Strategies for computing goal-oriented a posteriori error measures in non-linear elasticity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2002; 55:879–894 (DOI: 10.1002/nme.513)

Strategies for computing goal-oriented a posteriori errormeasures in non-linear elasticity

Fredrik Larsson, Peter Hansbo and Kenneth Runesson∗;†

Department of Solid Mechanics; Chalmers University of Technology; S-41296 G�oteborg; Sweden

SUMMARY

We investigate the characteristics and performance of goal-oriented a posteriori error measures for aclass of non-linear elasticity models, while restriction is made to small strain theory. The chosen errormeasure of the displacement �eld can be global or local (probing the chosen quantity in a speci�cspatial point). The error is computable with the aid of the solution of a dual problem whose datadepend on the error measure. The main thrust of the paper is to evaluate the performance of a fewdi�erent approximation strategies for computing the dual solution. The chosen strategies are comparedin terms of accuracy, ease of implementation, reliability and cost-e�ciency. A well-known numericalexample, the Cook’s membrane, is used for the numerical evaluations. Copyright ? 2002 John Wiley& Sons, Ltd.

KEY WORDS: adaptivity; error estimation; elasticity

1. INTRODUCTION

From an engineering viewpoint it is important to assess the quality of a computational proce-dure with respect to its capability of predicting quantities of engineering signi�cance with highdegree of accuracy. This means that goal-oriented error measures should be used in adaptive�nite element (FE) algorithms. In particular, pointwise estimates of the error in displace-ments and stresses (e.g. the von Mises equivalent stress) are usually of interest in engineeringpractice. However, until recently most algorithms have been based on global error measures,viz. the strain energy of the error. The main reason is that the mathematical theory forenergy measures is more straightforward and (in some sense) simpler to implement. Amongthe extensive literature we mention Babu�ska and Rheinboldt [1], Ladev�eze and Leguillon [2],R�uter and Stein [3], Carstensen and Funken [4]. It is also possible to use rather heuristic ap-proaches, such as those proposed by Zienkiewicz and co-workers [5], to achieve good resultsfor problems of the self-adjoint-type. A typical example is elasticity.

∗Correspondence to: Kenneth Runesson, Department of Solid Mechanics, Chalmers University of Technology,S-41296 G�oteborg, Sweden.

†E-mail: [email protected] 2 January 2001Revised 9 October 2001

Copyright ? 2002 John Wiley & Sons, Ltd. Accepted 19 December 2001

880 F. LARSSON, P. HANSBO AND K. RUNESSON

Recent developments to establish strategies for goal-oriented error computations rely heav-ily on the idea of solving a dual problem, which is suitably linearized around the knownFE-solution. In this fashion it is possible to account for the local residual and the transportof the error within the considered domain independently of the character (elliptic, parabolic,hyperbolic) of the problem. This idea, which was introduced by Johnson and co-workers (see,e.g. References [6–8]) and further developed by Rannacher and co-workers (see, e.g. Refer-ences [9, 10]), is now being exploited quite extensively from di�erent perspectives. Relatedwork has been carried out by, e.g. Cirak and Ramm [11], and Larsson et al. [12, 13]. Onestrategy, which is pursued by Prudhomme and Oden [14, 15], Ohnimus et al. [16], and R�uteret al. [17], is to aim for upper (and lower) bounds using the Cauchy–Schwarz’ inequalitybased on the energy product of the error and the dual solution. However, this approach canbe used only for self-adjoint problems, and no good estimation of the actual error can beexpected (since the aim is to establish guaranteed bounds). The most straightforward andgeneral approach is to exploit the (formally exact) error representation format in a directfashion. The main problem is then to compute the dual solution in an e�cient and economi-cal manner while retaining the characteristic features that are necessary in order to ensure ahigh predictive capability, say for the error transport.In this paper we �rst recall the general framework for a posteriori error computation for

