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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2000; 00:1–6 Prepared using nmeauth.cls [Version: 2002/09/18 v2.02] Variational time integrators for finite dimensional thermo-elasto-dynamics without heat conduction Pablo Mata and Adri´ an J. Lew Department of Mechanical Engineering, Stanford University, Stanford,CA 94305-4040, USA. SUMMARY This paper focuses on the formulation of variational integrators (VI) for finite dimensional thermo- elastic systems without heat conduction. The dynamics of these systems happen to have a Hamiltonian structure, after thermal displacements are introduced. It is then possible to formulate integrators by taking advantage of standard methods in VI. The class of integrators we construct have some remarkable features: (a) they are symplectic, (b) they exactly conserve the entropy of the system, or in other words, they exactly satisfy the second-law of the thermodynamics for reversible adiabatic processes, (c) they nearly exactly conserve the value of the energy for very long times, and (d) they exactly conserve linear and angular momentum. We first describe how to adapt any VI for classical mechanical systems to integrate adiabatic thermo-elastic ones, and then formulate three new types of integrators. The first class, based on a generalized trapezoidal rule, gives rise to two first-order, explicit integrators, and a second-order, implicit one. By composing then the two first-order integrators we construct a second-order, explicit one. Finally, we formulate a fourth-order, implicit integrator, which is a symplectic partitioned Runge-Kutta method. The performance of these new algorithms is showcased through numerical examples. Copyright c 2000 John Wiley & Sons, Ltd. key words: Time stepping scheme, variational integrators, symplectic momentum-conserving methods, thermo-elasticity. 1. Introduction Early approaches toward the creation of time integrators for the dynamics of deformable bodies consisted in constructing a discretization of the momentum balance equations, without * Please cite as: P. Mata and A.J. Lew, ”Variational time integrators for finite dimensional thermo-elasto- dynamics without heat conduction,” International Journal for Numerical Methods in Engineering, Vol. 88, No. 1, pp. 1-30, 2011, 10.1002/nme.3160 * Correspondence to: Durand 207, Department of Mechanical Engineering, Stanford University, Stanford,CA 94305-4040, USA. E-mail:[email protected]. Contract/grant sponsor: Department of the Army Research Grant; contract/grant number: W911NF-07-2-0027 Contract/grant sponsor: National Science Foundation Career Award; contract/grant number: CMMI-0747089 Received 3 July 1999 Copyright c 2000 John Wiley & Sons, Ltd. Revised 18 September 2002
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Page 1: Variational time integrators for finite dimensional …...Pioneering contributions in the development of the so-called conserving schemes are for example [3, 4], among others. Conspicuous

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2000; 00:1–6 Prepared using nmeauth.cls [Version: 2002/09/18 v2.02]

Variational time integrators for finite dimensionalthermo-elasto-dynamics without heat conduction∗

Pablo Mata and Adrian J. Lew∗

Department of Mechanical Engineering, Stanford University,Stanford,CA 94305-4040, USA.

SUMMARY

This paper focuses on the formulation of variational integrators (VI) for finite dimensional thermo-elastic systems without heat conduction. The dynamics of these systems happen to have a Hamiltonianstructure, after thermal displacements are introduced. It is then possible to formulate integratorsby taking advantage of standard methods in VI. The class of integrators we construct have someremarkable features: (a) they are symplectic, (b) they exactly conserve the entropy of the system, orin other words, they exactly satisfy the second-law of the thermodynamics for reversible adiabaticprocesses, (c) they nearly exactly conserve the value of the energy for very long times, and (d) theyexactly conserve linear and angular momentum. We first describe how to adapt any VI for classicalmechanical systems to integrate adiabatic thermo-elastic ones, and then formulate three new types ofintegrators. The first class, based on a generalized trapezoidal rule, gives rise to two first-order, explicitintegrators, and a second-order, implicit one. By composing then the two first-order integrators weconstruct a second-order, explicit one. Finally, we formulate a fourth-order, implicit integrator, which isa symplectic partitioned Runge-Kutta method. The performance of these new algorithms is showcasedthrough numerical examples. Copyright c© 2000 John Wiley & Sons, Ltd.

key words: Time stepping scheme, variational integrators, symplectic momentum-conserving

methods, thermo-elasticity.

1. Introduction

Early approaches toward the creation of time integrators for the dynamics of deformablebodies consisted in constructing a discretization of the momentum balance equations, without

∗Please cite as: P. Mata and A.J. Lew, ”Variational time integrators for finite dimensional thermo-elasto-dynamics without heat conduction,” International Journal for Numerical Methods in Engineering, Vol. 88, No.1, pp. 1-30, 2011, 10.1002/nme.3160∗Correspondence to: Durand 207, Department of Mechanical Engineering, Stanford University, Stanford,CA94305-4040, USA. E-mail:[email protected].

Contract/grant sponsor: Department of the Army Research Grant; contract/grant number: W911NF-07-2-0027

Contract/grant sponsor: National Science Foundation Career Award; contract/grant number: CMMI-0747089

Received 3 July 1999Copyright c© 2000 John Wiley & Sons, Ltd. Revised 18 September 2002

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2 PABLO MATA AND ADRIAN J. LEW

accounting for additional structure these equations might have had; see for example Ch. 9 of[1] or Ch. 9 and 10 of [2].Pioneering contributions in the development of the so-called conserving schemes are for

example [3, 4], among others. Conspicuous among these are the energy-momentum algorithmsby Simo et al. [5, 6], which conserve energy as well as linear and angular momentum. Thesehave been further extended to consider mechanical problems described by partial differentialequations, such as encountered in the nonlinear dynamics of solids [7], rods [8, 9], and shells[10]. The list of contributions in this area is long, with for example [11, 12, 13, 14, 15] amongothers.Variational integrators constitute a more recent approach toward the creation of structure-

preserving integrators. The construction of VI is rooted in the formulation of a discrete analogto Hamilton’s variational principle. These ideas were first developed in the context of integrablesystems in mechanics by Veselov [16] and Moser and Veselov [17], who constructed a suitableapproximation of the action integral named the discrete action sum. Stationary points of thisfunctional are then the discrete-in-time trajectories of the mechanical system, and can beproved to approximate the exact trajectories as the time-step goes to zero. Furthermore, theresulting integrator is symplectic, and if the discrete action sum preserves the symmetries of theoriginal mechanical system, then discrete versions of the conjugate momenta will be conservedby the discrete trajectories. This is also known as a discrete version of Noether’s theorem, seee.g. [18, 19, 20]. Finally, an appealing aspect of VI is that they display outstanding energybehavior without explicitly enforcing it. More precisely, the energy of the discrete trajectoriesremains close to its initial value for very long times, provided the time step is small enough,see e.g. [21, 22].A thorough account on symplectic methods can be found in [21, 22], and about the

essential aspects of VI in [18]. Applications and extensions of this basic theory to differentfields are numerous, such as to mechanical problems with multi-symplectic geometry [23], tothe construction of asynchronous integration methods in solid mechanics and field theories[19, 20, 24, 25, 26], to problems with contact [27, 28, 29, 30], problems with oscillatorysolutions [31], stochastic differential equations [32], constrained and forced problems [18], andto problems where the configuration space is a nonlinear manifold [33, 34, 35], among manyothers.As noted in [36], VI and energy-momentum schemes have been formulated based on

exploiting the underlying Hamiltonian structure of conservative problems. Some energy-momentum algorithms have been formulated for problems that do not posses this structure,such as involving plastic dissipation [37, 38] or, more recently, thermo-elastic materials withheat conduction [36]. The use of a discrete version of Lagrange-D’Alembert principle enablesthe extension of VI to include non-conservative generalized forces. This is advantageous inmildly dissipative systems, since by adopting a VI integrator for the conservative part verylittle numerical dissipation is included, and hence accurate solutions at relatively large timesteps are recovered, see [39]. However, when non-conservative forces play a dominant role inthe dynamics of the system, the use of Lagrange-d’Alembert principle does not seem to offerany particular advantages.The formulation of time integrators for thermo-mechanical systems presents a unique

set of requirements over purely mechanical systems. Of course, conserved quantitiesassociated to symmetries of the system should be respected, but what makes integratorsfor thermo-mechanical systems special is whether the computed trajectories satisfy the first

