an energy-entropy-consistent time stepping scheme …...an energy-entropy-consistent time stepping...

An energy-entropy-consistent

time steppingscheme for

finite thermo-viscoelasticity

Melanie Krüger1 ,Michael Groß &

Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies

Strong forms & energy laws

Weak forms & balance laws

Boundary conditions

Time discrete weak forms

Time discrete operators

Time discrete balance laws

Numerical studies

Fully closed system

Fully open system

Summary &outlook

1

A p p l i e dMechanics/Dynamics

An energy-entropy-consistenttime stepping scheme for


Melanie Krüger1, Michael Groß & Peter Betsch2

Chair of Applied Mechanics/Dynamics

Chemnitz University of Technology

March 12, 2014

1Chair of Computational Mechanics 2 Institute of MechanicsUniversity of Siegen Karlsruhe Institute of Technology

GAMM 2014 – Section S4: Structural mechanics

Acknowledgments: This research was provided by DFG grant GR 3297/2-1





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

2


Contents of the presentation

Outline

TitelOutlineOverviewMotions

of continua

Constitutive

laws

Balance of linear momentum

Balance of entropy

Strong Weak

forms forms

Boundary

conditions

Time discreteTime discrete

weak forms operators

Balance of total energy

Balance of Lyapunov function

Fully closed Fully open

systemsystem

Summary

& outlook

2©3©4©5©

6©

7©

8©

9©

10©

11© 12© 13©

14©

15©

16©

18© 20©

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





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

3


Overview of structure-preserving time integrators1Stuart & Humphries [1998], 2Hairer et al. [2006], 3Gonzalez [1996], 4Armero & Romero [2001], 5Marsden & West [2001], 6Romero [2010]

7Betsch & Steinmann [2000], 8Mohr et al. [2008], 9Bauchau & Bottasso [1999], 10Ober-Blöbaum & Saake [2013]

Characteristics

1 Preserve physical structures of (constraints on) solution spaces of

time evolution equations (ODE or PDE or DAE)

2 Examples: Geometric constraints, conservation laws (e.g. energy or linear

and angular momentum), balance laws (e.g. entropy, Lyapunov function).

Finite difference methods in time

1 Symplectic methods1,2

2 Energy-momentum time

stepping schemes3

3 Energy dissipative schemes4

4 Variational integrators5 (VI)

5 Energy-entropy consistent

(TC) integrators6 for

thermoelastic closed systems

Finite element methods in time

1 Higher-order accurate

symplectic methods7

2 Galerkin-based

energy-momentum methods8

3 Galerkin-based energy

dissipative methods9

4 Galerkin-based variational

integrators10 (GVI)





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

4


Introduction of considered continuum mechanicsMalvern [1969], Truesdell & Noll [1992], Marsden & Hughes [1994], Ogden[ 1997], Holzapfel [2000], Haupt [2002]

Motion of a single continuum

TXB0

TxBt

Vi

X x

B0 Bt

T

B

N

ϕ

F i Fe

ρc

p = J (ρtv) ◦ϕ= ρ0V

F = ∇ϕ

C = FT

F

C i = FTi F i

ρ = Jρc◦ϕ

Q

scr

s = Jsc◦ϕ

Θ = θ ◦ϕR = Jr ◦ϕ

Θ∞

Energy functionals

1 Total energy, total entropy & Lyapunov function

H = K + E S =

∫

B0

s dV V = H −Θ∞S

2 Kinetic energy & internal energy

K =

∫

B0

k(p) dV =

∫

B0

1

ρp · p dV E =

∫

B0

e(C , s,C i) dV





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

5


Constitutive laws of the considered continuumLu & Pister [1975], Miehe [1988], Lion [1997], Yu et al. [1997], Reese [2000], Holzapfel [2000], Vujosevic & Lubarda [2002], Heimes [2005], Hartmann [2012]

