Energy-consistent time-integration for dynamic finite deformation. thermoviscoelasticity. GAMM Annual Meeting,

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Tracy Arron Robbins
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1 for dynamic finite thermo-viscoelasticity Chair of Computational Mechanics University of Siegen GAMM Annual Meeting,

2 Thermo-viscoelastic double pendulum (closed system) Thermodynamically Consistent algorithm enhanced GENERIC format partitioned discrete gradients enhanced hybrid Galerin method continuous-discontinuous Galerin method algorithmic stresses

3 Structure preserving integrators great interest in the last two decades Energy balance long stability enhanced computational robustness exact reproduction of the physical structure - adiabatic - not restricted to adiabatic systems 3

4 Kinematics Thermoviscoelastic double pendulum: m q r q m 4 linear momenta: p = m v and π = p p principal stretches: λ = l L = c with l = q l = r rel. temperatures: ϑ = θ θ int. isotr. variable: c i c e = c ci

5 Physical structure 5 closed system - q, p, s, c i Total Energy: E = T (π ) + e (c, s, c i ) = Kin. energies: T = m π Temperatures: Internal energies: θ = e s e = θ s + ψ Lyapunov-function: L = E θ S Dissipation: L = D tot 0 open system - q, v, θ, c i Total Energy: E = ˆT (v )+ ˆψ (c, θ, c i ) = Kin. energies: +θ s (θ, c ) ˆT = m v Entropies: s = ˆψ θ Relative internal energies: ê = ϑ s + ˆψ Lyapunov-function: ˆL = ˆT + ê = Dissipation: ˆL = D tot 0

6 Semidiscrete equations Equations of motion: ṗ = (e + e ) q ṗ = (e + e ) q Thermal evolution equations: ṡ = κ q = p m q = p m ( ) θ + Dint θ θ Viscous evolution equations: ṗ = ( ˆψ + ˆψ ) q ṗ = ( ˆψ + ˆψ ) q ṡ = κ with D int = ψvis c i ( ) θ + Dint θ θ ċ i = 4 c ψ vis i V ċ i = 4 c ψ vis i vol c i V vol c i ċ i 6

7 Enhanced GENERIC framewor H.C. Öttinger (005) Initial value problem: ż = (L + L c i ) ze + (M + M c i ) zs z(t 0 ) = z 0 State vector: ] z = [q, q, p, p, s, s, c i, c i Degeneracy conditions: L zs = M ze = 0 Enhanced condition: ze M c i zs = ze L c i ze Consequences: Ė = 0 Ṡ 0 L 0 7

8 Gradients and matrices Gradients: [ e + e E =, q q e, p, p ], θ, θ, ci e, ci e m m S = [ 0, 0, 0, 0,,, 0, 0 ] Matrices: I I L + L ci = c i V vol c i V vol κ θ M + M ci = θ + Dint θ κ 0 0 κ κ θ θ + Dint θ

9 Discretization Initial value problem: z n+ z n = (L + L c i ) π T h n+ DπE+(M + M c i ) π T n+ DπS n Vector of invariants: ] π = [c, c, π, π, s, s, c i, c i Discrete conditions: L Dz S = M DzẼ = 0 DzẼ Mc i Dz S= DzẼ Lc i DzẼ Consequences using the TC integrator: and: E(z n+ ) E(z n ) = 0 S(z n+ ) S(z n ) 0 L(z n+ ) L(z n ) 9

10 Discrete equations Discrete eq. of motion: midpoint & part. discr. grad. q n+ q n = p h n m n+ (no sum) p n+ p n = h n L D c e q n+ + L p n+ p n = h n L D c e r n+ Discrete thermal evolution equations: s n+ s n h n = κ ( ) Dsj e j + Dint D s e D s e D c e r n+ part. discr. grad. Discrete viscous evolution equations: part. discr. grad. c in+ c in = 4 ci h n V n+ D ci e vol 0

11 Partitioned discrete gradients D c e = e ( ) ( ) c n+, s n+, c in+ e c n, s n+, c in+ + (c n+ c n ) e (c n+, s n, ci n) e (cn, s n, ci n) (c n+ c n ) D s e = e ( ) ( ) c n, s n+, c in+ e c n, s n, c in+ + (s n+ s n ) e (c n+, s n+, ci n) e (c n+, s n, ci n) (s n+ s n ) D ci e = e ( c n, s n, c in+ ) e (c n, s n, c n i ) ( c in+ c in ) + e ( c n+, s n+, c in+ ) e (cn+, s n+, c n i ) ( c in+ c in ) and: D int = D ci e c i n+ c in h n = 4 V vol c i n+ D ci e

