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Analytical and numerical solutions of dissipative systems
Analytical and numerical solutions of dissipativesystems
R. Kovacsin collaboration with
Peter Van, Tamas Fulop, Gyula Grof, Matyas Szucs
Department of Energy Engineering, BMEDepartment of Theoretical Physics, Wigner RCP,
andMontavid Thermodynamic Research Group
Budapest, Hungary
October 11, 2018.
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Analytical and numerical solutions of dissipative systems
Outline
Origin of dissipative systems:non-equilibrium thermodynamics
Structure of partial differential equations (PDE):parabolic vs. hyperbolic
Examples: heat conduction vs. kinetic theory
Benchmark: solve these PDEs with real boundary conditions
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Analytical and numerical solutions of dissipative systems
Thermodynamics: physical aspects
Stability: direction of physical processes
Constitutive equations: material behavior
Universality: independent of the material type
Realization: II. law of thermodynamics
Entropy existsEntropy is a potential function of state variables(e.g. T , p, and so on)
Thermodynamic stability ⇐⇒ entropy is concave
Increasing entropy ⇒ entropy production is non-negative
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Analytical and numerical solutions of dissipative systems
Thermodynamics: mathematical aspects
Resulting PDEquations:
Parabolic vs. hyperbolic ⇒ propagation speed
Structure: compatibility with other theories?
Boundary conditions: physically admissible solutions
Solution method: numerical or analytical?
Feedback: real or artificial phenomena?
Validation through EXPERIMENTS!
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Analytical and numerical solutions of dissipative systems
The mathematics is the light.5 / 54
Analytical and numerical solutions of dissipative systems
Motivation to extend the classical theory I.
Heat conduction
Paradox of the Fourier equation: infinite speed ofpropagation, parabolic type equation (”a”: thermal diffusivity)
∂tT = a∂xxT
First modification (1958): Maxwell-Cattaneo-Vernotte (MCV)equation → hyperbolic type, finite propagation speed →second sound, temperature: around 1-20 K
τq∂ttT + ∂tT = a∂xxT
TEST: heat pulse experiments
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Analytical and numerical solutions of dissipative systems
Heat pulse experiments I.
Arrangement for solids
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Analytical and numerical solutions of dissipative systems
Heat pulse experiments II.
Narayanamurti et al.: Experimental proof of second sound in Bi (’74)
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Analytical and numerical solutions of dissipative systems
Heat pulse experiments III.
Further propagation mode at low temperatures!Presence of second sound and ballistic propagation!
Jackson, Walker andMcNelly (1968):NaF experiments
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Analytical and numerical solutions of dissipative systems
Heat conduction: propagation modes
Diffusive propagation → Fourier: ∂tT = a∂xxT
Damped wave propagation (“Second sound”) →Maxwell-Cattaneo-Vernotte: τ∂ttT + ∂tT = a∂xxT
Ballistic propagation: goes with speed of sound.
Continuum theory: elastic wave propagation.Kinetic theory: phonon hydro, non-interacting particles,particle-wall interaction
An unified continuum theory is obtained!Speed of sound ⇒ mechanical coupling!
Leading theory: phonon hydrodynamics (kinetic theory)
Analogous with rarefied gases!
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Analytical and numerical solutions of dissipative systems
Theory?
Goal:SIMPLE
INDEPENDENT OF MICRO / MESO / MACROSTRUCTURE
EASY TO USE AND IMPLEMENT
→ NON-EQUILIBRIUM THERMODYNAMICSWITH INTERNAL VARIABLES
See also: Berezovski - Van: Internal Variables in Thermoelasticity(2017, Springer)
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Analytical and numerical solutions of dissipative systems
About the internal variables
Non-equilibrium description
Nonlocal extensions in space and time → generalizations
Rheology: Kluitenberg-Verhas bodyDual internal variables: beyond the Newton equationWave propagation: coupling between mechanical and thermaleffects
Non-equilibrium thermodynamics
Extension of entropy density and its fluxPre-assumption: only the tensorial orderII. law → evolution equation for internal variableCan be eliminated: does not require further physical knowledge⇒ Universality
Final (and simple) benchmark: EXPERIMENTS
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Analytical and numerical solutions of dissipative systems
Classical approach
Fourier’s law (heat conduction only)
entropy density: s = se(e), e = cT
entropy current: J i = qi/T
local equilibrium
Balance: internal energy → ρc∂tT +∇ · q = 0,Entropy production:σs = ρs + ∂iJ
i = ρ (∂ese) + ∂i(qi/T
)=
����− 1T ∂iq
i+ ���1T ∂iq
i +qi∂i1T = qi∂i
1T ≥ 0
Constitutive equation in 1D: q = −λ∂xT .Together with the energy balance: ∂tT = a∂xxT .