an abstract solid mechanics problem involving a non-self-adjoint di�erential operator. Themain thrust of this paper is to elaborate on cost-e�ective methods for computing good ap-proximations of the (exact) dual solution. Each method involves, as the starting point, thecomputation of a dual FE-solution on the same mesh as the primal problem. This FE-solutionis then improved in a postprocessing stage using spatially global or local strategies. We em-phasize that we aim for good error estimates in the sense that the e�ectivity index is closeto unity, whereas it is obvious that no guaranteed upper (or lower) bound can be obtained inthis fashion, since this would require knowledge of the exact dual solution. We then focus onthe (self-adjoint) non-linear elasticity problem with restrictions to small strains while employ-ing selected goal-oriented error measures. The di�erent methods are compared numericallyfor the Cook’s membrane problem, and their possible merits in terms of accuracy, case ofimplementation and reliability are discussed.

2. ERROR ESTIMATE USING A DUAL PROBLEM—ABSTRACT FORMAT

2.1. Derivation of exact error representation

In order to be complete, we repeat the general strategy in establishing the error representationfor an abstract problem as described by e.g. Eriksson et al. [8] and Rannacher [10]. We thusconsider the general non-linear PDE in space time

A(u) = f in � (1)

where A(u) is a non-linear operator on the domain � and f are given data. Equation (1) issubjected to the appropriate boundary and initial conditions.The standard variational format of (1) is the following: Find u∈V such that

a(u; u′) = F(u′) ∀u′∈V (2)

Copyright ? 2002 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 55:879–894

GOAL-ORIENTED A POSTERIORI ERROR MEASURES 881

where u is the exact (weak) solution, a(· ; ·) is a semi-linear form (linear in the secondargument), F(·) is a linear functional and V is the appropriate Sobolev space (with theappropriate regularity and boundary=initial conditions).In order to de�ne the corresponding FE-solution, we introduce the approximation space

Vh⊂V, where h represents the mesh function on �. We thus solve the problem: Find theapproximate solution uh∈Vh such that

a(uh; u′h)=F(u′h) ∀u′h∈Vh (3)

If it is assumed that uh is the exact solution (without rounding-o� errors, etc.) of (3) we mayde�ne the error e def= u − uh.We shall now introduce the appropriate secant form(s) of a(· ; ·). Upon using the Gateaux

derivative w.r.t. the �rst argument of a,

a′(u; u; u′) def=@@�a(u+ �u; u′)|�=0 (4)

we may construct the secant form

aT(u; uh;w; u′)def=

∫ 1

0a′(�u(s);w; u′) ds (5)

where �u(s) def= uh + se, s∈(0; 1). The forms a′(· ;w; u′) and aT(· ; · ;w; u′) are non-symmetrical(in general) but bilinear forms in w and u′. Clearly, aT is the tangent form used in Newtoniterations. Noting that

a′(�u(s); e; u′)=@@sa(�u(s); u′) (6)

we may use (5) and (6) to derive the residual

aT(u; uh; e; u′)= a(u; u′)− a(uh; u′)=F(u′)− a(uh; u′) ∀u′∈V (7)

where also (2) was used to give the last equality.From (3) we obtain the Galerkin orthogonality for general non-linear equations

aT(u; uh; e; u′h)=F(u′h)− a(uh; u′h)=0 ∀u′h∈Vh (8)

and this result will be used in the error analysis, cf. below.We shall now turn to the error computation. In order to retain maximal generality in the

formulation, we select the appropriate goal-oriented error measure E(u; uh) that can be any(Gateaux-di�erentiable) functional that satis�es the condition E(u′; u′)=0 for any u′. Hence,E(· ; ·) may be a norm of e as a special case. Upon introducing the ‘goal quantity’ Q(u), wemay de�ne E(u; uh) typically as one of the alternatives:

E(u; uh) =Q(u − uh)=Q(e) ‘quantity of the error’ (9)

E(u; uh) =Q(u)− (uh) ‘error of the quantity’ (10)



Using the Gateaux-derivative of E(· ; ·) w.r.t. the �rst argument,

E′(u; uh; u)def=

@@�E(u+ �u; uh)|�=0 (11)

we de�ne the secant form of E(u; uh) as follows:

J (u; uh;w)def=∫ 1

0E′(�u(s); uh;w) ds ∀w∈V (12)

which is a linear functional in w. We then obtain the relation

J (u; uh; e)=E(u; uh)− E(uh; uh)=E(u; uh) (13)

For given u; uh, we now associate with aT(· ; · ; v;w) the dual bilinear form a∗T(· ; · ; v;w) suchthat

a∗T(· ; · ;w; v)= aT(· ; · ; v;w) (14)

Clearly, in the case that the non-linear problem is self-adjoint, then aT = a∗T, i.e. aT is sym-metric in the last two arguments.We are now in the position to introduce the variational format of the dual problem: Find

�∈V such that

a∗T(u; uh;�; u′)= J (u; uh; u′) ∀u′ ∈V (15)

RemarkBy Riesz representation theorem it is possible to �nd j∈V such that

J (u; uh; u′)= (j(u; uh); u′) ∀u′ ∈V (16)

where (· ; ·) is the usual scalar product. Hence, the strong form of the dual problem can bewritten as

L∗(�) = j in � (17)

where L∗ is a linear (dual) di�erential operator that depends on u and uh.

Having computed �, we can use (13)–(15) to formulate the exact error representation asfollows:

E(u; uh)= aT(u; uh; e;�)= aT(u; uh; e;�− �h) ∀�h ∈Vh (18)

where the last equality was obtained through the Galerkin-orthogonality (8). Typically, wemay choose �h as some projection �� of � onto V, e.g. the nodal interpolant or theL2-projection. Finally, we use (7) to obtain from (18)

E(u; uh)=F(�− ��)− a(uh;�− ��)=NEL∑e=1E(u; uh)�e (19)

where E(u; uh)�e is the contribution to the error from element e.



2.2. Linearization of the dual problem

We shall next introduce certain approximations in (15), since a∗T and J involve the (unknown)exact solution u. We thus replace (15) by the approximate problem

a∗T(�; u′)= J (u′) ∀u′ ∈V (20)

where a∗T and J are de�ned as

a∗T(· ; ·)= a∗T(ua; uh; · ; ·) (21)

J (·)= J (uJ ; uh; ·) (22)

and we have introduced ua and uJ as approximations of u. In practice, we consider ua anduJ as ‘enhanced’ versions of uh.

RemarkThe ‘naive’ choice is ua= uJ = uh. To choose ua= uh is normally su�cient, whereby a

∗T is

simply the adjoint of the lastly computed aT that is used in the Newton iterations to computeuh, from (3). However, the choice of uJ is far more critical, and the choice uJ = uh may becompletely inadequate. For example, in the case

E(u; uh)= ‖e‖⇒ J (u; uh; ·)=(e‖e‖ ; ·

)(23)

then

J (·)=(uJ − uh‖uJ − uh‖ ; ·

)(24)

and it is obviously not possible to choose uJ = uh.

Finally, we note that the dual problem can be solved using the same FE-mesh as the primalproblem, i.e. �h ∈Vh is the solution of

a∗T(�h; u′h)= J (u′h) ∀u′h ∈Vh (25)

3. STRATEGIES FOR COMPUTING THE DUAL SOLUTION

3.1. Preliminaries

Considering the (exact) error representation formula, we note that a key issue is tocompute a good approximation of � (that is a better approximation than �h). We shall callthis �∈ V the ‘enhanced dual solution’. We also note that (in general) �� =�h althoughboth are in Vh. If V⊂V then

E(u; uh) ≈ F(�− ��)− a(uh; �− ��)=NEL∑e=1E(u; uh)�e (26)

Since we shall (later) consider the elasticity problem, we now turn to purely spatial problemswithout time dependence. Subsequently, we device three di�erent schemes for computing �



Figure 1. Quadratic (hierarchic) basis functions.�1 denotes the linear nodal values and �2 is ahierarchically added nodal value at the midpoint

of the interval.

