Error estimation and adaptive spatial discretisation for quasi-brittle failure

7/31/2019 Error estimation and adaptive spatial discretisation for quasi-brittle failure

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Error Estimation andAdaptive Spatial Discretisation

for Quasi-Brittle Failure

Tanyada Pannachet


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Error Estimation and

Adaptive Spatial Discretisationfor Quasi-Brittle Failure

Proefschrift

ter verkrijging van d e graad van doctor

aan de Technische Univer siteit Delft,

op gezag va n d e Rector Ma gnificus p rof. dr. ir. J. T. Fokkema,

voorzitter v an h et College van Promoties,

in het openbaar te verd edigen op dond erdag 19 oktober 2006 om 10.00 uur

door

Tanyada PANNACHET

Master of Engineering , Asian Institut e of Technology

geboren te Khon Kaen, Thailand


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Dit proefschrift is goedgekeurd d oor d e promotoren:

Prof. d r. ir. L. J. Sluys

Prof. dr. ir. H . Askes

Samenstelling promotiecommissie:

Rect or M ag nifi cu s Vo or zit ter

Prof. dr. ir. L. J. Sluys Technische Universiteit Delft, prom otor

Prof. dr. ir. H. Askes University of Sheffield, The United Kingdom, promotor

Prof. dr. K. Runesson Chalmers Tekniska H ogskola, Sweden

Prof. dr. ir. A. van Keulen Technische Universiteit Delft

Dr. P. Dez Universitat Politecnica de Catalunya, Spain

Dr. ir. R. H. J. Peerlings Technische Universiteit Eind hoven

D r. G . N . We lls Te ch n is ch e U n iv ersiteit D elft

Prof. dr. ir. J. G. Rots Technische Universiteit Delft , reservelid

Copyright c 2006 by Tanyada PannachetCover design: Theerasak Techakitkhachon

ISBN-10: 90-9021123-3

ISBN-13: 978-90-9021123-7


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Contents

1 Overview 1

1.1 P hy sica l, m o d el a n d d is cr et is ed p r ob le m s . . . . . . . . . . . . . . . . . . . . . 2

1.2 Q u alit y o f a fi nit e ele men t m esh . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Er ro r co nt ro l a nd m esh a d ap tiv it y . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 A d ap t iv e m o d ellin g o f q u as i-b rit tle fa ilu r e . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Th e con tin uo us cr ack m od el . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.2 Th e d isco nt in u ou s cr ack m od el . . . . . . . . . . . . . . . . . . . . . . . 11

2 Finite element interpolation 13

2.1 Ba sic settin gs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Element-based finite element s hape functions . . . . . . . . . . . . . . . . . . . 15

2.2.1 Non-hier ar chical (clas sical) s hape functions . . . . . . . . . . . . . . . . 16

2.2.2 H ie ra rch ica l sh ap e fu n ct io ns . . . . . . . . . . . . . . . . . . . . . . . . 172.2.3 Com parison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 N o d e-b ase d h ie ra rch ica l e nh a nce m en t . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 En h an ce men t t ech n iq u e . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.2 C hoices of polynomial enrichment functions . . . . . . . . . . . . . . . 22

2.4 Rem arks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 A posteriori error estimation 25

3.1 Discretisa tion er ror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 St an d a rd r es id u a l-t yp e e rr or e st im a tio n . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Boundar y conditions of the local err or equations . . . . . . . . . . . . . . . . . 29

3.3.1 Lo ca l N eu m an n co nd it io ns . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.2 Lo ca l D ir ich le t co nd it io ns . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Er r or estimation for non-unifor m inter polation . . . . . . . . . . . . . . . . . . 34

3.5 Er ro r a ss ess m en t in n o nlin ea r a n aly sis . . . . . . . . . . . . . . . . . . . . . . . 34

3.6 So me im p lem en ta tio na l a sp ect s . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6.1 Solu tio n m ap p in g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6.2 Ir re gu la r e le m en t co nn e ct iv it y . . . . . . . . . . . . . . . . . . . . . . . 38

3.7 Per for ma nce a na lyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.8 Rem arks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48


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viii Contents

4 Error estimation for specific goals 49

4.1 Q ua ntities of in terest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Se tt in g o f d u alit y a rg u men t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.1 Th e in flu en ce fu n ct io n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.2 Th e d u al p ro blem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 G oa l-o rien te d e rr or e st im a tio n . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.1 Se tt in g o f e rr or in t he g oa l q u an t it y . . . . . . . . . . . . . . . . . . . . 53

4.3.2 Er ro r a sse ss m en t in t h e d u a l p r ob le m . . . . . . . . . . . . . . . . . . . 55

4.3.3 C hoices of err or measur es in local domains . . . . . . . . . . . . . . . . 56

4.3.4 N o n lin ea r fi n it e e le m en t a n aly sis . . . . . . . . . . . . . . . . . . . . . . 57

4.4 N u mer ica l exa mp les . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.5 Rem arks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 Mesh adaptive strategies 67

5.1 M esh q u alit y a n d e n ha n ce m en t s tr at eg ie s . . . . . . . . . . . . . . . . . . . . . 68

5.1.1 A priori er ror estim ates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.1.2 Re m ar ks o n m e sh a d a pt iv e a lg or it h m s . . . . . . . . . . . . . . . . . . 69

5.2 A da ptiv e cr iteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.1 En er gy n or m b as ed a d a p tiv e cr it er ia . . . . . . . . . . . . . . . . . . . . 71

5.2.2 G oa l-o rie nt ed a d ap t iv e cr it er ia . . . . . . . . . . . . . . . . . . . . . . . 72

5.3 O ptim alit y cr it er ia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3.1 En er gy n or m b as ed o p tim a lit y cr it er ia . . . . . . . . . . . . . . . . . . . 74

5.3.2 G oa l-o rie nt ed o p tim a lit y cr it er ia . . . . . . . . . . . . . . . . . . . . . . 75

5.4 Sm oo th in g-b ase d m esh g ra d at io n . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4.1 M esh g ra d at io n st ra teg y . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4.2 A u xilia ry t ech n iq u es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.4.3 Exam ples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.5 Va ria ble t ra nsfe r a lg or it hm s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.5.1 Tr an sfe r o f st at e v ar ia ble s . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.5.2 Tr an sfe r o f p r im a ry v ar ia ble s . . . . . . . . . . . . . . . . . . . . . . . . 86

5.6 Rem arks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 Mesh adaptivity for continuous failure 91

6.1 Th e g ra d ien t-en h an ced d am a ge m od e l . . . . . . . . . . . . . . . . . . . . . . . 92

6.2 Er ror a na lyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.3 Cen tr al t ra nsv er se cr ack t est . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.3.1 P re lim in ar y in ve st ig at io n . . . . . . . . . . . . . . . . . . . . . . . . . . 966.3.2 M esh a d ap tiv e t est s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.4 Single-edge-notched (SEN) beam tes t . . . . . . . . . . . . . . . . . . . . . . . . 114

6.4.1 P re lim in a ry in v es tig at io n . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.4.2 M esh a d ap tiv e t est s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.5 Rem arks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127


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Contents ix

7 Mesh adaptivity for discontinuous failure 1317.1 P U-b ased co hesiv e z on e m od el . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.2 Er ror a na lyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.3 Cr ossed cr ack t est . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138



7.4 Th ree -p oin t b en d in g t est . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154



7.5 Rem arks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

8 Conclusions 165

A Critical survey on node-based hierarchical shape functions 173

A .1 Con verg en ce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

A .2 En fo rce m en t o f b ou n d a ry co nd it io n s . . . . . . . . . . . . . . . . . . . . . . . . 174

A .3 Lin ea r d ep en d en ce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Bibliography 181

Summary 187

Samenvatting 189

Propositions/Stellingen 191

Acknowledgement 193

Curriculum vitae 195


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CHAPTER

ONE

Overview

The finite element method is a numerical tool to approximate solutions to partial

differential equations, for instance those describing physical phenomena in engi-

neering. Accuracy of a finite element solution depends mainly on the discretisation

of the p roblem dom ain. Certainly, a m ore refined/ enriched discretisation imp roves

the ability of th e finite element analysis to app roximate th e exact solution.

