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COMPUTATIONAL MECHANICSNew Trends and Applications

S. Idelsohn, E. Onate and E. Dvorkin (Eds.)cCIMNE, Barcelona, Spain 1998

ADAPTIVE ANALYSIS OF SOFTENING SOLIDS USING ARESIDUAL-TYPE ERROR ESTIMATOR

P. Dez, M. Arroyo and A. Huerta

Departamento Matematica Aplicada IIIE.T.S. de Ingenieros de Caminos, Canales y Puertos

Universidad Politecnica de Cataluna

Campus Norte UPC, 08034 Barcelona, Spaine-mail: [email protected], web page:http://www.upc.es/ma3/lacan.html

Key words: Adaptivity, Error estimation, softening solids, regularized models

Abstract. This paper focuses on the numerical simulation of strain softening mechan-ical problems. Two problems arise: (1) the constitutive model has to be regular and(2) the numerical technique must be able to capture the two scales of the problem (themacroscopic geometrical representation and the microscopic behavior in the localizationbands). The Perzyna viscoplastic model is used in order to obtain a regularized softeningmodel allowing to simulate strain localization phenomena. This model is applied to qua-

sistatic examples. The viscous regularization of quasistatic processes is also discussed:in quasistatics, the internal length associated with the obtained band width is no longeronly a function of the material parameters but also depends on the bound value problem(geometry and loads, specially loading velocity).An adaptive computation is applied to softening viscoplastic materials showing strain lo-calization. As the key ingredient of the adaptive strategy, a residual type error estimatoris generalized to deal with such highly nonlinear material model.In several numerical examples the adaptive process is able to detect complex collapsemodes that are not captured by a first, even if fine, mesh. Consequently, adaptive strate-gies are found to be essential to detect the collapse mechanism and to assess the optimallocation of the elements in the mesh.

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1 INTRODUCTION

Adaptive mesh refinement in strain softening problems has received important at-tention in the last decade. The presence of two well-differentiated length scales in such

problems seems to indicate that adaptive remeshing strategies, in a general sense, arethe natural approach. Recall that the spatial interpolation of the primitive variablesmust describe both the macroscopic scale associated to the solid geometry and themicro-scale related to the shear band. For instance, [13] shows for two different models,that in the localization area element sizes must be one order of magnitude smaller thanthe internal length if errors under 5% are desired. Moreover, a priori knowledge of thelocation of the localization area is sometimes not possible.

Adaptivity in finite element computations requires three main ingredients. The firstone is an algorithm for increasing/decreasing the richness of the interpolation in aparticular area of the computational domain. For instance, a good mesh generator ifh-refinement [21,38] is used. Second, an error estimator or error indicator must be

employed to locate where there is a need for refinement/de-refinement. And, third, aremeshing criterion must be used to translate the output of the error analysis into theinput of the mesh generator, for instance, the distribution of desired mesh sizes.

These three steps are fundamental in adaptivity. However, only the second one willbe discussed here because it is, probably, the critical one in softening problems. Infact, [23] presented an excellent discussion on the difficulties that have deterred anextensive application of adaptive methods in the context of strain localization: 1) pathdependent constitutive equations, and 2) error estimation relying, in statics, on theellipticity of the equations which is lost at the inception of localization. Here, thesetwo difficulties are overcomed because: 1) an error estimator mathematically sound forpath dependent constitutive equations is employed, and 2) a well known regularizedmodel which precludes the loss of ellipticity is used. The main goal here, is to showthat adaptive mesh refinement based on estimating the actual error is now possiblefor regularized problems. Any model maintaining ellipticity after the inception of thelocalization could be used. Moreover, the numerical examples show that adaptivitybased on the error estimation is essential for accurate computations and, in certainproblems, for capturing a realistic physical behavior.

