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1 Peter Betsch An energy-entropy-consistent finite thermo-viscoelasticity 1 Peter Betsch Chair of Applied Chemnitz University of Technology March 1, Chair of Computational Mechanics Institute of Mechanics University of Siegen Karlsruhe Institute of Technology GAMM 014 Section S4: Structural mechanics Acknowledgments: This research was provided by DFG grant GR 397/-1

2 Contents of the presentation Outline Constitutive laws 5 Motions of continua 4 Overview 3 Outline Titel Peter Betsch Balance of linear momentum Balance of entropy Strong forms 8 Weak forms 11 Boundary 1 Time discrete 13 conditions weak forms Time discrete operators 7 10 Balance of total energy Balance of Lyapunov function Fully closed system 18 Fully open system 0 Summary & Goto strong forms Goto weak forms Goto balance laws Goto boundary conditions Goto time discrete weak forms Goto time discrete balance laws Goto numerical examples

3 Overview of structure-preserving time integrators 1 Stuart & Humphries [1998], Hairer et al. [006], 3 Gonzalez [1996], 4 Armero & Romero [001], 5 Marsden & West [001], 6 Romero [010] 7 Betsch & Steinmann [000], 8 Mohr et al. [008], 9 Bauchau & Bottasso [1999], 10 Ober-Blöbaum & Saake [013] Peter Betsch Characteristics 1 Preserve physical structures of (constraints on) solution spaces of time evolution equations (ODE or PDE or DAE) Examples: Geometric constraints, conservation laws (e.g. energy or linear and angular momentum), balance laws (e.g. entropy, Lyapunov function). 3 Finite difference methods in time 1 Symplectic methods 1, Energy-momentum time stepping schemes 3 3 Energy dissipative schemes 4 4 Variational integrators 5 (VI) 5 Energy-entropy consistent (TC) integrators 6 for thermoelastic closed systems Finite element methods in time 1 Higher-order accurate symplectic methods 7 Galerkin-based energy-momentum methods 8 3 Galerkin-based energy dissipative methods 9 4 Galerkin-based variational integrators 10 (GVI)

4 of considered continuum mechanics Malvern [1969], Truesdell & Noll [199], Marsden & Hughes [1994], Ogden[ 1997], Holzapfel [000], Haupt [00] Motion of a single continuum T N Q ϕ Θ Peter Betsch B 0 C i = F T i F i s = Js c ϕ R = Jr ϕ ρ = Jρ c ϕ X T XB 0 B F i F = ϕ C = F T F F e V i T xb t ρ c s c r x p = J(ρ tv) ϕ =ρ 0V Θ =θ ϕ B t Energy functionals 1 Total energy, total entropy & Lyapunov function H = K + E S = s dv V = H Θ S Kinetic energy & internal energy 1 K = k(p) dv = p p dv E = e(c, s, C i ) dv ρ 4

5 Peter Betsch 5 of the considered continuum Lu & Pister [1975], Miehe [1988], Lion [1997], Yu et al. [1997], Reese [000], Holzapfel [000], Vujosevic & Lubarda [00], Heimes [005], Hartmann [01] Thermo-viscoelastic motion and heat conduction 1 First Piola-Kirchhoff stress tensor P = F δe δc e = e ela (C) + e the (C, s) + e vis (C, C i ) Fourier s law of isotropic heat conduction Dissipative behaviour 1 Total dissipation Q = κ J C 1 Θ D tot = D cdu + D vis = 1 Θ Q Θ Viscous evolution equation Goto outline C i = V 1 δe (C i ) : δc i V 1 = 1 T Idev V dev 0 + δe : C i 0 δc i 0 V 1 (C i ) = C i V 1 C i 1 Ivol n dim V vol

