Advances in Adaptive Computational Methods in Mechanics (eBook)

P. Ladeveze, J.T. Oden (Herausgeber)

eBook Download: PDF

1998 | 1. Auflage
525 Seiten
Elsevier Science (Verlag)
978-0-08-052593-8 (ISBN)

Infos

Mastering modelling, and in particular numerical models, is becoming a crucial and central question in modern computational mechanics. Various tools, able to quantify the quality of a model with regard to another one taken as the reference, have been derived. Applied to computational strategies, these tools lead to new computational methods which are called adaptive. The present book is concerned with outlining the state of the art and the latest advances in both these important areas.

Papers are selected from a Workshop (Cachan 17-19 September 1997) which is the third of a series devoted to Error Estimators and Adaptivity in Computational Mechanics. The Cachan Workshop dealt with latest advances in adaptive computational methods in mechanics and their impacts on solving engineering problems. It was centered too on providing answers to simple questions such as: what is being used or can be used at present to solve engineering problems? What should be the state of art in the year 2000? What are the new questions involving error estimators and their applications?

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Mastering modelling, and in particular numerical models, is becoming a crucial and central question in modern computational mechanics. Various tools, able to quantify the quality of a model with regard to another one taken as the reference, have been derived. Applied to computational strategies, these tools lead to new computational methods which are called "e;adaptive"e;. The present book is concerned with outlining the state of the art and the latest advances in both these important areas.Papers are selected from a Workshop (Cachan 17-19 September 1997) which is the third of a series devoted to Error Estimators and Adaptivity in Computational Mechanics. The Cachan Workshop dealt with latest advances in adaptive computational methods in mechanics and their impacts on solving engineering problems. It was centered too on providing answers to simple questions such as: what is being used or can be used at present to solve engineering problems? What should be the state of art in the year 2000? What are the new questions involving error estimators and their applications?

Inhalt

Cover 1
Advances in Adaptive Computational Methods in Mechanics 4
Copyright Page 5
Contents 8
Preface 6
PART 1: ERROR ESTIMATORS AND ADAPTIVE COMPUTATIONAL METHODS FOR LINEAR PROBLEMS 12
Chapter 1. Recovery procedures in error estimation and adaptivity: Adaptivity in linear problems 14
Chapter 2. The relationship of some a posteriori error estimators 36
Chapter 3. A technique for a posteriori error estimation of h-p approximations of the Stokes equations 54
Chapter 4. A mathematical framework for the P. Ladevèze a posteriori error bounds in finite element methods 76
PART 2: MODELLING ERROR ESTIMATORS AND ADAPTIVE MODELLING STRATEGIES 90
Chapter 5. Adaptive finite element in elastoplasticity with mechanical error indicators and Neumann- type estimators 92
Chapter 6. A reliable a posteriori error estimator for adaptive hierarchic modelling 112
Chapter 7. A two-scale strategy and a posteriori error estimation for modeling heterogeneous structures 126
Chapter 8. A modelling error estimator for dynamic structural model updating 146
PART 3: LOCAL ERROR ESTIMATORS FOR LINEAR PROBLEMS 164
Chapter 9. A posteriori estimation of the error in the error estimate 166
Chapter 10. Bounds for linear-functional outputs of coercive partial differential equations: Local indicators and adaptive refinement 210
Chapter 11. On adaptivity and error criteria in meshfree methods 228
PART 4: ERROR ESTIMATORS FOR NON LINEAR TIME-DEPENDENT PROBLEMS AND ADAPTIVE COMPUTATIONAL METHODS 240
Chapter 12. A posteriori constitutive relation error estimators for nonlinear finite element analysis and adaptive control 242
Chapter 13. A review of a posteriori error estimation techniques for elasticity problems 268
Chapter 14. A posteriori error control and mesh adaptation for EE. models in elasticity and elasto- plasticity 286
Chapter 15. An adaptive finite element approach in associated and non-associated plasticity considering localization phenomena 304
Chapter 16. An a-adaptivity approach for advective-diffusive and fluid flow problems 320
Chapter 17. Adaptive strategy for transient/coupled problems. Applications to thermoelasticity and elastodynamics 336
Chapter 18. Error estimation and adaptive finite element analysis of softening solids 344
Chapter 19. Aspects of adaptive strategies for large deformation problems at finite inelastic strains 360
Chapter 20. Adaptive solutions in industrial forming process simulation 376
Chapter 21. Recovery procedures in error estimation and adaptivity: Adaptivity in non-linear problems of elasto-plasticity behaviour 394
PART 5: ADAPTIVE COMPUTATIONAL METHODS FOR 3D LINEAR PROBLEMS 422
Chapter 22. 3-D error estimation and mesh adaptation using improved R.E.P. method 424
Chapter 23. Adaptive methods and related issues from the viewpoint of hybrid equilibrium finite element models 438
Chapter 24. Error estimator and adaptivity for three-dimensional finite element analyses 454
PART 6: ERROR ESTIMATORS AND MESH ADAPTIVITY FOR VIBRATION, ACOUSTICS AND ELECTROMAGNETICS PROBLEMS 470
Chapter 25. Error estimation and adaptivity for h-version eigenfrequency analysis 472
Chapter 26. Error estimation and adaptivity for the finite element method in acoustics 488
Chapter 27. Error through the constitutive relation for beam or C° plate finite element: Static and vibration analyses 504
Chapter 28. A posteriori error analysis for steady-state Maxwell's equation 524
AUTHOR INDEX 538

