An energy consistent hybrid space-time Galerkin method for nonlinearthermomechanical problems

Michael Groß and Peter Betsch ∗

Chair of Computational Mechanics, Department of Mechanical Engineering, University of Siegen, Paul-Bonatz-Str. 9-11,D-57068 Siegen, Germany

In this paper, we present an energy consistent hybrid space-time discretisation for thermomechanical coupled continuumdynamics, which is directly derived from the total energy itself. In comparison to a standard time approximation, we obtainthe same advantages as for viscoelasticity [1], which are numerical stability and time step size independent balance laws aswell as qualitative solutions.

1 Constitutive Formulation

We start from the specific internal energy e(ηt, Ct) = [Θt − Θ∞] ηt + Ψ(Θt, Ct), which is the Legendre transform of thefree energy Ψ(Θt, Ct) with respect to the temperature Θt. In the internal energy, it is already taken into account that theequilibrium state is at environment temperature Θ∞. The internal energy therefore depends on the entropy ηt = −∂Ψ/∂Θt

and the right Cauchy-Green tensor Ct = F TtF t corresponding to the deformation gradient F t = ∇Xϕ

t. The first Piola-

Kirchhoff stress tensor P t = F tSt in the equation of motion then results from the second Piola-Kirchhoff stress tensorSt = 2∇CΨt. We restrict ourselves to isotropic finite classical thermoelasticity. The equilibrium part of the free energythen depends on the invariants of the right Cauchy-Green tensor and the non-equilibrium part depends additionally on thetemperature: Ψt = Ψ ela(ICt , IICt , IIICt) + Ψ the(Θt, ICt , IICt , IIICt). The material entropy flux Ht = −Kt∇XΘt/Θt

depends linearly on the temperature gradient∇XΘt and an isotropic material conductivity tensor K(Ct) = k0

√IIICt C−1

t .

2 Spatially Weak Formulation

We use a dissipative Poissonian formulation with the entropy ηt ∈ VT as a further state variable among the deformation ϕt ∈ Qand the momentum field πt = ρ0V t ∈ VQ. For all functionalsF ,G : X → R on this extended phase space X = Q × VQ × VT ,there exists a two-part bracket structure {{F ,G}} = {F ,G}− ≺ F ,G �. The first part is given by a skew-symmetricbracket {F ,G} =

∫B0

δF/δϕt · δG/δπt − δF/δπt · δG/δϕt which represents the Poissonian (conservative) part. Second,the symmetric semimetric bracket ≺ F ,G �=

∫B0

δ[δF/δηt]/δϕt · κ(∇ϕt)/Θt · δ[δG/δηt]/δϕt represents the dissipative

part, where κt = k0

√IIICt I denotes the isotropic conductivity tensor. This bracket structure defines spatially weak evolu-

tion equations F = {{F ,H}} with respect to the Hamiltonian H(ϕt, πt, ηt) =

∫B0

πt · V t − T (V t) + e(ηt,∇ϕt), the sum

of Legendre-transformed kinetic energy T and internal energy E . Considering the coordinate functionals Fϕ =∫

B0δπ · ϕ

t,

Fπ =∫

B0δϕ · πt and Fη =

∫B0

δΘ ηt corresponding to the state variables, we obtain the two spatially weak equations ofmotion

∫B0

δπ · ϕt=

∫B0

δπ · V t and∫

B0δϕ · πt = −

∫B0∇[δϕ] : P (Θt,∇ϕ

t) as well as the spatially weak entropy balance

∫B0

δΘ ηt =∫

B0∇[δΘ ] · H(Θt,∇ϕ

t). These weak evolution equations fulfil the balance laws on the interesting time inter-

val It = [0, T ]. First, we obtain total linear momentum conservation P(T )− P(0) =∫It{{P ,H}} = 0 having a transla-

tional symmetry, where P =∫

B0ξ · πt for all ξ ∈ VQ. Second, a rotational symmetry of the system leads to a conservation

L(T ) = L(0)of the total angular momentumL =∫

B0ξ · [ϕ

t× πt], and third, the balanceH(T ) −H(0) = −

∫ItDcon between

Hamiltonian and dissipation Dcon =∫B0∇Θt · Kt · ∇[lnΘt] by conduction of heat is fulfilled.

3 Space-time Weak Formulation

Weak equations on the space-time element Cα = Iα × B0, where Iα ∈ [0, 1] denotes the temporal master element, can bederived from the energy balances per time step. The kinetic energy balance Tα=1 − Tα=0 =

∫Cα

V α · πα and internal en-ergy balance Eα=1 − Eα=0 =

∫Cα

P α : ∇[ϕα] + [Θα − Θ∞ ]ηα have to be related by the space-time weak equations of motion.Thus, we obtain the condition

∫Cα

V α · πα

.= −∫Cα

P α : ∇[ϕα], where ◦ denotes differentiation with respect to α. The relationbetween the total energy balance Hα=1 −Hα=0 =

∫Cα

[Θα − Θ∞ ]ηα and the dissipation∫IαDcon during a time step with size

∗ Corresponding author: e-mail: betsch@imr.mb.uni-siegen.de, Phone: +49 271 740 2224, Fax: +49 271 740 2436

PAMM · Proc. Appl. Math. Mech. 6, 443–444 (2006) / DOI 10.1002/pamm.200610202

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

hn leads to the condition∫Cα

[Θα − Θ∞ ]ηα

.= −hn

∫Cα∇Θα · Kα/Θα · ∇Θα ≤ 0. In this way, we get the weak equations

of motion∫Cα

δπα · ϕα = hn

∫Cα

δπα · V α as well as∫Cα

δϕα · πα = −hn

∫Cα∇[δϕα] : P α(Θα,∇ϕα) and of entropy evolu-

tion∫Cα

δΘαηα = hn

∫Cα∇[δΘα] · H(Θα,∇ϕα). However, the energy consistency requires first, that the test functions δϕα,

