Comput Mech (2012) 50:603–610DOI 10.1007/s00466-012-0693-y

ORIGINAL PAPER

An analysis of the stress formula for energy-momentum methodsin nonlinear elastodynamics

Ignacio Romero

Received: 23 October 2011 / Accepted: 9 February 2012 / Published online: 29 February 2012© Springer-Verlag 2012

Abstract The energy-momentum method, a space–timediscretization strategy for elastic problems in nonlinear solid,structural, and multibody mechanics relies critically on a dis-crete derivative operation that defines an approximation ofthe internal forces that guarantees the discrete conservationof energy and momenta. In the case of nonlinear elastody-namics, the formulation for general hyperelastic materials isdue to Simo and Gonzalez, dating back to the mid-nineties.In this work we show that there are actually infinite sec-ond order energy-momentum methods for elastodynamics,all of them deriving from a modified midpoint integrator byan appropriate redefinition of the stress tensor at equilibrium.Such stress tensors can be interpreted as the solutions to localconvex projections, whose precise definitions lead to differ-ent methods. The mathematical requirements of such projec-tions are identified. Based on this geometrical interpretationseveral conserving methods are examined.

Keywords Energy-momentum · Time integration ·Geometric integration · Nonlinear elastodynamics

Ignacio—Romero Currently Visiting Faculty at the California Instituteof Technology.

I. RomeroE.T.S.I. Industriales, Technical University of Madrid,Madrid, Spain

I. Romero (B)José Gutiérrez Abascal, 2, 28006,Madrid, Spaine-mail: ignacio.romero@upm.es

1 Introduction

The design of robust space–time discretizations for transientproblems in mechanics has been a long-standing goal in theComputational Mechanics community (see, for example, [1],for an early account). In the late eighties and nineties, Simoand co-workers introduced the energy-momentum method, adiscretization scheme which preserved both the energy andthe momentum maps in Hamiltonian problems ([2–4], amongmany others). In the context of nonlinear elastodynamics, thefirst energy-momentum method was proposed by Simo andTarnow [5] for materials with Saint Venant-Kirchhoff consti-tutive relation. Only later, Simo and Gonzalez [6,7] extendedthe original idea to arbitrary hyperelastic materials by pro-posing a modified evaluation of the stress tensor. This workhas been the basis of numerous applications for the designof algorithms with application to conserving problems (forexample [8–12]) as well as non-conserving ([11,13–16]).

The formulation of conserving methods pivots on the ideaof a discrete gradient, a modified, albeit consistent, evaluationof the differential that evaluates exactly the internal energybalance in a time step. This concept goes back, at least, to Got-usso [17] (see also [18,19]), but it was first in [20,21] wherethe concept appears clearly defined for elastodynamics.

The discrete gradient formula of [6,7] is not uniqueand different alternatives have been proposed in the liter-ature [11,13,22,23]. In the current work we aim to showthat there are indeed infinite ways of obtaining second orderaccurate, energy and momentum preserving methods. Thisassertion follows from a geometrical interpretation of thestress formula as a projection of the midpoint stress onto acertain affine set. More importantly, not all projections ofthis type result in accurate methods, and we also present ananalysis which identifies the conditions under which suchminimizations are valid. We illustrate by means of examples

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that it is possible to construct conserving approximationswhich, despite their appealing simplicity, loose the secondorder accuracy.

The characterization of the conserving stress as a (con-vex) minimization problem leads to an infinite set of possiblemethods that preserve the accuracy order. It is argued that allof them are equivalent from the numerical analysis stand-point. It turns out that the simplest of all is the original oneof Simo and Gonzalez, and thus it is optimal in that sense.

The rest of the article is structured as follows. Section 2summarizes the boundary value problem of nonlinear elasto-dynamics, setting the notation for subsequent developments.In Sect. 3, space–time discretizations which employ finiteelements in space and a midpoint-type approximation intime are described. Among them, we single out the midpointrule and the energy-momentum method. In Sect. 4 we pres-ent the formulation of energy-momentum methods as localminimization problems, and we derive sufficient conditionsthat ensures the consistency of the attendant stresses. Theseresults are employed in Sect. 5 to formulate several con-serving stress formulas, but identifying some which fail topreserve the accuracy of the discretization. The article closeswith a summary of results in Sect. 6.