6 Peter Betsch 6 Strong forms derived from the balance laws Holzapfel [000], Marsden & Ratiu [000], Wriggers [001], Maugin & Kalpakides [00], Bargmann [008], Mata & Lew [011] Equations of motion 1 Balance of linear momentum p dv = B dv + Equations of motion Heat conduction equations 1 Entropy inequality principle D tot Θ dv = B 0 Heat conduction equations p ϕ T da = DivP + B = 1 ρ p δh δp with T = PN [ s R ] 1 dv + Q N da 0 Θ B 0 Θ s = 1 [ D vis + R DivQ ] Θ Θ = e s δh δs

7 Peter Betsch 7 Strong forms and the first law of thermodynamics Truesdell & Toupin [1960], Gurtin [1975], Kestin [1979], Silhavy [1997], Wilmanski [1998] Balance of total energy H/ = P mec + P the 1 Balance of mechanical energy [ δh δp p + δh ] δf : F dv = k/ p int Balance of thermal energy [ δh s + δh : C ] i dv δs δc i = p ent D vis ϕ B dv + ϕ T da eqns. of motion} {{ } P mec R dv Q N da heat cond. eqns. } {{ } P the Balance of Lyapunov function V = H Θ S V H s Θ dv V Goto outline = + total energy balance heat cond. eqn. Θ ϕ Θ Dtot dv + B dv + ϕ T da Θ Θ Θ Θ R dv Q N da ( 0) Θ Θ

8 Peter Betsch 8 Weak forms derived from the strong forms Willner [1990], Miehe [1993], Gonzalez [1996], Simo [1998], Reese [000], Romero [010a,010b], Krüger [01] Weak equations of motion (integration by parts) [ δϕ p ] + (δϕ) :δh dv = δϕ BdV + δϕ T da δf [ ϕ δp δh δp integration by parts Gauss theorem ] dv = 0 Weak heat conduction equations ( ) δθ δθ DivQ dv = Θ Q dv Θ [ δθ s +δθ Θ Goto outline δh : C ] i dv = δc i D vis [ δs Θ δh ] dv = 0 δs δθ Q N da Θ [ ( ) ] δθ δθ Θ R + Q dv Θ δθ Q N da Θ

9 Peter Betsch 9 Weak forms fulfill all balance laws (I) Balance of linear momentum Balance of entropy Θ [ Chooseδϕ(X, t) = c = const. [ [ ] ] p c B dv T da = 0 Balance of linear momentum ChooseδΘ(X, t) = Θ = const. D cdu /Θ [ { ( }}) ]{ s Dvis + R 1 Q dv + Θ Θ Balance of mechanical energy Entropy inequality principle with D tot 0 1 Θ Chooseδϕ(X, t) = ϕ(x,t) andδp(x, t) = p(x,t) k/ [{}}{ δh δp p p int P mec ] Q N da {}} ]{ {}} { + δh δf : F ϕ dv = B dv + ϕ T da Balance of mechanical energy = 0

10 Peter Betsch 10 Weak forms fulfill all balance laws (II) Balance of thermal energy ChooseδΘ(X, t) = Θ(X, t) andδs(x, t) = s(x,t) p ent D vis P the {[ }}{ δh s δs + δh : C ] { }} { i dv = R dv Q N da δc i Balance of total energy Balance of thermal energy Add balance of mechanical and thermal energy [ ] [ ] H ϕ ϕ = B + R dv + T Q N da Balance of Lyapunov function Balance of total energy Balance of total energy 1. ChooseδΘ(X, t) = Θ(X, t) Θ,δs(X, t) = s(x,t) and add the balance of mechanical energy (without entropy and total energy consistency possible, cp. ehg method). V H s or. = Θ dv Goto outline Balance of entropy

11 in the weak forms Babuska [1973], Kikuchi & Oden [1988], Farhat & Geradin [199], Glowinski et al. [1994], Stenberg [1995], Wriggers [00], Donea & Huerta [003], Hartmann et al. [008] TB 0 T(t) QB 0 Q(t) Peter Betsch Mechanical boundaries Θ Thermal boundaries Θ 11 B 0 ϕb 0 Mechanical boundary δϕ T da = δϕ ϕ =:0 B 0 T da + T ΘB 0 δϕ T(t) da Thermal boundary (with Lagrange multiplier technique) δθ δθ Q N da = δθλ Θ da Θ Θ Q(t) da Θ Q with δλ Θ [Θ Θ ] da = 0 λ Θ : Lagrange multiplier (outward normal entropy flux) Θ Goto outline