How ever, some qu estions arise, for examp le, whether the m esh used in the com-

putation is good enough to output an acceptably accurate result and, if not, how

fine it should be. Using a finer m esh also means an increased nu mber of unkn owns

that mu st be solved in the finite element compu tation. And, even though the ca-

pability of comp uters now adays is mu ch improved , the num erical models are also

becoming more complicated as the knowledge about the physical phenomena has

become m uch clearer than in the p ast.

The measurement of error information is the basis for an answer to the above

questions. Error information is an objective measure to assess wh ether th e used fi-

nite element mesh is of sufficient quality. Moreover, local error information and the

corresponding local criteria give the user some hints where in the mesh the discreti-

sation should be imp roved. This procedure of discretisation improvement is known

as mesh adaptivity. It can en han ce the efficiency of the discretisation enorm ously, es-

pecially in problems whose solutions need very fine discretisation only in a small

part, whereas coarse discretisation may be applied in the rest of the problem do-

main. A typical example of such a problem, to which this dissertation is devoted,

is the analysis of cracks. For quasi-brittle m aterials, cracks constitute small zones

where the mechanical nonlinear activity is concentrated, while the rest of the struc-ture behaves elastically. The cracking zones, normally not known a priori, require

a fine d iscretisation wh ereas the remainder of the stru cture can be an alysed with a

coarser discretisation. Thus, crack analysis can benefit from mesh adaptivity.

The aim of this chapter is to give a brief introdu ction to the w hole dissertation. We

will start with defining three levels of problems in preparation for the finite element


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2 Chapter 1 Overview

analysis as well as the corresponding errors that emerge during transitions from

one levelto another. Next, as the finite element solution relies essentially on how the

problem d omain is d iscretised, remarks abou t m esh d iscretisation in finite element

analysis w ill be ad dressed. Essentials about m esh adap tivity and error estimation,

as well as its applications in crack modelling, will end this chapter.

1.1 Physical, model and discretised problems

Generally, there are three d efined problems in nu merical comp utation. In p ractice,

th e physical problem to analyse must be defined as the first step. Due to the com-

plexity of the real physical problem, norm ally som e assump tions are mad e. These

assumptions may be, for instance, a 2D representation of the real 3D problem being

und er p lane stress/ plane strain conditions w ith the assumed material behaviourduring a loading process described by a certain constitutive relation. With those

assump tions, the ph ysical problem is now tran sformed into the model problem. In fi-

nite element m odelling, the problem d omain m ust then be discretised so that it can

be analysed numerically. At this stage, the problem becomes the discretised problem.

The boundary conditions are projected to the discretised domain and the forces

are distributed corresponding to the discretisation, resulting in so-called consistent

nodal forces.

Progressing from one problem to another leads to different types of error. The as-

sump tions made in the m odel problem to represent the physical problem cause the

so-called modelling error, wh ile the mapp ing of the m odel into the d iscretised do-

main brings about the discretisation error. While the mod elling error ind icates how

accurate the mathematical model is in representing the real physical problem, the

discretisation error indicates how accurate the discretisation is in approximating

the solution to th e mathem atical model. While the m odelling error is m easured bycomparing the m athematical m odel (model p roblem) with the experimental data

(physical problem), the discretisation error can be estimated by comp aring th e so-

lutions of the discretised p roblem with th ose of the model p roblem represented by

a very refined/ enriched discretisation . Even so, in real practice, it is not simpleat all to d istinguish betw een the mod elling error an d the d iscretisation error since

the answer to the constitutive relation (model problem) can generally not be de-

termined analytically but only numerically. And, via the finite element concept, the

discretisation of the model problem is unavoidable, whereby it becomes impossible

to separate modelling errors from discretisation errors. However, this constitutes a

dilemma in the transition from physical problems to model problems. In this dis-

sertation, we are concerned w ith the transition from mod el problems to d iscretisedproblems.

Here, we denote the solution to the mathematical model as the exact solution of the model problem.A refined discretisation is defined as a discretisation with an improvement regarding element sizes,

whereas an enriched discretisation denotes a discretisation with an improvement regarding interpola-

tion capability.


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1.2 Quality of a finite element mesh 3

In p articular, our main goal is to assess the error in finite element discretisation.

Thus, one of our assump tions is that the constitutive relations of material m od-

els used throughout this thesis are perfectly correct, i.e. they are a perfect repre-

sentation of the un derlying ph ysical p rocesses. The d iscretisation error, resulting

from the projection of the model quantities to the discretised domain, originates

from two sources, namely the inability to reprodu ce the geometric boundary of the

mod el problem an d th e inability to reprod uce the exact solution of the model prob-

lem. The error from the first source is actually a source of error that, for not too

complicated boundary geometries, is avoidable by carefully selecting suitable type

of finite elements. Thu s, our main focus will be on the second source of discretisa-

tion error.

1.2 Quality of a finite element mesh

As mentioned earlier, accuracy of the finite element solution depends on how the

mod el problem is d iscretised. Two main factors of the stand ard finite element d is-

cretisation ar e

size of the finite elements (the h-factor), and characteristic of the interpolation fun ctions, for instan ce the polynom ial ord er

(the p-factor).

Obviously, a sm aller element size m ay p rovide a better resolution of the exact so-

lution. However, the approximation also depends on how suited the interpolation

function, often based on piecewise polynom ials, is for d escribing th e exact solution.

Figure 1.1 shows how the finite element analysis approximates the exact solution

of an ordinary differential equation, which here is a quartic polynomial. Keeping

the interpolation function in linear form, a better resolution to the exact solution

can be obtained via the uniform refinement of the finite element mesh, the so-called

h-version finite element m ethod. Each smaller element h as to m odel a smaller seg-

ment of the exact solution. On the other hand , in the p-version finite element frame-

work, the ap proximation is imp roved by enr ichment of the interpolation functions.

Withou t chan ging the elemen t size, the resolution of the exact solution is well-fitted ,

especially when the order of interpolation polynomial approaches that of the ana-

lytical solution. Another imp ortant observation from this p roblem is th at, with the

same nu mber of d egrees of freedom, the p-version p rovides a better ap proximationto the exact solution than the h-version. This holds in particular for higher values

of the interpolation orders.

In this research, we focus on the p-version (cf. Chapt er 2) as well as the h-version

finite element m ethods. Although not as p opu lar, the p-version has some outstand -

ing advantages. Firstly, it provides accuracy improvement without changing the


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0 0.5 1

Coordinate, x

0

0.02

0.04

0.06

Solution,u(x)

2 linear elements (3 DOFs)

Exact

Approx.

0 0.5 1

Coordinate, x

0

0.02

0.04

0.06

Solution,u(x)


Exact

Approx.

0 0.5 1

Coordinate, x

0

0.02

0.04

0.06

Solution,u(x)

2 quadratic elements (5 DOFs)

Exact

Approx.

0 0.5 1

Coordinate, x

0

0.02

0.04

0.06

Solution,u(x)


Exact

Approx.

0 0.5 1

Coordinate, x

0

0.02

0.04

0.06

Solution,u(x)

2 cubic elements (7 DOFs)

Exact

Approx.

0 0.5 1

Coordinate, x

0

0.02

0.04

0.06

Solution,u(x)


Exact

Approx.

0 0.5 1

Coordinate, x

0

0.02

0.04

0.06

Solution,u(x)

2 quartic elements (9 DOFs)

Exact

Approx.

Figure 1.1 Comparison between h-extension (left colum n) an d p-extension (right column) of the p rob-

lem 2ux2

= 6x 2 3x , with bound ary conditions u(0) = 0 and u(1) = 0.


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1.3 Error control and mesh adaptivity 5

mesh configuration. Secondly, for p roblems with smooth solutions, the p-versionprovides a higher rate of convergence [93], i.e. the app roximate solution becomes

more accurate by increasing the polynomial degree than by ad ding th e same num -

ber of degrees of freedom via th e h-version. Thirdly, when the hierarchical p-version

(for example, [93,94]) is employed, each additional higher-order contribution does

not change any of the interpolation functions used in th e previous contribution. As

such, the stiffness m atrix for order p is embedded in the stiffness matrix for order

p + 1, reducing computational effort and improving the conditioning of the stiff-ness m atrix.