2 Nonlinear elastodynamics

We review the formulation of the initial boundary value prob-lem of nonlinear elastodynamics. To that purpose, let usdefineBo ⊂ R

3 to be the reference placement of a deformablebody with boundary ∂Bo = �g ∪ �h , where �g ∩ �h = ∅,and let its reference density be the function ρo : Bo → R

+.The deformation ϕ : Bo × [0, T ] → B ⊂ R

3 maps, at everyinstant t ∈ [0, T ], points X ∈ Bo to points x = ϕ(X, t) inthe current placement B := ϕ(Bo, t). The deformation gra-dient is defined as F := ∇Xϕ, and the right Cauchy-Greentensor as C := F T F .

The variational statement of the dynamic equilibrium isas follows. Let us assume that the body is subjected to bodyforces b : Bo × [0, T ] → R

3 and boundary tractions h :�h × [0, T ] → R

3. Additionally, homogeneous Dirichletboundary conditions are imposed on �g . The solution spacefor the finite strain problem is, at any instant t , the set V ofsmooth deformations satisfying the Dirichlet boundary con-ditions. See, e.g. [24], for additional details on the precisedefinition of the problem.

With these definitions, the weak solution ϕ ∈ V to theinitial boundary value problem satisfies

∫

Bo

S · F T ∇Xη dV +∫

Bo

ρoV · η dV =∫

Bo

b · η dV +∫

�h

h · η dA,

(1)

for all η ∈ V . In the previous equation, S is the second Piola-Kirchhoff stress tensor, the overdot denotes the material timederivative, and V is the material velocity defined as V := ϕ.The dot between vectors or tensors indicates their respectiveinner products.

In addition to the equilibrium Eq. (1), the deformation andmaterial velocity must satisfy the initial conditions

ϕ(X, 0) = ϕ(X) , V (X, 0) = V (X) , (2)

for all X ∈ Bo and some given functions ϕ, V : Bo → R3.

If the material is hyperelastic, the (symmetric) Piola-Kirchhoff stress is obtained through the relation

S := 2∂W (C)

∂C, (3)

where W = W (C) is the stored energy function of the mate-rial which, for simplicity, is assumed to be homogeneous.

Assuming that the external loading derives from a poten-tial Vext (ϕ), the total potential and kinetic energies of thebody are defined, respectively, as:

V (ϕ) :=∫

Bo

W (C) dV − Vext (ϕ) ,

K (V ) :=∫

Bo

ρo ‖V ‖2 dV . (4)

Under the previous hypothesis, the total energy E(ϕ,V ) :=V (ϕ) + K (V ) remains constant for all time, and thus equalto its initial value.

3 Discretization

The solution to the initial boundary value problem describedin Sect. 2 might be approximated with a spatial and timediscretization. In this section we describe two of such dis-cretizations, both of them based on finite elements in spaceand finite differences in time, as often done for problems incomputational solid and structural dynamics (see, for exam-ple, [25]).

To discretize the evolution equations in time, consider apartition of the time interval [0, T ] into subintervals [tn, tn+1]with 0 = to < t1 < t2 < · · · < tN = T and �tn = tn+1 − tn .Likewise, to introduce a spatial discretization, let the body Bo

be partitioned into a finite element triangulation defining afinite dimensional space of functions Vh ⊂ V , piecewiselinear in each element. Based on this time partition, the con-figurations and velocities at time tn are approximated by finiteelement functions ϕh

n and V hn , respectively, both in Vh .