12 Peter Betsch 1 ( nd order accurate) Simo & Tarnow [199], Simo & Hughes [1998], Betsch & Steinmann [00], Ibrahimbegovic & Mamouri [00], Armero [006], Romero [010], Hesch & Betsch [011] Equations of motion [ δϕ n+1 p δp n+1 [ ] ϕ t P H dv = 0 p t + (δϕ n+1) : P H F Heat conduction equations [ δθ n+1 s δs n+1 t +δθn+1 ] dv = [ ] Θ n+1 P H dv = 0 s P H Θ n+1 C i : C ] i dv = t + Local and boundary time evolutions 1. Viscous evolution. Boundary condition δϕ n+1 B 1 dv + [ Θ ( δθn+1 Θ n+1 T δϕ n+1 T 1 da ) ] Q 1 +δθn+1 R 1 dv Θ n+1 δθ n+1 Q 1 da δθ n+1λ n+1 da + Θ Q n+1 C i + t V 1 (C in+ 1 ) : P H = 0 C i δλ n+1 [Θ n+1 Θ ] da = 0 Θ

13 Peter Betsch 13 ( nd order accurate) Milne-Thomson [1933], Greenspan [1973], LaBudde & Greenspan [1974], Simo & Gonzalez [1993], Gonzalez [1996a, 1996b], Sansour et al. [004], Bourdin & Cresson [01] Single-variable function discrete derivative of f (z) f z f {k, s, p,ϕ, C i } z {t, s, p, C, C i } {,, :} = f (z n+ 1 ) + z Special case 1: Scalar-valued z: Special case : Quadratic f : f (z n+1 ) f (z n ) f (z n+ 1 ) [z n+1 z n ] z [z n+1 z n ] [z n+1 z n ] [z n+1 z n ] f t = f (z n+1) f (z n ) t n+1 t n k p = k(p 1 n+ p ) 1 ρ p n+ 1 Multi-variable functional (partitioned) discrete derivatives Goto outline P H p P H F P H s = k p = F n+ 1 = P H C i = with k(p) and e(c, s, C i) [ e C + e sn,c C in [ 1 e s + e C n,c s i [ n+1 1 e + e C i C i C n,sn sn+1,c i n+1 C n+1,c in C n+1,sn+1 ] ] ]

14 Peter Betsch 14 (I) Balance of linear momentum Balance of entropy Chooseδϕ n+1 (X) = c = const. [ [ ] ] p c t B 1 dv T 1 da = 0 T Time discrete balance of linear momentum ChooseδΘ n+1 (X) = Θ = const. [ [ s Dtot Θ t 1 + R ] 1 Q 1 dv + λ n+1 da da = 0 Θ n+1 Θ Θ Q n+1 Balance of mechanical energy Time discrete entropy inequality principle with D tot 1 Chooseδϕ n+1 (X) = ϕ(x) t andδp n+1 (X) = p(x) t [ K + 1 E t t + E ] ϕ sn,c t = in sn+1,c t B 1 dv + ϕ t T 1 da i n+1 T Time discrete balance of mechanical energy 0

15 Peter Betsch 15 (II) Balance of thermal energy ChooseδΘ n+1 (X) = Θ n+1 (X),δs n+1 (X) = s(x) t andδλ n+1 (X) =λ n+1 (X) [ 1 E t + E C n,c t + E i n+1 C n+1,c t + E ] in C n,sn t C n+1,sn+1 = R 1 dv Θ λ n+1 da + Q 1 da Θ Q Balance of total energy Time discrete balance of thermal energy Add time discrete balance of mechanical and thermal energy [ ] H ϕ = t t B 1 + R ϕ 1 dv + t T 1 da Θ λ n+1 da + Q 1 da T Θ Q Time discrete balance of total energy Balance of Lyapunov function (Lagrange multiplier will be eliminated) 1. ChooseδΘ n+1 (X) = Θ n+1 (X) Θ,δs n+1 (X) = s(x) t,δλ n+1 (X) =λ n+1 (X) and add the time discrete balance of mechanical energy (cp. ehg method) s or. = Θ V t Goto outline H t Time discrete balance of total energy t dv Time discrete balance of entropy

