Finite element verification in the case of missing data

Sylvain Pavot, Eric Florentin1, Philippe Rougeot, Laurent Champaney

LMT-Cachan (ENS Cachan / CNRS / UPMC / PRES UniverSud Paris)61, avenue du President Wilson, 94235 Cachan, France

Abstract

The objective of this work is to obtain an a posteriori estimate of the discretization error

of a reference problem which, in our case, is a linear elasticity problem solved using the

finite element method. Classically, one assumes that both the reference problem and its

discretized solution are known. The originality of the present work is that some of the

boundary conditions are considered to be missing or incomplete to assess the error. The

proposed method is particularly suited to industrial situations where only the finite element

solution is available. We present and illustrate two possible strategies. The error estimator

used is the error in constitutive relation.

Keywords: Boundary conditions - Finite elements - Construction admissible field - Error

Introduction

Numerical simulation has become an indispensable tool in industry. In the specific case

of mechanical engineering, such tools enable one to predict the response of a structure to

solicitations which may influence dimensioning choices without resorting to costly testing of

prototypes. However, these numerical simulations involve the use of a discretized version of

an initially continuous mathematical model (finite element analysis). Therefore, they lead

to only an approximation of the exact solution of the reference problem. In an industrial

context, the results of a finite element calculation must often satisfy certain quality require-

ments. The purpose of error estimation is to evaluate the distance between the exact solution

1corresponding author : eric.florentin@lmt.ens-cachan.fr

Preprint submitted to Finite Elements in Analysis and Design February 28, 2013

and an approximate solution of the problem. Methods for evaluating the global error due

to the problem’s discretization have been available for several years. Different methods have

been developed for linear elasticity problems. Subsequently, these techniques were extended

to more complex problems. A state-of-the-art review of these methods can be found in

(Ladeveze and Pelle, 2005; Babuska and Strouboulis, 2001; Pares et al., 2006). In some

cases, such as in aeronautical design offices, great emphasis is put on the conservative aspect

of these error estimators. Methods based on the concept of error in constitutive relation lead

to a guaranteed upper bound of the actual error.

This, however, requires a perfect knowledge of the reference model. In practice, the ref-

erence problem is not always completely defined. More precisely, we make the assumption

that the missing data was available to produce the solution (in the computation) but not

anymore to assess the error. In particular, because of file interchange formats or specific

loading situations, the boundary conditions may not be perfectly known. From a mathe-

matical point of view, this situation presents little interest because if the discretization error

is not defined correctly it is difficult to estimate it. However, from a practical standpoint

(especially in an industrial context), this problem occurs regularly.

This issue has already been addressed in preliminary works on error in the stochastic

case. The idea presented in Florentin and Dıez (2012) is to model the lack of knowledge by

using random variables. The work presented here goes one step further and assumes that

some boundary conditions can be missing.

Section 1 reviews the equations of the reference problem, the constitutive relation error

method and a technique for building admissible fields. In Section 2, we develop recovery tech-

niques for the reference model which are necessary in order to calculate admissible fields for

imperfectly known models. Section 3 presents some results obtained using these techniques.

Finally, Section 4 presents an alternative calculation method based on the introduction of

an intermediate problem called a “child” problem.

2

Figure 1: The reference model

1. Error in constitutive relation

1.1. The reference problem

Let us consider a linear elastic static problem under the assumption of small transforma-

tions. This type of calculation, which is easily performed, represents a large proportion of

the calculations routinely carried out by engineers. The structure is defined in a domain Ω

bounded by ∂Ω. The structure is submitted to a prescribed displacement ud over a part ∂1Ω

of the boundary ∂Ω, a prescribed volume force fd

within Ω and a prescribed surface force

density F d over ∂2Ω = ∂Ω− ∂1Ω. Hence the reference problem: find the displacement field

u (M) and the stress field σ (M), defined at all points of structure Ω, which satisfy:

• the kinematic admissibility equations:

u ∈ U , u|∂1Ω = ud (1)

