The background of error estimation and adaptivity in finite element computations

Transcription

1 Comput. Methods Appl. Mech. Engrg. 195 (2006) The background of error estimation and adaptivity in finite element computations Olgierd C. Zienkiewicz 1 School of Engineering, University of Wales, Swansea, UK Received 29 June 2004; accepted 10 July 2004 Abstract The paper summarizes the present state of the art concerning error estimation and adaptive remeshing. Recent work shows that methods based on recovery result on most accurate error estimators. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Error estimation; Adaptive remeshing 1. Introduction This paper is being written to open the special issue of this journal concerned with reporting the major achievements of the conference called ADMOS held in Göteborg, Sweden, from September 29 to October 1, It is appropriate when considering the present state of the art of any subject, to be informed about its background and origins. Error estimation in numerical computations is obviously as old as the numerical computations themselves. The very first paper by Richardson [1] published in 1910 is the first paper for practical computations utilizing finite differences which presents a general procedure by which errors can be estimated. The original Richardson method starts from the fact that errors in the solution of any finite difference scheme usually depend of some power of the mesh size used (or today the size of finite elements). The order of the error is thus usually given and depends as well on the solution method adopted. In the Richardson procedure a set of meshes, which nest within each other, are used to provide the solution, then with a knowledge of the convergence rate the errors can be estimated. 1 Unesco Professor, CIMNE, UPC, Barcelona, Spain /$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi: /j.cma

2 208 O.C. Zienkiewicz / Comput. Methods Appl. Mech. Engrg. 195 (2006) Unfortunately, the procedures used by Richardson and his early followers are not easily adaptable to the computer age. At the present stage the time and resources available for resolution of the problem on a finer mesh do not exist as generally all the funds available for computation have been used in the first solution. What is therefore required is to find the error in the existing solution which certainly cannot be repeated on a mesh of half the size. 2. Residual and recovery based methods 2.1. Error estimation The work, originally introduced by Babuška and Rheinboldt [2,3], to estimate errors, considers local residuals of the numerical solution. By investigating the residuals occurring in a patch of elements or even in a single element it becomes possible to estimate the errors which arise locally, usually in such norm as the energy norm. The original approaches for error estimation and indeed for correcting these by adaptive refinement started by simply looking at the elements with largest error and dividing these to achieve some acceptable accuracy. Since this early work many players entered the field and today the procedures available for error estimation are essentially reduced to two kinds. These are, first, the residual error estimators continuing the original suggestions of Babuška but the emphasis is now on use of self-equilibrating patches. Here the work of Ainsworth and Oden [4,5] and others is important. Such residual approaches have gained universal mathematical aplomb and for some years formed the basis of acceptable error estimators. However, in the second approach, an alternative was possible and this was suggested by myself and my collaborator Dr. J.Z. Zhu. This is the use of a recovery process to obtain more accurate representation of the unknowns. In the original process [6], introduced in 1987, simple averaging and the so-called L 2 projection recovery of Brauchli and Oden [7] were used to estimate errors. Although these methods were quite successful, better recovery procedures were sought. It was only in 1992that a very accurate recovery process based on superconvergence was achieved [8 10]. This so-called superconvergent patch recovery (SPR) method led to very accurate recovered values which generally converged at a higher rate than the original solution and which gave a solid basis for error estimation. In the first uses of the SPR the standard process was to find, in a patch corresponding to the solution surrounding each node or element, a set of points at which the accuracy was higher such as the superconvergence points and then to pass a least squares approximation of the function of the stresses or gradients (as in the case of thermal problems) through such values. With the knowledge of these more accurate stresses or gradients of the solution, the energy norm could easily be estimated and indeed so could the local values of stresses and gradients. Today the methodology is widely used, industrially, following the same practice but a variant of it is sometimes desirable. In that variant the superconvergent values are taken not as the stresses or gradients of the solution, but on the values of the original functions themselves. It is known for instance that superconvergence will always occur in these values for triangular elements at the vertex nodes, whereas in the same triangular elements such superconvergence for the stresses or gradients does not exist. Therefore for triangular elements it is preferable to include in the patch superconvergence of the displacements or the basic function itself and then to find its gradients by establishing a suitable polynomial of higher order. This procedure of obtaining the recovered values is currently widely used and many papers by Wiberg (who is a co-editor of this issue of the Journal) have contributed very much to this recovery methodology [11 15]. Although both the recovery and residual based procedures have proven their worth it was necessary to establish which of the particular methods proved most accurate. To test these procedures Babuška together