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 3

(energy conservation) and second (non-decreasing entropy of an isolated system) laws ofthermodynamics. More precisely, for an isolated system the energy of the system should remainconstant, and the entropy of the system should not decrease. If additionally the system isadiabatic (no heat conduction) then the entropy should remain constant. Integrators whosetrajectories satisfy this property are sometimes said to be thermodynamically consistent.Thermo-elastic systems are likely the simplest thermo-mechanical systems. They are

also relevant systems for some applications. This is why these are the systems of choicetowards testing ideas for the formulation of thermodynamically consistent integrators, seee.g. [36, 40, 41, 42, 43]. Of course, a crucial difficulty is that thermo-elastic systemswith an arbitrary constitutive behavior for heat conduction do not posses a Hamiltonianstructure. In particular, and crucially, when Fourier’s law of heat conduction is adopted theresulting evolution equations do not derive from an autonomous Lagrangian (or equivalently,an autonomous Hamiltonian). Alternative variational principles other than a Hamiltonianstructure have been proposed instead, e.g., [44, 45, 46, 47, 48, 49, 50].A particularly appealing formulation of thermo-elasticity is obtained by introducing the

concept of thermacy, or thermal displacements, created apparently early in the last centuryand extensively discussed by Maugin [51, 52, 53] and Green and Naghdi [54, 55, 56, 57]. In thisformulation the thermal displacements are analogs to the mechanical displacements. Similarly,the temperatures, as the time derivatives of the thermal displacements, are analogs to thevelocities. Furthermore, the entropy is the conjugate momentum to the temperature, inasmuchthe linear momentum is the conjugate momentum to the velocities. For adiabatic thermo-elasticity, i.e., no heat conduction, the introduction of the thermal displacements unveils aHamiltonian structure for the evolution equations. The Hamiltonian structure can be stillconserved by considering a different type of heat conduction, which depends on the gradientof the thermal displacements rather than the temperature, analog to the dependence of thestress on the gradient of the displacements or strains [52, 54]. Of course, such structure is lostwhen classical Fourier’s heat conduction is considered, as mentioned earlier.A number of different integration algorithms have been proposed for thermo-elastic systems,

see [58, 59, 60, 61, 62, 63, 64, 65, 66] to name a few. Closely related to the discussion in thispaper are the integration algorithms by Romero [36, 67]. These are energy-momentum methodsthat are thermodynamically consistent even with heat conduction. Notably, this contributionsuggests that energy-momentum algorithms can be adapted to a more general class of systemsother than those with a Hamiltonian structure: systems based on the so-called GENERICformalism [68].Time-integration algorithms for thermo-elastic systems with the Hamiltonian structure

considered here have been proposed by Bargmann and Steinmann [40, 41, 42]. In [43] anincremental formulation, in part obtained with the help of an adequate discretization ofHamilton’s principle, is introduced. However, the resulting integration schemes do not preserveany type of constant of motion of the system, and no further advantage is taken of theHamiltonian structure of the problem.As a starting point towards the formulation of variational integrators for bodies made of

thermo-elastic materials, in this paper we formulate VI for a class of finite-dimensional thermo-elastic systems. These are systems formed by point masses connected by nonlinear thermo-elastic springs. This class of systems share many features with the finite-dimensional thermo-mechanical systems obtained after introducing a finite element discretization of a body. Inthis paper we restrict our attention to the adiabatic case in which there is no heat conduction.

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4 PABLO MATA AND ADRIAN J. LEW

The introduction of the special heat conduction formulated by Green and Naghdi [54] is rathersimple, and we shall discuss it elsewhere. The methodology to include Fourier’s heat conductionis, however, less clear.The formulation of VI is made possible by taking advantage of the Hamiltonian structure

unveiled by the introduction of the thermal displacements. We review the basic aspects of thistheory in §2, and show examples of the type of thermo-elastic systems under consideration in§3. We see, for example, that in the absence of heat conduction the constancy of the entropyof each spring is a consequence of the symmetry of the system upon rigid translations of eachone of the thermal displacements. We therefore identify that, for adiabatic evolutions, theLagrangian of classical mechanics is the Lagrangian resulting from a Routh reduction of thissymmetry §2.3.2.We construct the integrators by utilizing the tools and results available from the theory

of variational integrators, which we review in §4.1. The remarkable aspects of the resultingintegrators are:

1. The algorithms are symplectic in an extended phase space that now includes the thermaldisplacements and the entropies.

2. The discrete Euler-Lagrange equations are discrete analogs to Newton’s second-lawand, most importantly in this context, the second-law of thermodynamics for reversibleadiabatic evolutions of the system. Therefore, the constancy of the entropy of each springis strongly imposed as one of the equations that define the algorithm. This has to becontrasted to other possible ways to define the discrete thermodynamic evolution of thesystem, such as by imposing the conservation of energy or first law of thermodynamics.

3. The energy of the system is nearly exactly conserved for very long times, as it is thecase with variational integrators for classical mechanics. Consequently, the first law ofthermodynamics is satisfied without explicitly imposing it, only as a consequence of thesymplecticity of the integrators.

4. The discrete trajectories exactly conserve the momenta conjugate to the symmetries ofthe system, such as linear and angular momentum, and in this case, the entropy of eachspring as well.

We construct three classes of time integrators. However, before that, we describe how toadapt variational integrators for classical mechanical systems to thermo-elastic ones withoutheat conduction. This procedure is based on the reduced Lagrangian obtained from thetranslational symmetry of the thermal displacements, see §4.2. In this case, if the initialvalue of the entropy of each spring is adopted as a parameter, then the a Lagrangian fora classical mechanical system is recovered. Therefore, any standard variational integrator forclassical mechanics can be utilized to integrate its trajectories. As an example, we reproducethe central differences or Newmark’s second-order explicit discrete Lagrangian, but others suchas Asynchronous Variational Integrators [20] function equally well. The distinguishing aspectsof these integrators are that they heavily rely on the fact that the entropy is constant fortheir formulation, and that no approximation of the temperature or thermal displacements isneeded. The temperature is computed a posteriori given the discrete mechanical displacementsand the parametric values for the entropies of the springs.The new integrators we construct are instead based on the thermo-elastic Lagrangian. In

§4.3.1 we formulate integrators through a generalized trapezoidal rule, the most elementarytype of variational time integrator. Among the integrators in this class there is one that

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 5

is second-order but implicit, and two that are first-order and explicit, identified as the twovariants of symplectic Euler applied to this system [69]. In a purely mechanical problem thesecond-order integrator would be explicit as well. However, a crucial difference in the thermo-elastic setting is that in general the Lagrangian does not depend quadratically on the thermalvelocities (or temperatures), as it does in classical mechanics. Consequently, the formulationof a second-order explicit integrator is not so straightforward. To address this issue, in §4.4 weturn instead to the standard procedure in time integration of elevating the order of an algorithmthrough composition, see e.g. [22]. By composing the two first-order, explicit algorithms weobtain a second-order, explicit algorithm. The composition algorithm happens to be a non-trivial rephrasing in terms of the thermal displacements of Newmark’s explicit second-orderalgorithm adapted to adiabatic thermo-elastic systems, as described above. Consequently, thecomposition provides a way to obtain this last algorithm directly from the thermo-elasticLagrangian, instead of resorting to a priori prescribing that the entropy is constant.

Along the way (§4.3.2), we construct a fourth-order implicit integrator by making use ofpiecewise quadratic polynomials to describe the mechanical and thermal trajectories in time.This is also a symplectic partitioned Runge-Kutta method, see [18].

2. Lagrangian and Hamiltonian formulations

2.1. Description of the systems

We consider the dynamics of systems formed by a spatial arrangement of N masses connectedby M thermo-elastic springs, such as those shown in Fig. 1. Despite their simplicity,these systems retain important features of continuum mechanics systems, such as geometricnonlinearities and coupled thermo-elasticity in finite dimensions.

Figure 1. (a) Triple thermo-elastic pendulum. (b) Spatial net.

Following some of the ideas in [52], it is possible to obtain the dynamics of these systemsfrom Hamilton’s principle. This formulation relies crucially on the introduction of the socalled thermal displacements of the system. These thermal displacements play the role ofthe mechanical displacements but on the thermal evolution of the system. The temperaturesof each one of the springs are recovered as the thermal velocities, or the time derivatives of thethermal displacements. We introduce these concepts next.

The spatial position of the masses at time t is described by means of the generalized

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6 PABLO MATA AND ADRIAN J. LEW

displacementsq(t) = (q1(t), .., qP (t)) ∈ Q, (1)

where Q is the mechanical configuration manifold. The generalized velocities follow as v = q,and (q, q) ∈ TQ. Additionally, each thermo-elastic spring is assigned a thermal displacement

Φi ∈ R such that, for any time evolution Φi(t), the empirical temperatures θi(t) follow as

θi =dΦi

dt, i ∈ 1, ...,M. (2)

We denote all thermal displacements and (empirical) temperatures of the system by

Φ = (Φ1, ...,ΦM ) and θ := (θ1, ..., θM ), (3)

where Φ ∈ RM and (Φ, θ) ∈ TRM . The configuration of the system is fully specified as a point

(q,Φ) ∈ G := Q × RM , the configuration manifold of the system. Similarly, points in state

space are given by (q,Φ, q, θ) ∈ TG := TQ× TRM .The thermo-elastic behavior of each spring is assumed to be described by a Helmholtz free-

energy functionAi(q, θ

i), i ∈ 1, ...,M,

so that the Helmholtz free energy of the system A : TG → R follows as

A(q, θ) =

M∑

i=1

Ai(q, θi). (4)

For simplicity, we shall henceforth assume that for each value of the mechanical displacementsq the function A(q, ·) is convex. Of course, this precludes important phenomena such as somephase transitions, but additional care should be exercised to analyze algorithms in that case.