The study of localization in solids has received much attention and a number ofdifferent approaches have been devised to overcome the difficulties encountered duringits analysis. A possible approach is to consider the limit problem and consequently,

jumps in the displacement field across surfaces (strong discontinuity approach). In

fact, it emanates from classical perfect plasticity where discontinuous displacements areunderstood in a distributional sense, see for instance [30,2,17]. Another possible ap-proach is to regularize the problem precluding any discontinuity in the displacementfield. These models bear the same fundamental property: an internal length is intro-duced to limit the thickness of the localization band. Among these are the micropolarconstitutive models [10], non-local models [7,27], gradient dependent models [11,18],

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and rate-dependent models [32,8]. The present study is developed in the context ofregularized models. Here, a simple rate-dependent model is employed [26] because themain focus is on adaptivity.

The issue of a mathematically sound error estimator for path dependent constitutive

equations is crucial in this study. Adaptivity in softening problems has been associatedto error indicators [25,37,28]. Error indicators are based on heuristic considerationswhile error estimators approximate a measure of the actual error in a given norm. Inthis work, a tool for assessing the error measured in the energy norm is proposed. Theobtained approximation to the error is asymptotically exact, that is, tends to the actualerror if the element size tends to zero [4,5]. In that sense, this tool is an error estimator.

2 PERZYNA VISCOPLASTICITY MODEL

As noted previously, the Perzyna viscoplastic model is used for regularization [26,31].Thus, regularization is associated to viscous effects in the inelastic range. For clarity,

this model is briefly reviewed in this section.In small-strain viscoplasticity, the strain rate is decomposed into an elastic strainrate e and a viscoplastic strain rate vp,

= e + vp,

and the stress rate is obtained as

= C : ( vp),

where C is the elastic moduli tensor. Eq. (1) and (2) are very similar to those of

elastoplasticity, with plastic strain replaced by viscoplastic strain vp

.An expression for vp is needed. For associative flow, the Perzyna model [26] takes

vp =

f

0

Nf

,

where and N are the material parameters of the model, f is the Von Mises yieldfunction which depends on the yield stress [15] This yield stress is assumed to be alinear function of the equivalent viscoplastic strain , according to

= 0 + h,

where 0 is the initial yield stress and h is the hardening (for h > 0) or softening (forh < 0, used here) parameter. The symbol f means that

f =

f if f 00 if f 0

.

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In contrast to elastoplasticity, stress states outside the yield surface (i.e., with f > 0)are admissible in viscoplasticity. Combining Eq. (3) and (5) shows that the viscoplasticstrain rate vp is, in general, non-zero. On the other hand, for stress states inside or onthe yield surface, vp = 0 and the strain rate is purely elastic.

Figure 1. One dimensional rheologic scheme of Perzyna type viscoplastic model (N = 1).

Figure 1 shows the one-dimensional rheologic scheme of the Perzyna model for thecase N = 1 [30]. The viscous effect is represented by the damper in the right, withviscosity

0

/, and softening is represented by the element on the left, which symbolizesthe linear evolution of the yield stress, Eq. (4).

The Perzyna viscoplastic model has been used as a regularization technique in tran-sient dynamic processes [31,8]. Here, it will be employed to regularize a quasistaticproblem.

For dynamic analysis (i.e. inertia effects are considered) the width of the localizationband can be predicted a priori as a function of material parameters [31]. It is inde-pendent of the loading velocity. The rationale is that the stress waves (those producingloading and unloading paths) travel at a celerity which depends only on material param-eters. Thus, in dynamic analysis, the band width depends on the stress wave celerityand the viscous effects.

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0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40

X1.E-4

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

X1.E6

Figure 2. Influence of the loading velocity in the response of the Perzyna model in a quasistatic

case (reaction versus imposed displacement)