1.3 Error control and mesh adaptivity

Due to limitation of computer capacity, not all information describing the actualcontinuum model can be included in the finite element computation. And even

though a m ore refined/ enriched discretisation is a better representation of the con-

tinuu m model, it requires higher compu tational cost accordingly. As a solution

to this problem, one should set a balance between accuracy and computational

cost. An acceptably accurate solution that does not require outrageous computa-

tion should be the rule for practical applications.

Types of error assessment

To m easure the accuracy of th e finite element solution, it is necessary to assess an

error quantity, which results from the finite element discretisation. Basically, there

are two types of error estimation procedures available, namely a priori an d a pos-

teriori error estimators. The a priori estimate provides general information on the

asymptotic behaviour of the d iscretisation errors but is n ot d esigned to give an ac-

tual error estimate for a specific given m esh, geometry and loading conditions. On

the other hand, the a posteriori estimate measures the actual error at the end of a

specific computation and can be exploited to drive a subsequent mesh adaptivity

procedure.

In this context, following [43], we distinguish between error estimation an d error

indication based on objectivity of th e outp ut quan tity. The error ind ication d oes not

provide objective information about the exact error, but gives some hints where

the solution may need a more refined/ enriched discretisation. Based on heuristic

observations, we can actually predict in which regions of the problem domain er-

rors are likely to occur based on the p roblem geometry and the solution itself. For

example, errors are always concentrated at sharp corners of the problem domain,wh ere point loads are prescribed, and w here there is an abrup t change in bound ary

conditions; in other w ords, errors concentrate wh ere high gradients of the solution

occur.

The geometric representation of a problem with complex geometry may change slightly the mesh con-figuration d uring the p-extension. However, this type of p roblems will not be stu died in this thesis.


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Mesh enrichment

Mesh refinement

Mesh gradation

(radaptivity)

Original

(hadaptivity)

(padaptivity)

Figure 1.2 Some m esh adap tive schemes u sed in this research.

However, as the error indication directly links available quantities to error in-

formation, it needs to be derived for each material model and is rather restricted

to the assumptions based on types of the problem to be analysed. In contrast, the

standardised and mathematically founded error estimation can be applied to ev-

ery problem for any m aterial mod el without any (major) reformu lation. In spite of

being computationally more expensive than the indication, the error information

obtained from the error estimation is objective and can be exploited with optimality

criteria in designing an op timal mesh . We employ, by such reasons, an error estimator

in this research study.

Error estimators

Basically, the a posteriori error estimators can be categorised into tw o m ain classes.

The recovery-type error estimator s (for examp le, [106, 107]) measu re the smooth ness

of stresses between adjacent elements. Since the methods do not require solving the

error equations, they are simple and more preferable in many practical problems.

How ever, there are not so m any cases reported in [106,107] that show superconver-

gence. On an isotropic meshes or those w ith mixed element m eshes, the analysis is

hindered by an apparent lack of superconvergence properties. Also the recovery-type estimator is not proven to converge in n onlinear problems.

In contrast, the residual-type estimators, although related somehow to the

recovery-type [103], do not d epend on the sup erconvergence properties. Thus, they

Superconvergence property belongs to some points wh ere a very accurate solution can be obtained.They are usually the qu adratu re points [105].


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1.3 Error control and mesh adaptivity 7

Real Problem

Adap

tiveprocess

Model Problem

Mesh Discretisation

Finite element analysis

Reliability check

Acceptable results

Discretisation

Error

Modelling

Error

Numerical

Error

Yes

No

Figure 1.3 Standard procedure for adaptive finite element computation.

can be app lied to a w ider var iety of p roblems. The m ethods (for example, [12]) de-

termine the error by calculating the residu al of the finite element solutions in each

local space. We have chosen a residu al-type error estimator in this stu dy. Followin g

the idea in [29], homogeneous Dirichlet conditions are imp osed in the error equa-

tion defined by forming patches of several elements. The method is applied for

estimating the error in energy norm (cf. Chapter 3), as well as the error in a local

quantity of interest (cf. Chapter 4).

Mesh adaptivity

Once the error information is obtained, the finite element mesh can be ad apted ac-

cordingly. The mesh should be imp roved where the local error exceeds the accept-

able limit (controlled by refinement criteria also known as adaptiv e criteria). Thereare many techniques for local mesh imp rovement, for instance, mesh refinement (h-

adaptivity), mesh enrichment (p-adaptivity), mesh gradation (r-adaptivity), mesh su-

perposition (s-adaptivity) or combinations of any tw o.

In th is d issertation, w e consider only th ree ad aptive techniques (cf. Figure 1.2).

By applying h-adaptivity, it is p ossible to d esign an op timal mesh based on an op-


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timality criterion, which is formu lated from the a priori convergence assumption (cf.

Chapter 5). On the other hand , finding a p recise balance between acceptable error

levels and computational costs via p-adaptivity m ay imply that fractional polyno-

mial degrees must be used, which is not a feasible option. Hence, in p-adaptivity,

the interpolation is enriched hierarchically by one ord er at a time. Without add ing

any extra d egrees of freedom, r-adaptivity can be a comp romising alternative to h-

adap tivity. And w ithout solving any equ ation, a smoothing based r-adaptive tech-

nique based on the weighted Laplace smoothing is introduced and investigated in

Chapter 5.

Figure 1.3 shows the standard adaptive procedu re in the finite element analy-

sis. In the sam e figure, the d ashed box rou ghly ind icates the scope of this research,

wherein the discretisation error is the only error under consideration. Although it is

difficult to neglect involvement of the n um erical error (e.g. floating point error) in

this study, its contribution is assumed to be very m arginal as comp ared to the d is-cretisation error. All detailed information about mesh adap tive aspects, includ ing

th e transfer of state variables for nonlinear analysis, is add ressed in Chapter 5.

1.4 Adaptive modelling of quasi-brittle failure

In this dissertation, error estimation and mesh adaptivity are applied to problems

involving stationary and propagating cracks. The focus is on materials such as con-

crete, rock,ceramics and som e m atrix comp osites, which show so-called quasi-brittle

behaviou r. Unlike perfectly brittle materials, qu asi-brittle ma terials do not lose their

entire strength immediately after the maximum strength is exceeded but instead

gradually lose their material strength and show the so-called strain-softening phe-

nomenon (cf. Figure 1.4). Softening stress-strain relations show a drop of stress af-

ter the applied load exceeds the material strength (peak point). In fact, microcracks

are initiated in the material before the stress in the material reaches its maximum

strength [89]. However, the material is still able to carry loads to an extent. Up on

further loading, these microcracks will then join together to form a dominant crack

line which will lead to failure of the specimen.

Another p henomen on that occurs d uring th e fracture process is strain localisation

(cf. Figure 1.5). When the material loses its ability to carry load, the affected part

shows increasing d isplacement gr adients. Ultimately, when a complete rup ture h as

occurred and the m aterial is separated into d istinct pieces, the d isplacement grad i-

ent has transformed into a d isplacement jump .

Basically, there are tw o m ain assum ptions to m odel the fracture m echanism oc-

curring in these qu asi-brittle materials. The first class consists of continuous crackmodels, in which the material deterioration is accounted for in a smeared way. The

stress field and strain field remain continuous during the entire fracture process

resulting from a gr adu al degradation in m aterial prop erties.

Discontin uous crack m odels can be regarded as the second class. In these models,

the failure m echanism is presented by means of geometrical discontinuities in the


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1.4 Adap tive modelling of quasi-brittle failure 9

Figure 1.4 Softening phenomenon.

material domain. Cracking takes place when stresses in the materials, in any di-

rection, exceed th e m aximu m quan tity that th e m aterial can resist in that direction.

Such discontinuities imply that materials have separated parts and a jump in the

displacement field can be found in th e zone wh ere discontinuities exist. Figure 1.5

shows th e d ifference between crack representations of the two assump tions in the

context of a three-point bending test.