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Analytical and numerical solutions of dissipative systems
Generalizations of entropy density and entropy flux
Extended Thermodynamics:
entropy density: s(e, qi ) = se(e)− m12 qiq
i
entropy current: J i = qi/T
qi : heat flux ≡ vectorial internal variable
a hope of hyperbolic equations
Constitutive equation in 1D:τ∂tq + q = −λ∂xT .
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Analytical and numerical solutions of dissipative systems
Kinetic theory: phonon hydrodynamics
Instead of solving the Boltzmann eq. (6+1 D)
Momentum series expansion + truncation closure
∂u〈n〉∂t
+n2
4n2 − 1c∂u〈n−1〉
∂x+ c
∂u〈n+1〉
∂x=
0 n = 0− 1
τRu〈1〉 n = 1
−(
1τR
+ 1τN
)u〈n〉 2 ≤ n ≤ N
It requires at least N=30 momentum equation to approximate thereal propagation speeds.
Coupling between different tensorial order quantities!
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Analytical and numerical solutions of dissipative systems
Generalizations of entropy density and entropy flux
Non-equilibrium thermodynamics with current multiplier
entropy density: s(e, qi ,Q ij) = se(e)− m12 qiq
i − m22 QijQ
ij
entropy current: J i = bi j qj
qi : heat flux ≡ vectorial internal variable
Q jk ≡ tensorial internal variable: pressure!
bi j : current multiplier
→ coupling between qi and Q ij ⇒ CONSEQUENCE:
parabolic equations → simplified to hyperbolic ones
LET US SEE THE EXAMPLE!
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Analytical and numerical solutions of dissipative systems
Generalization of heat conduction
Rigid body → ρ = const.
Entropy production: σs = s + ∂iJi ≥ 0 in 1 spatial dimension:(
b − 1T
)∂xq + (∂xb −m1∂tq) q − (∂xB −m2∂tQ)Q + B∂xQ ≥ 0
Linear relations between thermodynamic fluxes and forces, isotropy:
m1∂tq − ∂xb = −l1q,m2∂tQ − ∂xB = −k1Q + k12∂xq,
b − 1
T= −k21Q + k2∂xq.
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Analytical and numerical solutions of dissipative systems
Generalization of heat conduction
Rigid body → ρ = const.
Entropy production: σs = s + ∂iJi ≥ 0 in 1 spatial dimension:(
b − 1T
)∂xq + (∂xb −m1∂tq) q − (∂xB −m2∂tQ)Q + B∂xQ ≥ 0
Linear relations between thermodynamic fluxes and forces:
m1∂tq − ∂xb = −l1q,m2∂tQ − ∂xB = −k1Q + k12∂xq,
b − 1
T= −k21Q +���k2∂xq.
Compatibility withkinetic theory,hyperbolic eq.
⇒ SAME structure as 3 momentum eqs. of phonon hydro.
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Analytical and numerical solutions of dissipative systems
Ballistic-conductive system, tested on NaF experiments!
ρc∂tT + ∂xq = 0,
τq∂tq + q + λ∂xT + κ∂xQ = 0,
τQ∂tQ + Q + κ∂xq = 0.
Properties of the structure
Thermo-mechanical coupling:
Q ij → current density of the heat flux → pressure
Ballistic-conductive:τqτQ∂tttT + (τq + τQ)∂ttT + ∂tT = a∂xxT + (κ2 + τQ)∂txxT
Hyperbolicity is checked: eigenvalues of Aij , F ≡ 0
∂tui + Aij∂xu
i + B ijui = F ,
±√κ2+τQ√τqτQ
; 0→ strict hyperbolicity19 / 54
Analytical and numerical solutions of dissipative systems
Solution: numerical method I.