Thermo-viscoelastic motion and heat conduction

1 First Piola-Kirchhoff stress tensor

P = 2 FδE

δCe = eela(C) + ethe(C , s) + evis(C ,C i)

2 Fourier’s law of isotropic heat conduction

Q = −κ J C−1∇Θ

Dissipative behaviour

1 Total dissipation

Dtot = Dcdu + Dvis = −1

ΘQ · ∇Θ

︸︷︷︸

≥0

−δE

δC i

:∂C i

∂t︸︷︷︸

≥0

≥ 0

2 Viscous evolution equation

∂C i

∂t= −V̄−1(C i) :

δE

δC i

V̄−1(C i) = C i V

−1C i

V−1 =

1

2V devI

devT

+1

ndimV volI

vol

Goto outline





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

6


Strong forms derived from the balance lawsHolzapfel [2000], Marsden & Ratiu [2000], Wriggers [2001], Maugin & Kalpakides [2002], Bargmann [2008], Mata & Lew [2011]

Equations of motion

1 Balance of linear momentum∫

B0

∂p

∂tdV =

∫

B0

B dV +

∫

∂B0

T dA with T = PN

2 Equations of motion∂p

∂t= DivP + B

∂ϕ

∂t=

1

ρp ≡δH

δp

Heat conduction equations

1 Entropy inequality principle∫

B0

Dtot

ΘdV =

∫

B0

[∂s

∂t−

R

Θ

]

dV +

∫

∂B0

1

ΘQ ·N dA ≥ 0

2 Heat conduction equations

∂s

∂t=

1

Θ

[Dvis + R −DivQ

]

Θ =∂e

∂s≡δH

δs





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

7


Strong forms and the first law of thermodynamicsTruesdell & Toupin [1960], Gurtin [1975], Kestin [1979], Silhavy [1997], Wilmanski [1998]

Balance of total energy ∂H/∂t = Pmec + Pthe

1 Balance of mechanical energy∫

B0

[δH

δp·∂p

∂t︸︷︷︸

∂k/∂t

+δH

δF:∂F

∂t

]

︸︷︷︸

pint

dV =︸︷︷︸

eqns. of motion

∫

B0

∂ϕ

∂t·B dV +

∫

∂B0

∂ϕ

∂t·T dA

︸︷︷︸

Pmec

2 Balance of thermal energy∫

B0

[δH

δs

∂s

∂t︸︷︷︸

pent

+δH

δC i

:∂C i

∂t

]

︸︷︷︸

−Dvis

dV =︸︷︷︸

heat cond. eqns.

∫

B0

R dV −

∫

∂B0

Q ·N dA

︸︷︷︸

Pthe

Balance of Lyapunov function V = H −Θ∞S

∂V

∂t≡

∂H

∂t︸︷︷︸

total energy balance

−Θ∞

∫

B0

∂s

∂t︸︷︷︸

heat cond. eqn.

dV

∂V

∂t= −

∫

B0

Θ∞Θ

Dtot dV +

∫

B0

∂ϕ

∂t·B dV +

∫

∂B0

∂ϕ

∂t·T dA

+

∫

B0

Θ−Θ∞Θ

R dV −

∫

∂B0

Θ−Θ∞Θ

Q ·N dA (≤ 0)

Goto outline





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

8


Weak forms derived from the strong formsWillner [1990], Miehe [1993], Gonzalez [1996], Simo [1998], Reese [2000], Romero [2010a,2010b], Krüger [2012]

Weak equations of motion (integration by parts)∫

B0

[

δϕ ·∂p

∂t+∇(δϕ) :

δH

δF

]

dV =

∫

B0

δϕ ·B dV +

∫

∂B0

δϕ ·T dA

∫

B0

δp ·

[∂ϕ

∂t−δH

δp

]

dV = 0

Weak heat conduction equations

−

∫

B0

δΘ

ΘDivQ dV =

︸︷︷︸

integration by parts

Gauss theorem

∫

B0

∇

(δΘ

Θ

)