For quasistatic analysis processes (i.e. inertia terms are neglected) time does not ap-pear explicitly in the momentum balance. However time is still an independent variableof the problem because it is present in the constitutive relations, Eqs. (1)-(3). Time hasstill its physical meaning and it is not, as in rate-independent elastoplasticity, a loadingparameter (i.e. a pseudo-time). Notice that for quasistatic problems the loading veloc-ity has a crucial influence on the material response, see Figure 2 for a one-dimensional

example. Thus, from a physical point of view, viscoplasticity in quasistatics is still asound regularization technique (i.e. viscous effects and the associated regularization arestill present). Section 4 illustrates this issue with several examples. In fact, viscoplas-ticity is an obviously sound regularization technique in a quasistatic analysis becausea quasistatic problem is simply a dynamic problem where inertia terms are below thethreshold of computational accuracy. Viscoplasticity is not employed here as a numer-ical strategy to avoid imposing the plastic consistency condition. It is assumed to bethe real material behavior. It is not used as a numerical relaxation technique [24]designed to get the elastoplastic solution. In fact, for softening models, the steady statelimit (after zero loading velocity is reached) of a viscoplastic analysis does not coincidewith an elastoplastic solution.

Nevertheless, in quasistatics, the velocity governing the loading and unloading is nolonger the stress wave celerity but the external load velocity. Thus, the band width inquasistatics is not governed exclusively by material parameters, it depends also on theload velocity. In fact, in [20] it is demonstrated that for one-dimensional problems theload velocity plays a crucial role in the resulting band width.

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Another difference between dynamic and quasistatic viscoplastic analysis is relatedwith the need and the influence of imperfections. It is well known that dynamic vis-coplastic problems do not need an imperfection to induce localization, reflecting orincoming stress waves are usually employed to initiate localization. Quasistatic vis-

coplastic problems in simple domains, however, do need of an imperfection to inducelocalization. In any case, both in dynamics and quasistatics, if an imperfection exists,the resulting band width may be affected by the imperfection in its vicinity, see [31,36]for dynamic cases and [3] or Section 4 for quasistatics.

3 ADAPTIVITY BASED ON ERROR ESTIMATION

The key feature of an adaptive strategies is the error assessment. Once the approx-imate solution is computed, it has to be decided if this solution is accurate enough orif a new mesh must be generated in order to obtain an improved solution. The errorassessment fournishes the global error measure (a number) and its distribution in the

domain (a function or a set of numbers). The error information is used to decide ifremeshing must be performed and, if it must be, to determine the element size of thenew mesh in each part of the domain.

The error assessment can be done using error estimators or error indicators. Whileerror estimators fournish objective error measures in a suitable error norm, error indi-cators use heuristic criteria, mainly based on interpolation error estimates, to obtainqualitative information about the error. Error indicators are usually explicitely com-puted, they may be used to identify the zones where the mesh must be finer and theyare cheaper than error estimators which typically require to solve a set of local prob-lems. However, the information given by error indicators cannot be used as an objectiveerror assessment tool because the selection of the error indicator is very sensitive to theanalysed problem.

Here, adaptivity is driven by the residual tipe a posterior error estimator introducedin [12]. This estimator has been generalized to nonlinear problems, precludes the usualdrawbacks of residual type estimators (computation of the flux jumps and flux splitingprocedure) and gives good results in the analysed examples.

Once the error is estimated, the adaptive procedure requires a remeshing criterionin order to generate the input for the mesh generator. The error estimator furnisheslocal measures of the error in each element, that is, ehk for k = 1, . . . This set ofnumbers describes the spatial distribution of the error. The input of a mesh generatoris a distribution of desired element size in the computational domain. Generally, this is

described by the desired element size in each element of the current mesh, that is, hkfor k = 1, . . . Thus, a remeshing criterion is required to translate ehk into hk.Different remeshing criteria have been defined [39,22,19,12] leading to quite different

optimal meshes. This is because the underlying optimality criteria are different. In fact,all these remeshing criteria tend to equidistribute the error in some sense. The choiceof the error function that has to be uniform is related with the underlying optimality

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criterion. In the examples presented in section 4, two remeshing criteria are used andit is shown how the obtained meshes are very different.

In the following, the quadrilateral mesh generator developed by Sarrate [29] is used.This mesh generator supplies excellent unstructured meshes, both ensuring the pre-

scribed element size and the regularity of the elements.