In standard finite element computation, in order to deal with complicated ma-

terial models, the finite element mesh must be properly designed a priori. Such

mesh design has to rely on information before the computation. The mesh may

be designed based on information such as the regions wherein the stresses may

concentrate or where the material/ geometrical imperfections are. These guidelines

are not always obvious in p ractice and the designed mesh d oes not always guar-

antee app ropriate results du ring cracking processes. Apparently, the need of mesh

adap tivity becomes of great importance in crack prop agation analyses.

1.4.1 The continuous crack model

Continuous crack models can be implemented u sing either the concept ofplasticity

or the concept ofcont inuu m damage mechanics. In this research, the gradient-enhanced

damage model [70] is used for mesh adap tivity in a continuous crack concept. Dam-

age occurs in the part of m aterial dom ain wh ere the stress cannot be sustained fully

anymore. As a regularised continuum, the gradient-enhanced damage model con-

verges properly up on refinem ent of the finite element d iscretisation.

Error estimation, as well as error indication, has been applied in problems with

softening phenomena. Some outstanding works, employing the residual-type er-

ror estimation an d h-adaptivity in softening media such as viscoplastic or nonlocal

dam age mod els, can be found in [28, 78]. In these w orks, the error estimation takesplace at the end of the analysis. Thus, the mesh is ad apted based on the final er-

ror distribution. This refined mesh is then used to restart the whole analysis from

scratch. As the error is n ot m easured during comp utation, there is a possibility that

the failure mechanism obtained is incorrect. It was shown in [6] that crack paths

may be d ifferent for diffent meshes and the ad aptive process must be up dated d ur-


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Continuous Discontinuous

Figure 1.5 Representation of crack and corresponding strain localisation. Strain localisation occurswhen the tensile strength is exceeded in the bottom part of the beam (top). The resulting crack can be

modelled with a continuous (bottom left) or a discontinuous (bottom right) crack concept.

ing the nonlinear computation in order to make sure that the solution path is cor-

rect.

An alternative to the use of expensive error estimation in d riving mesh ad aptivity

is the u se of inexpensive error ind ication. In [6,90], an error ind icator is d erived

from the critical wave length in the damage model. The desired element sizes are

defined as functions of the dam age level and are successfully ap plied in h-, r- and

hr-adap tivity. How ever, in [10,11,69], it is sugg ested that it is as imp ortan t to assess

the error both in the linear regime (where no damage exists) and the nonlinear

regime (where there exists damage). Without damage and localised strain fields,

error estimation may be a suitable choice to drive the adaptive process in the earlier

computational steps (the linear elastic part), whereas the error indicator is used

wh en th e solution p resents nonlinearity. To sup port this idea, it is claimed in [25]

that the error estimate [49] becomes less significant in the localisation region as

dam age grows and stresses tend to vanish. By app lying the error estimation du ring

the w hole computation, w e can verify these statements.As h-adaptivity leads to changes in mesh configuration, the finite element anal-

ysis needs reformu lating the shape functions, stiffness matrix and force vectors. A

challenging alternative is the use of richer interpolation, or p-adap tivity. In this con-

tribution, we investigate p erforman ces ofp-adaptivity in combination w ith simple

mesh gradation applied to problems with strain localisation. A slightly modified


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1.4 Adap tive modelling of quasi-brittle failure 11

version of the error estimation in [29] is chosen in this study as it can be easily ap-

plied to problems with non-uniform higher-order interpolations while still being

well-integrated with the optimality criterion in designing the element sizes. Perfor-

mance of the ad aptive mod els will be investigated in Chapter 6.

1.4.2 The discontinuous crack model

As the terms cracking an d rupture already imply, introduction of discontinuity as

a result of material failure seems to be natural. Unlike the continuous modelling,

the fracture criterion of this concept is defined separately from the constitutive re-

lations. Discontinu ities in the material domain are mod elled by introdu cing a jum p

either in the displacement field (the so-called strong discontinuity) or the strain

field (the so-called weak discontinuity).

A classical app roach to mod el a crack is to ad apt the finite element m esh accord-ing to geometrical change due to crack propagation. It then requires a continuous

change of the topology of discretisation (i.e., remeshing process), which is compu-

tationally laborious and comp licated. An alternative ap proach is to place interface

elements of zero width in the finite element m esh [79]. How ever, since the d irection

of crack growth is not known a priori, small elements are needed to allow a jump

in the displacement field in a ran ge of p ossible directions of cracking, resulting in

an expensive comp utation.

Without restriction to mesh alignment, the crack can be m odelled in a m uch sim-

pler way. It is shown in [5,63,85] that modelling cracks within elements is possible

by both weak and strong discontinuity assumptions. Via the introduction of inter-

nal degrees of freedom, the discontinuous contribution is solved on the element

level and the displacement jump can be modelled without being restricted to theunderlying mesh. The method is known as the embedded discontin uit y approach. An-

other recent d evelopment is to model the displacement d iscontinuity by simply

adding extra nodal degrees of freedom via the partition of unity (PU) [14], which

is a basic property of the finite element interpolation. This PU-based finite element

method, also known as the ext ended fin ite element method (XFEM) [21, 32,58, 101], is

more robust in implementation than the embedd ed d iscontinuity app roach. As ex-

tra degrees of freedom, the enhanced functions are solved at the global level and

do not involve modification at the element level, thus preserving symmetry of the

Figure 1.6 Discontinuity m odelling based on en richment via the p artition of unity.


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global stiffness m atrix.

Although, via the PU concept, the jum p in th e displacement field can be modelled

without any restriction to the underlying finite element mesh (cf. Figure 1.6), the

resolution of the discretisation along the cracked element still needs to be ensured.

It has been observed in [99] that a too coarse discretisation may lead to a rough

global response. Even without the oscillations, the response and the resulting crack

path may not be sufficiently accurate. So far, without any research investigating

mesh requirement in the PU-based discontinuity model, it is hardly certain that

the propagation of a discontinuity leads to an acceptable level of accuracy. We will

investigate intensively the discretisation aspect of the discontinuous crack model

in Chapter 7.


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CHAPTER

TWO

Finite element interpolation

The finite element method is a num erical tool for approximating solutions of

boundary value problems, which are usually too complicated to be solved by an-

alytical techniques. As its name implies, the method employs the concept of sub-

dividing the model problem into a series of finite elements over which variational

formulations are set to construct an app roximation of the solution.

The finite element approximation relies mainly on the interpolation via piecewise

polynomials over a set of finite elements. As m entioned before, the introd uction of

higher-order interpolation functions (also called shape functions) is one technique

to achieve a better approximation to the solutions of the problem and is our main

motivation for this study.

The higher-order interpolation can be constructed either based on the so-called

Lagrange (non-hierarchical) elements, or based on adding hierarchical counterparts.

Two types of hierarchical shape functions, formulated based on elements and

nodes, as well as some critical aspects are presented in this chapter.

In the first part of this chapter, we attempt to give a short introduction of stan-

dard finite element analysis, and m ove on to th e formulations of higher-order shape

functions in the second part, as this concept will subsequently be used in so-called

p-elements in the rest of the thesis.

2.1 Basic settings

Let be a bounded domain with the boundary . The bound ary consists of theDirichlet boundary d and the Neumann boundary n for w hich d n = an dd n = . For a p roblem in statics, we try to fin d the un known solution u of thevariational bound ary value p roblem


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14 Chapter 2 Finite element interpolation

(v ) : (u) d =n

v g d+

v q d (2.1)

wh ich can be written in terms of d erivatives of trial and test functions, u an d v, a s

(v ) : D : (u) d =n

v g d+

v q d (2.2)

The test function v is any arbitrary fun ction in the Sobolev space V, wh ich is definedby V := {v H1(); v = 0 on d}. Moreover, (v) := v an d (u) := D : urepresent strains and stresses, g represents the traction forces along the boundary

n an d q denotes the body forces in the domain . The Galerkin weak form of a

linear problem can also be written as

B(u, v) = F(n )(v ) + F()(v ) = F(v), v V (2.3)wh ere the term B(, ) is a symmetric positive-definite bilinear form, correspondingto the left-hand-side of Eq. (2.2), while F(n ) an d F() refer to the first and thesecond terms of the right-hand-side, respectively.