·Q dV −

∫

∂B0

δΘ

ΘQ ·N dA

∫

B0

[

δΘ∂s

∂t+δΘ

Θ

δH

δC i

:∂C i

∂t

]

︸︷︷︸

−Dvis

dV =

∫

B0

[δΘ

ΘR +∇

(δΘ

Θ

)

·Q

]

dV

−

∫

∂B0

δΘ

ΘQ ·N dA

∫

B0

δs

[

Θ−δH

δs

]

dV = 0

Goto outline





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

9


Weak forms fulfill all balance laws (I)


Choose δϕ(X , t) = c = const.

c ·

[∫

B0

[∂p

∂t−B

]

dV −

∫

∂B0

T dA

]

︸︷︷︸


= 0

Balance of entropy

Choose δΘ(X , t) = Θ∞ = const.

Θ∞

[∫

B0

[∂s

∂t−

Dvis + R

Θ−

Dcdu/Θ︷︸︸︷

∇

(1

Θ

)

·Q

]

dV +

∫

∂B0

1

ΘQ ·N dA

]

︸︷︷︸

Entropy inequality principle with Dtot ≥ 0

= 0

Balance of mechanical energy

Choose δϕ(X , t) = ∂ϕ(X,t)∂t

and δp(X , t) = ∂p(X,t)∂t

∫

B0

∂k/∂t︷︸︸︷[δH

δp·∂p

∂t+

pint

︷︸︸︷

δH

δF:∂F

∂t

]

dV =

Pmec

︷︸︸︷∫

B0

∂ϕ

∂t·B dV +

∫

∂B0

∂ϕ

∂t·T dA

︸︷︷︸






Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

10


Weak forms fulfill all balance laws (II)

Balance of thermal energy

Choose δΘ(X , t) = Θ(X , t) and δs(X , t) = ∂s(X,t)∂t

∫

B0

pent

︷︸︸︷[δH

δs

∂s

∂t+

−Dvis

︷︸︸︷

δH

δC i

:∂C i

∂t

]

dV =

Pthe

︷︸︸︷∫

B0

R dV −

∫

∂B0

Q ·N dA

︸︷︷︸



Add balance of mechanical and thermal energy

∂H

∂t=

∫

B0

[∂ϕ

∂t·B + R

]

dV +

∫

∂B0

[∂ϕ

∂t·T −Q ·N

]

dA

︸︷︷︸


Balance of Lyapunov function

1. Choose δΘ(X , t) = Θ(X , t)−Θ∞, δs(X , t) = ∂s(X,t)∂t

and add the

balance of mechanical energy (without entropy and total energy consistency possible, cp. ehG method).

or 2.∂V

∂t=

∂H

∂t︸︷︷︸


−Θ∞

∫

B0

∂s

∂tdV

︸︷︷︸

Balance of entropy

Goto outline





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

11


Boundary conditions in the weak formsBabuska [1973], Kikuchi & Oden [1988], Farhat & Geradin [1992], Glowinski et al. [1994], Stenberg [1995], Wriggers [2002], Donea & Huerta [2003], Hartmann et al. [2008]

Boundary conditions

T̄(t)Q̄(t)

B0B0

∂TB0

∂ϕB0

∂QB0

∂ΘB0

Mechanical boundaries Thermal boundaries

Θ∞ Θ∞

Mechanical boundary∫

∂B0

δϕ ·T dA =

∫

∂ϕB0

δϕ︸︷︷︸

=:0

·T dA +

∫

∂TB0

δϕ · T̄(t) dA

Thermal boundary (with Lagrange multiplier technique)∫

∂B0

δΘ

ΘQ ·N dA =

∫

∂ΘB0

δΘλΘ dA−

∫

∂QB0

δΘ

ΘQ̄(t) dA

with ∫

∂ΘB0

δλΘ [Θ−Θ∞] dA = 0 λΘ : Lagrange multiplier (outward normal entropy flux)

