On the consistent discretization in time of nonlinear thermo-elastodynamics

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1 of nonlinear thermo-elastodynamics Chair of Computational Mechanics University of Siegen GAMM Annual Meeting,

2 Thermodynamic double pendulum

3 Structure preserving integrators Great interest in the last two decades Energy balance Long stability Enhanced computational robustness - adiabatic - not restricted to adiabatic systems 3

4 Kinematics Thermodynamic double pendulum: q m r m q linear momenta: p i = m i v i and π i = p i p i lengths of the springs: λ i = c i with c = q q c = r r relative temperatures: ϑ i =θ i θ 4

5 Poisson - Lagrange Poissonian formalism Total Energy: E = T i (π i ) + e i (c i, s i ) i= Kin. energies: T i = m i π i Temperatures: Internal energies: θ i = e i s i e i =θ i s i +ψ i Lyapunov-function: L = E θ S Dissipation: L = D cdu 0 Lagrangian formalism Total Energy: E = T i (v i ) + ˆψ i (c i,θ i ) i= Kin. energies: T i = m i v i Entropies: s i = ˆψ i θ i Relative internal energies: ê i =ϑ i s i + ˆψ i Lyapunov-function: V = T i + ê i i= Dissipation: V = D cdu 0 5

6 Semidiscrete equations First equations of motion: Second equations of motion: ṗ = (e + e ) q ṗ = (e + e ) q q = p m q = p m ṗ = ( ˆψ + ˆψ ) q ṗ = ( ˆψ + ˆψ ) q Thermal evolution equations: ṡ =κ ( ) θ θ ṡ =κ ( ) θ θ 6

7 GENERIC framework Initial value problem: ż t = L(z t ) E(z t ) + M(z t ) S(z t ) z(t 0 ) = z 0 State vector: ] z = [q, q, p, p, s, s Degeneracy conditions: L(z t ) S(z t ) = M(z t ) E(z t ) = 0 Consequences: Ė(z t ) = 0 Ṡ(z t ) 0 7

8 Gradients and matrices Gradients: E = S = [ f ν f ν, f ν, p [ ] 0, 0, 0, 0,, Poisson matrix and friction matrix:, p ],θ,θ m m 0 0 I I 0 0 I L =, M = 0 8 κ θ 0 I θ κ κ κ θ θ

9 Discretization Initial value problem: z n+ z n h n Vector of invariants: Degeneracy conditions: = L π T DẼ +M π T D S π = [ c, c,π,π, s, s ] L(z n, z n+ )DS(z n, z n+ ) = M (z n, z n+ )DE(z n, z n+ ) = 0 Consequences using the TC integrator: E(z n+ ) E(z n ) = 0 S(z n+ ) S(z n ) 0 and: L(z n+ ) L(z n ) 9

10 Discrete equations First discrete equations of motion: midpoint rule q in+ q in h n = m i p in+ (no sum) Second discrete equations of motion: part. discr. grad. p n+ p n = D c e q h n+ + D c e r n+ n p n+ p n = D c e r h n+ n Discrete thermal evolution equations: part. discr. grad. ( ) s in+ s in θ j =κ h n θ i 0 with: θ = D s e θ = D s e

11 Partitioned discrete gradient Second discrete equations of motion: D ci e i = e i(c in+, s in+ ) e i (c in, s in+ ) (c in+ c in ) Discrete thermal evolution equations: D si e i = e i(c in+, s in+ ) e i (c in+, s in ) (s in+ s in ) + e i(c in+, s in ) e i (c in, s in ) (c in+ c in ) + e i(c in, s in+ ) e i (c in, s in ) (s in+ s in )

12 Weak forms ] State vector: z = [q, q, v, v,θ,θ First weak equations of motion: tn+ t n δṗ i v i = Second weak equations of motion: tn+ t n m i v i δ q i = Weak thermal evolution equations: tn+ t n tn+ t ( ) n+ [s i,t=t0 ] + [si enh ] δθ i,t=t ṡ i δθ i = t n t n δṗ i q i ˆψ q i δ q i cg method cg method modified dg method t n+ with: [s enh i ] = ϑ + i t=t0 ( [ϑ i,t=t0 ] s i,t=t0 + [ ˆψ i,t=t0 ] t n κ ( ) θj δθ i θ i )

13 Discrete equations First discrete eq. of motion: q in+ q in = v in+ h n Second discrete eq. of motion: midpoint rule algo. stresses m v n+ v n h n = S q n+ + S r n+ m v n+ v n h n = S r n+ Discr. thermal evolution eq.: two-point Gaussian quadr. rule ê + in êin + θ + in θ (s i n+ s i + n ) κ [ ]( ) ξl θ + jn+ξ h l n (s i n+ s i + n ) ξ l θ + = 0 l= i n+ξl with: c i = c in+ andθ + i n+ξl = ( ξ l )θ + i n +ξ l θ in+ 3

14 Stress approximation Constraints: G i (Ŝi ) = ê in+ ê + i n ê in+ Ŝi (c in+ c in ) Minimization of the functional: F i (Ŝi,µ i ) =µ i G i (Ŝi ) + Ŝ i algorithmic stresses S i = S i + Ŝi with S i = ˆψ i c i : yields: ê in+ ê i + n [ s in+ s + ] i n ϑ + in+ S i = c in+ c in V n+ V n + h n 0 D cdu = 0 4

15 Parameters Initial state: [] [] q 0 = m p 0 0 = Ns θ 0 = 380 K m 0 = kg [ ] [ ] q 0. = m p = Ns θ = 30 K m = kg Free energy: ˆψ i = K i log ( λi λ 0 i with (i [, ]): ) ( ) [ ( )] λi θi β i ϑ i log λ 0 + k i ϑ i θ i log i θ λ 0 = m K i = 0000 J κ = 0 W K θ = 300 K λ 0 = m k i = 000 J K β i = 0. J K 5

16 Reference solution 6 temperature E x θ θ length of q L x

17 Standard integrators temperature E x 05 h= h=0.009 h=0.0 3 h= θ θ length of q L x 05 h=0.006 h= h=0.0 h=

18 Structure preserving integrators 8 temperature E x 05 h=0.006 h= h=0.0 h= θ θ length of q L x 04 h=0.006 h= h=0.0 h=

19 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 Outlook Implementation and comparison of both integrators for a thermodynamical continuum Enlargement of the GENERIC system for unisolated driven systems 9

20 Thank you for your attention! Questions?! 0 The authors thank the German Research Foundation (DFG) for financial support (GR 397/-)

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

Energy-consistent time-integration for dynamic finite deformation. thermoviscoelasticity. GAMM Annual Meeting, for dynamic finite thermo-viscoelasticity Chair of Computational Mechanics University of Siegen GAMM Annual Meeting, 9.04.0 Thermo-viscoelastic double pendulum (closed system) Thermodynamically Consistent