2.2. Lagrangian formulation

We now obtain the equations of motion of the system from Hamilton’s principle. To this end,the Lagrangian of the system, L : TG → R is constructed as

L(q, q,Φ, θ) = K(q, q)− A(q, θ), (5)

where the kinetic energy K : TG → R is given by

K(q, q) =1

2q ·m(q)q. (6)

Here m(q) is the symmetric and positive definite mass matrix of the system, which ingeneralized coordinates can be configuration-dependent.Once the Lagrangian has been defined, the equations of motion of the system follow as the

Euler-Lagrange equations of Hamilton’s principle. Even though this is standard, we includethe derivation below for completeness.We consider the set C of all (smooth enough) trajectories (q(·),Φ(·)) : [ta, tb] → G of the

system, as well as the action S : C → R

S[q(·),Φ(·)] =

∫ tb

ta

L(q, q,Φ, θ)dt. (7)

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 7

Hamilton’s principle states that the trajectory of the system is a stationary point of theaction under all variations in C that leave the end-points of the trajectory fixed, i.e.,δq(ta) = δq(tb) = 0 and δΦ(ta) = δΦ(tb) = 0. The trajectory (q(·),Φ(·)) should then satisfythe Euler-Lagrange equations

∂L

∂qI−

d

dt

(

∂L

∂qI

)

= 0, I ∈ 1, ..., P, (8a)

∂L

∂Φi−

d

dt

(

∂L

∂Φi

)

= 0, i ∈ 1, ...,M. (8b)

Particularizing the above equations for the Lagrangian (5) gives the following Euler-Lagrangeequations:

d

dt(mq) =

1

2(∇qm) : q⊗ q−∇qA = ∇qL, (9a)

d

dt(∇θA) = 0. (9b)

To define the trajectory, these equations need to be supplemented with appropriate boundaryconditions: (i) q(ta) = qa, (ii) Φ(ta) = Φa, (iii) q(ta) = va and (iv) θ(ta) = θa.

Several remarks are appropriate:

i. Equation (9a) is the statement of Newton’s second law for a thermo-elastic system. Inthis case the Helmholtz free energy function is the potential for the thermo-elastic forcesacting on the system, i.e.,

f = −∇qA(q, θ). (10)

ii. A classical result from thermodynamics (e.g. [70]) identifies the Helmholtz free energyfunction as a potential for the vector of entropies as well, i.e.,

η(q, θ) = −∇θA(q, θ). (11)

Therefore, the Euler-Lagrange equation (9b) implies that the entropy of each spring staysconstant in time. In other words, the invariance of the entropy in each spring is obtainedas a stationarity condition for the action. The total entropy of the system is obtained as

ηt =

M∑

i=1

ηi.

iii. Given η and q, it is possible to solve (11) for the vector of temperatures of the system.

We denote the resulting function with θ(q,η). Therefore, if the vector of entropies of thesystem at time ta is η0, the temperatures of the spring at any time t are computed as

θ(t) = θ(q(t),η0). (12)

The function θ is well-defined due to the assumed convexity of A with respect to θ.iv. Equation (9b) is the statement of the second law of thermodynamics for reversible,

adiabatic evolutions.v. Additionally, if the coordiantes q are such that the resulting mass matrix is configuration

independent, Eqs. (9a) and (9b) reduce to

mq = −∇qA,

η = 0.

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8 PABLO MATA AND ADRIAN J. LEW

2.3. Hamiltonian formulation

Alternatively, it is possible to work directly with the Hamiltonian point of view of the problem.The Hamiltonian H : T∗G → R follows as the Legendre transform of the Lagrangian in thevelocity and temperature variables, namely,

H(q,p,Φ,η) = infq,θ

p · q+ η · θ − L(q, q,Φ, θ)

, (13a)

and carrying out the minimization yields the following definition of the mechanical and thermalmomenta:

p = ∇qL = mq, (14a)

η = ∇θL = −∇θA. (14b)

Notice that the thermal momentum coincides with the constitutive entropy (see (11)) andtherefore, the use of the concept of thermal displacements establishes a complete analogybetween the mechanical and thermal parts of the problem, as summarized in Table I. Thephase space T

∗G has then coordinates (q,Φ,p,η).

Table I. Relation between mechanical and thermal variables.

Mechanical q q p

Thermal Φ θ η

The Hamiltonian can be compactly written after introducing the internal energy U(q,η), i.e.,the Legendre transform of the Helmholtz free energy with respect to the vector of temperatures

A(q, θ(q,η)) + η · θ(q,η) = U(q,η) =M∑

i=1

Ui(q, ηi), (15)

where Ui = Ai + ηiθi, i ∈ 1, ...,M is the internal energy of each spring. The internal energythen constitutes a potential for the internal forces and the temperatures, namely,

f = −∇qU(q,η), θ = ∇ηU(q,η) = θ(q,η). (16)

The Hamiltonian can then be explicitly written as

H(q,p,Φ,η) = K(q,p) + U(q,η), (17)

and Hamilton’s equations of motion become

q = ∇pH(q,p,Φ,η) = m−1p, (18a)

Φ = ∇ηH(q,p,Φ,η) = ∇ηU(q,η), (18b)

p = −∇qH(q,p,Φ,η) = −∇q(K(q,p) + U(q,η)), (18c)

η = −∇ΦH(q,p,Φ,η) = 0. (18d)

These equations are equivalent to the Euler-Lagrange equations (9a)-(9b).Finally, if we denote a point in T

∗G with z ≡ (q,Φ,p,η), then it is possible to rewrite theseequations in canonical form as

z = JJJ∇zH(z), (19)

where JJJ is the canonical symplectic matrix [71].

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 9

2.3.1. Conservation properties. Due to the fact that the problem possesses a Hamiltonianstructure, a number of conserved quantities can be identified.

(i) Energy conservation.The Hamiltonian, or the total energy, is constant throughout any trajectory of the system,satisfying in this way the first law of thermodynamics. This can be seen as either a consequenceof the time-translation symmetry of the system, or by directly computing the time derivativeof H along solution trajectories:

H = ∇zH(z) · z = ∇zH(z) · JJJ∇zH(z) = 0, (20)

since JJJ is skew-symmetric.

(ii) Symplecticity of the flow.Hamiltonian systems generate flows in T

∗G that are symplectic, i.e., for each time t the map(q(0),p(0),Φ(0), θ(0)) 7→ (q(t),p(t),Φ(t),η(t)) is symplectic, so it preserves area elements inT∗G (see [22] pp. 184–185, and [19]). In this case the canonical symplectic form preserved by

the flow is

Ω = dqI ∧ dpI + dΦi ∧ dηi, (21)

where repeated indices in the same term indicate sum over all mechanical or thermalcoordinates.

(iii) Symmetries: conservation of total linear and angular momentum.Noether’s theorem states that each symmetry of the Lagrangian leads to a conjugateconserved momentum. In particular, the invariance of the Lagrangian under the action ofrigid displacements or rigid rotations in space lead to the conservation of the linear or angularmomenta of the system, respectively.

(iv) Conservation of each spring’s entropy.In addition to the possible symmetries upon the action of rigid body motions, the Lagrangian in(5) is also symmetric under translations of each thermal displacement. The conjugate conservedmomentum is the entropy of each spring. We expand on this next.

Clearly the conservation of the entropy of each spring along trajectories is a directconsequence of the Euler-Lagrange equation (9b). To see this as a symmetry of the Lagrangian(5), we note that

L(q, q,Φ, θ) = L(q, q,Φ+ ǫ, θ) (22)

for all ǫ = ǫ1, . . . , ǫM ∈ RM .

Therefore, it is clear that

d

dǫiL(q, q,Φ+ ǫ, θ) =

∂L

∂Φi= 0. (23)

Thus it follows from (8b) that

d

dt

∂L

∂Φi= ηi = 0, i ∈ 1, ...,M.

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10 PABLO MATA AND ADRIAN J. LEW

2.3.2. The simplest entropy-conserving Lagrangian. Taking into account that Eq. (9b)ensures entropy conservation, it is possible to write a Lagrangian L

η : TQ → R expressedin terms of η0 regarded as a vector of constant parameters i.e.