Goto outline





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

12


Time discrete weak forms (2nd order accurate)Simo & Tarnow [1992], Simo & Hughes [1998], Betsch & Steinmann [2002], Ibrahimbegovic & Mamouri [2002], Armero [2006], Romero [2010], Hesch & Betsch [2011]

Equations of motion∫

B0

δpn+1 ·

[∆ϕ

∆t−

∆PH

∆p

]

dV = 0

∫

B0

[

δϕn+1 ·∆p

∆t+∇(δϕn+1) :

∆PH

∆F

]

dV =

∫

B0

δϕn+1 ·B 12

dV +

∫

∂TB0

δϕn+1 · T̄ 12

dA

Heat conduction equations∫

B0

δsn+1 ·

[

Θn+1 −∆PH

∆s

]

dV = 0

∫

B0

[

δΘn+1∆s

∆t+δΘn+1

Θn+1

∆PH

∆C i

:∆C i

∆t

]

dV =

∫

B0

[

∇

(δΘn+1

Θn+1

)

·Q 12

+δΘn+1

Θn+1R 1

2

]

dV

+

∫

∂ΘB0

δΘn+1 λn+1 dA +

∫

∂QB0

δΘn+1

Θn+1Q̄ 1

2dA

Local and boundary time evolutions

1. Viscous evolution∆C i

∆t+ V̄

−1(C in+ 1

2

) :∆PH

∆C i

= 0

2. Boundary condition

∫

∂ΘB0

δλn+1 [Θn+1 −Θ∞] dA = 0





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

13


Time discrete operators (2nd order accurate)Milne-Thomson [1933], Greenspan [1973], LaBudde & Greenspan [1974], Simo & Gonzalez [1993], Gonzalez [1996a, 1996b], Sansour et al. [2004], Bourdin & Cresson [2012]

Single-variable function discrete derivative of f (z)

f ∈ {k, s,p,ϕ,C i} z ∈ {t, s,p,C ,C i} ⊙ ∈ { , ·, :}

∆f

∆z=∂f (zn+ 1

2)

∂z+

f (zn+1)− f (zn)−∂f (zn+ 1

2)

∂z⊙ [zn+1 − zn ]

[zn+1 − zn ]⊙ [zn+1 − zn][zn+1 − zn]

Special case 1: Scalar-valued z:∆f

∆t=

f (zn+1)− f (zn)

tn+1 − tn

Special case 2: Quadratic f :∆k

∆p=∂k(pn+ 1

2)

∂p≡

1

ρpn+ 1

2

Multi-variable functional (partitioned) discrete derivatives

∆PH

∆p=

∆k

∆pwith k(p) and e(C , s,C i)

∆PH

∆F= Fn+ 1

2

[

∆e

∆C

∣∣∣∣sn ,C in

+∆e

∆C

∣∣∣∣sn+1,C in+1

]

∆PH

∆s=

1

2

[

∆e

∆s

∣∣∣∣Cn ,C in+1

+∆e

∆s

∣∣∣∣Cn+1,C in

]

∆PH

∆C i

=1

2

[

∆e

∆C i

∣∣∣∣Cn ,sn

+∆e

∆C i

∣∣∣∣Cn+1,sn+1

]

Goto outline





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

14


Time discrete balance laws (I)


Choose δϕn+1(X) = c = const.

c ·

[∫

B0

[∆p

∆t−B 1

2

]

dV −

∫

∂TB0

T̄ 12

dA

]

︸︷︷︸

Time discrete balance of linear momentum

= 0

Balance of entropy

Choose δΘn+1(X) = Θ∞ = const.