3 with a team of researchers led by Professor Strouboulis at the University of Texas at Austin, produced a general methodology. This methodology is essentially based on what I like to call the Babǔska patch test in which arbitrary fields of finite elements, assumed to be repeatable in space, are subjected to the next higher order basic polynomial than contained in the original shape functions. Obviously if the finite elements used include quadratics any solution which is quadratic will be exactly returned through the numerical computation and again equally the main errors will be concentrated only in the cubic terms. Thus it is necessary, in a repeatable sense, to impose only cubic polynomials on the patch and test whether errors arise and, if so, their magnitude. By doing this it is easy to establish whether the procedures capture errors exactly or simply show the errors for inexact solutions. Here Babuška proposed to value each error estimator with a robustness index and after very many tests concluded that the robustness index was definitively optimal for the recovery methods, with the residual methodology giving much poorer results. This work is reported in a series of papers [16 20] and in the book available in 2001 [21] Adaptivity O.C. Zienkiewicz / Comput. Methods Appl. Mech. Engrg. 195 (2006) Once the error estimator process has been achieved it is natural to seek methods by which the design can be improved. This process is not as simple as first implied by the work of the pioneers who simply considered a subdivision of elements in which error indicators were too large. There should be a logical way in which adaptivity can proceed. This is something to which attention was only sparingly given in the early days. It seems to me that we now have a method of obtaining a solution which gives certain accuracy with such norms as energy. The whole process today would be summarized as follows. First the errors in each element would be estimated, if for instance energy norm is sought, and the total energy norm error for the whole problem will be established. Once the decision is taken that the total error permissible should be less than a certain value, it is a simple matter to search the design field for a new solution in which the total error satisfies this requirement. The simplest process is to try to achieve an equal error distribution between all elements as with such equal error distribution the results are most economical [6,22 25]. Considering the total permissible error, the error can be divided between the existing number of elements used and it is then found whether more elements need to be used. Generally this will be the case but at the same time, in some other areas of the problem we may be over generous with subdivision. For this reason it is best in our experience to redesign the mesh completely and use the results of such a calculation to gain a new subdivision. Of course this is simple, if such requirements as those placed by mathematicians where the total energy norm has to be limited to some value, is used. It is much more difficult to provide answers for other quantities of interest such as local stresses, groups of particular forces, etc. How to get this is obviously something that occupies minds of mesh designers and I should not discuss it further, except to mention that in many cases we are not concerned by overall errors or errors in such norms as energy but wish to look at another kind of norm and possibly to a local value of some quantity, forgetting all the others. Much progress is today being achieved in solution of problems of goal oriented adaptivity or estimation in the error on quantities of interest. And this by itself will form a subject of some papers at this conference [26 29] and I anticipate new ideas will be presented. 3. Bounds on quantities of interest Although we have shown that excellent estimators of errors exist today, many researchers are striving to know that these estimators are not only close but that they are bounded. The strain energy was one of the first quantities in which bounds could be established. Here the classical work of Fraeijs de Veubeke in the mid-1960s is of vital importance [30,31].

4 210 O.C. Zienkiewicz / Comput. Methods Appl. Mech. Engrg. 195 (2006) It was quickly realized by Fraeijs de Veubeke that the standard (displacement) procedures from which structural analysis usually started would provide a lower bound of the strain energy contained in the structure and thus the solution always underestimated the value of strain energy. He therefore sought procedures which could solve the same structural problem by concentrating on so-called complementary energy this would allow the establishment of solutions in which the strain energy would always be overestimated. This process proved very difficult as equilibrating solutions have to be established at all stages. I had the opportunity of collaborating with Fraeijs de Veubeke and suggested a possible way, useful for many two dimensional problems, by the use of stress functions and slab analogy [32]. Nevertheless the methodology did not succeed as a practical way of providing the bounds of strain energy in an actual analysis. Much later when the residual method was being applied to determine error in structural analysis it was realized that once again opportunity would arise for establishing bounds. The residuals are nothing else but a measure by which the numerical solution fails to satisfy the differential equations of the problem. By using a local solution, generally based on single elements, the error can be estimated locally and the total error be obtained by combining these estimates for all elements. The completely independent solution for the displacement, stresses, etc. established by the residuals provides the measure of the error. This solution can be carried out in a number of ways and here a departure from the original, say classical displacement, method can be made. As many local problems are solved, it is assumed the total error is a combination of local (patch) solutions. It seems that one of the first to extend the concept of establishing upper bounds is Kelly in 1984 [33] and in his subsequent work. He endeavored to obtain solutions for the residual placed as a load with equilibrating methodologies. Two similar alternative approaches, though stemming from completely different origins, were proposed by Ladevèze [34,35] and almost simultaneously by Bank and Weiser [36]. These were later adopted by Ainsworth and Oden forming a large part to their book [37]. The adaptive methodologies so far produced are very effective when the basic quantity of interest is a simple one, such as strain energy or the energy norm. However, when the quantity of interest is more localized and if for instance it is the displacement at some part of the structure, rather than an overall measure of stresses as it is in the case of energy, different procedures arise and pure examination of energy errors does not suffice or it is not very selective in showing how to correct the answers. For this reason much effort has been given in recent years to discussing the possibilities of bounds and optimizing the manner in which such localized adaptive goals of analysis can be solved. Much of the present work in this field concentrates on such methodologies and indeed many papers presented later in this special issue deal with these problems. The first to give attention to the possible extension of norms was a series of papers presented by Babuška and Miller in 1984 [38 40]. These papers laid the foundations for much of the work started some 10 years later and which today occupies much of our interest. Here it appears that the first full extension of the methodology is due to Peraire who published in 1997 with many papers following [41 46]. By extending the ideas introduced in the Babuška and Miller papers, Peraire et al. show that whatever the quantity of interest is, it is always possible to establish an adjoint problem which can be solved on the same mesh of the original problem but now with different loads for dealing with the accuracy. Such adjoint problems may be called differently and names such as extraction problems and dual problems are also used. Although many people are now entering the field and the methodology has been followed by Oden and Prudhomme in a series of papers, it appears that only Peraire so far has extended the approach to non self-adjoint problems such as fluid dynamics [43,45,46]. Here again, the extension of Babuška and Miller goes much further and I believe that a number of papers included in this issue are following this line of attack. Clearly, much remains to be done and the range of possibilities to appear is quite large. The equilibrated methods have always provided upper bounds for such quantities as strain energy, etc. Similar position of the bounding occurs if we look at the energy in the adjoint approaches. However, some interest now goes back to placing satisfactorily the lower bounds and thus bracket the solutions. Of course,