Lη0(q, q) := K(q, q)− U

η0(q), (24)

where

Uη0(q) = U(q;η0) =

M∑

k=1

Uk(q; η0k).

Alternatively, Lη0 can be seen as the Lagrangian obtained after a Routh reduction of thesymmetry on Φ.

Eq. (24) implies that neither Φ nor θ play an essential role in the dynamics, andthe temperature in each thermo-elastic spring is obtained from (162). The Euler-Lagrangeequations of the reduced system are

d

dt(mq)−

1

2(∇qm) : q⊗ q = −∇qU

η0 , (25)

as expected.

3. Examples

We next give two simple examples of the thermo-elastic systems under consideration.

3.1. Thermo-Elastic pendulum

Let (r, ϕ) be the polar coordinates describing the kinematics of the thermo-elastic pendulumshown in Fig. 2. Additionally, the system has associated a thermal displacement Φ. The

m,

ϕ

φ

r

Figure 2. Thermo-elastic pendulum.

corresponding kinetic energy is K = 12m(r2 + (rϕ)2) while the Helmholtz free energy of the

spring is assumed to be

A(r, θ) =E

2r0(r − r0)

2 − β(θ − θ0)(r − r0)−C0

2θ0(θ − θ0)

2 − (θ − θ0)η0, (26)

where θ and θ0 are the current and reference temperature, η0 is a reference entropy, r0 is areference length, E is the elastic modulus, C0 a specific heat and β is a parameter coupling thethermal and mechanical behaviors. Therefore, from (10), (11) and (15) the following relations

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 11

are obtained:

fr = −∂A

∂r= −

E

r0(r − r0) + β(θ − θ0), (27a)

η = −∂A

∂θ=

C0

θ0(θ − θ0) + β(r − r0) + η0, (27b)

θ = θ(r, η) =θ0C0

(

η − η0 − β(r − r0))

+ θ0, (27c)

U(r, η) =E

2r0(r − r0)

2 +θ02C0

(

η − η0 − β(r − r0))2

+ ηθ0. (27d)

Eqs. (27a) and (27b) are the constitutive relations for the modulus of the radial force andentropy in the spring, which couple the mechanical and thermal parts. Eq. (27c) is the inverseof (27b) for each r and determine the temperature. Finally, Eq. (27d) is the internal energy.The configuration manifold is G = R

2 × R, which corresponds to the position of the massand the thermal displacement. The E-L equations (9a) and (9b) become

r − rϕ2 +E

mr0(r − r0)−

β

m(θ − θ0) = 0, (28a)

d

dt(mr2ϕ) = 0, (28b)

C0

θ0θ + βr = 0, (28c)

which have to be complemented with appropriate initial conditions. The second and thirdequations are statements of the conservation of the angular momentum and entropy,respectively.The corresponding conjugate momenta are pr = mr, pϕ = mr2ϕ and η in (27b). The

Hamiltonian function is given by

H =1

2m(p2r +

p2ϕr2

) + U(r, η), (29)

and then the balance equations in Hamiltonian form are obtained as

r =∂H

∂pr=

prm

, ϕ =∂H

∂pϕ=

pϕmr2

, Φ =∂H

∂η= θ, (30a)

pr = −∂H

∂r=

p2ϕmr3

+ fr, pϕ = −∂H

∂ϕ= 0, η = −

∂H

∂Φ= 0. (30b)

Eqs. (30b2) and (30b3) are the conserved quantities associated to the invariance of theLagrangian with respect to rigid body rotations in the plane and rigid body thermaldisplacements.

3.2. Chain of N masses

The next example consists of a chain of N masses connected in series by means of thermo-elastic springs, sketched in Fig. 3.In the following qi ∈ R

3 denotes the position of mass i, and by convention, q0 is a fixedpoint in R

3. Each spring has a thermal displacement Φi and empirical temperature θi. The

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12 PABLO MATA AND ADRIAN J. LEW

m1

m2

m3

qq

q

0

1

2

3

q0

ϕ1

ϕ2 ϕ

3

Figure 3. Chain of N masses connected in series by means of thermo-elastic springs.

position vector, the vector of thermal displacements and the vector of temperatures of thesystem are denoted by

q = (q1, ...,qN ) ∈ R3N , Φ = (Φ1, ...,ΦN ) ∈ R

N and θ = (θ1, ..., θN ) ∈ RN ,

respectively. The configuration manifold is G ≡ R(3+1)N and the Helmholtz free energy of each

spring is assumed to be

F(l, θ) =c

2ln2(

l/l0)

− β(θ − θ0) ln(

l/l0)

+ C0

(

θ − θ0 − θ ln(

θ/θ0))

, (31)

where l/l0 is the quotient between the length of the spring and a reference length l0, c andθ0 are an elastic constant and a reference temperature, β is a thermo-mechanical couplingparameter and C0 is the heat capacity.Then, the following relations are obtained:

fl = −∂F

∂l= −

c

lln(l/l0) +

β

l(θ − θ0), (32a)

η = −∂F

∂θ= β ln(l/l0) + C0 ln(θ/θ0), (32b)

θ = θ0 exp

[

η − β ln(l/l0)

C0

]

, (32c)

U(l, η) =c

2ln2(l/l0) + βθ0 ln(l/l0) + C0θ0

(

exp[

C−10

(

η − β ln(l/l0))]

− 1)

. (32d)

In this way, the Helmholtz free energy, internal energy and entropy of the system are

A(q, θ) =

N∑

i=1

F(li, θi), U(q,η) =

N∑

i=1

U(li, ηi) ηt =

N∑

i=1

ηi,

where lj = ‖qj − qj−1‖, j = 1, ..., N . Here ‖ · ‖ is the Euclidean norm in R3.

The kinetic energy K : TQ → R is given by

K(q) =1

2

N∑

j=1

mj qj · qj , (33)

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 13

and the Lagrangian follows as

L(q, q,Φ, θ) = K(q)− A(q, θ) =

N∑

j=1

[

1

2mj q

j · qj − F(lj , θj)

]

. (34)

The Euler-Lagrange equations for each spring and mass of the system are

mj qj = −∇qjA, (35a)

ηj = 0, (35b)

for j ∈ 1, ..., N, which have to be supplemented with adequate initial conditions on themechanical displacements, velocities and temperatures.

4. Variational integrators

Variational integrators provide a methodology for constructing integrators which automaticallyhave a number of properties: (i) they are symplectic, (ii) they exactly preserve the momentaassociated to symmetries, (iii) and they have excellent longtime energy behavior (see[19, 20, 23, 39]). In this section we begin by reviewing the method, and then use it to constructintegrators in two ways: (i) by adapting integrators for classical mechanical systems to themo-elastic ones in the absence of heat conduction, and (ii) by directly constructing variationalintegrators for the thermo-elastic Lagrangian (5). In the first case, as an example we adaptNewmark’s explicit second–order algorithm. In the second case, a polynomial interpolationof the mechanical and thermal displacements is combined with Lobatto quadrature rules toconstruct discrete Lagrangians. The resulting numerical maps correspond to the standardLobatto IIIA-IIIB symplectic partitioned Runge-Kutta methods, as explained in §3.6.6 of[18]. Finally, we utilize composition to elevate the order of two first-order, explicit variationalintegrators into a second-order, explicit one, and find that we recover the adapted Newmark’sexplicit second-order algorithm. For simplicity, throughout this section we assume that themass is independent of the configuration.

4.1. Summary of the method

We begin by partitioning the time interval of interest [ta, tb] into equally spaced time intervalsof length h = (tb − ta)/Nt, and set tk = ta + kh for k = 0, . . . , Nt. In the following, we use [·]k

to denote the numerical approximation to the time varying quantity [·] evaluated at tk.The method is based on constructing a discrete Lagrangian Ld, which approximates the

action integral over the time interval [tk, tk+1] as

Ld(zk, zk+1) ≈

∫ tk+1

tkL(z(t), z(t))dt, (36)

where z(t) : [tk, tk+1] → G is the solution of the Euler-Lagrange equations satisfying z(tk) = zk

and z(tk+1) = zk+1, in a sense made precise in [18]. The discrete Lagrangian then leads to theconstruction of the discrete action sum

Sd(z0, ..., zNt) =

Nt−1∑

k=0

Ld(zk, zk+1). (37)

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14 PABLO MATA AND ADRIAN J. LEW

The integrator then follows from a discrete version of Hamilton’s principle, namely the variationof the action sum δSd should be equal to zero among all variations of z1, ..., zNt−1. Thestationarity conditions, or discrete Euler-Lagrange (DEL) equations, are then

D2Ld(zk−1, zk) + D1Ld(z

k, zk+1) = 0, k ∈ 1, ..., (Nt − 1), (38)

where DiLd denotes the partial derivative of Ld with respect the ith slot. The momentum ofthe system Π is defined by

Πk = −D1Ld(zk, zk+1), (39)

for k = 0, 1, . . . , Nt − 1, from where the DEL equations can be equivalently expressed as

Πk+1 = D2Ld(zk, zk+1), (40)

for k = 0, 1, . . . , Nt − 1. Equations (39) and (40) constitute what is called the position-momentum form of the algorithm. They define a map (zk,Πk) 7→ (zk+1,Πk+1), whichconstitutes the integrator. The method consist in solving (39) for zk+1, and then replacingit in Eq. (40) to obtain Πk+1.The integrator is initialized by setting (z0,Π0) = (z(t0),Π(t0)). If only the initial velocity

z(t0) is known instead, then Π0 is computed from

Π0 = ∇zL(z(t0), z(t0)).