In order to approximate the continuous variable u, a numerical computation

mu st be performed . The d iscretised system of equations

B(u(h,p), v (h,p)) = F(v (h,p)), v (h,p) V(h,p) (2.4)is solved in the finite element space V(h,p), where V(h,p) V. The subscripts h an dp denote the finite element analysis using element size h and polynomial order p.

As a result, the solution u(h,p) is an ap proximation to the exact function, u. The ap -

proximate solution u(h,p) V(h,p) and the test function v (h,p) V(h,p) are d iscretisedas

u(h,p) =n

i=1

i a i = a , v (h,p) =n

j=1

j cj = c (2.5)

via the use of basis functions (also known as shape functions) i an d j of the trial

(unk now n solution) and th e test functions, respectively. Sub stituting th e discretised

fields u(h,p) an d v (h,p) back into Eq. (2.4) results in a system of discretised equ ations

n

j=1

Ki jaj = fi , i = 1,2, .., n, n := nu mber of nod es (2.6)

where Ki j :=

B(j,i), fi :=

F(i), and aj denotes the approximate solutions

correspond ing to the shap e function j. Eq. (2.6) can be r ewritten in a mat rix formas

Ka = f (2.7)

where K denotes th e stiffness matrix of the linear system, a represents the vector

containing the unknowns and f denotes the force vector.


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2.2 Element-based finite element shape fun ctions 15

...

x 3yx4

x 3

x 2

1

x

x 2y 2 x y 2

x 2y x y2

x y

y

y 2

y 2

y 3

Quadratic

Lin ear

Cubic

Quartic

Quadratic

Lin ear

Element

TriangularQuadrilateral

Element

Figure 2.1 Comp lete 2D polynom ial terms d escribed by Pascals triangle [18].

2.2 Element-based finite element shape functions

The finite element shape functions are characterised by two basic features, as sug-

gested in [105], which are the continuity requirement and the so-called partition-of-unity prop erty. The latter p roperty su ggests that

n

i=1

i(x) = 1, x (2.8)

allowing the description of rigid body motions. Importantly, the shape functions

should n ot perm it straining of an element w hen n odal displacements are caused bya rigid body d isplacement.

The finite element interpolation is fund amentally set in a piecewise polynomial

format. To ensure the convergence of the approximation, it has been suggested that

the shap e functions should contain complete polynomials, wh ich can be described

in the Pascal triangle shown in Figure 2.1. Basically, there are two categoriesof polynomial-based interp olation functions, nam ely the non-hierarchical functions

and the hierarchical functions. The key difference between the tw o schemes is h ow

the polynomial bases are u pgraded to higher-order levels. While higher-order

shape functions in the non-hierarchical scheme are completely different from the

lower-order bases, the hierarchical scheme hierarchically adds the higher-order

contributions and retains the lower-order bases withou t an y reformulation. Details

of the two versions will be described in this section.

The differential equations studied here are all second-order. Hence, C0-continuity is required (that is,interelement continuity of the unknowns but not of derivatives of the unknowns).

The quadrilateral elements referred to in Figure 2.1 are the so-called Lagrangian elements. The quadri-lateral serendipity elements use a subset of the Lagrangian elements polynom ials.


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p = 1

p = 2

p = 4

p = 3

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

Figure 2.2 One dimensional shape functions for a non-hierarchical element.

2.2.1 Non-hierarchical (classical) shape functions

The classical finite element approach employs the so-called Lagrange polynomials

introducing the local interpolation function by p rescribing values at n odal p oints.

The app roach is a direct extension of the classical Lagran ge interp olation. The inter-

polation is based on fitting values at nodal points. For one-dimensional problems,

the shape functions containing polynom ials of degree p are of the general form

(1D ,p)i

() =

p

j=1

( j)p

j=1;i=j

(i j)(2.9)

where i, i = 1, 2, ..., p + 1, denotes a set of nodal coordinates in th e finite elementmod el. With some manip ulations, the one-dimensional functions can be extend ed

to generate higher-dimensional functions such as 2D quadrilateral and 3D brick

elements.The comp utation of the shap e functions in the given form obviously requires the

reconstruction of the shape functions once it is upgraded to higher orders, which

implies that the stiffness matrix must be completely recomputed. As our mesh

It is noted here that the shape functions for triangular and pyramid elements can also be formed differ-ently, in terms of area coordinates or barycentric coordinates. (See, for example, [93].)


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p = 3

p = 1

p = 2

p = 4

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

-1 0 1-1

0

1

Figure 2.3 One d imensional shape functions for a hierarchical element based on Legendre polynom ials.

adaptive technique includes p-adaptivity, having to recomp ute all stiffness m atrix

components everytime the mesh is up graded can be an unp referable feature.

2.2.2 Hierarchical shape functionsUnlike in the classical version, higher-order shape functions can be extended by

add ing an extra set of functions wh ile the existing functions are p reserved, i.e. span

of(p)i

is contained in span of(p+1)i

, in the hierarchical approach. Some exam-

ples of hierarchical interpolations are those based on Legendre polynomials (for ex-

ample, [93]), Chebychev polynomials (for examp le, [98]) and Hermite polynomials [61].

Also, Lagrange shape functions can be reformulated in the hierarchical form (for

example, [27]), where the hierarchical degrees of freedom can be referred to as tan-

gential derivatives of various orders at the midside n odes.

In this w ork, we focus on the u se of Legendre polynom ials since they possess or-

thogonality implying no linear dependence between the polynomial functions [105]

and , hence, a sparse d ata structure as compared to the u se of classical shape func-

tions. The Legendre basis also provides consistent element conditioning numberwh en the polynomial order is increased, thus leading to a smaller nu merical round -

off error and a faster convergence in nonlinear analysis, as compared to other

bases [35].

The Legendre interpolation function is based on the Legendre polynomials,

which originally are solutions to Legendres differential equations. The polynomial


26/203


of degree p may be expressed using Rodrigues formula

Pp() = (2pp!)1

dp

dp

(2 1)p

. (2.10)

In add ition to the stand ard linear shap e functions (vertex modes), the hierarchical

enrichment including edge and internal m odes are defined in the interval 1 1 a s

1D ,p

i() :=

2p 1

2

1

Pp1(t) d t =1

2(2p 1)

Pp() Pp2()

(2.11)

for p 2. The m ain d ifference between th e standard finite element shape fun ctionsand Legend re shape functions, given in Figur es 2.2 and 2.3, can be clearly observed.

Similar to the stand ard finite element interpolation, the higher-dimensional func-tions are based on products of one-dimensional functions and Legendre polyno-

mials [93]. Another set of combinations, in forming th e higher-dimensional set of

shape fun ctions, has been suggested in [24], with th e improvem ent of sparsity and

conditioning of the stiffness matrix.

2.2.3 Comparison

It is noted that, in the hierarchical app roach, the h igher-order d egrees of freedom ,

known as edge modes and internal modes (also known as bubble modes), are not

based on nodes. Figure 2.4 compares how the two interpolation schemes work.

While the non-hierarchical version (based on Lagrange p olynomials) interpolates

values at nodes, the hierarchical version (here, based on Legendre polynomials)interpolates values at the primary nodes as well as values corresponding to

add itional h igher-order interpolation functions. Du e to such characteristic, the

following difficulties obviously emerge:

(A) Enforcement of constraints

The standard element shape functions have a superiority over the hierarchical

functions when it comes to constraint enforcement. Possessing the Kronecker delta

property (i.e. i(xj) = i j, where i an d j refer to nodes), either external constraints(i.e. prescribed values of pr imary variables) or internal constra ints (i.e. the relation-

ship between different d egrees of freedom) can be simply imposed at nodes. In

contrast, the enforcement of constraints in the h ierarchical approach causes some

difficulties due to the obsence of nod es on ed ges. Direct imp osition can be app lied

only in case of constant or linear constraints (p 1). In that case the edge shapefunctions at the corresponding edge are dropped out (i.e. zero-value prescribed)

and the linearly varying constraints are directly prescribed at nodes, which exist

only at the vertices of an element in the hierarchical approach.