• the static admissibility equations:

σ ∈ S, σ|∂2Ω.n = F d (2)

σ ∈ S, div (σ) + fd

= 0 (3)

• and the constitutive relation:

σ = Kε (u) . (4)

3

where U is the set of the fields u which are regular, and S is the set of the fields σ which

are symmetrical and regular (i.e. with finite energy). ε(u) represents the linearized strain

tensor: ε(u) = 12

(grad(u) + grad(u)t). Now let Uad,0 be the space of the fields which are

kinematically admissible to zero:

Uad,0 = u | u ∈ U , u|∂1Ω = 0 (5)

and Sad,0 the space of the fields which are statically admissible to zero:

Sad,0 = σ | σ ∈ S, div(σ) = 0, σ|∂2Ω.n = 0 (6)

1.2. The discretized problem

In practice, the analytical expression of the exact solution of model (uex, σex) is often

unknown. Therefore, one calculates an approximate solution by introducing a discretized

problem based on a weak formulation of the model’s equations. This leads to a pair (uh, σh)

which is the solution of the discretized problem. This solution is defined from a set of

approximate displacements Uh ⊂ U .

Thus, the finite element problem consists in seeking the pair (uh, σh) which satisfies:

• the kinematic admissible equations:

uh ∈ Uh, uh|∂1Ω = ud, (7)

• the equilibrium equation:

σh ∈ S,∫

Ω

Tr [σh.ε (u∗h)] dΩ =

∫Ω

fd.u∗hdΩ +

∫∂2Ω

F d.u∗hdΩ, ∀u∗h ∈ Uad,0 ∩ Uh, (8)

• and the constitutive relation:

σh = Kε (uh) . (9)

4

1.3. The error in constitutive relation

The difference between the solution of the finite element problem (uh, σh) and the exact

solution of the mathematical problem (uex, σex) is called the discretization error. This error

is due mainly to the approximation concerning the equilibrium equations, the discretization

of the geometry or the approximation concerning the loading.

For the purpose of dimensioning the structure, it is important to be able to quantify that

error and, if possible, to obtain an upper bound. A global measure of the error is defined by:

e = ‖σex − σh‖K−1,Ω (10)

where σex is the stress field which is the exact solution of the reference problem and σh is

the approximate solution obtained using the finite element method. The energy norm is

introduced based on Hooke’s tensor K:

‖•‖2K−1,Ω = (•, •)K−1,Ω =

∫Ω

Tr[•K−1•

]dΩ. (11)

To estimate e, we use an error estimator based on the constitutive relation, introduced by

Ladeveze. A measure of the non-verification of the constitutive relation by an admissible

pair (u, σ) is introduced:

e2rdc (u, σ) = ‖σ −Kε (u)‖2

K−1,Ω . (12)

A pair (u, σ) is said to be admissible if u satisfies Equations (1) and if σ satisfies Equations

(2) and (3). Since field uh satisfies (7), it necessarily satisfies (1); therefore, one generally

chooses u = uh. The construction of field σ is reviewed in Section 1.4. The error in consti-

tutive relation satisfies the Prager Synge relation Prager and Synge (1947) which leads to

the following upper bound :

e2rdc > e (13)

This upper bound of the global error can be used to obtain bounds of the local quantities

thanks to the introduction of a dual problem Becker and Rannacher (2001). The contribu-

tions of this error can also be used to adapt the mesh Florentin et al. (2005).