Figure 2. Local problem on one single element.

in the case that Vh consists of piecewise linear functions on triangular meshes (in 2D) thatemploy the three corner nodes of each triangle. We shall de�ne V as piecewise quadraticfunctions on triangular meshes (in 2D) that employ three side nodes in addition to the cornernodes. Finally, we de�ne the fractional space V = {v∈ V, v = 0 at each corner node inMh}, i.e. V=Vh�V. The corresponding basis functions of V are shown schematically inFigure 1.

3.2. Enhanced FE-space (basic method)

Find �= �1 + �2, with �1 ∈Vh and �2 ∈V, such that

a∗T(�1 + �2; u′1 + u

′2)= J (u

′1 + u

′2) ∀u′1 ∈Vh ∀u′2 ∈V (27)

If �i are the nodal representations of �i (i=1; 2), we may formulate (27) in matrix form asfollows: [

K11 K12K21 K22

][�1

�2

]=

[j1

j2

](28)

We note that only �2 = � − �� is of interest, although also �1 =�� is part of thesolution. Obviously, the cost of solving (28) is signi�cantly higher than that of obtaining theFE-solution �h.

RemarkIn the case of hyperelasticity, or any other self-adjoint problem, the tangent form will besymmetric and we will have a∗T = aT as a special case.

3.3. Enhanced FE-space with ‘decoupled’ solution

We introduce the approximation �1≈�h and solve for �2 ∈V such that

a∗T(�2; u′2)= J (u

′2)− a∗T(�h; u′2) ∀u′2 ∈V (29)

or, in matrix form,

K22�2 = j2 −K21�h (30)



Figure 3. Local problem on element patch.

This corresponds to solving the enhanced FE-problem after introducing ‘�xed supports’ atall corner nodes in Mh. The cost of solving (29) is of the order of that of obtaining theFE-solution �h.

3.4. Locally enhanced FE-space

We shall reconsider (29) and aim at computing �2 approximately on each element, or on apatch of elements. Two di�erent assumptions are considered:Method A: One single element with ‘free’ boundaries is considered at a time. Hence, we

compute �e2 ∈Ve, e=1; 2; : : : ; NEL, such that

a∗T; e(�e2 ; u

′2)= J (u

′2)− a∗T; e(�h; u′2) ∀u′2 ∈Ve (31)

where Ve is the restriction of V to a single element �e. It is noted that (31) involvesunknowns at the three side nodes of each element, which is illustrated in Figure 2.Method B: A patch �e-patch is formed of each element and all its neighbours. We now

compute �e2 ∈Ve-patch, e=1; 2; : : : ; NEL, such that

a∗T; e-patch(�e2 ; u

′2)= J (u

′2)− a∗T; e-patch(�h; u′2) ∀u′2 ∈Ve-patch (32)

where Ve-patch is the restriction of V to �e-patch, which homogeneous boundary conditionson its boundary @�e-patch. Also (32) involves unknowns at the three side nodes of each element,as shown in Figure 3.When �e2 has been computed for all e (=1; 2; : : : ; NEL) using one of the methods, then

the global �2 is computed by weighted averaging of �e2 from two adjacent elements at each

side node.

RemarkAt �rst glance it might appear as if only the local error (and not the error transport) isaccounted for when the error is of the local type, i.e. it represents a single point or (at themost) a particular element. However, error transport is included in �h (the linear part of� that is solved globally) which a�ects �2 through the second term on the right-hand sideof (32).