In real app lications, there hard ly exist p roblems with constraints of higher-order

functions. H owever, if necessary, special techniques su ch as Lagrange m ultipliers


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u1u2 u1

u2

u3u1

u1

u2

u2u4u3

u4u3u1

u2u5

u2

P2(xp) d3

u1

u2u1

P3(xp) d4

u2

P4(xp) d5

u1

Linear

LegendreLagrange

Quartic

Cubic

Quadratic

xp

xp

xp

Figure 2.4 Higher-order interpolation based on standard isoparametric element and hierarchical ele-ment based on Legendre polynomials.

or a Penalty formulation m ay be ap plied (for examp le, [65]), leading to a mod ified

Galerkin w eak form.

(B) Modelling of geometrical data

The standard p-elements are able to d escribe the m odel geometry via the h igher-

order shape functions by relocating the edge nodes. A complex geometry, however,

brings some comp lications in the hierarchical p-version as the edge nodes d o not

exist. As a remedy, geometrical map ping v ia linear/ quad ratic parametric mapp ing

functions [93] or the so-called blending functions [36] is suggested. The blending

functions can be flexibly selected, thu s allowing an accurate representation of

various configurations.

(C) Compatibility of the hierarchical modes between adjacent element sDue to the C0-continuity requirement, it is necessary that the interpolation fun c-

tions (shape functions) between ad jacent elements are compatible. Using th e stan-

dard shape functions, nodes at shared edges (in case of 2D problem) and shared

faces (in case of 3D problem) have id entical values of the p rimary un known , ensur-

ing compatibility of the corresponding shape functions. In contrast, the hierarchical


28/203


Figure2.5 Example of interelement comp atibility of the hierarchical edge m ode: (left) wrong d efinitionand (right) correct definition.

version does not have a physical definition of the mod es at shared parts of elements.

A p roblem of incompatibility m ay occur. In Figure 2.5, we illustrate this problem

using the edge mode that is added for upgrading an element from quadratic order

to cubic order. Obviously, the edge shape functions may not be continuous over

the interelement boun dary if the edge m ode is separately defined for each element

(cf. Figure 2.5 (left)). This is due to the fact that asymmetric shape functions do notappear in pairs, as in Figure 2.2. Nevertheless, with careful consideration, the edge

mod e can be properly defined u sing an app ropriate node ordering rule as in Figure

2.5 (right).

2.3 Node-based hierarchical enhancement

In the last d ecade, the so-called m eshless method s [22] have gained p opu larity d ue

to their ability in avoiding a complicated remeshing procedures in adaptive finite

element analysis. How ever, they possess some limitations. For examp le, the mesh-

less shape functions (e.g. the m oving Least-Squ ares (MLS) app roximation [50]) are

normally much more computionally expensive than the conventional finite ele-

ment interpolation. Furthermore, most meshless shape functions do not possessthe Kronecker delta property, implying that the approximation function does not

pass through data points, thus leading to difficulties in imposing essential bound-

ary cond itions [46,59,65].A nd , since they are meshless, difficulties due to nu merical

integration arise [22]. Instead of being specified on elements, the quadrature points

are then located in the newly created background cells, which may not conform the

domain (or subdomain) geometry. Such treatment brings about quadrature errors

and a background of integration cells destroys the meshless nature of these interpo-

lations.

By such shortcomings, attention has been concentrated on how to improve the

existing finite element mod els with th e strong p oints of the m eshless method s. Ex-

amples of some attempts are the cloud-based finite element method [62], the par-

tition of unity finite element method [57], the generalised finite element method

[91,92], the special finite element method [13] and the new hierarchical finite ele-

ment method [94], which are all based on th e same concept: nod al enrichmen t via

the p artition of un ity p roperty of the fin ite element shape functions (cf. Eq. (2.8)).

Based on the finite element hat functions, computational cost is much reduced asDue to its shape, linear finite element shape functions are also known as hatfunctions.


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2.3 Node-based hierarchical enhancement 21

compared to the u se of MLS shape functions.

The node-based enrichment technique inherits the strong points of meshless

techniques while it retains the strong points of the FEM. For instance, the choice

of enrichment functions is much more flexible as compared to the element-based

hierarchical enrichment. The technique also concentrates hierarchical degrees of

freedom at n odes thus providing a sparser band structure of the stiffness matrix

than the one in the trad itional app roach. Moreover, the Kronecker d elta prop erty of

the finite element interpolation introd uces straightforward imposition of bound ary

constraints, while numerical integration based on element structure is automatic

and conformed to the element domains resulting in better accuracy in the numer-

ical analysis. The technique can be imp lemented easily and efficiently withou t in-

trodu cing an y complicated arran gement of hierarchical mod es at edges and inside

elements as in the element-based hierarchical p-version finite element method .

2.3.1 Enhancement technique

The enhancement of the finite element shape functions to higher-order polynomi-

als being added through the partition-of-unity property has been applied in many

studies (for example, [54,91,92,94,102]). The scheme avoids the use of ad ditional

nodes in the domain to enrich the polynomial order of the shape function and can

be considered as a h ierarchical class ofp-enrichment. In particular, for the approxi-

mant u, it is written that

u =n

i=1

i

m i+1

j=1

(i)j

a(i)j

(2.12)

where, at node i, (i)1 = 1 always and the corresponding degree of freedom a

(i)1

represents rigid-body movement. The enhancement can be added hierarchically.

To reveal th is pr oper ty, Eq. (2.12) can also be w ritten as

u =n

i=1

i

a i +

m i

j=1

(i)jb

(i)j

(2.13)

which distinguishes between the existing interpolation functions (corresponding to

a i with i = 1,2,.. , n) and th e add itional enrichment functions (corresponding to b(i)j

with i = 1,2, .., n an d j = 1,2, .., m i). Here, n an d m i are num ber of nodes and extra(non-unity) terms for node i, denotes the interpolation function of the discrete

primary unknown a, contains enhancement terms and b refers to a set of extradegrees of freedom that is introdu ced th rough the partition of unity prop erty of the

finite element shape fun ction. Note th at the interp olation of th e degrees of freedom

b is not set by alone but rather through the produ cti(i)j

.

It is noted here that Eq. (2.13) is equivalent to Eq. (2.12) assuming that {} ={} + {1}, i.e. span of the enrichm ent fun ction (x) (cf. Eq. (2.12)) comprises one


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extra component representing rigid body movement, i.e. the unity component, in

add ition to the span of the enrichment fun ction (x)(cf. Eq. (2.13)).

2.3.2 Choices of polynomial enrichment functions

In order to construct a set of shape functions based on higher-order interpolation,

one should realise that the resulting shape functions should possess the complete

polynomial property to guarantee convergence of the finite element solutions and

satisfy the continuity requirement. Generally, the polynomial enrichment functions

are added through nodal shape functions, which are basically of linear order (i.e.

only vertex shap e fun ctions exist). In such a case, to obtain higher-ord er shape func-

tions, one may specify a set of enrichment functions at node i, according to the

polyn omial term s in Pascals triang le (cf. Figure 2.1) as

(i)(12) = {2i ,ii , 2i } (2.14)

(i)(13) = {2i ,ii , 2i ,3i ,2i i ,i2i , 3i } (2.15)

(i)(14) = {2i ,ii , 2i ,3i ,2i i ,i2i , 3i ,4i ,3i i,2i 2i ,i3i , 4i } (2.16)

to upgrade linear shape function to quadratic, cubic and quartic order, respectively.

The subscript (j k) here refers to an upgrade from polynomial degree j to poly-nomial degree k, and the sup erscript (i) refers to the enr ichm ent fun ction associatedwith node i. It is worth noting that the enrichment functions are added hierarchi-

cally, i.e.

(i)(1p+1) (

i)(1p+2) (

i)(1p+3) . . . (

i)(1p+) (2.17)

As such, the resulting shape functions can be viewed as a specific type of hierarchi-

cal shape functions.

The fun ctions = (x) an d = (y) must be chosen such that the aforemen-tioned continuity requ irement is satisfied. An example is the one prop osed in [94],

i.e.

i = (x x i) an d i = (y y i) (2.18)where x an d y represents the global Cartesian coordinates of a point in the dom ain.

Corresponding to the enrichment at node i, x i an d y i denote th e global coordinates

of the nodal point. The choice represents the distance of any point to the to-be-enriched node, providing the continuity of the enrichment functions throughout

the domain. The enrichment functions (i)j

increase in magnitude with increasing

distance from the associated nod e i. How ever, the enrichment is cut off at the en d

of the element edge du e to the m ultiplication w ith the existing finite element shape

function i, which equals zero in all elements not adjacent to n ode i.