Initial conditions
All fields are homogeneous at t=0.
Boundary conditions ⇐⇒ discretization method
Only for the field of heat flux ⇐⇒ given by the experiments
q(t, x = 0) =
{1− cos(2π · t
tpulse) if 0 < t ≤ tpulse
0 if t > tpulse
Shifted fields:One goes from x = 0to x = 1, the others
shifted by ∆x2 .
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Analytical and numerical solutions of dissipative systems
Solution: numerical method II.
Explicit finite differences: quick, requires stability conditions !
Forward in time → first order only :(
(Weak) Consistency is checked: lim ∆t,∆x → 0⇒ ∂t , ∂x
Stability conditions: Neumann + Jury conditions
Neumann: φnj = ξne ikj∆x , ξ: growth factor; |ξ| ≤ 1Jury: what about the roots of the characteristic polynomial (f (ξ))?
Conditions:
1 f (ξ = 1) ≥ 0,
2 f (ξ = −1) ≥ 0 if n is even; f (ξ = −1) ≤ 0 if n is odd,
3 |a0| ≤ an and so on...
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Analytical and numerical solutions of dissipative systems
Solution: numerical method III.
For higher order polynomials this method comes ”crazy”...1 f (ξ = 1) ≥ 0,2 f (ξ = −1) ≥ 0 if n is even; f (ξ = −1) ≤ 0 if n is odd,3 |a0| ≤ an,
4 |b0| > |b2|, where b0 =
∣∣∣∣ a0 a3
a3 a0
∣∣∣∣ and b2 =
∣∣∣∣ a0 a1
a3 a2
∣∣∣∣and so on...
Coefficients for ballistic-conductive equation:
a3 = 1
a2 =∆t
τq+
∆t
τQ− 3
a1 =∆t2
τqτQ+ 3−
2∆t
τq−
2∆t
τQ−
(κ2∆t2
τqτQ∆x2+
∆t2
τq∆x2
)(2 cos(k∆x)− 2)
a0 =∆t
τq+
∆t
τQ−
∆t2
τQτq− 1 +
(κ2∆t2
τqτQ∆x2−
∆t2
τq∆x2
(∆t
τQ− 1
))(2 cos(k∆x)− 2)
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Analytical and numerical solutions of dissipative systems
Solution: numerical method IV.
Implicit method: robust stabilityMixed: Crank-Nicholson-type (Θ-method → linear combination)
E.g.: balance equation for internal energyρc∆t
(T k+1i − T k
i
)=
− 1∆x
((1−Θ)(qki+1 − qki ) + Θ(qk+1
i+1 − qk+1i )
)Θ = 0: completely explicitΘ = 1: completely implicitΘ = 1/2: Cranck-Nicholson-type discretization
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Analytical and numerical solutions of dissipative systems
The ballistic-conductive model - Solutions
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Analytical and numerical solutions of dissipative systems
Comparison: phonon hydro I.
Phonon hydrodynamics (RET) vs NET+IVRequire at least N=30 eqs., but only 3 is used vs 3 eqs.
No. of fitted parameters: 2 relaxation time vs. 2+1 parametersSolved on semi-infinite region vs real domain
Relative amplitudes: false vs trueSummary: RET results are more like model testing than fitting;
Wrong: heat pulse length, sample size, thermal conductivityIs the RET model appropriate? Can not be decided.
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Analytical and numerical solutions of dissipative systems
Comparison: phonon hydro II.
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Analytical and numerical solutions of dissipative systems
Comparison: phonon hydro III.
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Analytical and numerical solutions of dissipative systems
Comparison: hybrid phonon gas theory
Hybrid phonon gas model of Y. Ma vs NET+IVLongitudinal signal: artificial extension vs simplified model
Fitted parameters: 2 relaxation time vs 2+1Boundary conditions: no information vs effective cooling
Wrong thermal conductivity, no information about the others.