A convergence result on variational integrators in [18] roughly states that if the approximationof the action in (36) is of order hr+1, r > 1, and some standard smoothness conditions on L

are satisfied, then zT/h → z(T ) and ΠT/h → Π(T ) as h → 0 with an error that decays at leastas hr. Therefore, higher-order approximations in (36) result in correspondingly higher-orderintegrators.

4.2. Classical mechanics integrators for thermo-elastic systems without heat conduction

Perhaps the simplest strategy for the construction of time-integrators for adiabatic thermo-elastic systems is to use the reduced Lagrangian (24). Due to the fact that in this case theLagrangian is parametrized in terms of the entropies of the springs, the constancy of eachspring’s entropy is automatically satisfied.Since for each value of η the reduced Lagrangian is that of a classical mechanical system,

the mechanical displacements can be approximated by any variational integrator for classicalmechanics. The approximate temperatures are computed a posteriori, once the approximatedisplacements are known.We illustrate these ideas with the adaptation of the central differences or Newmark’s explicit

second-order algorithm to this setting. The discrete Lagrangian is:

Lη0

d (q0,q1) = h

[

Kd(q0,q1)−

1

2Uη0(q0)−

1

2Uη0(q1)

]

, (41)

where

Kd(q0,q1) =

1

2

(

q1 − q0

h

)

·m

(

q1 − q0

h

)

. (42)

The DEL equations (38) are then

m

h2(qk+1 − 2qk + qk−1) = ∇qU

η0(qk), k ∈ 1, ..., Nt − 1, (43)

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 15

which, as mentioned earlier, correspond to the central difference approach for finite dimensionalmechanical problems.The position-momentum form of the algorithm given by (39) and (40), which defines the

conjugate momenta, follows as

pk =m

h(qk+1 − qk)−

1

2h∇qU

η0(qk), (44a)

pk+1 =m

h(qk+1 − qk)−

1

2h∇qU

η0(qk+1). (44b)

The algorithmic steps for each update are:

(i) Solve (44a) for qk+1.(ii) Update pk+1 using (44b).(iii) Compute the temperature as

θk+1 = ∇ηU(qk+1,η0) = θ(qk+1,η0).

which only adds the last step to a classical integrator.The resulting algorithms are the trivial extension of variational integrators to adiabatic

thermo-elasticity. Among other characteristics it is worth mentioning that

• It is simple to construct discrete Lagrangians so that the resulting integrators conservelinear and angular momentum, if the exact trajectories do. Additionally, all variationalalgorithms are symplectic and display very good long term energy behavior, with theHamiltonian given by (17). The central differences scheme above has all of these features.

• Higher order algorithms can be easily constructed as well, see [18].• Even when the resulting algorithms inherit these desirable characteristics, they heavily

make use of the fact that η is constant. Consequently, it is not possible to extend themto problems including heat conduction (ηi > 0).

4.3. Variational integrators based on the thermo-elastic Lagrangian

In the following we concentrate on the formulation of variational time integrators based on theLagrangian (5), so we discretize both mechanical and thermal displacements. A key motivationto do it is that they open a door for a future extension of these ideas to the case with heatconduction.

4.3.1. Generalized trapezoidal rule. We consider first the perhaps most commonly useddiscrete Lagrangians, namely, Lαd given by

Lαd = αL0d + (1− α)L1d, (45)

where α ∈ [0, 1] and

L0d(q

0,q1,Φ0,Φ1) = h

[

Kd(q0,q1)− A

(

q0,Φ1 −Φ0

h

)

]

, (46a)

L1d(q

0,q1,Φ0,Φ1) = h

[

Kd(q0,q1)− A

(

q1,Φ1 −Φ0

h

)

]

. (46b)

with Kd given by (42).

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16 PABLO MATA AND ADRIAN J. LEW

Then, the following expressions for the DEL equations are obtained:

m

h2(qk+1 − 2qk + qk−1) = αfk+ + (1− α)fk−, (47a)

αΓk+ + (1− α)Γ(k+1)− = αΓ(k−1)+ + (1− α)Γk−. (47b)

where

fk+ = −∇qA

(

qk,Φk+1 −Φk

h

)

,

fk− = −∇qA

(

qk,Φk −Φk−1

h

)

,

Γk+ = −∇θA

(

qk,Φk+1 −Φk

h

)

,

Γk− = −∇θA

(

qk,Φk −Φk−1

h

)

.

Equation (47a) is a discrete analog to Newton’s second law for mechanical displacements, while(47b) establishes a discrete form of entropy conservation.The position-momentum form is

pk =m

h(qk+1 − qk)− αhfk+ (48a)

pk+1 =m

h(qk+1 − qk) + (1 − α)hf (k+1)−. (48b)

ηk = αΓk+ + (1− α)Γ(k+1)− (48c)

ηk+1 = αΓk+ + (1− α)Γ(k+1)−, (48d)

where ηk and ηk+1 are the approximations to the entropy at tk and tk+1, respectively. Explicitexpressions for (47a), (48a) and (48b) for the case of a configuration-dependent mass matrixare given in Appendix 7.It follows from (47b), (48c) and (48d) that

ηk+1 = ηk, (48e)

which is a statement of algorithmic entropy conservation.Different integrators are obtained for each value α:

1. Setting α = 0 or α = 1 yield first-order algorithms with explicit updating rules for q in(47a). The entropy equation (47b) together with (48d)reduce to

ηk = −∇θA

(

qk+1,Φk+1 −Φk

h

)

, (49)

for α = 0, and to

ηk = −∇θA

(

qk,Φk+1 −Φk

h

)

, (50)

for α = 1. These updating rules are also explicit, as we discuss later. These algorithmsare sometimes called symplectic Euler.

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 17

2. If 0 < α < 1, both q and Φ have to be updated by means of simultaneously solving (47a)and (47b). These are all first-order algorithms, except for the one obtained for α = 1/2,which is second order. In this last case the entropy equation (47b) together with (48d)reduce to

ηk = −1

2

[

∇θA

(

qk,Φk+1 −Φk

h

)

+∇θA

(

qk+1,Φk+1 −Φk

h

)]

.

Remark. In connection with the higher-order integrators described later in §4.3.2, theintegrators constructed in this section can be seen as a result of choosing continuous trajectoriesfor (q,Φ) which restricted to [tk, tk+1] are affine, together with quadrature rules withquadrature points tk and tk+1. With this perspective, the velocity and temperature vectorsare piecewise constant in the each time interval, but not necessarily continuous across timeinterval boundaries, see Fig. 4. In fact, it is possible to define the two one-sided limits

limt→tk+

(

q(t), θ(t))

= (qk+, θk+) and limt→tk−

(

q(t), θ(t))

= (qk−, θk−),

which serve to motivate and interpret the notation fk± and Γk± above.

Figure 4. The integrators in this section can be constructed by assuming a piecewise affine evolutionof the thermal and mechanical displacements, sketched in (a), together with suitable quadrature rules.Under these conditions, the temperatures are piecewise constant, and possibly discontinuous across

time-interval boundaries, as sketched in (b).

Notes on implementation. A practical way to implement the algorithm is through theposition-momentum form. Each time step consists in: Given (q,Φ,p,η)k, α ∈ [0, 1] and h, find(q,Φ,p,η)k+1 such that (48a) to (48d) are satisfied up to a certain tolerance. This involves

i. Determining qk+1 and Φk+1 by solving the (implicit) relations given in (48a) and (48c).If implicit, the solution of these equations can be carried out with a Newton-Raphsonmethod.

ii. Computing pk+1 and ηk+1 by using (48b) and (48d).iii. If of interest, compute θk+1 = ∇ηU(q

k+1,ηk+1).

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18 PABLO MATA AND ADRIAN J. LEW

As mentioned earlier, explicit algorithms are obtained for the α = 0, 1. We detail next thesteps for the update in these cases.