5

1.4. Construction of the statically admissible stress field

While the admissible displacement field can be obtained directly, the construction of σ

requires a specific recovery step based on the finite element solution σh and the data of the

problem. Different paths can be used see e.g. (Pares et al., 2008; Parret-Freaud et al., 2010;

Maunder and Moitinho de Almeida, 2012; Kempeneers et al., 2010)

The construction we use here consists of two steps:

• Construction of nodal densities F h

The nodal densities F h are defined along the edge Γ belonging to ∂E, the set of the

edges of each element. These densities are calculated to be in equilibrium with the

problem’s boundary conditions, i.e. they satisfy:

F h = F d over ∂2Ω (14)∫E

fd.U∗hdE +

∫∂E

ηΓEF h.U

∗hdΓ = 0 ∀U∗h solid body motion of E. (15)

where ηΓE = ±1 in order to guarantee the continuity of the loads along edge Γ of an

element ∂E. (ηΓEi.ηΓEj

= −1 for two adjacent elements Ei and Ej) Ladeveze and Pelle

(2005). Field F h can be built in several ways. For this work, we chose what is called

the “improved method”, see (Florentin et al., 2011) for more details.

• Construction of σ from F h

Once field F h has been calculated, the field of the admissible stresses is built element

by element. Analytical methods Ladeveze and Pelle (2005) can be used. One can

also use a numerical method. For example, Babuska et al. (1994) showed that the

construction of the stresses in each element through the resolution of a higher-order

finite element calculation leads to good results. Thus, one has an implicit link between

stress σ and F h:

σ = l(F h

)(16)

where l is a linear mapping which corresponds to the resolution of an elastic problem for

each element. The details of the construction of l can be found in (Florentin et al., 2011;

Florentin and Lubineau, 2010).

6

2. Recovery of the reference model

In practice, in the case of industrial calculations, the problem is rarely completely defined.

Some information is often missing. The data were available to produce the finite element

solution but not anymore to estimate the discretization error. In particular, one must know

∂1Ω, ∂2Ω, Fd or ud in order to use the error in constitutive relation presented in Section 1.3.

One option would be to redefine a reference problem based on the finite element results and

whatever data are available.

In this section, we will study (and propose solutions for) several cases which occur fre-

quently, from the ideal case in which everything is known to the critical situation in which

all the boundary conditions are unknown. We will show that the adjunction of known in-

formation improves the effectiveness of the error calculation. The objective is to recover

the incomplete reference model by approaching reality as closely as possible. The different

Cases ([C1] trough [C4]) are presented in Sections 2.1 through 2.4. These different cases

correspond to different amount of missing data. This classification brings up simple conclu-

sions: the quality of estimation is increased when using available data (when available) in

the construction. In other hand, taking into account different cases (data available or not)

increase the complexity of the algorithm developed. The numerical results corresponding to

these cases grouped in Section 3 illustrate these conclusions.

2.1. The case of a completely known model – [C1]

First, we consider the ideal case in which the model is completely known for verification.

In this situation, one can use the classical methods, here we use the one developed in Florentin

et al. (2011) ; this is the classical case.

2.2. The case of a partially degraded model: no knowledge at all of Fd – [C2]

Now, we consider the case in which the loading quantity is imperfectly known, or not

known at all. Industrial problems involve specific boundary conditions. For example, bound-

ary conditions as resultants : traction F d are not known in each point of ∂2Ω but are often

7

given as : ∫∂2Ω

F ddΩ and/or

∫∂2Ω

x ∧ F ddΩ

or as 2D-1D transitions which is equivalent.

Other situations as elastic foundation or symmetries, periodicities are also possible. It

is possible to develop a strategy for each case to determine the error. For such situations,

one would need to implement all the boundary conditions into the calculation codes used

to determine the admissible loading. In practice, this is difficult because it would require

an ad hoc construction for each boundary condition available in the code. With the code

changing with each new version, it would mean developing the associated construction for

each upgrade of the code. In this case, an other possibility is to forget the type of boundary

condition and construct it as if it was missing, which is a very simple way to implement it

in an industrial code.