4 NUMERICAL EXAMPLES

Figure 3. Rectangular specimen with one centered imperfection

Two examples are presented in this section. Both of them reproduce the compression

of a plane strain rectangular specimen. In order to induce the strain localization inthe specimen, some imperfection must be introduced. Typically, these imperfectionsare introduced by either a weaker element or a geometric imperfection [31,8]. Here,geometric imperfections are used: circular openings inside the material create a weakerzone in the specimen. The difference between the two presented examples is the numberand the location of these circular openings. In the first example the specimen has one

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centered circular opening and, consequently, the two axes of symmetry allow to studyonly one fourth of the specimen (see Figure 3). In the second example, the specimenhas two circular openings symmetric with respect to the center. That allows to studyonly one half of the specimen (see Figure 4). In both examples the tests are driven by

imposing the velocity at the top of the specimen.

Figure 4. Rectangular specimen with two symmetric openings

4.1 Rectangular specimen with one imperfection

As shown in Figure 5, the collapse mechanism in this example is formed by two

strain localization bands and softening behavior is observed. In Figures 6 and 7 crosssections of equivalent inelastic strain and Von Mises stress are shown. It can be seenthat strain grows in the localization band along the loading process, but Von Misesstress remains almost constant once the failure mechanism is developed. Recall that allthe deformation is localized along the band. Once the band is developed, the specimenbehaves like two sliding rigid blocks: the material inside the band is practically flowing.

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0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

X1.E-2

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

X1.E8

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4

X1.E-2

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

X1.E8

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

X1.E-2

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

X1.E8

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4

X1.E-2

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

X1.E8

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

X1.E-2

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

X1.E8

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4

X1.E-2

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

X1.E8

Figure 7. Evolution of the Von Mises equivalent stress (solid line) and

the yield stress (dashed line) in the loading process

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Figure 8. Influence of the imperfection size: contours and profiles of the inelastic strain.

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Figure 9. Remeshing process using Li-Bettesss criterion, without pollution errors,

for a prescribed accuracy of 0.5%

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Figure 10. Remeshing process using Li-Bettesss and considering pollution errors

for a prescribed accuracy of 0.5%:

succession of meshes and estimated error distributions

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Figure 11. Remeshing process using ULA criterion and considering

pollution errors for a prescribed accuracy of 0.5%:

succession of meshes, accuracy distributions and comparison with LB final mesh

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Figure 12. Remeshing process using Li-Bettesss for a prescribed

accuracy of 1.5%: succession of meshes and estimated error distributions

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This simple example has been used to study the influence of the imperfection size andthe loading velocity in the resulting localization band. First, for a given load velocity,three problems have been examined. The reference problem has an imperfection ofradius R = 2.5mm as shown in Figure 3. Two other problems have been studied with

radii Rsmall = 1.25mm and Rlarge = 5mm. As shown in Figure 8, the specimen with thelarge imperfection has the same band width than the reference one. On the contrary,the specimen with the small imperfection has a narrower band in the neighborhoodof the imperfection. However, far enough from the imperfection, the width of theband is practically the same as in the other cases. In fact, if the loading process withthe small imperfection continues, when the imposed displacement is large enough (e.g. = 0.19mm, see Figure 8) the band width is practically the same as the one obtainedwith larger imperfections. Thus, the imperfection size plays a role in the determinationof the band width only if it is small. Moreover, this influence is only local and disappearsat large load levels.

Second, the specimen with R = 2.5mm has been studied with different loading ve-locities. As expected, when the load velocity decreases, the viscoplastic solution tendsto the elastoplastic one. Thus, if loading velocity is very small the band width is verynarrow and, from a practical point of view, the viscoplastic solution has the same prob-lems as the elastoplastic: 1) pathological mesh dependence and 2) an almost verticalsoftening branch with the associated convergence difficulties.

An adaptive procedure has been used. First the computations are carried out with acoarse almost uniform mesh. Then the error is estimated at the end of the loadingprocess. Using the estimated error distribution, a remeshing criterion and a meshgenerator, a new mesh is created and the computations are carried out from scratch.This is repeated until the estimated error is below some acceptability requirements.