δπα and the derivatives ϕα, πα have to be elements of the same space and second, that the test function δΘα and the func-tion ϑα = Θα − Θ∞ have to be elements of the same space. Therefore, in a finite element (fe) approximation of the entropybalance, that results in a jump because thermal initial conditions can be fulfilled only weakly. Considering again the energy bal-ances per time step, this jump leads to the internal energy balance Eα=1 − Eα=0− = [[E ]]α=0 +

∫Cα

P α : ∇[ϕα] + [Θα − Θ∞ ]ηα

and therefore in the total energy balance Hα=1 −Hα=0− =∫B0

[[e]]α=0 +∫Cα

[Θα − Θ∞ ]ηα

.= hn

∫Cα∇Θα · Hα ≤ 0. Hence,

the energy consistency requires∫B0

δΘα=0/[Θα=0 − Θ∞][[e]]α=0 +∫Cα

δΘαηα = hn

∫Cα∇[δΘα] · Hα in a fe approximation

of the entropy balance. Applying numerical quadrature, we need an energy consistent stress approximation. We add tothe approximation of the physically based stress tensor Sα an algorithmic stress tensor Salg

αwhich fulfil the internal en-

ergy balance G(Salgα ) = eα=1−eα=0+−

∫Iα

[Sα+Salgα ] :Cα/2+Θαηα as constraint. A closed form expression can be derived

by a constrained minimisation of the Lagrange functional L(Salg, λ) = λG(Salgα ) +

∫Iα

Salgα :Salg

α /2 where λ denotes the La-

grange multiplier. It results in a weighted time derivative Salgα = [2G(0)/

∫Iα

Cα :Cα]Cα of the right Cauchy-Green tensor.We approximate the fields with Lagrange polynomials MJ in time and NA in space. Therefore, we obtain the trial func-tions ϕh

α =∑k+1

J=1

∑nnode

A=1 MJ NA xAJ and V h

α =∑

J,A MJ NA V A

J as well as Θhα =

∑J,A MJ NA ΘA

J . The test functions

δϕhα =

∑kI=1

∑A MI NA δxA

I and δπhα =

∑I,A MI NA δπA

I corresponding to the equations of motion have to be formu-

lated with reduced Lagrange polynomials MI in time, however, the test function of the weak entropy evolution has to beapproximated with the same Lagrange polynomials as in the trial function: δΘh

α=

∑J,A MJ NA δϑA

J. The approximated de-

formation directly generates the approximation F h

α =∑

J MJ F J of the deformation gradient and the natural approximationsCh

α = [F h

α]T F h

α and Sh

α = 2 ∂Ψ(Θhα , Ch

α)/∂C of the strains and the stresses, respectively. We obtain the approximationηh

α =∑

J MJ η(ΘhJ ,∇ϕh

J) of the entropy additionally with the approximated temperature field. In the energy consistentmethods, however, we apply the incrementally objective assumed strain approximations C

h

α =∑

J MJ [F h

J ]T [F h

J ] in timeand the corresponding stresses S

h

α = 2 ∂Ψ(Θhα , C

h

α)/∂C depending on these strains.

4 Numerical Example

We consider a free motion of a heated propeller initiated by an initial translation velocity v0 = 2ex and an initial angularvelocity ω0 = −0.7ez . The temperature at the blade tips is held fixed at Θ∞ = 298.15. The free energy is a thermallyextended Neo-Hookean with the material parameters listed in the figure below. Here, we apply linear finite elements in time(k = 1) and 4-node finite elements in space. We compare the new hybrid methods (ehG, hG) with energy consistent jumpterm and continuous methods (eG, cG) which avoid jump terms. The ‘e’ refers to the use of the algorithmic stress tensor. TheehG-method and the eG-Method show a similar steady decreasing total energy H, however the eG-Method needs a smallertime step size while the dissipation is high. The reason is that the ehG-method is energy consistent, however the eG-Method isnot (compare the energy balance conditions). Without the algorithmic stress tensor, both methods show only a time-averageddecrease of total energy depending on the time step size. Both momentum maps are conserved by all these methods.

−15 −5 5 15 25 35 45 55 65−15

−10

−5

0

5

10

15

290

295

300

305

310

315

320

x

y

0 5 10 15 202.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5x 10

4

Time

Tota

lene

rgy

hn = 0.4

ehG(1) method

0 5 10 15 202.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5x 10

4

Time

Tota

lene

rgy

hn

=0.

1 hn = 0.4

eG(1) method

λ = 30000µ = 7500

ρ = 8.93

β = 10−4

c0 = 100

k0 = 10000 5 10 15

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5

3.6x 10

4

Time

Tota

lene

rgy

divergence

hn = 0.2

hn

=0.

1

hG(1)/cG(1) method

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2x 10

−8

Time

Tota

lene

rgy

bala

nce

cond

ition hn = 0.4

ehG(1) method

Newton tolerance

0 5 10 15 200

1

2

3

4

5

6

7

8

9

10

Time

Tota

lene

rgy

bala

nce

cond

ition

hn = 0.4

hn

=0.

1

eG(1) method

References

[1] Groß M. and Betsch P. Galerkin Methods in Time for Semi-Discrete Viscoelastodynamics Proc. Appl. Math. Mech. 5 (2005), 397-398.

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Section 7 444