Therefore, let us focus on missing information concerning the magnitude of the prescribed

load. In this case, the error can no longer be calculated using the classical method Florentin

et al. (2011). Indeed, the model is considered as not completely known for assessing the

error. The incomplete model can be described using

• the kinematic admissibility equations:

u ∈ U , u|∂1Ω = ud (17)

• the static admissibility equations:

σ ∈ S, σ|∂2Ω.n =? (18)

σ ∈ S, div (σ) + fd

= 0 (19)

• the constitutive relation:

σ = Kε (u) . (20)

To calculate the admissible field σ, one tries to approximate the actual model, which is

partially known, as closely as possible. An important point is the definition of the actual

8

model. We consider that the actual model is the one used to compute the finite element

solution which is known. At the end of the finite element calculation, for different reasons,

we miss information. Here, the information missing is Equation (18). Then the idea is to

recover the actual boundary limit which has been forgotten from the finite element uniquely.

In order to do that, one uses the results of the finite element calculation (displacements uh

and stresses σh), which are known. Two approaches can be considered. The first approach

consists in using the results of the finite element code directly to build the missing boundary

condition, denoted (F d1). The second solution consists in seeking the boundary conditions

which, after optimization, come as close as possible to the finite element stress, denoted F d2.

2.2.1. Direct construction of the missing data: F d1

One seeks an approximate value of the load given by the finite element solution of the

problem. If the mesh is sufficiently refined, one has: σh.n ' F d. In any case, after con-

vergence, one has: σh.n = F d. With an arbitrary mesh, in general, the equilibrium is not

satisfied: σh.n 6= F d. However, this is a very simple approximation which can be retained as

a first try for a simple recovery of our model.

Thus, the incomplete equation (18) is approximated by:

σ ∈ S, σ|∂2Ω.n = F d1 (21)

with:

F d1 = σh.n (22)

The values of σh used over ∂2Ω are obtained by interpolation from the values of the finite

element stresses at the Gauss points. The choice of the interpolation is very important. We

use here the classical interpolation using finite element shape functions.

Remark: An other interesting point of view is to use nodal reaction easy computable dur-

ing the finite element analysis. This choice is not retained in this study because the nodal

reactions are not always available.

9

2.2.2. Optimum balanced construction of the missing data: F d2

In this second approach, one does not build the unknown load directly from stress σh,

but one tries to get as close to it as possible. However, equilibrium of the recovered load is

enforced. This leads to a minimization with constraint. The model thus recovered is more

consistent in terms of equilibrium. Equation (18) is approximated by:

σ ∈ S, σ|∂2Ω.n = F d2 (23)

with:

F d2 = σ?|∂2Ω.n (24)

The stress field σ? is recovered using a global minimization under constrain of equilibrium:

σ? = argmindiv(σ)+fd=0

‖σ − σh‖K−1,Ω (25)

The minimization (25) that conduce to recover σ?. It is done in 2 steps following the path

described section 1.4 for σ:

• Construction of nodal densities F?

These nodal densities F?

are defined along the edge Γ belonging to ∂E, the set of the

edges of each element like F?. But these densities are calculated to be in equilibrium

only with the body force field fd, i.e. they satisfy:∫

E

fd.U∗hdE +

∫∂E

ηΓEF

?.U∗hdΓ = 0 ∀U∗h solid body motion of E. (26)

• Construction of σ? from F?

The analytical or numerical solving of div(σ) + fd = 0 over each element conduces to

the same implicit link between stress σ? and F?.

σ? = l(F?)

(27)

The main difference between the classical method for construction of σ (see section 1.4) and

the construction of σ?, is that equation (14) is not enforced here. As far as F d is not known

10

over ∂2Ω, this constraint is naturally not enforced. The main drawback is that σ? is not

admissible. It is in equilibrium in the volume but not with the unknown boundary limits.

Along an element edge Γ, the nodal densities F?

are chosen linear, which corresponds to the

simplest choice:

F h|Γ =∑i∈IΓ

riΓλiΓ (28)

where λiΓ are the linear shape functions associated with edge Γ, and IΓ the associated set

nodes of the edge Γ being considered.