When α = 0, we compute

qk+1 = qk + hm−1pk, (51a)

Φk+1 = Φk + h∇ηU(

qk+1,ηk)

, (51b)

pk+1 = pk + hf (k+1)−, (51c)

ηk+1 = ηk. (51d)

which follow from (49) and (16).

By rewriting f (k+1)− in terms of (qk+1,ηk) in (51c), we recover the classical structure ofthe symplectic-Euler A method as presented in [21, 22, 69] and revisited in the contextof VI for classical mechanics in [18]. More precisely, we have that

f (k+1)− = −∇qA

(

qk+1,Φk+1 −Φk

h

)

= −∇qU(

qk+1,ηk)

, (51e)

where we have used the expression for h−1(Φk+1 −Φk) obtained from (51b).

When α = 1, we compute

Φk+1 = Φk + h∇ηU(

qk,ηk)

, (52a)

pk+1 = pk + hfk+, (52b)

qk+1 = qk + hm−1pk+1, (52c)

ηk+1 = ηk. (52d)

which follow from (50) and (16). Analogously, by rewriting fk+ in (52b) in terms of(qk,ηk), taking advantage of the expression for h−1(Φk+1 − Φk) obtained from (52a),it is possible to recognize the classical structure of the symplectic-Euler B method[18, 21, 22, 69].

Error analysis and convergence. The convergence of these algorithms for α = 0, 1, 1/2is proved in [18, pp. 402], by computing the order of these discrete Lagrangians. Of course,some standard conditions on the smoothness of A need to be satisfied. As stated earlier, whenα = 0, 1 only first order convergence is obtained, while second-order convergence happens forα = 1/2. From the results therein, or by simply computing the consistency error, it is not hardto prove that the algorithms are first order for all other values of α as well.

4.3.2. Higher-order algorithms. The formulation of higher order variational integrators isstraightforward, as detailed in [18]. This methodology is equivalent to constructing bothcontinuous Galerkin methods in time and a family of symplectic partitioned Runge-Kuttamethods (see theorem 2.6.1 and 2.6.2 in [18]). The main idea consists in considering continuoustrajectories that are polynomials of a certain degree in each time interval. The discreteLagrangian is obtained by numerically integrating its exact counterpart for each discretetrajectory. This is accomplished by selecting an appropriate set of quadrature points andweights.

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 19

More precisely, let PK([0, h]) be the space of polynomials of degree K in [0, h], and

tgng

g=1 ⊂ [0, h] and wgng

g=1 be a set of quadrature points and suitable quadrature weightsfor polynomials of degree K, respectively. The class of variational integrators considered herestem from the discrete Lagrangian

Ld(z0, z1) = inf

z(t)∈[PK([0,h])]NG

z(0)=z0, z(h)=z1

[ ng∑

g=1

wgL(

z(tg), z(tg))

]

.

where NG is the dimension of the configuration manifold G.We now formulate time integrators by adopting second degree polynomials in each time

interval, so trajectories in G can be written as

Φi(tk + τ) = N1(τ)Φ

k +N2(τ)Φk+ 1

2 +N3(τ)Φk+1,

q(tk + τ) = N1(τ)qk +N2(τ)q

k+ 12 +N3(τ)q

k+1 ,(53)

for each k = 0, 1, . . ., where τ ∈ [0, h] and Ni : [0, h] → R, i ∈ 1, 2, 3, are the set ofbasis functions for P

2([0, h]) which, for convenience, satisfy Ni(τj) = δij for τj ∈ 0, h/2, h.For the quadrature we chose a full Lobatto rule with weights h

6 ,4h6 , h

6 and quadraturepoint coordinates 0, h/2, h. In (53), (qk,Φk) ∈ G for k = 0, 1/2, 1, 3/2, . . ., which are thecoefficients that define the continuous piecewise polynomial trajectory in time.To construct the discrete Lagrangian, we first define the unoptimized discrete Lagrangian

Lud(q

0,q12 ,q1,Φ0,Φ

12 ,Φ1) =

h

6(L0 + 4L

12 + L

1), (54)

where

L0 = Kd(v

0)− A(q0, θ0),

L12 = Kd(v

12 )− A(q

12 , θ

12 ),

L1 = Kd(v

1)− A(q1, θ1),

and

θ0 = h−1(4Φ12 − 3Φ0 −Φ1), θ

12 = h−1(Φ1 −Φ0), θ1 = h−1(Φ0 − 4Φ

12 + 3Φ1),

v0 = h−1(4q12 − 3q0 − q1), v

12 = h−1(q1 − q0), v1 = h−1(q0 − 4q

12 + 3q1),

(55)

which denote the values of the temperatures and velocities at the quadrature points τ =0, h/2, h. The discrete Lagrangian follows as

Ld(q0,Φ0,q1,Φ1) = inf

q12 ,Φ

12

Lud(q

0,q12 ,q1,Φ0,Φ

12 ,Φ1). (56)

The conditions for the infimum in (56) in the time interval [tk, tk+1] are

4m

h2(qk − 2qk+ 1

2 + qk+1) = fk+12 , (57a)

Γk+ − Γ(k+1)− = 0, (57b)

while the discrete Euler-Lagrange equations then follow as

m

h2(−qk+1 + 8qk+ 1

2 − 14qk + 8qk− 12 − qk−1) =

1

2(fk+ + fk−), (57c)

Γ(k+1)− − 4Γk+ 12 − 3Γk+ + 3Γk− + 4Γk− 1

2 − Γ(k−1)+ = 0, (57d)

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20 PABLO MATA AND ADRIAN J. LEW

where

fk+ = −∇qA

(

qk,−3Φk + 4Φk+ 1

2 −Φk+1

h

)

fk− = −∇qA

(

qk,Φk−1 − 4Φk− 1

2 + 3Φk

h

)

,

fk+12 = −∇qA

(

qk+ 12 ,

Φk+1 −Φk

h

)

,

Γk+ = −∇θA

(

qk,−3Φk + 4Φk+ 1

2 −Φk+1

h

)

,

Γk− = −∇θA

(

qk,Φk−1 − 4Φk− 1

2 + 3Φk

h

)

,

Γk+ 12 = −∇θA

(

qk+ 12 ,

Φk+1 −Φk

h

)

.

Together, (57a)-(57d) provide enough equations to solve for (qk+ 12 ,qk+1,Φk+ 1

2 ,Φk+1) given

(qk− 12 ,qk,Φk− 1

2 ,Φk). Expressions for the case of configuration dependent mass can be foundin appendix 7.The conjugate momenta are given by

pk =m

3h(8qk+ 1

2 − 7qk − qk+1)−h

6fk+, (58a)

pk+1 =m

3h(qk − 8qk+ 1

2 + 7qk+1) +h

6f (k+1)−, (58b)

ηk =3

6Γk+ −

1

6Γ(k+1)− +

4

6Γk+ 1

2 , (58c)

ηk+1 =3

6Γ(k+1)− −

1

6Γk+ +

4

6Γk+ 1

2 . (58d)

Subtracting Eq. (58c) from (58d) and using (57b) it is clear that

ηk+1 = ηk, (59)

and therefore, the numerically computed entropy is exactly conserved.

Regarding the order of convergence, for an arbitrary quadrature rule the convergence ratemight only be cubic, but since this is a symmetric method (i.e., equal to its adjoint, see [22]),then the convergence rate should be even. The algorithm is fourth-order.

Notes on implementation. The algorithm consists in determining (qk+1,Φk+1), and

(pk+1,ηk+1) given (qk,Φk) and (pk,ηk). To this end, the midpoint variables (qk+ 12 ,Φk+ 1

2 )have to be simultaneously solved for. The solution procedure is summarized in the followingtwo steps:

(i) Solve simultaneously (57a), (57b), (58a) and (58c) for (qk+ 12 ,Φk+ 1

2 ,qk+1,Φk+1) with,for example, a Newton-Raphson scheme.

(ii) Use (58b) and (58d) to update pk+1 and ηk+1.

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 21

4.4. Composition methods

In previous sections we have formulated time integrators presenting second or higher accuracyorder which are always implicit. The only explicit integrators we formulated so far were thosein section 4.3.1 after selecting α = 0, 1, and these are only first-order accurate. For someproblems explicit algorithms exhibiting second order of accuracy are very attractive.

An alternative to formulate a second-order algorithm is by composing a first-order integratorwith its adjoint, see [22, Ch. 2]. More precisely, an integrator defines a map Fh

Ld: T∗G → T

∗G,

parameterized by h, such that (zk+1,Πk+1) = FhLd(zk,Πk). The idea then is to construct

another integrator defined by the map

Fh = Fh/2Ld

F∗h/2Ld

, (60)

where F∗hLd

is the adjoint method to FhLd. For VI the following relation [18] holds between a

method and its adjoint

F∗hLd

= FhL∗d, (61)

where

L∗d(z

0, z1, h) = −Ld(z1, z0,−h), (62)

for all zk ≡ (qk,Φk) ∈ G, k = 0, 1. In (62) the arguments of Ld have been augmented, nowincluding h to highlight its role in the computation of the discrete Lagrangian of the adjointmethod.