Θ∞

[∫

B0

[

∆s

∆t−

Dtot12

+ R 12

Θn+1dV +

∫

∂ΘB0

λn+1 dA−

∫

∂QB0

Q̄ 12

Θn+1dA

]

︸︷︷︸

Time discrete entropy inequality principle with Dtot12

≥ 0

= 0


Choose δϕn+1(X) = ∆ϕ(X)∆t

and δpn+1(X) = ∆p(X)∆t

∆K

∆t+

1

2

[

∆E

∆t

∣∣∣∣sn ,C in

+∆E

∆t

∣∣∣∣sn+1,C in+1

]

=

∫

B0

∆ϕ

∆t·B 1

2dV +

∫

∂TB0

∆ϕ

∆t· T̄ 1

2dA

︸︷︷︸

Time discrete balance of mechanical energy





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

15


Time discrete balance laws (II)


Choose δΘn+1(X) = Θn+1(X), δsn+1(X) = ∆s(X)∆t

and δλn+1(X) = λn+1(X)

1

2

[

∆E

∆t

∣∣∣∣Cn,C in+1

+∆E

∆t

∣∣∣∣Cn+1,C in

+∆E

∆t

∣∣∣∣Cn,sn

+∆E

∆t

∣∣∣∣Cn+1,sn+1

]

=

∫

B0

R 12

dV − Θ∞

∫

∂ΘB0

λn+1 dA +

∫

∂QB0

Q̄ 12

dA

︸︷︷︸

Time discrete balance of thermal energy


Add time discrete balance of mechanical and thermal energy

∆H

∆t=

∫

B0

[∆ϕ

∆t·B 1

2+ R 1

2

]

dV +

∫

∂TB0

∆ϕ

∆t· T̄ 1

2dA−Θ∞

∫

∂ΘB0

λn+1 dA +

∫

∂QB0

Q̄ 12

dA

︸︷︷︸

Time discrete balance of total energy

Balance of Lyapunov function (Lagrange multiplier will be eliminated)

1. Choose δΘn+1(X) = Θn+1(X)−Θ∞, δsn+1(X) = ∆s(X)∆t

, δλn+1(X) = λn+1(X)

and add the time discrete balance of mechanical energy (cp. ehG method)

or 2.∆V

∆t=

∆H

∆t︸︷︷︸

Time discrete balance of total energy

− Θ∞

∫

B0

∆s

∆tdV

︸︷︷︸

Time discrete balance of entropy

Goto outline





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

16


Free rotating insulated thermo-viscoelastic disc

Geometry

no boundary conditions

Θ∞ = 300 K

ri = 0.8 m

ra = 2 m

h = 0.4 m

ωz = 0.33 s−1

x

yz

Reference configuration

t = 0 s

310 K

320 K

330 K

340 K

350 K

360 K

370 K

380 K

ETC integrator with ∆t = 0.02 s

t = 5 s 380 K

370 K

360 K

350 K

340 K

330 K

320 K

310 K

t = 10 s 380 K

370 K

360 K

350 K

340 K

330 K

320 K

310 K

t = 15 s 380 K

370 K

360 K

350 K

340 K

330 K

320 K

310 K

t = 20 s 380 K

370 K

360 K

350 K

340 K

330 K

320 K

310 K

t = 25 s 380 K

370 K

360 K

350 K

340 K

330 K

320 K

310 K

t = 30 s 380 K

370 K

360 K

350 K

340 K

330 K

320 K

310 K





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

17


Free rotating insulated thermo-viscoelastic disc

Midpoint rule (∆t = 0.1 s)

t = 25 s

310 K

320 K

330 K

340 K

350 K

360 K

370 K

380 K

ETC integrator (∆t = 0.1 s)

t = 25 s

310 K

320 K

330 K

340 K

350 K

360 K

370 K

380 K

Midpoint rule ETC integrator

t [s]

H[J

]

×104

4

3

20 10 20 30

0.020.08

0.10

t [s]

H[J

]

×104

4

3

20 10 20 30

0.020.08

0.10

t [s]

S[J

/K

]