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Analytical and numerical solutions of dissipative systems
What about on room temperature?
Role of material heterogeneities?! Let’s see the experiments!
Arrangement of the measurement, DEE BME
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Analytical and numerical solutions of dissipative systems
Samples I.
Capacitor: periodic layered structure, thickness = 3 mm
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Analytical and numerical solutions of dissipative systems
Samples II.
5 different rock samples, e.g. limestone, thickness = 1.5 mm
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Analytical and numerical solutions of dissipative systems
Samples III.
2 different metal foam samples, inclusions: mm scale; thickness =5 mm
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Analytical and numerical solutions of dissipative systems
Over-diffusive phenomenon I.
Measurement at room temperature, metal foam sampleτq∂ttT + ∂tT = a∂xxT + κ2∂txxT , Fourier equation
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Analytical and numerical solutions of dissipative systems
Over-diffusive phenomenon II.
Measurement at room temperature, metal foam sampleτq∂ttT + ∂tT = a∂xxT + κ2∂txxT , Guyer-Krumhansl equation,κ2/τq > a!! Enhanced diffusion?! Parabolic equation!
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Analytical and numerical solutions of dissipative systems
Properties of Guyer-Krumhansl equation
τ∂tq + q = −λ∂xT + κ2∂xxq andρc∂tT + ∂xq = 0.
It is applicable for second sound (damped wave) andover-diffusive phenomena.
Parabolic equation.
Hierarchy: Fourier + its time derivativeτ∂ttT + ∂tT = a∂xxT + κ2∂txxT .
Fourier resonance condition: κ2
τ = a.
κ2
τ < a: under-damped region, “wave-like” propagation.
κ2
τ > a: over-damped region.
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Analytical and numerical solutions of dissipative systems
Analytical solution of Guyer-Krumhansl equation I.
Temperature representation:τ∂ttT + ∂tT = a∂xxT + κ2∂txxT .
“Hybrid”: T and q together.Advantageous for numerical methods.
Heat pulse experiment: q(t, x = 0) and q(t, x = L) are given⇒ let us use only the q!
Heat flux representation: τ∂ttq + ∂tq = a∂xxq + κ2∂txxq .
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Analytical and numerical solutions of dissipative systems
Analytical solution of Guyer-Krumhansl equation II.
Let it be dimensionless.
τ∂ttq + ∂tq = ∂xxq + κ2∂txxq and
BC:
q(x = 0, t) = q0(t) =
{ (1− cos
(2π · t
τ∆
))if 0 ≤ t ≤ τ∆,
0 if t > τ∆,
and q(t, x = 1) = 0 (adiabatic).
IC: homogeneous, q(t = 0, x) = 0, ∂tq(t = 0, x) = 0.Split the solution: section I. → 0 < t < τ∆, section II. → τ∆ ≤ t.
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Analytical and numerical solutions of dissipative systems
Analytical solution of Guyer-Krumhansl equation III.
q(x , t) = w(x , t) + v(x , t), (1)
where
w(x , t) := q0(t) +x
L
(qL(t)− q0(t)
)=(
1− x
L
)q0(t), (2)
after substitution:
τq(w + v) + w + v = w ′′ + v ′′ + κ2(w ′′ + v ′′). (3)
τq v + v = v ′′ + κ2v ′′ − f (x , t), (4)
where f (x , t) = w + τqw .Preserves the IC for v , but now the BC is zero: v(x = 0, t) = 0.
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Analytical and numerical solutions of dissipative systems
Analytical solution of Guyer-Krumhansl equation IV.
Separation of variables: “classical” but works,
v(x , t) = ϕ(t)X (x), (5)
and assume that X (x) (for f = 0) solves the inhomogeneous case(f 6= 0) too.
Xn(x) = sin(nπ
Lx)
(6)
with eigenvalues
βn =(nπ
L
)2. (7)
Calculating the Fourier series of f , and solving
τqϕn +(1 + κ2b
)ϕn + bϕn = −fn(t) (8)
ϕn(t) = Y1nex1t + Y2ne
x2t + A sin(gt) + B cos(gt). (9)
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Analytical and numerical solutions of dissipative systems
Analytical solution of Guyer-Krumhansl equation V.