For the discrete Lagrangian given in (45) and the algorithms given in (51a) to (52d) thefollowing relations are obtained

(L0d)∗ = L

1d and (L1d)

∗ = L0d, (63)

and taking into account (60) and (61), the following second-order maps can be constructed

F01

h = F(h/2)

L0d

F∗(h/2)

L0d

= F(h/2)

L0d

F(h/2)

L1d

, (64)

F10

h = F(h/2)

L1d

F∗(h/2)

L1d

= F(h/2)

L1d

F(h/2)

L0d

. (65)

The explicit form for the F01

h map is given by

Φk+ 12 = Φk +

h

2∇ηU(q

k,ηk), (66a)

pk+ 12 = pk −

h

2∇qA

(

qk,Φk+ 1

2 −Φk

h/2

)

, (66b)

qk+1 = qk + hm−1pk+ 12 , (66c)

Φk+1 = Φk+ 12 +

h

2∇ηU(q

k+1,ηk), (66d)

pk+1 = pk+ 12 −

h

2∇qA

(

qk+1,Φk+1 −Φk+ 1

2

h/2

)

, (66e)

ηk+1 = ηk. (66f)

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22 PABLO MATA AND ADRIAN J. LEW

The discrete Lagrangian for this integrator is

L01d (q0,q1,Φ0,Φ1) = inf

(q,Φ)12

[

h

2L0d(q

0,q12 ,Φ0,Φ

12 , h/2) +

h

2L1d(q

12 ,q1,Φ

12 ,Φ1, h/2)

]

. (67)

The equations for F10

h are obtained following the same procedure, and from the discreteLagrangian that results from commuting the order in which L

0d and L

1d are used to approximate

the action over [0, h2 ] and [h2 , h]. Even though F01

h and F10

h differ only in the order of application

of the maps F(h/2)

L0d

and F(h/2)

L1d

, they in fact constitute different algorithms of the same order.

The algorithm F01

h coincides exactly with that presented in §4.2 as it can be verified afteridentifying the relations (66a) and (66d) with the Legendre transform given in (162) andreplacing in the rest of the equations of the method. So this section provides the formulationof that algorithm directly from the thermo-elastic Lagrangian, without a priori using the factthat the entropy is constant in time.It is interesting to note that in the context of a separable Lagrangian such the one for classical

mechanics in Cartesian coordinates, the discrete Lagragians (67) and (45) with α = 1/2 areexactly the same ones. This is why in that case the composition of the two symplectic Eulermethods results in Newmark’s explicit second-order algorithm, also known as Stormer-Verlet[69]. This is not the case for thermo-elastic Lagrangians, because the free-energy depends onthe thermal velocities.

5. Numerical examples

In this section two numerical examples showcase the performance and main features of theformulated algorithms are presented.

5.1. Three-dimensional motion of a chain of thermo-elastic springs and masses

The first example correspond to the simulation of the dynamics of a chain of three thermo-elastic springs and masses in R

3. See Fig. 5. The free-energy of all springs is that in (31),with constants β = 0.2, C0 = 5.0, l0 = 1, c = 100 and θ0 = 300. The initial momenta (incomponents in a Cartesian basis) and initial temperature are summarized in Table II. Theinitial positions are those shown in Fig. 5.

Figure 5. Chain composed by three thermo-elastic springs and masses.

5.1.1. Order of convergence of the algorithms. We first examine the numerical order ofconvergence of the algorithms introduced in sections 4.3.1, 4.3.2 and 4.4. Henceforth we shall

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 23

Mass m p1(0) p2(0) p3(0)1 1.0 20.0 10.0 10.02 2.0 10.0 20.0 10.03 1.5 0.0 10.0 0.0

Spring θ(0)1 800.02 1200.03 400.0

Table II. Mechanical properties, initial temperature and Cartesian components of the initial momentaof the system.

refer to them as GTR for generalized trapezoidal rule, HO4 for higher-order, and CO2 forcomposition, respectively. For the GTR algorithms we set α to 0.0, 0.5, 1.0, as discussedearlier, and label them GTR-0, GTR-1/2 and GTR-1.We simulated the dynamics of this system with these algorithms and evaluate errors at

time t = 1, with the goal of numerically verifying their order of convergence. The errors atthis time were measured for positions, momenta and temperatures separately. We show therate of convergence by computing the square root of the sum of the square of the error ineach Cartesian component of each mass, for the algorithmic positions qh(1) or algorithmicmomenta ph(1). For the algorithmic temperatures θh(1) (obtained with (18b)), we computethe square root of the sum of the square of the error in the temperature of each spring instead.In the absence of an analytical solution, we computed a reference solution with HO4 by settingh = 10−4. This time step yields the precise computation of 10 decimal digits in all variables, sowe adopted it as a proxy for the exact solution, and errors were computed with respect to it.Simulations were carried out for all or a subset of the following time steps, depending on theintegrator: 0.1, 0.05, 0.025, 0.0125, 0.00625, 0.003125, 0.0015625, 0.00078125 and 0.000390625.Figures 6 to 8 show the convergence curves. As expected, the curves show first-order

convergence in all variables for GTR-0 and GTR-1, second-order convergence for GTR-1/2and CO2, and fourth-order convergence for HO4. A remarkable feature is that the accuracyof the implicit algorithm GTR-1/2, and of the explicit algorithm CO2, seem to be the same.This might be a good reason to prefer the latter, CO2. The higher-order algorithm HO4 ismore accurate than the rest for all tested time steps.Regarding the computational performance, Fig. 9 shows the computing time as a function

of the the magnitude of the error in position at t =100, for each one of the algorithms.Analogous results are obtained when the computing time is plotted as a function of the errorsin momentum or temperature. No particular emphasis has been placed in optimizing the codefor each algorithm, so these results could change upon a more careful implementation.As expected, for lower accuracy simulations the lower order algorithms (GTR-0, GTR-1,

GTR-1/2 and CO2) are somewhat more advantageous when compared against the higher orderalgorithm HO4. However, if higher accuracy is required, HO4 becomes a lot more convenient.Finally, notice that throughout the whole range of computing times CO2 is more accuratethan GTR-1/2, which makes it more attractive.

5.1.2. Long-term energy behavior and conserved quantities. We next showcase the behaviorthe algorithms for long times. We simulate the evolution of the system up to t = 100, with atime step length h = 0.01, so a complete simulation involves 104 time steps.We first show parts of the trajectories and temperatures, as computed with CO2. Figures 10

and 11 show the projections of the motion of each mass in the e1–e2 and e1–e3 planes, whereeii=1,2,3 is the Cartesian basis in Fig. 5. The motion of the masses is clearly non-trivial and

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24 PABLO MATA AND ADRIAN J. LEW

Err

or

in p

ositio

n

1e-10

1e-08

1e-06

1e-04

1e-02

1e+00

1e+021e+02

Time step1e-03 1e-02 1e-01

4.00

1

2.00

1

1

GTR-1/2

CO2

GTR-0

GTR-1

HO4

0.97

Figure 6. Convergence curve for the positions.

Error in m

omentum

1e-10

1e-08

1e-06

1e-04

1e-02

1e+00

1e+021e+02

Time step1e-03 1e-02 1e-01

1

1

3.99

1

0.97

2.00

GTR-1/2

CO2

GTR-0

GTR-1

HQ4

Figure 7. Convergence curves for the linear momenta.

fully three-dimensional, with very large stretches of each one of the springs which induces largefluctuations of temperature and potential energy in the system. The evolution of the formeris shown in Fig. 12.

In following the results obtained with CO2 is compared with those from HO4. Of course,by design both the entropy and the angular momentum of the system are conserved up tomachine precision. This is shown in Figures 13 and 14. In both figures the curves obtainedfrom each one of the two algorithms superposes exactly.

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 25

Err

or

in t

em

pera

ture

1e-08

1e-06

1e-04

1e-02

1e+00

1e+021e+02

Time step1e-03 1e-02 1e-01

1

0.951

2.00

1

4.08

1 CO2

GTR-0

GTR-1

HO4

GTR-1/2

Figure 8. Convergence curves for the temperatures.

Error in position

1e-08

1e-06

1e-04

1e-02

1e+00

1e+021e+02

Computing time1e-01 1e+00 1e+01 1e+02

GTR-1/2

GTR-0, GTR-1

CO2

HQ4

Figure 9. Relation between computational cost and error in position.