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3.1 A class of space–time discretizations

We consider next a family of space time integrators which,given the deformation ϕh

n ∈ Vh and velocity V hn ∈ Vh , find

the deformation and velocity at time tn+1 by solving

ϕhn+1 − ϕh

n

�tn= V h

n+ 12

,

⟨V h

n+1 − V hn

�tn, ρo δϕh

⟩=

⟨bn+ 1

2, δϕh

⟩+

⟨hn+ 1

2, δϕh

⟩h

−⟨S,F T

n+ 12∇Xδϕh

⟩, (5)

for every δϕh ∈ Vh . In the previous expressions, and below,we have employed the standard notation zn+ 1

2to denote the

convex combination (zn+zn+1)/2, where z might be a scalar,a vector, or a tensor. Also, the bracket pairings denote the L2

inner products:

〈a, b〉 :=∫

Bo

a · b dV , 〈a, b〉h :=∫

�h

a · b dA , (6)

for any pair of scalars, vectors, or tensors a, b.To complete the space–time discretization, the update for-

mulas (5) need to be supplemented with an expression for thestress tensor S.

3.2 Remarks

(i) It was shown in [5] that any algorithm of the family (5)preserves the linear and angular momenta for any sym-metric stress tensor S.

(ii) All the methods in (5) are second order accurate, aslong as the stress formula employed provides a secondorder approximation to the tensor

S(∇T

X ϕ(X, tn+ 1

2

)∇Xϕ

(X, tn+ 1

2

)). (7)

3.3 The midpoint rule

The midpoint rule applied to the spatially discrete equationsof nonlinear elastodynamics defines the solution ϕh,V h ∈Vh × Vh to be the one that verifies (5) with S = SM P and

SM P := S(C

(Fn+ 1

2

)), Fn+ 1

2:= ∇Xϕh

n+ 12

. (8)

This is a second order accurate Runge-Kutta method. Sincethe stress approximation is symmetric, it preserves the lin-ear and angular momentum. In general, the method does notpreserve the total energy of the solution.

3.4 The energy-momentum method

The energy-momentum method, as proposed in [6,7], isanother method of the class defined in (5) which ensures,in addition to linear and angular momentum conservation,strict energy conservation. This energy is defined to be thesum of the kinetic and potential energies, defined respectivelyat time tn as

Kn :=∫

Bo

ρo

2

∥∥∥V h∥∥∥2

dV ,

Vn :=∫

Bo

W(∇T

X ϕhn ∇Xϕh

n

)dV − Vext

(ϕh

n

). (9)

To find an appropriate expression for the stress tensor thatcan guarantee energy conservation, the balance of energy inthe algorithm (5) is analyzed. If there is no external forc-ing, and the deformation variations are chosen as δϕh =ϕh

n+1 − ϕhn , the Eq. 52 reads

⟨V h

n+1 − V hn , ρo

ϕhn+1 − ϕh

n

�tn

⟩= −

⟨S,

1

2(Cn+1 − Cn)

⟩,

(10)

where we have used the notation

Cn := ∇TX ϕh

n ∇Xϕhn, Cn+1 := ∇T

X ϕhn+1 ∇Xϕh

n+1 . (11)

Using (5)1, the left hand side of (10) evaluates to the incre-ment of kinetic energies

Kn+1 − Kn = 1

2

∫

Bo

ρo

(∥∥∥V hn+1

∥∥∥2 −∥∥∥V h

n

∥∥∥2)

dV , (12)

and the right hand side to the (negative) increment of potentialenergies

− Vn+1 + Vn = −∫

Bo

(W (Cn+1) − W (Cn)) dV , (13)

provided that the relation

W (Cn+1) − W (Cn) = S · 1

2(Cn+1 − Cn) (14)

holds. This equation mimics the continuum relation W =S · 1

2 C and is a sufficient condition for energy conservation.A tensor S is said to be an approximation of 2 ∂W

∂C satisfy-ing the directionality property when (14) holds for any twotensors Cn,Cn+1.

The formulation of consistent stress formulas that satisfythe directionality condition was addressed in a series of worksduring the nineties by Simo and co-workers. For the simpleSaint Venant-Kirchhoff hyperelastic model, it was shown in[5] that the simple definition

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S = Sm , (15)

with

Sm := S(Cm) , and Cm := (Cn + Cn+1)/2 . (16)

provided a consistent, conserving approximation. We notethat, in general, Sm �= SM P .