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2.4 Remarks 23

In [33], the choice

i =(x x i)

hian d i =

(y y i)hi

(2.19)

was chosen to weigh the enrichment function with hi, the diameter of the largest

finite element sharing nod e i. Obviously, this format p rovides an im proved version

of the format in Eq. (2.18) as a better conditioning number of the resulting stiffness

matrix is obtained.

In ad dition to th e enrichment functions p resented above, it is also possible to se-

lect other types of polynomials such as harmonic polynomials as p resented in [91]

and [57]. This set of polynomials has the advantage that its dimension grows lin-

early with p olynomial order, whereas the set of full polynomials from FEM grows

quadratically.

2.4 Remarks

In this chapter, finite element shape functions have been introduced in various

forms. In spite of their non-physical meaning, the hierarchical shape functions gain

more popularity in the mesh adaptive studies as the existing shape functions are

preserved thus avoiding that the whole stiffness matrix system must be recalcu-

lated. This at tractive feature facilitates p-adaptive analysis to be carried out in this

thesis.

The hierarchical enhancement of the fin ite element shape fun ctions can be intro-

du ced in an element-based fashion or a node-based fashion. Despite the attractive

features of the nod e-based h ierarchical extension, one serious p roblem that makesthe method unattractive is linear dependence of the resulting shap e functions, which

leads to unsolvability of the discretised equations. This is discussed in detail in Ap-

pend ix A. Although there are some techniques to overcome such shortcoming, w e

do not wish to complicate the present work unnecessarily. Therefore, in this thesis,

we will emp loy only the element-based hierarchical shape functions.


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CHAPTER

THREE

A posteriori error estimation

An important component of finite element adaptive analysis is how to assess the

local error accurately. This error information normally gives a clue where and to

wh ich extent some parts of the mesh should be enhanced so that the finite element

analysis can provide acceptably accurate and cost effective results. As such, the so-

called a posteriori error estimators, which ap proximate the actual error at the end of

the calculation step, p lay an importan t role in ensuring reliability of finite element

models. The error information, which is the focus in this research work, refers to

the error th at is caused by inadequate discretisation in th e finite element analysis,

and it is also know n as the discretisation error.

This chapter starts with a mathematical definition of the discretisation error in

the finite element method, which is usually measured in terms of an energy norm.

Then, w e ad dress some basic ideas about the standard residual-type error estima-

tion, which later leads to th e formulation of the simple error estimator used in this

research. The chapter end s with some investigations about p erforman ces and som e

critical comm ents about the m ethod.

3.1 Discretisation error

The d iscretisation error, e, is defined as

e := u u(h,p) (3.1)i.e. the difference between the exact solution to the mathematical model, u, and thefinite element solution, u(h,p). Here, we assume that the error that comes from the

numerical process, known as the numerical error, is marginal in comparison to the

error in the d iscretisation p art, and thus can be neglected.

App arently, the error e in Eq. (3.1) cannot be comp uted directly since the exact

solution u is generally unkn own . Nevertheless, as a more refined/ enriched d iscreti-


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26 Chapter 3 A posteriori error estimation

sation gives a better approximation to the actual solution u, we can closely repre-

sent the actual solution u by a very fine discretisation (so-called reference m esh), via

h-extension an d/ or p-extension , for example.The finite element solution from the refined/ enriched system u(h, p), obtained

from solving the reference discretised problem

B(u(h, p) , v (h, p)) = F(v (h, p)) v (h, p) V(h, p) (3.2)

is now denoted as a reference to the actual solution u. As a consequence, the dis-

cretisation err or, defin ed in Eq. (3.1), is app roximated by

e u(h, p) u(h,p) =: e(h, p) (3.3)

The approximation involved in Eq. (3.3) is sufficiently accurate because the actualsolution u is much closer to the solution from the refined system u(h, p) than to the

primary solution u(h,p).

In order to p rovide a prop er measurement of global and elemental error, the dis-

crete error should be measured in a well-defined norm. A classical option, also em-

ployed in this contribution, is the measurement of error in an energy norm defined

as

e :=

B(e, e) =

k

Bk(e, e) =

k

e2k

(3.4)

where the subscript k denotes the error contribution obtained from the elemental

level. The global estimation is obtained by summing up the elemental contribu-

tions. The global error measure e is used in consideration whether or not the finiteelement solution is acceptably accurate. As well, the elemental error measure of the

element k,

ek :=

Bk(e, e) (3.5)

is necessary in driving the mesh adaptive process (See Chapter 5).

3.2 Standard residual-type error estimation

Basicallly, a posteriori error estimators can be categorised in tw o main group s

namely the recovery ty pe a n d th e residual type. As aforementioned in Chapter 1,the residual-type error estimators are employed in th is research. The m ethods, pi-

oneered by the work of Babuska and Rheinboldt [12], determine th e error by cal-

culating the residual of the finite element solutions in each local space. Without

The mesh may be either refined (h-extension) or enriched (p-extension). It is not necessary that bothfactors are enh anced to form the reference solution.


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3.2 Standard residual-type error estimation 27

Error indication Error estimation

Implicit estimationExplicit estimation

Local Neumann type Local Dirichlet type

Error assessment

Residual type Recovery type

Figure 3.1 Error assessment techniques in finite element analysis. Note that the double-bounded boxrefers to the type used in this research.

relying on the superconvergence property of some sample points in the problem

domain as in the recovery type, the residual-type error estimators can be applied

to a w ider variety of problems, including n on-homogeneous higher-order interpo-

lation or even nonlinear solution control, which are in the scope of this research.

The standard residual-type error estimation can be formulated either explicitly or

imp licitly. Whereas the explicit version employs the residuals in the current app rox-

imation directly, the implicit version uses the residuals indirectly via a set of local

algebraic equations. Obviously, th e im plicit v ersion, in comparison to the explicitversion, requires more computational effort in solving an additional set of equa-

tions. The bigger effort, however, pays for the approximate error function, which is

subsequently measured in a quantified n orm. This error estimate p rovides more ac-

curate information than those from the explicit version that relies on the inequality

setting [4,97]. Figure 3.1 show s an overview of error assessment techn iques used in

finite element analysis.

In this research, we concentrate on the imp licit error estimation . The method con-

sists of three components, i.e.

a set of error equ ations,

a reference discretisation, and

a local comp utational framework.Basically, the set of error equations is formulated based on residuals in a global

computational framework. Without the known exact solutions, the residuals are

estimated by setting the reference discretisation via either h-extension, p-extension,

or any other mesh improvement approaches. Finally, the computational costs in-


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volved with the reference discretisation can be reduced importantly by replacing

the solution of a global system w ith the solution of a series oflocal problems.

Setting of error equations based on residuals

The residual-type error estimator, as its name implies, approximates the error based

on residuals, i.e. the amount by which the finite element solution fails to satisfy

pointwise the equ ilibrium equation in th e math ematical mod el. The finite element

solution u(h,p) V(h,p) is obtained by solving the set of equations

B(u(h,p), v (h,p)) = F(v (h,p)) v (h,p) V(h,p) (3.6)In order to estimate the error of the finite element solution, we recall the set of

equations of the reference system . That is,B(u(h, p) , v (h, p)) = F(v (h, p)) v (h, p) V(h, p) (3.7)

is to be solved and used as a close representative to the actual m odel.

The d ifference between Eq. (3.6) and Eq. (3.7), and using Eq. (3.3) leads to a set of

error equations

B(e(h, p) , v (h, p)) = Ru (v (h, p)) = F(v (h, p)) B(u(h,p), v (h, p))v (h, p) V(h, p) (3.8)

with the boundary condition that e = 0 on d. The residual Ru , which is basedon the p rimary unknown u, can be interpreted as a fictitious load by which the

app roximate solution d eviates from the actual solution.

Setting of local computational framework

In fact, one can estimate the error of a finite element model by comparing the fi-

nite element solutions obtained from th e original m esh to those from th e enhanced

mesh. This however requires a large amount of computation and makes no sense.