16 Free rotating insulated thermo-viscoelastic disc Peter Betsch Geometry no boundary conditions ω z = 0.33 s 1 y z r i = 0.8 m x h = 0.4 m r a = m Θ = 300 K ETC integrator with t = 0.0 s 380 K t = 5 s t = 10 s 370 K 360 K 350 K 340 K 330 K 30 K 380 K t = 15 s t = 0 s 370 K 360 K 350 K 380 K 370 K 360 K 350 K 340 K 330 K 30 K 380 K 370 K 360 K 350 K Reference configuration 380 K t = 0 s 370 K 360 K 350 K 340 K 330 K t = 5 s 340 K 330 K 30 K 380 K 370 K 360 K 350 K 340 K 330 K t = 30 s 340 K 330 K 30 K 380 K 370 K 360 K 350 K 340 K 330 K 30 K 30 K 30 K 16

17 Free rotating insulated thermo-viscoelastic disc Peter Betsch 17 Midpoint rule ( t = 0.1 s) 380 K t = 5 s 370 K 360 K 350 K 340 K 330 K 30 K ETC integrator ( t = 0.1 s) 380 K t = 5 s 370 K 360 K 350 K 340 K 330 K 30 K H [J] S [J/K] V [J] 4 3 Midpoint rule H [J] S [J/K] V [J] ETC integrator

18 Rail-bound partly uninsulated thermo-elastic disc Peter Betsch Geometry Θ = K ϕb 0 r a = 1.5 m y z v = 0 ms 1 x ω z = 7 s 1 r i = 0.5 m h = 1 m insulated boundaries ΘB 0 Reference configuration t = 0 s 305 K MPR ( t = 0.01 s) ETC ( t = 0.01 s) y [m] y [m] t = 1. s x [m] t =. s x [m] t = 4.4 s 305 K 300 K 95 K 305 K 300 K 95 K y [m] y [m] t = 1. s x [m] t =. s x [m] t = 4.4 s 305 K 300 K 95 K 305 K 300 K 95 K 330 K y [m] K 300 K y [m] K 300 K K x [m] 95 K x [m] 95 K

19 Rail-bound partly uninsulated thermo-elastic disc Peter Betsch 19 Midpoint rule ( t = s) y [m] t = 1.54 s 305 K 300 K K x [m] ETC integrator ( t = s) y [m] t = 1.54 s 305 K 300 K K x [m] H [J] S [J/K] V [J] 4 3 Midpoint rule H [J] S [J/K] V [J] ETC integrator

20 Peter Betsch 0 1 Maugin & Kalpakides [00], Bargmann [008] Zielonka [006], Zielonka, Ortiz & Marsden [008], Mata & Lew [011] 3 Ibrahimbegovic, Chorfi & Gharzeddine [001], Hartmann, Quint & Arnold [008], Birken et al. [010], Moore [011], Gleim & Kuhl [013] Summary 1 Goal: The ETC integrator fulfills the balance of linear momentum AND entropy the balance of total energy AND Lyapunov function also with Dirichlet and Neumann boundary conditions. Algorithmic basis: A spatially weak formulation, which fulfills ALL balance laws for STANDARD finite element spaces 3 Algorithmic key: A time discretisation with time discrete differential operators which satisfy the discrete version of the fundamental theorem of calculus. 4 Benefit: A transient simulation with an improved numerical stability for large time steps, and physically consistent solutions (NO hour-glassing, NO waves) Outlook 5 Replacing Θ by thermal displacement t α in the weak formulation 1 mixed variational integrator physical consistency with time adaptivity 3