For arbitrary hyperelastic models, the average stress Sm

does not satisfy directionality. A stress formula that veri-fies (14) for an arbitrary stored energy function, while pre-serving the second order accuracy of the midpoint rule, wasproposed in [6,7], and reads:

SSG := Sm + �Wn − Sm · Zn

‖Zn‖Zn

‖Zn‖ , (17)

with

Zn := 1

2(Cn+1 − Cn) ,

�Wn := W (Cn+1) − W (Cn) . (18)

In the previous equation, and for the rest of the article,the norm ‖·‖ on second order tensors denotes the standard(Frobenius) norm defined as

‖T ‖ := (T · T )1/2 =⎛⎝ 3∑

i, j=1

Ti j Ti j

⎞⎠

1/2

. (19)

The correction term to the average stress is of size O(�t2

)for any energy-bounded motion

{(ϕh

n,V hn )

}n and any time

step size. In the limit Cn+1 → Cn the stress formula (17)coincides with (15).

4 A generalized energy-momentum stress

Since its publication, expression (17) has become the basisfor the formulation of energy-momentum methods in SolidMechanics. However, this is not the only symmetric stressformula that satisfies consistency and directionality for gen-eral hyperelastic models, as we will show in this section.To make the statement precise let us start by defining whatwe understand by an energy-momentum method.

Definition 1 An energy-momentum method is a space–timediscretization of the form (5) with a stress tensor approxima-tion that:

1. Is symmetric,2. Verifies the directionality property,3. Is a second order approximation of SM P for every

energy-bounded solution(ϕh

n,V hn

).

Notice that this definition is equivalent to saying that theresulting discretization preserves the conservation laws oflinear and angular momentum, as well as energy. Addition-ally, we restrict our definition to second order methods, which

is the highest accuracy attainable by a discretization of theform (5).

4.1 Energy-momentum from perturbations of the originalformula

The three conditions required for second order accuracy, aswell as conservation of linear and angular momenta, andenergy are summarized in the definition. It follows imme-diately that there is an infinite number of energy-momentummethods as the following result shows

Lemma 1 If SM P is any second order approximation toSM P , the stress formula

S∗ := SM P + �Wn − SM P · Zn

‖Zn‖Zn

‖Zn‖ (20)

defines an energy-momentum method.

Proof The tensor S∗ is obviously symmetric, is a secondorder perturbation of SM P by hypothesis, and is trivial toverify that it satisfies directionality, hence defining an energy-momentum method. ��

Remarks 1 1. As a particular choice in the approximation(20), SM P could be chosen to be equal to SM P giving, ingeneral, a different energy-momentum method than (17).

2. When the material model is of the Saint Venant-Kirch-hoff type, the stress approximation (20) does not sim-plify to the formula (15), unless SM P = Sm , makingthe expression (17) more attractive than others.

3. Another reason for preferring expression (17) over (20)is that only stress formulas based on Cm and not on

C(Fn+ 1

2

)are known to preserve the exact relative equi-

libria. See [26,8].

4.2 Geometric interpretation of the energy-momentumstress expression

Lemma 1 presents a trivial generalization of the standardstress formula (17). We present now other extensions whichrely on the geometric interpretation of the conserving stress.In particular, we study expressions for the tensor S whichare obtained from the deformations Cn and Cn+1.

For that, assuming Cn and Cn+1 are given, let us definethe affine set

En, n+1 = {T ∈ Sym, T · Zn = �Wn} (21)

where Sym is the set of second order, symmetric tensors,not necessarily consistent with the midpoint stress. Sincethe tensors in En, n+1 satisfy directionality, conserving stressapproximations must be projections of Sm onto the hyper-plane En, n+1. These are obtained by solving the minimumdistance problem

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Comput Mech (2012) 50:603–610 607

min.S∈Sym1

2|||S − Sm |||2

s.t. S ∈ En, n+1 (22)

for some norm ||| · ||| on Sym.By construction, the solution to the program (22) is a stress

tensor that leads to a conserving method. However, this algo-rithm need not be an energy-momentum method in the senseof Definition 1, since the second order accuracy might belost in the projection. Hence, the design of a valid energy-momentum method entails only the choice of an appropriatenorm in (22). There is much freedom in the choice of normwhich can, in particular, depend on the deformation itself.We study next how such a norm can be chosen.