12 Wea forms M. Groß (009) State vector: z = [q, q, v, v, θ, θ, c i, c i ] Wea equations of motion: cg method tn+ t n tn+ t n δṗ v = m v δ q = tn+ Wea thermal evolution equations: t n+ t n+ [[ê i,t0 ] ϑ + δθ +,t 0 + ṡ δθ = t0 t n t n t n tn+ t n [ κ δṗ q ˆψ q δ q modified dg method ) ] + Dint δθ ( θj θ Wea viscous evolution equations: t n+ ψ vis t n+ δċ i = c i V vol ci t n t n θ li vis δċ i cg method

13 Discrete equations Discrete eq. of motion: midpoint rule & algo. stresses q n+ q n m v n+ v n h n m v n+ v n h n h n = v n+ = Σ c q n+ L n+ + Σ c r n+ L n+ = Σ c r n+ L n+ Discr. thermal evolution eq.: two-point Gaussian quadr. rule [ê i,t0 ] + s ϑ + rat t0 [ ] [ ( ) ] = hn ξl θjn+ξl D int ξl κ + ξ l θ n+ξl s rat l= Discr. viscous evolution eq.: c in+ c in h n = 4 V vol c i n+ ci ψ vis n+ θ n+ξl with: s rat = (s n+ s + n ) midpoint rule 3

14 Stress approximation Σ Constraints: G ( ˆΣ ) = h n (ên+ ê + n ) ê ˆΣ l = 0 Minimization of the functional: F ( ˆΣ, λ m ) = λm G ( ˆΣ ) + ˆΣ algorithmic stresses Σ = Σ + ˆΣ with Σ = c ˆψ c : Σ = h n (ê n+ ê + n ) + D int ϑ n+ h n (s n+ s + n ) l with: l and: = c n+ c n+ c n h n D int α l vis = Σ vis α l vis α = c i n+ c in+ c in h n 4

15 Stress approximation Σ vis Constraints: vis G ( ˆΣ ) = D int l= Minimization of the functional: D int ξl vis F ( ˆΣ ξl, λ m ) = λ m G ( ˆΣ ) + algorithmic dissipation D int ξl D int ξl ( D = D int int n+ ξl + l= ϑ ξl θ ξl l= ( l= = D int ξl + ˆΣ vis ξl l vis ξl : ( ϑξl D int ξ + D int θ ξl l vis ξl ξ ) ) ϑ ξl θ ξl ˆΣvis ξl l vis ξl = 0 ) ˆΣ vis ξl l vis ξl 5

16 6 Free energy function Free energy: with: and: ψ com ψ vis = µe ψ = = ψ com + ψ vis + ψ the = µ (c ln c ) + ψ vol (c ) ( ce ln ) c e + ψ vol (c e ) ψ the = ( ϑ θ ln θ θ ) ψ vol (c ) = λ β ϑ ψ vol (c ) J [ ln c + ( c ) ] [ ln ce + ( c e ) ] ψ vol (c e ) = λe Parameters: µ = [750, 75] J µ e = 750 J = 500 J K λ = [3000, 30] J λ e = 3000 J β = J K V vol = 50 Js θ = 300 K κ = 0 W K

17 Orbits Reference solution Standard integrators, t = 0.09s z z y x ehg &, t = 0.09s y t = s x z y x 7

18 Reference solution temperature - θ θ θ dissipation - D total conduction internal energy - E.354 x lyapunov - L.5 x

19 Standard integrators temperature - θ θ, t=0.0s θ, t=0.05s θ, t=0.09s θ, t=0.0s θ, t=0.05s θ, t=0.09s D tot t=0.0s t=0.05s t=0.09s energy - E.353 x 05 t=0.0s t=0.05s.358 t=0.09s lyapunov - L 3 x 04 t=0.0s t=0.05s t=0.09s

20 Structure preserving integrators temperature - θ θ, t=0.0s θ, t=0.05s θ, t=0.09s θ, t=0.0s θ, t=0.05s θ, t=0.09s D tot t=0.0s t=0.05s t=0.09s ehg-e TC-E.354 x x 05 t=0.0s t=0.05s.35 t=0.09s lyapunov - L.5 x 04 t=0.0s t=0.05s.4 t=0.09s

21 Conclusions Implementation of the - partitioned discrete gradient Implementation of the - algorithmic stresses Comparison of structure preserving integrators with standard integrators by means of a double pendulum Outloo Implementation of a thermoviscoelastic continuum for the enhanced GENERIC system Enlargement of the enhanced GENERIC system to unisolated driven systems

22 Than you for your attention! Questions?! The authors than the German Research Foundation (DFG) for financial support (GR 397/-)

On the consistent discretization in time of nonlinear thermo-elastodynamics

On the consistent discretization in time of nonlinear thermo-elastodynamics of nonlinear thermo-elastodynamics Chair of Computational Mechanics University of Siegen GAMM Annual Meeting, 3.03.00 0000 0000 Thermodynamic double pendulum Structure preserving integrators Great interest