Firstly, two series of adapted meshes following the Li and Bettess [19] optimalitycriterion are presented. This remeshing criterion seeks uniform error distribution (ekconstant for every k). In the adaptive procedure of Figure 9 the error is estimated onlylocally (interior and patch estimate), in the series shown in Figure 10 pollution errorsare also taken into account. The goal in both examples is to obtain an error below the0.5%. Discretizations corresponding to the local estimate have less elements than theones obtained considering the pollution error. This is because the local error is lowerand, consequently, a mesh with less elements suffices. However, the final distributionsof elements are very similar. In fact, the pollution error is only relevant in the firstmesh which is coarse and roughly uniform. Once the discretization is refined where

it is needed (in particular in the vicinity of the singularities) the pollution effects areattenuated and become negligible. Considering pollution error does not make a bigdifference in the final results. This result was expected given the strong ellipticity ofthe problem. Figure 10 shows also the evolution of the error distribution along theadaptive procedure. The distribution of the error tends to be uniform, as expected.

In Figure 11 the adaptive procedure is carried out using a different remeshing cri-

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terion. Here, the Relative Local (RL) optimality criterion introduced in [12] which isbased on obtaining a uniform distribution of local relative errors (ek/uk constantfor every k) is used. In Figure 11 the accuracy maps (local relative error distribution)for the final mesh in Figure 10 is also shown. Using the RL criterion the accuracy is

almost uniform over the domain but it requires more elements than the Li and Bettesscriterion. This result was expected because the Li and Bettess criterion is optimal inthe sense that it produces the mesh with less elements given a globally acceptable error.The RL criterion guarantees uniform accuracy, even locally, but it has a larger cost.

Note that the numerical examples confirm that the problem has been regularizedbecause the behavior of the solution does not vary along the remeshing process. Theerror estimation is robust in the sense that the adaptive procedure converges (the finalmesh has the desired error).

4.2 Rectangular specimen with two imperfections

The mechanism of failure in this case is much more complex. In fact, it dependsstrongly on the position of the circular openings. Two cases are examined with differenthorizontal gaps between the openings (see Figure 4). In this case only the Li & Bettessremeshing criterion has been used [19].

Example 2.a (distant openings)

If the horizontal distance between the circular openings is large enough, the behavioris similar to the previous case. One shear band is developed aligned with the twoopenings. The remeshing process (see Figure 12) leads to a mesh with a large numberof elements concentrated along a single shear band. Figure 13 shows the general behaviorof the solution: the softening force-displacement curve is similar to the previous caseand the equivalent inelastic strain is concentrated along the shear band, both in theoriginal and the final meshes of the remeshing process. That is, the captured collapsemechanism is the same in both meshes.

Example 2.b (close openings)

On the contrary, if the circular openings are closer, the behavior of the solution is

much more complex and the original mesh is not able to reproduce such a mechanism.

Figure 14 shows the succession of meshes in this case. It is worth noting that, inthe final mesh, according to the concentration of elements, two bands are developed.In fact, the resulting bands are not aligned with the imperfections (as in example 2.a),but have opposite inclination.

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0.00 0.20 0.40 0.60 0.80 1.00 1.2

X1.E-4

0.00

0.20

0.40

0.60

0.80

1.00

1.20

X1.E7

0.00E+00

7.00E-04

1.40E-03

2.10E-03

2.80E-03

3.50E-03

4.20E-03

4.90E-03

5.60E-03

6.30E-03

7.00E-03

7.70E-03

8.40E-03

9.10E-03

Figure 13. General solution for example 2a: Reaction versus imposed displacement,

deformation of mesh 5 and inelastic strain contours for meshes 0 and 5

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Figure 14. Remeshing process using Li-Bettesss for a prescribed accuracy of 1.5%:

succession of meshes and estimated error distributions

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Figure 15. Numerical bifurcation in the first meshes:

mesh deformation amplified 40 times and equivalent inelastic strain contours

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Figure 16. Two consecutive failure mechanisms for the final mesh:

mesh deformation amplified 40 times and equivalent inelastic

strain contours at two moments of the loading history

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Indeed Figure 15 shows how the computed equivalent inelastic strain and the defor-mation evolve along the remeshing process. Only after two remeshing steps the meshcaptures two bands. In the previous meshes the discretization is not accurate enoughand only one band is completely developed. Since large deformations are considered,

once the first band evolves enough, the kinematic mechanism associated with this bandlocks: then a second band appears as a new deformation mode with less energy. Fig-ure 15 shows also how the force-displacement curves for meshes 0 and 1 are qualitativelydifferent from meshes 2 to 5. In fact the shape of the force-displacement curve for meshes2,3,4 and 5 are practically identical and have two inflections in the descending branch.In Figure 16 it is shown how the inflections correspond to the formation of a new fail-ure mechanism. The solution given by the last mesh is obviously more accurate thanthe original one because the energy of deformation (area under the force-displacementcurve) is lower. In fact, since the error is controlled in energy norm, one can be sure thatthe actual curve, associated with the exact solution, is not too far from the obtainedcurve (the error in energy norm is less than 1.5% and, consequently the difference of

the area under the curves is less than 1.5%). The first meshes are not able to reproducethe behavior of the actual solution because the elements along the later band (whichdevelops in a further stage of the loading process) are too large and, consequently, thediscretization is too stiff. Then, the size of the elements in this zone does not allow theinception of softening. On the contrary, once the remeshing process introduces smallenough elements along the second band, the second mechanism can be captured.

Thus, this example demonstrates that adaptivity based on error estimation is anessential tool for the determination of a priori unpredictable final solutions. Withoutthis adaptive strategy, the initial mesh (mesh 0 in figure 14) and the resulting solutioncould be regarded as correct, and the second mechanism would not be detected.

5 CONCLUDING REMARKS

The Perzyna viscoplastic model has been used in order to obtain a regularized soft-ening model allowing to simulate strain localization phenomena. This model has beenapplied to quasistatic examples, where inertia terms are negligible. The viscous regu-larization of quasistatic processes has been discussed: the rate effects are still presentand regularize the problem. However a difference between the dynamic and the qua-sistatic cases must be mentioned: in quasistatics, the internal length associated withthe obtained band width is no longer only a function of the material parameters butalso depends on the boundary value problem (geometry and loads, specially loading

velocity).An adaptive computation has been successfully applied to softening viscoplastic ma-

terials showing strain localization. As the key ingredient of the adaptive strategy, aresidual type error estimator has been generalized to deal with such highly nonlinearmaterial model. Moreover, this estimator has been designed in order to account forpollution errors. However, as expected, the pollution errors have been found to be neg-

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ligible, specially in the refined meshes, with elements concentrated in the vicinity of thesingularities.

In the numerical examples, the adaptive process is shown to be able to detect complexcollapse modes that are not captured by a first, even if fine, mesh. This is specially

interesting for softening localization problems, where small variations in the geometryof the problem may induce very different mechanical behavior. In this situation thelocation of the localization band cannot be predicted a priori. For instance, in one ofthe examples, the first mesh is not able to reproduce the two consecutive mechanismscaptured with the final mesh. The second mechanism is associated with a second shearband appearing in a further stage of the loading process. However, if adaptivity isnot used, the first mesh would be regarded as correct and one never would detect thesecond mechanism. In fact, even if some heuristic remeshing is done, based on thesolution obtained with the first mesh, the mesh would not be refined along the secondshear band, that is, where it is needed to capture the second mechanism. On thecontrary, if a remeshing strategy based on the error distribution is used, the elementsare concentrated along this second band and the new mesh is able to reproduce the twomechanisms. Consequently, adaptive strategies based on error estimation are essentialto detect the collapse mechanism and to assess the location of the elements in an optimalmesh.

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