4.3 A general class of energy-momentum methods

The following result states a general condition that guar-antees that the solution to the projection problem (22) is asecond order accurate stress tensor:

Theorem 1 Let ||| · ||| : Sym → R+ ∪{0} be a norm derived

from an inner product 〈〈〈·, ·〉〉〉 in the standard way:

|||T ||| := 〈〈〈T ,T 〉〉〉1/2 , for all T ∈ Sym. (23)

If this norm is uniformly equivalent to the Frobenius normfor every energy bounded solution, then the solution to theconvex program (22) defines an energy-momentum methodin the sense of Definition 1.

Proof By construction, the solution to (22) yields a conserv-ing scheme, so it remains to prove that the approximationis second order accurate. For that, let C1, C2 > 0 be twoconstants such that

C1 ‖T ‖ ≤ |||T ||| ≤ C2 ‖T ‖ , (24)

for every T ∈ Sym. Then, if S∗ denotes the unique solutionto the projection problem (22), using the properties of normsand inner products:

|||S∗ − Sm |||2 = 〈〈〈S∗ − Sm,S∗ − SSG + SSG − Sm〉〉〉≤ 〈〈〈S∗ − Sm,S∗ − SSG〉〉〉

+〈〈〈S∗ − Sm,SSG − Sm〉〉〉 . (25)

To proceed we note that, from the projection theorem (see,e.g., [27]), the tensor S∗ − Sm is orthogonal with respect tothe 〈〈〈·, ·〉〉〉 inner product with all tensors parallel to the affineset En, n+1. Since both S∗ and SSG are in En, n+1, their differ-ence is parallel to this set and the first term of the right handside vanishes. The remaining term can be bounded, using theCauchy-Schwartz inequality, as

〈〈〈S∗ − Sm,SSG − Sm〉〉〉 ≤ |||S∗ − Sm ||| |||SSG − Sm |||.(26)

Combining expressions (25) and (26) we obtain

|||S∗ − Sm ||| ≤ |||SSG − Sm ||| . (27)

This is the best approximation property of the projection S∗among all tensors in En, n+1. Finally, using the equivalenceof norms and this last result, we obtain the bound:

‖S∗ − Sm‖ ≤ C−11 |||S∗ − Sm |||

≤ C−11 |||SSG − Sm |||

≤ C2

C1‖SSG − Sm‖ . (28)

Since the constants on the right hand side are independentof the solution, we conclude that the difference S∗ − Sm isof the same order as the correction term SSG − Sm of theoriginal energy-momentum method, and thus the projectionbased on ||| · ||| is a second order approximation of Sm . ��

Remarks 2 1. When the norm in the projection prob-lem (22) does not derive from an inner product the solu-tion still exists always and it is unique, since the objec-tive function of the program (22) is convex and the con-strained set affine. This happens, for example, when thenorm employed in the minimization is the 1-norm, the2-norm, or the ∞-norm on Sym.

2. For norms that do not derive from an inner product, The-orem 1 does not apply, since its proof depends on theorthogonality of the approximation error to the constraintset.

4.4 A class of stress projections

Equations (22) define the problem of minimum distance pro-jection of a point on a closed convex set, and this programhas a unique solution. To find it, the norm must be specified.As stated above, there are infinite norms and each of themresults in a different projection. However, the subset of normson Sym that derive from an inner product can all be writtenin the form

|||T ||| := (T · M T )1/2 , for all T ∈ Sym, (29)

where M is a symmetric, positive definite fourth order tensor.This re-formulation of the norm allows to find a closed formsolution to the program (22). For the rest of the article wewill assume that the norm ||| · ||| is always derived from aninner product and is thus of the form (29).