One can build a column vector rΓ = (riΓ)i∈IΓ by adding the riΓ together. One can also build

the column vector rΩ = (rΓ)Γ∈Ω which combines the set of the rΓ for all the edges Γ of the

structure Ω being considered. Then, the energy ‖σ − σh‖K−1,Ω of the problem (25) can be

expressed as a quadratic form of rΩ:

‖σ − σh‖2K−1,Ω = ‖σ‖2

K−1,Ω − 2 (σ, σh)K−1 + ‖σh‖2K−1,Ω (29)

= rtΩLrΩ − 2btrΩ + ‖σh‖2K−1,Ω (30)

Using notation rΩ, the constrain (26) writes in a linear form MrΩ = s.

argminMrΩ=s

(1

2rtΩLrΩ − btrΩ

)(31)

Minimization (31) is carried out exactly thanks to the introduction of Lagrange multipliers,

which leads to rΩ and, therefore, to σ?.

2.3. The case of a partially degraded model: partial lack of knowledge of Fd – [C3]

In some specific situations, certain data may be well-known. Quite frequently, even

when some of the applied loads are not known well, there are free edges and simple, known

solicitations over part of the boundary of the structure. This available information must be

taken into account rather than approximated to assess error.

In this case, Equation (18) of the problem is divided into two parts:

σ ∈ S, σ|∂2kΩ.n = Fd (32)

σ ∈ S, σ|∂2uΩ.n =? (33)

11

where ∂2kΩ corresponds to the zone where the loads are known, and ∂2uΩ to the zone

where the loads are unknown.

Remark: In practice, the zone where the applied loads are known corresponds to the free

edges; then, one writes Equation (18):

σ ∈ S, σ|∂2kΩ.n = 0 (34)

σ ∈ S, σ|∂2uΩ.n =? (35)

Of course, this is a particular case of the previous situation which could be addressed

in the same way. However, it is wiser to take into account the maximum amount of known

information for the recovery of the actual model. In this case, the applied loads are divided

into two parts. One part is ∂2kΩ and the other is an uncertainty zone ∂2uΩ where the exact

value of Fd is only approximated. The approximation of load F d over part ∂2uΩ can be

carried out through Fd1 or Fd2 as described in the previous section.

2.4. The case of a fully degraded model – [C4]

In some cases (e.g. partial data transfer, code deficiencies, exotic boundary conditions,...)

the model transmitted to the error code may be degraded to the point that only the finite

element mesh, the stress field and the displacements are available, but no data concerning

the boundary conditions are known. ∂1Ω and ∂2Ω are also unknown.

In this case, in order to calculate the error, since the initial model is completely degraded,

one rebuilds an approximate model which contains some generalized boundary conditions

over its edges. As before, one recovers the model by approximating the actual boundary con-

ditions as closely as possible using a field obtained by optimization. Thus, the admissibility

equation of the approximate problem becomes:

σ ∈ S, σ|∂Ω.n = Fd2 (36)

Remark: In this case, approximation F d1 is not applicable because the poor recovery achieved

does not lead to equilibrium. In particular, the global equilibrium of the structure, which is

a necessary condition for a solution to exist, would not be guaranteed.

12

3. Numerical results

This section presents the numerical results obtained for two sample problems using the

different recovery methods described in Sections 2.1 to 2.4. The first problem is a holed

plate, i.e. a 2D academic test. The second example is a 3D part taken from an industrial

calculation. For both examples, the exact model discretization error eex was obtained by

carrying out a finite element analysis based on a highly refined mesh, leading to a quasi-exact

solution. Classically, the effectiveness index is defined as the ratio of the estimated error ec

to that exact error:

η =eceex

(37)

3.1. The 2D test case

Let us consider the 2D problem of a holed plate which is fixed along one side and loaded

along the opposite side. The mathematical model was discretized into 604 linear elements

to obtain the numerical solution (Figure 2). In order to find the exact error, we calcu-

lated a quasi-exact solution using a mesh with 17, 032 quadratic elements. With the exact

discretization error thus obtained, we calculated the effectiveness indices of the estimators

for various situations, denoted [C1], [C2], [C3] and [C4], corresponding to the situations de-

scribed in Sections 2.1, 2.2, 2.3 and 2.4 respectively. Both quasi-exact solution and coarse

finite element are computed with the exact boundary limits which are known at this step.