5 O.C. Zienkiewicz / Comput. Methods Appl. Mech. Engrg. 195 (2006) lower bounds of zero values have been used and this is rather defeatist because they do not provide a tight specification. It is my belief that more precise bounds are required and here the work of Díez, who himself is a co-editor of this issue, and Huerta seems to lead the way [47 49]. Although the bounding of quantities of interest involve quite sophisticated mathematics, it is possible that simpler approaches will evolve in general engineering practice, thus leading possibly to a parallel with the main error estimators which have been so satisfactorily introduced for industry in general. There is apparent need for such error estimators in fluid dynamics and in many other problems. We look forward to some interesting work being presented for this in the future. 4. Closure Clearly the concept of bounds and local solutions can be extended to other quantities of interest beyond strain energy, such as may arise for instance in determining a local displacement, a maximum stress, a fracture mechanics quantity, etc. Indeed in extending the procedures to non self-adjoint problems in which symmetry does not exist the concept of residual is paramount, however, it does not need to be the only way of establishing the kind of error bound. As clearly the residuals of the original solution need not be calculated to obtain the best answers, already it has been shown that much better estimates of error and accuracy are obtained by the process of recovery. And it seems to me we should endeavour to combine the achievements of a good recovery process with the possibilities of bounding by local solutions. It is clear to me that recovered solution values could be used in any procedure to find a set of residuals corresponding to the recovered solution and not to the original numerical one and, thus, to establish bounds on the recovered solution which has been shown to be very precise. Here, incidentally, an energy norm bound obtaining local solutions now would be more efficacious than starting from the so-called equilibrated-residual basis. The reader will have observed that in many papers included in this symposium the question of bounds and of extending the possibilities beyond the simple strain energy is possible. Some of the work concerning fluid mechanics should be of special interest here as clearly in such problems a bound no longer exists in the energy norm way. References [1] L.F. Richardson, The approximate arithmetical solution by finite differences of physical problems, Trans. Roy. Soc. (London) A 210 (1910) [2] I. Babuška, C. Rheinboldt, A-posteriori error estimates for the finite element method, Int. J. Numer. Methods Engrg. 12(1978) [3] I. Babuška, C. Rheinboldt, Adaptive approaches and reliability estimates in finite element analysis, Comput. Methods Appl. Mech. Engrg (1979) [4] M. Ainsworth, J.T. Oden, A procedure for a posteriori error estimation for h p finite element methods, Comput. Methods Appl. Mech. Engrg. 101 (1992) [5] M. Ainsworth, J.T. Oden, A unified approach to a posteriori error estimation using element residual methods, Numerische Mathematik 65 (1993) [6] O.C. Zienkiewicz, J.Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Numer. Methods Engrg. 24 (1987) [7] H.J. Brauchli, J.T. Oden, On the calculation of consistent stress distributions in finite element applications, Int. J. Numer. Methods Engrg. 3 (1971) [8] O.C. Zienkiewicz, J.Z. Zhu, The superconvergent patch recovery (SPR) and adaptive finite element refinement, Comput. Methods Appl. Mech. Engrg. 101 (1992) [9] O.C. Zienkiewicz, J.Z. Zhu, The superconvergent patch recovery (SPR) and a posteriori error estimates. Part 1: The recovery technique, Int. J. Numer. Methods Engrg. 33 (1992)

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