There is obviously little value in estimating the error of a coarse d iscretisation by

solving a global system of equations according to an enhanced discretisation. The

computational costs involved with the error estimation would far outweigh those

involved with solving for u(h,p), while at the same time an imp roved solution u(h, p)is already provided. By virtue ofu(h, p) , the solution u(h,p) has become redun dant,

and so has e(

h, p)

. In contrast, an efficient calculation of e(

h, p)

should involve local

(rather than global) solutions ofu(h, p) .

Since Eq. (3.8) is defined globally, it requires a large amount of computer re-

sources. In order to avoid this, the local spaces Vk, k = 1, 2, ..., n an d Vk V,Again, it is not necessary that both h an d p factors are enh anced to form the reference solution. How ever,

at least one factor need s to be up graded to form the reference system of equations.


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3.3 Boun dary cond itions of the local error equations 29

are defined and the residual-based error is computed in each local space. That is,

instead of solving Eq. (3.8), we, instead , solve a set of local equ ations

Bk(e(h, p) , v (h, p)) = Ruk(v (h, p)) v (h, p) Vk(h, p) (3.9)

wh ere the local residu al is defined as

Ruk(v (h, p)) = Fk(v (h, p)) Bk(u(h,p), v (h, p)) +

k\(kn )

u

nkv (h, p) d (3.10)

As a result from the integration by p arts on each local d omain, the add itional

contribution, w hich is the last term on the right-hand side of Eq. (3.10), represents

the norm al d erivatives (or flux) on the interelement boundary k as well as on

d and cancels in the global system of equations. Note that the contribution of the

norm al derivative on element ed ges on n , i.e.kn

unk

v d is included in Fk(v )as defined earlier.

To obtain the error associated with the primary unknowns u(h,p), the local error

equations (cf. Eq. (3.9)) must be solved. It is then necessary to d efine a prop er set

of bound ary cond itions of these local problems. We w ill ad dress this su bject in the

next section.

3.3 Boundary conditions of the local error equations

As mentioned in the last section, a key ingredient in solving local error equations is

setting the boundary conditions to be prescribed in Eq. (3.9). Taken from the global

finite element setting, the only Dirichlet boundary condition defined in each localspace k is

e = 0 on k d (3.11)This is because the primary unknown u is exactly p rescribed on the Dirichlet

boundary d. Obviously, add itional bound ary conditions for the local problems are

needed.

Basically, there are tw o subclasses of the implicit residual error estimation, d e-

pending on how the boundary conditions are defined in the local problems. While

th e Neumann-type error estimation prescribes the Neu mann conditions in the local

problems, the Dirichlet-type error estimation imposes the local Dirichlet conditions,

see also the ov erview in Figure 3.1. Some basic ideas abou t the tw o ap pro aches will

be presented in this section.

3.3.1 Local Neumann conditions

The imposition of the non-hom ogeneous flux bou nd ary conditions (local Neum ann

conditions), represented b y the last term of Eq. (3.10), may b e set via the simp le flu x


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averaging technique as

u(h,p)

nku(h,p)

nk

=

1

2nk

(u(h,p))k + (u(h,p))k

on k k (3.12)

The considered edge of an element k is shared by another (adjacent) element de-

noted as k. The introduction of two distinct indices k an d k allows to describejum p s of th e nor mal flu xes at th e in terelem en t bo u nd ar y.

The simple averaging has been criticised for being ad-hoc and fails to respect

the basic requirement for the local p roblem to be w ell-posed. Some researchers [3,

48] have proposed a new modification, the so-called equilibrated flux approach by

setting the equilibration condition

Fk(v

) Bk(u

(h,p), v

) + k\(kn )

u

nkv d

=0 (3.13)

where v = 1 and v = are selected for zeroth-order equilibration an d first-orderequilibration conditions, respectively. And with the consistency condition

u

nk+

u

nk= 0 on k k (3.14)

the error equations are well-posed on the regular subspace and the resulting error

estimator w ill provide a guar anteed up per bou nd of the exact error.

It should be noted that imposing only Neumann boundary conditions in the lo-

cal problems is not sufficient. It is necessary to impose a proper set of Dirichlet

conditions to eliminate the zero energy mod es (rigid bod y m odes), leading to solv-

ability of the equations. Obviously, the Dirichlet conditions described in Eq.(3.11)are not sufficient for solving the local problems that are not attached to the Dirichlet

bound ary. To overcome this p roblem, one may reformu late the local problem over

a redu ced subspace where the zero energy m odes have been factored out [1, 2,16].

3.3.2 Local Dirichlet conditions

Modelling of the equilibrated residual fluxes at the interelement boundaries gen-

erally requires high computational effort. To avoid such complicated computation,

the local Neumann boundary conditions in conventional element residual method

may b e replaced by a set of local Dirichlet cond itions. The m ethod app roximates lo-

cal errors without the necessity to compute the flux jump, thus the computational

cost can be significantly diminished. However, this assumption leads to a lower

bound estimate that is often not of a good quality.

An im prov emen t of the ap pr oach has been p rop osed by D ez et al. [29]. In th eir

app roach, an ad ditional set of local error equations is introdu ced to h elp improving

quality of the error estimate computed based on the elemental basis. The error func-

tion can be app roximated by solving a set of local problems w hose spaces overlap.

These local (patch) spaces must be selected in such a way that


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3.3 Boun dary cond itions of the local error equations 31

(I) Internal residual estimate (2) Boundary jump recovery

Figure 3.2 Two steps of residu al-based error compu tation in the local Dirichlet-type framework basedon local h-refinement [29].

= kp a tk , i.e. all patches together cover the w hole problem d omain, and p at

ip a t

j= , i.e. a patch p a t

imu st overlap partly at least one other patch

p at

jthat is in the neighbourhood. The overlapping of patches depends on

how the reference mesh is chosen.

For a local space (or patch) p a t

k, a set of homogeneous boundary condi-

tions is defined by suppressing error components as zero on the local boundary

p a t

k\(p at

k n ). The error estimate can then be obtained by find ing Vk

where Vk

:= {v H1(p a tk

); = 0 on d p a tk } fromBk((h, p) , v (h, p)) = Ruk(v (h, p)) = Fk(v (h, p)) Bk(u(h,p), v (h, p))

v (h, p) Vk(h, p) (3.15)temporarily neglecting the last term appearing in Eq.(3.10). The space Vk =s u p p (

p at

k), thus Vk V. In the original work [29], this first estimate to e(h, p)

is computed elementwise (i.e. the local space is based on one element) and denoted

as the interior estimate.

Since the estimated error is suppressed to zero on the inter-patch boundaries,

the obtained error solution is a poor approximation to the exact error. It is then

necessary to enrich the first patch solutions by a set of patches overlapping the local

space. Let l be the local space that overlaps k, find another error estimate U

l, where U

l:= {v H1(p a t

l); = 0 on d p a tl } from another boundary

value p roblem

Bl((h, p) , v (h, p)) = Rul (v (h, p)) = Fl (v (h, p)) Bl (u(h,p), v (h, p))v (h, p) Ul (h, p) (3.16)

The second estimate is based on the collection of parts in surrounding elements

to form each patch overlapping the element domain (thus interior domain) and is


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( A )

( B )

( C )

Figure3.3 How the local Dirichlet method works: (A) interior estimation , (B) patch estimation beforeorthogona lity setting and (C) patch estimation after the orth ogonality setting . The exact errors andthe estimated errors are shown in solid and dashed lines, respectively. The filled circles denote nodal

points in the one-dimensional problem domain.

called the patch estimate [29]. Similar to the interior estimate , the local errors on

l\(l n ) are prescribed to zero. This p atch estimate provides information ofthe error caused by the residu al fluxes on the elemental bound aries. See Figure 3.2

for an illustration of the tw o-step error comp utation.

To combine componen ts from different patches, the contributions an d must

be adjusted to satisfy the Galerkin orthogonality property,that is

Bl((h, p) ,(h, p)) =

0 o n l . Retrieving the interior estimate which is projected onto l , the patch solu-tion in each l can be recalculated as

(h, p)

= (h, p) Bl((h, p) ,(h, p))Bl((h, p) ,(h

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