Then the error is assessed for different level of knowledge of the data corresponding to Cases

[C1], [C2], [C3] and [C4]. With this definition of the exact error, the effectiveness indices

give information about quality of data recovering technique.

[C1] [C2]− Fd1 [C2]− Fd2 [C3]− Fd1 [C3]− Fd2 [C4]− Fd2

η 1.031 1.251 0.912 1.036 1.004 0.912

Table 1: The values of the effectiveness indices for the 2D problem

The results are given in Table 1. One can observe that the best approximation of the

exact error (i.e. the one whose indicator is closest to 1) corresponds to the calculation of

13

Figure 2: The model for the 2D test case

the admissible field Fd2. One can also note, by a general comparison of situations [C2]

and [C3], that the introduction of known data improves the estimator. In other words, the

introduction of known data brings the recovered model closer to the actual model. Hence

the importance of taking into account the maximum amount of known information.

One can observe that some effectiveness indices are less than 1. A consequence of the

approximations made in recovering the model is the loss of the conservative aspect of the

constitutive relation error estimator. Even though the loss of the mathematical proof is

unavoidable, in practice, results leading to effectiveness indices greater that 1 would be

highly desirable. This conservative aspect, even if it is only heuristic, is very important for a

use in design offices. Therefore, we propose an alternative method which is designed to lead

to an upper bound, even if this property is heuristic.

3.2. The 3D test case

Let us consider now the 3D industrial calculation based on a part of Ariane V launcher.

The structure was built-in over Zone A, a complex load was applied over Zone B. The finite

element model was discretized into 6912 linear elements (Figure 3). Method [C1] is not

available for this type of boundary limits in the code we develop. This is often the case with

14

Figure 3: The model for the 3D test case

industrial complex models.

[C1] [C2]− Fd1 [C2]− Fd2 [C3]− Fd1 [C3]− Fd2 [C4]− Fd2

η - 0.980 0.823 1.001 0.977 0.823

Table 2: The values of the effectiveness indices for the 3D problem

4. Recovery of a child model

In the previous study, the objective was clearly to recover a model which is as close as

possible to the unknown actual model for purpose of verification. Obviously, since the method

involves approximations beyond one’s control, the guaranteed nature of the constitutive error

estimation is lost. However, in some cases, it is desirable to obtain results with a guaranteed

quality, even if this guarantee is only heuristic, i.e. the conservative aspect of the proposed

estimation remains unproven (unguaranteed), but can be assessed numerically for practical

cases.

In this section, we propose a different approach. We introduce a new model, which is

not necessarily close to the unknown initial model (with no missing data), but for which

it is possible to derive a solution with a guaranteed associated error. Then, thanks to the

15

Modèle'de'référence'

Modèle'de'fils' problème'éléments'finis'

(ex, uex)

(h, uh)(, u)

em

ec

Reference model

Child model Finite element model

Figure 4: The error between models

concept of model error, we introduce a distance which is the model error between this new

model and the unknown model (with missing data).

4.1. The child model

The child model is a model based on the same geometry and the same material as the

initial model, in which the boundary conditions alone differ from the reference model.

The equations of the child model are:

• the admissibility equations:

u ∈ U , u|∂Ω = ud = uh (38)

• the equilibrium equations:

σ ∈ S, div (σ) + fd

= 0 (39)

• the constitutive relation:

σ = Kε (u) . (40)

This model, called the “child” model, is defined based on the finite element solution of the

reference model.

16

It is interesting to note that if one chooses the same discretization space Uh as for the

reference problem defined in Section 1.1, the child problem has the same finite element

solution (uh, σh ).