3.5. Error bounds based on the enhanced dual solution

We shall now consider the case of a self-adjoint, i.e. symmetric, problem and neglect thelinearization error of the dual solution. For the case of hyperelasticity (to be investigated later)the linearized dual problem (both a(· ; ·) and E(· ; ·)) is dependent on data, error measure, etc.It is recalled that we can compute the approximate error estimate

E(u; uh)= aT(e; �− �) ∀�∈Vh (33)

based on the approximate dual solution �∈ V. Furthermore, neglecting the linearization errorwe obtain the exact error representation

E(u; uh)= aT(e;�− �) ∀�∈Vh (34)

based on the exact dual solution �∈V that ful�lls

a∗T(�; v)= J (v) ∀v∈V (35)

We thus obtain the di�erence between the exact and the approximate dual solution as

E − E= aT(e; e�); e�def= �− � (36)

Since we consider the case when aT(· ; ·) is symmetric, Cauchy–Schwartz inequality can beused to give

|E(u; uh)− E(u; uh)|6‖e‖T‖e�‖T; ‖ · ‖T def=√aT(· ; ·) (37)

For instance, the norms in (37) can be estimated from the solution to local Neumann problems,cf. Reference [16]. We thus obtain the lower and upper bounds

E(u; uh)− ‖e‖T‖e�‖T6E(u; uh)6E(u; uh) + ‖e‖T‖e�‖T (38)

RemarkIncreasing the space V means that the bound becomes narrower due to the fact that the norm‖e�‖T will decrease. Hence, the extra computational e�ort that is required to obtain a goodestimate E (of E) will also provide a sharper bound for this estimate. In the special casethat no enhanced solution � is computed whatsoever, then e�= e�

def= �−�h and E(u; uh)=0,which means that (38) simply represents the pure bound that was exploited in References[14, 16]. Finally, we remark that the bounds are never guaranteed in practice. One reason isthat linearizations are introduced in the dual problem (whenever the primary problem or theerror measure is nonlinear). Another reason is that the local Neumann problems that representthe energy norm estimates are not solved exactly.

4. A MODEL PROBLEM OF NON-LINEAR ELASTICITY

4.1. The primal problem

We shall consider a body occupying the domain � with boundary =D ∪N, with D ∩N = ∅.Dirichlet and Neumann boundary conditions are imposed on D and N, respectively. The state



of an elastic body is de�ned by the displacement �eld u(x), x∈�, with the strain operatorU(u)= (∇⊗ u)sym. The stress � and the corresponding elastic tangent sti�ness modulus tensorEe in the case of a hyperelastic material response are given as

�[U]= @�[U]@U ; Ee[U]= @

2�[U]@U⊗ U (39)

where �[U] is the strain energy per unit volume. In the special case of isotropic hyperelasticity,the most general format of � and Ee can be expressed in terms of �=�(i1; i2; i3) with theinvariants ik

def= tr(�k), k=1; 2; 3. Here, we shall consider the situation where it is possible toobtain a complete decoupling of the volumetric and deviatoric stress–strain relations, whichis achieved if we set �=�vol(i1) +�

dev(j2) with j2def= tr(�2dev). This gives

�= d�vol

di1T+ 2 d�

dev

dj2Udev =�octT+

�oct�oct

Udev (40)

where

�octdef= 1

3 tr�=f�(�oct); �octdef= 1

3 i1 (41)

�octdef=

1√3|�dev|=f�(�oct); �oct

def=1√3(j2)1=2 (42)

and f�(�oct) and f�(�oct) can be chosen arbitrarily as long as they are monotonically increasing.We may rewrite (40) as follows:

�=2GsUdev + 3Ks�octT (43)

where the secant sti�ness coe�cients Gs(�oct) and Ks(�oct) are de�ned by

Gs =f�2�oct

; Ks =f�3�oct

(44)

The corresponding tangent sti�ness tensor Ee is obtained as

Ee=2GsIsymdev +

23(Gt −Gs)Udev�oct ⊗

Udev�oct

+ KtT⊗ T (45)

where the tangent sti�ness moduli Kt(�oct) and Gt(�oct) are de�ned as

Gt =f′�

2; Kt =

f′�

3with f′(U) def=

@f@�

(46)

To be speci�c, we choose

f�=2G�oct ; f�=3Kt0�0 ln(1 +

|�oct|�0

)�oct|�oct| Gt =G; Kt =K10

11 + |�oct|=�0 (47)

i.e. the shear response is linear, whereas the power law that de�nes the volumetric responsemay represent carbon-�lled rubber. Here, G, Kt0, and �0 are material parameters. In this case,



we may simplify (44)–(46) as follows:

�=2GUdev + 3Ks�octT (48)

Ee=2GIsymdev + KtT⊗ T (49)

The non-linear elasticity problem may now be formulated as follows in terms of equilibriumand the constitutive relation:

−∇ · �= f with �= �[U(u)] in � (50)

u=0 on D (51)

T def= � · n=Tp on N (52)

where f is the prescribed volume load and Tp are prescribed surface tractions.In order to establish the corresponding variational format, we de�ne V= {u∈ [H 1(�)]3;

u=0 and D}. Moreover, we obtain

a(u; u′) def=∫��[U(u)] : U(u′) dV (53)

F(u′) def=∫�f · u′ dV +

∫NTp · u′ dS (54)

The corresponding tangent forms are

aT(u; uh;w; u′)= a∗T(u; uh;w; u′)=

∫�U(w) :EeT(u; uh) : U(u′) dV (55)

where

EeT(u; uh)def=

∫ 1

0Ee[U(�u(s))] ds (56)

which leads to the choice

aT(w; u′)=∫�U(w) :Ee[U(uh)] : U(u′) dV (57)

In order to solve the primal FE-problem

a(uh; u′h)=F(u′h) ∀u′h ∈Vh (58)

we use Newton’s method: For given u(k)h , �nd u(k+1)h = u(k)h + �uh by solving �uh from

a(k)T (�uh; u′h)= − [a(u(k)h ; u′h)− F(u′h)] ∀u′h ∈Vh (59)

where a(k)T is evaluated from (57) at u(k)h .



4.2. Error measures and dual problem

We shall consider a few goal quantities, for which we consider the ‘error in quantity’:

(1) L2-norm (global and non-linear quantity):

Q(u) = ‖u‖L2 (60)

J (u′) =(

uh‖uh‖L2

; u′)

(61)

(2) Mean value of displacement along part of the boundary N1 (global and linear quan-tity):

Q(u) =1

m(N1)

∫N1t · u dS (62)

J (u′) = J (· ; · ; u′)= 1m(N1)

∫N1t · u dS (63)

where t is the unit tangent vector on N.(3) Equivalent stress in spatial point XB (local and non-linear quantity):

Q(u)= [�e[U(u)]]x=XB ; �edef=

√32|�dev| (64)

J (u′)=[3�dev(uh)2�e(uh)

: Ee(uh) : U(u′)]x=XB

(65)

5. ADAPTIVE STRATEGY

The strategy is based on the goal to equidistribute the error between the elements, i.e. wetry to achieve that the (absolute) error contribution from each element is equal. After eachcomputation (with inadequate accuracy) an h-re�nement of selected elements is performedusing the Rivara algorithm for triangular elements. To achieve the equidistribution, all elementswith a contribution less or equal to the fraction � of that of the highest contribution arere�ned. Furthermore, to make fewer computations in the case that many of the elements havesigni�cant error contributions, we also re�ne each element whose contribution is larger thanthat of TOL=N , where N is the number of elements and TOL is the given tolerance. Thus,we re�ne all elements with the error contribution

|E(u; uh)�e |¿min

�maxe |E(u; uh)�e |;

TOLN

∑e|E(u; uh)�e |∣∣∣∣∑

eE(u; uh)�e

∣∣∣∣

(66)

where E�e , is the estimated error contribution from element e. Note that the tolerance hasbeen scaled with the ratio of the sum of absolute values to the absolute value of the sum oferror contributions. Hence, the cancellation of errors from di�erent elements leads to adopting



Figure 4. Cook’s membrane with a distributed shear load.

Figure 5. Load–displacement relation. Figure 6. E�ectivity index for the error inquantity Q(u)= ‖u‖L2 .

a higher tolerance for the sum of absolute contributions than the tolerance for the sum oforiginal contributions.