The solution of the projection (22) can be found by findingthe stationary points of the Lagrangian L : Sym × R → R

defined by

L(S, λ) = 1

2|||S − Sm |||2 − λ (S · Zn − �Wn) . (30)

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608 Comput Mech (2012) 50:603–610

Using the norm definition (29), the Lagrangian can by writtenas

L(S, λ)= 1

2(S− vSm) · M (S−Sm)−λ(S · Zn −�Wn),

(31)

whose stationarity conditions are

0 = M (S − Sm) − λZn ,

0 = S · Zn − �Wn . (32)

Solving the first equation for the stress we obtain

S = Sm + λ M−1Zn , (33)

which, replaced in the second equation allows to find theLagrange multiplier,

λ = �Wn − Sm · Zn

Zn · M−1 Zn. (34)

Finally, inserting the value of this multiplier in (33) we obtainthe final value of the projected stress

S = Sm + �Wn − Sm · Zn

Zn · M−1 ZnM

−1 Zn . (35)

The stress tensor defined by the previous equation belongsto En, n+1 and hence, it can be used to formulate an energyand momentum conserving method, if it is a second orderapproximation to Sm . This can be verified with the aid ofTheorem 1.

Remarks 3 If the tensor M coincides with the identity onSym, the expression (35) simplifies to (17), the originalenergy-momentum formula by Simo and Gonzalez. Thus, weconclude from the previous geometric interpretation that theoriginal conserving stress is the closest tensor to Sm that sat-isfies directionality, when distance is measured in the Frobe-nius norm.

4.5 A restricted class of stress formulas

We have shown that, given an appropriate norm, a stress ten-sor can be formulated which results in an energy-momentummethod. Next we analyze a specific class of norms whichare particularly simple. For that, let M ∈ Sym be positivedefinite and define the fourth order tensor M : Sym → Symto be the linear map

M T = MTM . (36)

Using such fourth order tensor to construct a norm we define,for all T ∈ Sym, its norm

||T ||M = (MT · TM )1/2 . (37)

As before, if this norm is “close” to the Frobeniusnorm, the convex problem (22) defines an energy-momentumstress. The advantage of the norm (37) with respect to (29)

is that in the former it suffices to study the second order ten-sor M to ascertain the validity of the norm. This is shown inthe following result:

Corollary 1 Let M ∈ Sym be uniformly bounded fromabove and below for every energy bounded solution, i.e.,there exist two positive constants A1, A2, independent of thesolution, such that

A1 ≤ ‖M‖2 ≤ A2 . (38)

Then the norm (37) defines an energy-momentum methodthrough the minimization (22).

Proof It suffices to prove that the norm induced by M isuniformly equivalent to the norm ‖·‖ and the result followsfrom Theorem 1.

Since the tensor M is symmetric, positive definite, itseigenvalues, denoted {λa}3

a=1 are real and positive. It isstraightforward to verify that, for every T ∈ Sym,

||T ||2M = MT · TM =3∑

a,b=1

λa λb Tab Tab , (39)

where Tab are the components of the tensor T in the principalbasis of M . Then

||T ||M ≤ maxa

λa

⎛⎝ 3∑

a,b=1

Tab Tab

⎞⎠

1/2

= ‖M‖2 ‖T ‖ ≤ A2 ‖T ‖ ,

||T ||M ≥ mina

λa

⎛⎝ 3∑

a,b=1

Tab Tab

⎞⎠

1/2

=∥∥∥M−1

∥∥∥−1

2‖T ‖ ≥ A−1

1 ‖T ‖ . (40)

Hence, the || · ||M is uniformly equivalent to the Frobeniusnorm on Sym and the results of Theorem 1 apply. ��

The practical consequence of this results is that, in orderto define an energy-momentum method, it suffices simplyto define a symmetric, positive definite tensor M , as longas this tensor is well conditioned for all time step sizes andenergy bounded motions. This opens the door to the designof different conserving methods. Some examples, in additionto the original method, are discussed in Sect. 5.

5 Analysis of three possible stress formulas

We conclude by using the results of Sect. 4 to analyze thevalidity of three stress expressions, leading all to conservingmethods, although not necessarily energy-momentum algo-rithms.

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5.1 Example 1

A general class of relatively simple conserving stress for-mulas can be obtained by selecting M = (Cm)γ , γ ∈ R.In particular, the value γ = 1 was suggested in [13], where itis argued that the contra-contravariant nature of C−1

m infersthe stress approximation (35) with the correct tensorial char-acter.