Finally the total angular momentum at tk is computed as

Ak :=

N∑

i

(qi × pi)k,

and the energy of the system by

Hk =

1

2pk ·m−1pk + U(qk,ηk).

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26 PABLO MATA AND ADRIAN J. LEW

mass 2

-50

0

50

q1-40 -20 0 20 40

mass 3

-50

0

50

q1-40 -20 0 20 40

mass 1

q2

-50

0

50

q1-40 -20 0 20 40

Figure 10. Trajectories of the masses in the e1–e2 plane. Here qi is the coordinate along the ei direction.

mass 2

-20

0

20

q1-40 -20 0 20 40

mass 3

-20

0

20

q1-40 -20 0 20 40

mass 1

q3

-20

0

20

q1-40 -20 0 20 40

Figure 11. Trajectories of the masses in the e1–e3 plane. Here qi is the coordinate along the ei direction.

Temperature

2e+02

4e+02

6e+02

8e+02

1e+03

1e+03

1e+03

Time0 20 40 60 80 100

θ1

θ2

θ3

Figure 12. Evolution of the temperatures of the springs through 10,000 time steps.

The total energy trajectories computed with CO2 and HO4 is shown in Figure 15. Whenexamined at full scale, Fig. 15(a), it is evident that (i) the energy is essentially conserved byboth algorithms, and (ii) both algorithms are perceived as identical at this scale. Only whenthe vertical scale is small enough can the differences between the second- and fourth-order

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 27

Entropy

0

2

4

6

8

10

12

14

Time0 20 40 60 80 100

η1η2η3ηt

Figure 13. Evolution of the entropy of the springs through 10,000 time steps. The total entropyηt = η1 + η2 + η3 is shown as well.

Angular momentum

-40

-20

0

20

40

60

80

Time

0 20 40 60 80 100

A1

A2

A3

Figure 14. Evolution of each one of the Cartesian components of the angular momentum of the systemthrough 10,000 time steps.

algorithms be appreciated, see Fig. 15(b). It is at this scale that it is possible to appreciate thesmall energy oscillations characteristic of symplectic integrators. The energy oscillations forthe HO4 algorithm are only apparent once a yet smaller scale is selected for the energy axis,see Fig. 15(c).

5.2. The dynamics of a simple thermo-elastic net

To conclude, we show the behavior of the CO2 algorithm in a slightly different example, thethermo-elastic system shown in Fig. 16. This examples also showcases the conservation oflinear momentum. Each spring has the constitutive behavior defined by (31), with C0 = 5,β = 0.2, l0 = 1 and θ0 = 300. Initially the springs form an equilateral triangle with side length1, with the masses located at the vertices. The stiffness of each spring, as well as its initialtemperature and the initial momentum of each mass are shown in Table III. The simulationwas performed up to a final time t = 10, with time step h = 0.001.

Projections of the three-dimensional trajectories of mass 1 in the e1–e3 and e1–e2 planes areshown in Fig. 17, where eii=1,2,3 is the Cartesian basis shown in Fig. 16. Figure 18 shows thetime histories of temperature and strain in spring 2, with strain defined as ln(l/l0). Finally,Fig. 19 displays the evolution of the system’s energy, total entropy and total linear and angularmomentum about e2. The last three are conserved within machine precision, and the energy

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28 PABLO MATA AND ADRIAN J. LEW

Total energy

7.0e+03

7.5e+03

8.0e+03

8.5e+03

Time0 50 100

CO2

HO4

(a)

Total energy

7,983

7,984

7,985

7,986

Time42 43 44 45

CO2

HO4 (b)

Total energy

7,983.328

7,983.330

7,983.332

7,983.334

Time42 43 44 45

HO4 (c)

Figure 15. (a) Evolution of the energy of the system through 10,000 time steps, shown at a full scalein the energy axis. (b) The fact that the energy oscillates around a constant value can only be seenby selecting a smaller scale for the energy axis. The oscillations in the higher-order algorithm need a

yet smaller scale to be apparent, as shown in (c).

Figure 16. Thermo-elastic net for the second numerical example.

Mass m p1(0) p2(0) p3(0)1 1.0 0.0 0.0 -7.02 1.0 0.0 0.0 7.03 2.0 0.0 5.0 0.0

Spring c θ(0)1 400.0 500.02 500.0 1000.03 400.0 500.0

Table III. Mechanical properties and initial temperature of each spring, and mass and Cartesiancomponents of the initial momentum of each mass.

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 29

oscillates around a constant value for the duration of the simulation.

q3

-1

-0.5

0

0.5

1

q1

-1 -0.5 0 0.5 1

q2

0

5

10

q1

-1 -0.5 0 0.5 1

Figure 17. Projections of the trajectory of mass 1 in the e1-e3 and e1-e2 planes. Here qi denotes thecoordinate along the ei-direction.

Strain

0

0.2

0.4

0.6

0.8

Time0 2 4 6 8 10

Temperature

970

980

990

1,000

Time0 2 4 6 8 10

Figure 18. Evolution of the strain and the temperature in the thermo-elastic spring 2.

Conserved quantity

-10

-5

0

5

10

15

Time

0 2 4 6 8 10

e2-component of the

angular momentum

Entropy

Energy

e2-component of the

linear momentum

Figure 19. Numerical conservation of the energy and momenta of the system.

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30 PABLO MATA AND ADRIAN J. LEW

6. Conclusions

The construction of variational integrators for adiabatic thermo-elastic systems followed astandard procedure. These same ideas will soon be extended to continuum media, with thespecial heat conduction of Green and Naghdi [54]. The challenge of extending these structure-preserving integrators to the presence of Fourier’s heat conduction remains. A key propertywe would like to satisfy in that case is that, if the thermo-elastic system is isolated from theexternal world, so that there are no heat sources or sinks, then the energy of the system shouldbe conserved. This has been explicitly enforced in the energy-momentum method in [36]. Sincefor this case the equations no longer have a Hamiltonian structure, then the good long-termenergy behavior of variational integrators cannot be directly exploited.

7. Appendix

The explicit expression for the DEL equations given in (47a) for the case of configuration-dependent mass matrix is

1

h2

(

mk+αqk+1 − (mk+α +mk−1+α)qk +mk−1+αqk−1)

=

=1

2h2∇qm

k : Υk+1−α + (1 − α)fk− + αfk+, (68)

wheremk+α = αmk + (1 − α)mk+1,Υk+1−α = αΥk+1 + (1− α)Υk,Υk+1 = (qk+1 − qk)⊗ (qk+1 − qk).

The corresponding mechanical momentum maps are

pk = h−1mk+α(qk+1 − qk) + α[

hfk+ + h−1∇qmk : Υk+1

]

, (69a)

pk+1 = h−1mk+α(qk+1 − qk)− (1− α)[

hf (k+1)− + h−1∇qmk+1 : Υk+1

]

. (69b)

Similarly, the explicit form of Eqs. (57c) for the case of configuration-dependent mass is

−maqk+1 + 4mbq

k+ 12 −mcq

k + 4mk− 1

2

d qk− 12 −meq

k−1 =

=1

2∇qm

k : (Θk+ +Θk−) + h2(fk+ + fk−), (70)

where

ma = 3mk − 4mk+ 12 + 3mk+1,

mb = 3mk +mk+1,

mc = mk+1 + 4mk+ 12 + 18mk + 4mk− 1

2 +mk−1,

md = mk−1 + 3mk,

me = 3mk−1 − 4mk− 12 + 3mk,

Θk+ = (−3qk + 4qk+ 12 − qk+1)⊗ (−3qk + 4qk+ 1

2 − qk+1),

Θk− = (qk−1 − 4qk− 12 + 3qk)⊗ (qk−1 − 4qk− 1

2 + 3qk).

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VARIATIONAL INTEGRATORS FOR THERMO-ELASTO-DYNAMICS 31

Moreover, the analogue of (57a) is given by

mfqk − (mf +mg)q

k+ 12 +mgq

k+1 = h2fk+12 +

1

2∇qm

k+ 12 : Θk+ 1

2 , (71)

where

mf = 3mk +mk+1,

mg = mk + 3mk+1,

Θk+ 12 = (qk+1 − qk)⊗ (qk+1 − qk).

Additionally, explicit expressions for the mechanical momenta are

pk =1

6h

[

−mhqk + 4mgq

k+ 12 +miq

k+1 − h2fk+ −1

2∇qm

k : Θk+]

,

pk+1 =1

6h

[

−miqk − 4mgq

k+ 12 +mjq

k+1 + h2f (k+1)− +1

2∇qm

k+1 : Θ(k+1)−]

.(72a)

where

mh = 9mk + 4mk+ 12 +mk+1,

mi = −3mk + 4mk+ 12 − 3mk+1,

mj = mk + 4mk+ 12 + 9qk+1.

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