6. COMPUTATIONAL RESULTS

We evaluate the performance of the di�erent strategies of computing the dual solution forCooke’s membrane, which is depicted in Figure 4. The single loading is a distributed shearload of uniform intensity along the boundary N1.The typical non-linear load–displacement relation T2|x∈N1 versus u2|x=XA is shown in

Figure 5. For the chosen material parameters, Kt0 = 100, �0 = 11000 and G=100, �ve Newton

steps were required to achieve a converged result if the whole load Tp was applied in onesingle step.For each choice of goal function Q(u) we show the e�ectivity index, the convergence

(based on the estimated error) for di�erent types of mesh-re�nement and the �nal mesh w.r.t.a given tolerance; cf. Figures 6–14. The e�ectivity index � is de�ned as

�=E(u; uh)E(u; uh)

(67)

where E is the estimated error, whereas E is the true error of the FE-solution uh.



Figure 7. Convergence for the (estimated) errorusing adaptive and uniform mesh-re�nement. Theerror in the quantity Q(u)= ‖u‖L2 is considered.

Figure 8. The �nal, eighth, adaptive mesh set bythe tolerance (Q(u) − Q(uh))=Q(u)61% for the

quantity Q(u)= ‖u‖L2 .

Figure 9. E�ectivity index for the error inquantity Q(u)= (1=m(N1))

∫N1t · u dS.

Figure 10. Convergence for the (estimated) er-ror using adaptive and uniform mesh-re�nement.The error in the quantity Q(u)= (1=m(N1))∫

N1t · u dS is considered.

For the local goal function �e[U(u)]x, we also show the e�ect of di�erent linearizations ofJ in the dual problem, cf. Figure 15. It is noted that for simple tangent evaluation around theFE-solution, uh, the e�ectivity index becomes quite poor, while an improved J based on animproved solution, uJ , gives e�ectivity close to unity. Thus, the di�erence in e�ectivity inFigure 15 shows the in�uence of the linearization error.



Figure 11. The �nal, eighth, adaptive mesh setby the tolerance (Q(u) − Q(uh))=Q(u)61% forthe quantity Q(u)= (1=m(N1))

∫N1t · u dS.

Figure 12. E�ectivity index for the error inquantity Q(u)= [�e[U(u)]]x=XB .

Figure 13. Convergence for the (estimated) er-ror using adaptive and uniform mesh-re�nement.The error in the quantity Q(u)= [�e[U(u)]]x=XB

is considered.

Figure 14. The �nal, 12th, adaptive mesh set bythe tolerance (Q(u)−Q(uh))=Q(u)60:5% for the

quantity Q(u)= [�e[U(u)]]x=XB .

The e�ectivity index � is de�ned as

�=E(u; uh)E(u; uh)

(68)

where E is the estimated error, whereas E is the true error of the FE-solution uh.



Figure 15. E�ectivity index for the error in quantity Q(u)= [�e[U(u)]]x=XB using di�erentlinearizations in the dual problem.

7. CONCLUSIONS AND FUTURE WORK

In this paper we have compared di�erent approaches for the numerical evaluation of thedual problem, that is an essential ingredient in goal-oriented error estimation and adaptivity.The most reliable method considered is to use a higher-order approximation in an auxiliarycomputation; unfortunately, this approach is prohibitively expensive, e.g. in the case of time-dependent problems, so simpli�ed versions must be considered. Of the simpli�ed version wehave considered, we prefer the patch-based method: it is as accurate as the other simpli�edmethods, yet it carries a low computational overhead. We emphasize that the choice of ap-proximation of the dual problem is more important for control of ‘di�cult’ quantities, suchas pointwise stresses, where the dual problem may be less smooth.Future work will be concerned with time-dependent non-linear problems. Here, the basic

unanswered questions are how to store information about the primal and dual solutions, andwhether physical damping properties (leading to ‘memory loss’ in time) can be used to sim-plify the evaluation of the computational error. We will also consider strategies for evaluatingthe linearization error for more strongly non-linear problems.

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Strategies for computing goal-oriented a posteriori error measures in non-linear elasticity

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