Textauszug

Recovery Procedures in Error Estimation and Adaptivity: Adaptivity in Linear Problems

O.C. Zienkiewicza; B. Boroomanda; J.Z. Zhub    a University of Wales, Swansea, UK
b U.E.S. Inc. Annapolis, USA

ABSTRACT

The procedures of error estimation using stress or gradient recovery were introduced by Zienkiewicz and Zhu in 1987 [3] and with the improvement of recovery procedures (introduction of Superconvergent Patch Recovery SPR) since that date the authors have succeeded in making these error estimators the most robust of those currently available. Very recently introduced methods of recovery such as REP (Recovery by Equilibrium in Patches) allow general application.

In this paper the general idea of using recovery based error estimation in adaptive procedure of linear problems is explained through which the latest developments in recovery techniques is described. The fundamental basis of a ‘patch test’ introduced by Babuška et al [11]-[14] is explained and applied to the recovery based error estimators using both SPR and REP.

1 INTRODUCTION

Two types of procedures are currently available for deriving error estimators. They are either Residual based or Recovery based.

The residual based error estimators were first introduced by Babuška and Rheinboldt in 1978 [1] and have been since used very effectively and further developed by many others. Here substantial progress was made as recently as 1993 with the introduction of so called residual equilibration by Ainsworth and Oden [2].

The recovery based error estimators are, on the other hand, more recent having been first introduced by Zienkiewicz and Zhu in 1987 [3]. Again these were extensively improved by the introduction of new recovery processes. Here in particular the, so called, SPR (or Superconvergent Patch Recovery) method introduced in 1992 by the same authors [4]-[5] has produced a very significant improvement of performance of the Recovery based methods. Many others attempted further improvement [7]-[9] but the simple procedure originally introduced remains still most effective.

In this paper we shall concentrate entirely on the Recovery based method of error estimation. The reasons of this are straightforward:

(i) the concept is simple to grasp as the approximation of the error is identified as the difference between the recovered solution u* and the numerical solution uh; thus the estimate

e*||=||u*−uh||

  (1)

in any norm is achieved simply by assuming that the exact solution u can be replaced by the recovered one.

(ii) as some recovery process is invariably attached to numerical codes to present more accurate and plausible solutions, little additional computation is involved;

(iii) if the recovery process itself is superconvergent, it can be shown [5] that the estimator will always be asymptotically exact (we shall repeat the proof of this important theorem in the paper);

(iv) numerical comparisons on bench mark problems and more recently by a ‘patch test’ procedure introduced by Babuška et al [11]-[14] have shown that the recovery procedures are extremely accurate and robust. In all cases they appear to give a superior accuracy of estimation than that achievable by Residual based methods.