According to Corollary 1, the choice M = (Cm)γ leadsto a valid energy-momentum method if (Cm)γ is uniformlybounded from above and below. To prove this we note that,for sufficiently small time steps, the approximation (Cm)γ

is arbitrarily close to Cγ . The norm of the deformation ten-sor is itself bounded from below and above for a arbitraryenergy bounded motions due to the coerciveness of the storedenergy function and its behavior as det C → 0. See, e.g.,[24, p. 148].

5.1.1 Example 2

It has been suggested in the literature [22] that a simplescaling of the average stress leads to an energy-momentummethod. More precisely, the stress definition

S := �Wn

Sm · ZnSm , (41)

is a symmetric tensor that satisfies directionality by construc-tion. It can be shown, however, that this appealing expressiondoes not define a proper energy-momentum method. A directargument follows when analyzing the size of the scaling fac-tor, which can not be uniformly bounded. Additional insightcan be obtained from the analysis of Sect. 4 by consideringthe β-dependent minimization problem

min.S∈Sym1

2||S − Sm ||2β ,

s.t. S ∈ En, n+1 (42)

where the norm employed is

||T ||2β := T · [N ⊗ N + β(I − N ⊗ N )] T , (43)

for every T ∈ Sym, with N = Sm/ ‖Sm‖, and β > 0.This minimization program is of the form (22), with an

objective function that is an anisotropic norm (squared),weighting differently the difference S − Sm in the directionof N and its orthogonal complement. By increasing β, thestress S is forced to align with Sm . In the limit, β → ∞, thetensor S must be in the direction of Sm and (41) is recoveredas the minimizer.

Since the norm || · ||β derives from an inner product, The-orem 1 applies, and to check the validity of the approxima-tion (41) as a conserving stress the uniform equivalency ofthe norm (43) has to be verified in the limit β → ∞. To show

that this is not the case, let T ∈ Sym be such that ‖T ‖ = 1and T · Sm = T · N = 0. Then

||T ||2β=T · [N ⊗ N + γ (I − N ⊗ N )

]T=β ‖T ‖2 =β,

(44)

which can be arbitrarily larger than ‖T ‖ = 1. We concludethat as β → ∞ the norm || · ||β fails to be uniformly equiva-lent to the Frobenius norm, providing geometrical insight asfor why expression (41) is not accurate enough.

5.1.2 Example 3

The choice M = Zn is attractive because, if it were valid, itdetermines a stress formula of the form

S = Sm + 1

3(�Wn − Sm · Zn)Z−1

n , (45)

much simpler than the standard expression (17). However,the norm of Zn can not be uniformly bounded for arbitrary(energy bounded) solutions, and thus its induced norm doesnot result in a second order accurate integration scheme.

6 Summary and conclusions

We have shown that there exist an infinite number of secondorder, energy-momentum methods for nonlinear elastody-namics. The crucial part for their formulation is the definitionof a certain stress approximation satisfying various condi-tions which enforce the conservation properties, while pre-serving the second order accuracy, for every energy boundedsolution. While it is relatively straightforward to formulateconserving methods, the analysis reveals that it might be atthe expense of spoiling the accuracy of the integrator.

To establish a constructive proof of the existence of infi-nite conserving, and consistent, methods we have cast theproblem of formulating conserving stresses as that of a localconvex program. In such a picture, a conserving stress can beinterpreted as the closest projection of an average stress ontothe set of stresses with the right energy balance. The keyof the analysis turns out to be the choice of the projectionoperator, which must be uniformly well-behaved.

Based on the analysis, we have identified stress approx-imations leading to conserving schemes which are secondorder accurate, in some cases, and lose this precision, inothers. It remains to elucidate which of all (valid) energy-momentum methods is optimal. From the point of view ofaccuracy and stability, all the second order energy-momen-tum methods are equivalent. The simplest, however, that wehave found is the original energy-momentum method pro-posed in [6], thus becoming the optimal from the implemen-tation point of view.

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610 Comput Mech (2012) 50:603–610

Acknowledgments The author acknowledges partial financial sup-port from the Spanish Ministry of Science and Innovation under grantDPI2009-14305-C02-02 and the Caja Madrid Foundation for a mobilitygrant funding his stay at the California Institute of Technology during2011/12, where he currently works as Visiting Faculty.

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