100

105

110

0 10 20 30

0.020.08

0.10

t [s]

S[J

/K

]

100

105

110

0 10 20 30

0.020.08

0.10

t [s]

V[J

]4000

3000

2000

10000 10 20 30

0.020.08

0.10

t [s]

V[J

]

4000

3000

2000

10000 10 20 30

0.020.08

0.10





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

18


Rail-bound partly uninsulated thermo-elastic disc

Geometry

insulated boundaries

Θ∞ = 298.15 K

ri = 0.5 m

ra = 1.5 m

h = 1 m

ωz = −7 s−1

x

y

z v = 20 ms−1

∂ϕB0

∂ΘB0

Reference configuration

t = 0 s

295 K

330 K

305 K

310 K

MPR (∆t = 0.01 s) ETC (∆t = 0.01 s)

t = 1.2 s

295 K

300 K

305 K

310 K

0.5

1.0

0

1.5

−0.5

−1.0

−1.5

23 24 25x [m]

y[m

]

t = 1.2 s

295 K

300 K

305 K

310 K

0.5

1.0

0

1.5

−0.5

−1.0

−1.5

23 24 25x [m]

y[m

]

t = 2.2 s

295 K

300 K

305 K

310 K

0.5

1.0

0

1.5

−0.5

−1.0

−1.5

43 44 45x [m]

y[m

]

t = 2.2 s

295 K

300 K

305 K

310 K

0.5

1.0

0

1.5

−0.5

−1.0

−1.5

43 44 45x [m]

y[m

]

t = 4.4 s

295 K

300 K

305 K

310 K

0.5

1.0

0

1.5

−0.5

−1.0

−1.5

87 88 99x [m]

y[m

]t = 4.4 s

295 K

300 K

305 K

310 K

0.5

1.0

0

1.5

−0.5

−1.0

−1.5

87 88 99x [m]

y[m

]





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

19


Rail-bound partly uninsulated thermo-elastic disc

Midpoint rule (∆t = 0.014 s)

t = 1.54 s

0.5

1.0

0

1.5

−0.5

−1.0

−1.5 295 K

300 K

305 K

310 K

29.5 30 30.5 31 31.5 32

x [m]

y[m

]

ETC integrator (∆t = 0.014 s)

t = 1.54 s

0.5

1.0

0

1.5

−0.5

−1.0

−1.5 295 K

300 K

305 K

310 K

29.5 30 30.5 31 31.5 32

x [m]

y[m

]

Midpoint rule ETC integrator

t [s]

H[J

]

4

4

3

2

10 5 10 15 20

0.010.0120.014

×10

t [s]

H[J

]

4

4

3

2

10 5 10 15 20

0.010.0120.014

×10

t [s]

S[J

/K

]

00 5 10 15

20

20

40

60

80

0.010.0120.014

t [s]

S[J

/K

]

00 5 10 15

20

20

40

60

80

0.010.0120.014

t [s]

V[J

]4

3

2

10 5 10 15 20

0.010.0120.014

×10

1.44

1.46

1.48

t [s]

V[J

]

4

0 5 10 15 20

0.010.0120.014

×10





Peter Betsch2

Introduction

Outline & overview

Motion of continua

Constitutive laws

Theoretical studies



Boundary conditions




Numerical studies

Fully closed system

Fully open system

Summary &outlook

20


Summary & outlook1Maugin & Kalpakides [2002], Bargmann [2008] 2Zielonka [2006], Zielonka, Ortiz & Marsden [2008], Mata & Lew [2011]3Ibrahimbegovic, Chorfi & Gharzeddine [2001], Hartmann, Quint & Arnold [2008], Birken et al. [2010], Moore [2011], Gleim & Kuhl [2013]

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.

2 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 ∂tα in the weak formulation1

; mixed variational integrator2; physical consistency with time adaptivity3

an energy-entropy-consistent time stepping scheme …...an energy-entropy-consistent time stepping...

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