Kantartzis, I.T.Rekanos “FDTD Method for Wave Propagation in Havriliak-Negami Media Based on Fractional Derivative Approximation”. In: IEEE Trans Magn. 53(6) (2017) P.Bia et al. “A novel FDTD formulation based on fractional derivatives for dispersive Havriliak–Negami media”. In: Signal Processing , 107 (2015) 312–318 M.F.Causley, P.G.Petropoulos, “On the Time-Domain Response of Havriliak-Negami Dielectrics”. In: IEEE Trans. Antennas Propag. , 61(6) (2013) 3182–3189 M.F.Causley, P.G.Petropoulos, and S. Jiang “Incorporating the Havriliak-Negami dielectric model in the FD-TD method”. In: J. Comput. Phys. , 230 (2011), 3884–3899. Main problems: Define time-domain operator for HN Discretize the operator for simulations R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 7 / 25

Operators in the time domain 1 � � γ P ( t ) = E ( t ) ˆ (( i ωτ ) α + 1) γ ˆ τ α 0 D α P ( ω ) = E ( ω ) ⇐ ⇒ t + 1 ??? � � γ ? τ α 0 D α How to define the fractional pseudo-differential operator t + 1 R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 8 / 25

Operators in the time domain 1 � � γ P ( t ) = E ( t ) ˆ (( i ωτ ) α + 1) γ ˆ τ α 0 D α P ( ω ) = E ( ω ) ⇐ ⇒ t + 1 ??? � � γ ? τ α 0 D α How to define the fractional pseudo-differential operator t + 1 Combination of fractional operators � t � − t � � � γ = exp � 0 D αγ τ α 0 D α ατ α 0 D 1 − α · τ αγ ατ α 0 D 1 − α t + 1 · exp t t t Useful for theoretical investigations R.R.Nigmatullin and Y.E.Ryabov “Cole–Davidson dielectric relaxation as a self–similar relaxation process”. In: Physics of the Solid State 39.1 (1997) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 8 / 25

Operators in the time domain 1 � � γ P ( t ) = E ( t ) ˆ (( i ωτ ) α + 1) γ ˆ τ α 0 D α P ( ω ) = E ( ω ) ⇐ ⇒ t + 1 ??? � � γ ? τ α 0 D α How to define the fractional pseudo-differential operator t + 1 Expansion in infinite series ∞ � γ � � γ = � � 0 D α ( γ − k ) τ α 0 D α τ α ( γ − k ) t + 1 t k k =0 No satisfactory for error control V.Novikov et al. “Anomalous relaxation in dielectrics. Equations with fractional derivatives”. In: Mater. Sci. Poland 23.4 (2005) P.Bia et al. “A novel FDTD formulation based on fractional derivatives for dispersive Havriliak–Negami media”. In: Signal Processing , 107 (2015) 312–318 R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM 2018 8 / 25