For section II, the IC are
qII (x , t = 0) = qI (x , t = τ∆), qII (x , t = 0) = qI (x , t = τ∆).(10)
with t = t − τ∆ with homogeneous BC.Separating the variables again...
qII (x , t) = γ(t)X (x), (11)
X (x) is the same.
γn(t) = C1ner1n t + C2ne
r2n t , (12)
with
r1,2 =1
2τq
(−1− βnκ2 ±
√(1 + βnκ2)2 − 4τqβn
). (13)
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Analytical and numerical solutions of dissipative systems
Analytical solution of Guyer-Krumhansl equation VI.
C1n + C2n = ϕn(t = τ∆),
C1nr1n + C2nr2n = ϕn(t = τ∆). (14)
Temperature distribution is determined from the energy balance:
τ∆T + q′ = 0,⇒ T = − 1
τ∆q′ = − 1
τ∆
∞∑n=1
Γn(t)nπ
Lcos(nπ
Lx),
(15)
T = − 1
τ∆
t∫0
∞∑n=1
Γn(α)nπ
Lcos(nπ
Lx)dα, (16)
where Γn(t) could be ϕn or γn depending on which section isconsidered.
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Analytical and numerical solutions of dissipative systems
Analytical solution of Guyer-Krumhansl equation VII.
IC for T: T (t = 0, x) = 0 for section I. (Dimensionless!)For section II: TII (x , t = 0) = TI (x , t = τ∆). Convergence:
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Analytical and numerical solutions of dissipative systems
Analytical solution of Guyer-Krumhansl equation VIII.
The rear side temperature history considering τq = κ2 = 0.02,using 40 terms. Fourier solution!
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Analytical and numerical solutions of dissipative systems
Analytical solution of Guyer-Krumhansl equation IX.
The rear side temperature history considering τq = 0.02, κ2 = 0,using 200 terms. MCV (under-damped) solution!
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Analytical and numerical solutions of dissipative systems
Analytical solution of Guyer-Krumhansl equation X.
The rear side temperature history considering τq = 0.02, κ2 = 0.2,using 10 terms. Over-damped solution!
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Analytical and numerical solutions of dissipative systems
Finite element solution of Guyer-Krumhansl equation
Implementing in COMSOL. Fourier solution:
Needs ca. 20 s. Our code: runs under 1 s.46 / 54
Analytical and numerical solutions of dissipative systems
Finite element solution of Guyer-Krumhansl equation
Implementing in COMSOL. MCV solution:
Needs ca. 45 s. Our code: runs under 1 s.47 / 54
Analytical and numerical solutions of dissipative systems
Finite element solution of Guyer-Krumhansl equation
Implementing in COMSOL. GK solution:
FALSE! Needs ca. 65 s. Our code: runs under 1 s.48 / 54
Analytical and numerical solutions of dissipative systems
Finite element solution of ballistic equations
Implementing in COMSOL.
Needs ca. 3 hours! Our code: runs under 1 s.49 / 54
Analytical and numerical solutions of dissipative systems
How it works for conservative systems?
Pure elastic body:conservative system with velocity v and deformation ε,(dimensionless) ε = −v ′ and v = −ε′. Shifted fields:
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Analytical and numerical solutions of dissipative systems
How it works for conservative systems?
Shifted fields + semi-implicit Euler (symplectic in time):
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Analytical and numerical solutions of dissipative systems
Summary
Thermodynamics = stability.Theory for dissipation: see the work of Peter Van.(Example: Dissipative Lotka-Volterra.)
Thermodynamics: hierarchy of parabolic and hyperbolic eqs.
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Analytical and numerical solutions of dissipative systems
Summary
Under development:
Numerical solutions: shifted (staggered) in space +symplectic in time.
Analytical solutions: other methods, more applicable formulae.
Physics: understanding non-equilibrium thermodynamics.
Working on practical problems: experiments, industrialapplications.
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Analytical and numerical solutions of dissipative systems
Thank you for your kind attention!
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