It is of interest to remark that in many cases it is possible to devise a Residual method which has an identical performance to a particular recovery process. This indeed are first noted by Rank and Zienkiewicz in 1987 [15] but later Ainsworth and Oden [16] observed that this occurs quite frequently. In a recent separate paper Zhu [17] shows that:

(v) for every Residual based estimator there exists a corresponding Recovery based process. However the reverse in not true. Indeed the Recovery based methods with optimal performance appear not to have an equivalent Residual process and hence, of course, the possibilities offered by Recovery methods are greater.

In this paper we shall describe in detail the SPR based recovery as well as a new alternative REP process which appears to be comparable in performance.

With error estimation achieved the question of adaptive refinement needs to be addressed. Here we discuss some procedures of arriving at optimal mesh size distribution necessary to achieve prescribed error.

2 SOLUTION RECOVERY AND ERROR ESTIMATION

In what follows we shall be in general concerned with the numerical solution of problems in which a differential equation of the form given as:

TDSu+b=0

  (2)

has to be solved in a domain Ω with suitable boundary condition on:

Ω=Γ

  (3)

In above S is a differential operator usually defining stresses or fluxes as

=DSu

  (4)

where D is a matrix of physical parameters.

We shall not discuss here the detail of the finite element approximation which can be found in texts [18]. In there the unknown function u is approximated as:

≈uh=Nu¯

  (5)

which results in approximate stresses being:

h=DSuh=DBu¯

  (6)

In above

=Nxii−1−3andB=SN

  (7)

are the spatially defined shape-functions.

The solution error is defined as the difference between the exact solution and the numerical one. Thus for instance the displacement error is:

u=u−uh

  (8)

and the stress error is

σ=σ−σh

  (9)

at all points of the domain. It is, however, usual to define the error in terms of a suitable norm which can be written as a scalar value

=||eu||=||u−uh||

  (10)

for any specific domain Ω. The norm itself specifies the nature of the quantity defined. The well known energy norm is given, for instance, as

e||E=∫Ωσ−σhTD−1σ−σhdΩ12

  (11)

With the Recovery process we devise a procedure which gives, by suitable postprocessing of uh and σh, the values of u* and orσ* which are (hopefully) more accurate and we estimate the norm of the error as:

e||≈||e*||=||u*−uh||

  (12)

In the case of energy norm we have:

e*||E=∫Ωσ*−σhTD−1σ*−σhdΩ12

  (13)

The effectivity index of any error estimator is defined as:

=||e||este

  (14)

or in the case of recovery based estimators:

*=||e*||e

  (15)

A theorem proposed by Zienkiewicz and Zhu [5] shows that for all estimators based on recovery we can establish the following bound for the effectivity:

−e˜e<θ*<1+e˜e

  (16)

In above e is the actual error (viz Equation (10) and (11)) and ˜ is the error of the recovered solution i.e.

e˜||=||u−u*||

  (17)

The proof of the above theorem is straight forward, if we rewrite Equation (12) as:

e*||=||u*−uh||=||u−uh−u−u*||=||e−e˜||

  (18)

Using now the triangle inequality we have:

e||−||e˜||≤||e*||≤||e||+||e˜||

  (19)

from which the inequality (16) follows after division by ||e||. Two important conclusions follow:

(1) that any recovery process which result in reduced error will give a reasonable error estimator and, more importantly,

(2) if the recovered solution converges at a higher rate than the finite element solution we shall always have asymptotically exact...

Produkt-Details

• Erscheint lt. Verlag: 23.6.1998
• Sprache: englisch
• Themenwelt: Informatik ► Grafik / Design ► Digitale Bildverarbeitung
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• ISBN-10: 0-08-052593-8 / 0080525938
• ISBN-13: 978-0-08-052593-8 / 9780080525938