As in the case of reference problem 1.1, one can define the concept of admissibility to

zero. The spaces Uad,0 and Sad,0 are expressed as:

Uad,0 = u | u ∈ U , u|∂1Ω = 0 (41)

and

Sad,0 = σ | σ ∈ S, div(σ) = 0 (42)

4.2. Decomposition of the discretization error

The discretization error ‖e‖2K,Ω which we seek to estimate is divided into two parts:

e2 = ‖σex − σex‖2K−1,Ω + ‖σex − σh‖2

K−1,Ω (43)

The term emod = ‖σex − σex‖K−1,Ω corresponds to the model error. The term e =

‖σex − σh‖K−1,Ω corresponds to the discretization error of the child model.

The proof is straightforward if one notes that:

(σex − σex, σex − σh)K−1,Ω = 0 (44)

Indeed:

(σex − σex) ∈ Sad,0 (45)

and

(σex − σh) = Kε (uex − uh) , (uex − uh) ∈ Uad,0 (46)

4.3. Upper bound

Here, our objective is to estimate the two terms emod and e introduced previously.

• estimation of e

The term e can be estimated directly because it corresponds to a discretization error for

a problem (the child problem) which is completely known and whose discretized solution

17

(σh = σh) is easily available. One can apply the constitutive relation estimation techniques,

build an admissible field for this problem and obtain the error ‖σh − ˆσh‖K−1,Ω. Therefore,

the estimate of term e corresponds to a strict upper bound:

e ≤ ‖σh − ˆσh‖K−1,Ω (47)

• estimation of emod

A classical way to estimate the model error between two exact quantities ∆ex is to use

their discrete representations ∆h:

∆ex ≈ ∆h (48)

Here, we choose σh to represent σex (because it is an easily available quantity) and we choose

ˆσh to represent σex. Thus, the estimate proposed for emod is:

emod ≈ ‖σh − ˆσh‖K−1,Ω (49)

Remark: the choice of σh to represent σex would have been possible, but was not retained

because it would have led to an estimate of the model error equal to zero. This estimation

would have systematically underestimated a nonzero model error.

Then, we obtain the following approximation of our global error e:

e / 2.‖σh − ˆσh‖K−1,Ω (50)

Remark: Of course, this upper bound is not guaranteed. However, one can assume that

the approximation introduced (49) tends to produce an upper bound:

emod / ‖σh − ˆσh‖K−1,Ω (51)

Indeed, the proof is straightforward in the limit case where ∂1Ω = ∂Ω, in which e = e and

emod = 0 while ‖σh− ˆσh‖K−1,Ω is nonzero. Otherwise, unfortunately, there is no hard-and-fast

proof.

18

4.4. Numerical results

Let us revisit the examples of Section 3.

4.4.1. The 2D test case

The error in the discretized model of the holed plate was calculated using the method

presented in Section 4. We obtained an error value with an effectiveness index η equal to

1.27. Contrary to the previous cases, that value does not violate the heuristic guarantee of

the method. However, it is farther from the exact error than the values obtained using the

other estimation methods.

4.4.2. The 3D test case

The error in the discretized model of the industrial 3D part was also calculated based on

the child model. The effectiveness index η is equal to 1.162. The conclusion are the same

that on the previous 2D academical example.

5. Conclusion

We proposed two estimation methods dedicated to the industrial situation where data

are missing. The first method leads to effectiveness indices which are close to 1, i.e. errors

which are close to the exact value, but with a tendency to underestimate it. The second

method, in practice, lead to effectiveness indices which are systematically greater than 1.

This work clearly emphasized the need to take into account the known data. This should

serve as a reminder of the importance of the transmission of data between CAD and calcu-

lation codes Hamri et al. (2010).

This work falls within the scope of the simulation of linear elasticity problems. It could

be followed by an extension to assemblies of parts involving contact and friction (Coorevits

and Bellenger, 2004; Bellenger and Coorevits, 2005; Champaney et al., 2008).

Acknowledgments

This work was carried out as part of project ANR-09-COSI-012-01, ROMMA sponsored by

ANR (the French National Research Agency).

19

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