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15 Adaptive finite element refinement 15.1 Introduction In the previous chapter we have discussed at some length various methods of recovery by which the finite element solution results could be made more accurate and this led us to devise various procedures for error estimation In this chapter we shall be concerned with methods which can be used to reduce the errors generally once a finite element solution has been obtained As the process depends on previous results at all stages it is called adaptive Such adaptive methods were first introduced to finite element calculations by BabuSka and Rheinbolt in the late ~ ’Before > ~ proceeding further it is necessary to clarify the objectives of refinement and specify ‘permissible error magnitudes’ and here the engineer or user must have very clear aims For instance the naive requirement that all displacements or all stresses should be given within a specified tolerance is not always acceptable The reasons for this are obvious as at singularities, for example, stresses will always be infinite and therefore no finite tolerance could be specified The same difficulty is true for displacements if point or knife edge loads are considered The most common criterion in general engineering use is that of prescribing a total limit of the error computed in the energy norm Often this error is required not to exceed a specified percentage of the total energy norm of the solution and in the many examples presented later we shall use this criterion However, using a recovery type of error estimator it is possible to adaptively refine the mesh so that the accuracy of a certain quantity of interest, such as the RMS error in displacement and/or RMS error in stress (see Chapter 14, Eqs (14 loa) and (14.10b)), satisfy some user-specified criterion We should recognize that mesh refinement based on reducing the RMS error in displacement is in effect reducing the average displacement error in each element; similarly mesh refinement based on reducing the RMS error in stress is the same as reducing the average stress error in each element Here we could, for instance, specify directly the permissible error in stresses or displacements at any location Some investigators (e.g., Zienkiewicz and Zhu3) have used RMS error in stress in the adaptive mesh refinement to obtain more accurate stress solutions Others (e.g., Oiiate and Bugeda4) have used the requirement of constant energy norm density in the adaptive analysis, which is in fact equivalent to specifying a uniform distribution of RMS error in stress in each element We note that the recovery type of error estimators are particularly useful 402 Adaptive finite element refinement and convenient in designing adaptive analysis procedures for the quantities of interest As we have already remarked in the previous chapter we will at all times consider the error in the actual finite element solution rather than the error in the recovered solution It may indeed be possible in special problems for the error in the recovered solution to be zero, even if the error in the finite element solution itself is quite substantial (Consider here for instance a problem with a linear stress distribution being solved by linear elements which result in constant element stresses Obviously the element error will be quite large But if recovered stresses are used, exact results can be obtained and no errors will exist.) The problem of which of the errors to consider still needs to be answered At the present time we shall consider the question of recovery as that of providing a very substantial margin of safety in the definition of errors Various procedures exist for the refinement of finite element solutions Broadly these fall into two categories: The h-refinement in which the same class of elements continue to be used but are changed in size, in some locations made larger and in others made smaller, to provide maximum economy in reaching the desired solution The p-refinement in which we continue to use the same element size and simply increase, generally hierarchically, the order of the polynomial used in their definition It is occasionally useful to divide the above categories into subclasses, as the hrefinement can be applied and thought of in different ways In Fig 15.1 we illustrate three typical methods of h-refinement The first of these h-refinement methods is element subdivision (enrichment) [Fig 15.1(b)] Here refinement can be conveniently implemented and existing elements, if they show too much error, are simply divided into smaller ones keeping the original element boundaries intact Such a process is cumbersome as many hanging points are created where an element with mid-side nodes is joined to a linear element with no such nodes On such occasions it is necessary to provide local constraints at the hanging points and the calculations become more involved In addition, the implementation of de-refinement requires rather complex data management which may reduce the efficiency of the method Nevertheless, the method of element subdivision is quite widely used The second method is that of a complete mesh regeneration or remeshing [Fig 15.1(c)] Here, on the basis of a given solution, a new element size is predicted in all the domain and a totally new mesh is generated Thus a refinement and derefinement are simultaneously allowed This of course can be expensive, especially in three dimensions where mesh generation is difficult for certain types of elements, and it also presents a problem of transferring data from one mesh to another However, the results are generally much superior and this method will be used in most of the examples shown in this chapter For many practical engineering problems, particularly of those for which the element shape will be severely distorted during the analysis, adaptive mesh regeneration is a natural choice The final method, sometimes known as r-refinement [Fig 15.1(d)], keeps the total number of nodes constant and adjusts their position to obtain an optimal Introduction 403 Fig 15.1 Various procedures by h-refinement 404 Adaptive finite element refinement appro xi ma ti or^.'-^ While this procedure is theoretically of interest it is difficult to use in practice and there is little to recommend it Further it is not a true refinement procedure as a prespecified accuracy cannot generally be reached We shall see that with energy norms specified as the criterion, it is a fairly simple matter to predict the element size required for a given degree of approximation Thus very few re-solutions are generally necessary to reach the objective With p-refinement the situation is different Here two subclasses exist: One in which the polynomial order is increased uniformly throughout the whole domain; One in which the polynomial order is increased locally using hierarchical refinement In neither of these has a direct procedure been developed which allows the prediction of the best refinement to be used to obtain a given error Here the procedures generally require more resolutions and tend to be more costly However, the convergence for a given number of variables is more rapid with p-refinement and it has much to recommend it On occasion it is possible to combine efficiently the h- andp-refinements and call it the hp-refinement In this procedure both the size of elements h and their degree of polynomialp are altered Much work has been reported in the literature by Babugka, Oden and others and the interested reader is referred to the In the next two sections, Sec 15.2 and 15.3, we shall discuss both the h- and thep-refinements In Sec 15.3we also include some details of the very simple and yet efficient hp-refinement process introduced by Zienkiewicz, Zhu and Gong.’’ 15.2 Some examples of adaptive h-refinement 15.2.1 Mesh regeneration procedures In the introduction to this chapter we have mentioned several alternative processes of hadaptivity and we suggested that the process in which the complete mesh is regenerated is in general the most efficient Such a procedure allows elements to be de-refined (or enlarged) as well as refined (made smaller) and invariably starts at each stage of the analysis from a specification of the mesh size defined at each nodal point of the previous mesh Standard interpolation is used to find the size of elements required at any point in the domain This interpolation helps in the refinement subsequently Indeed at the starting point an initial mesh need not include the boundaries of the problem as it will be used only to interpolate the sizes required in the domain during the process of mesh generation However, after this first stage of analysis as the refinement proceeds the mesh sizes will be specified at the nodes of the last mesh In Chapter of this book, where we discussed mapping, we also discussed various possible mesh generators These did not allow a mesh size variation of the refined kind to be specified In adaptivity it is very important to be able to define quite precisely the element size or density of mesh so that a minimum number of elements can be used The generators which can this have been developed since the mid-1980s The was applied to aerospace engineering and fluid first of these by Peraire et Some examples of adaptive h-refinement 405 mechanics calculations Its basis is the frontal method of mesh generation developed originally by Cavendish2’ and Lo22 and the original generator was made available only for triangular elements Later such generators were generalized to include tetrahedral elements in three-dimensional space.23Today both triangular and tetrahedral generators form the basis of most adaptive codes Extension to quadrilateral and hexahedral elements is by no means easy First, procedures for generating quadrilateral elements in two dimensions have been devised The work of Zhu and Z i e n k i e w i c ~ ~and ~ ’ ~Rank ’ et a1.26>27 has to be noted The procedures are based on the joining of two triangles into a quadrilateral at different stages of the mesh generation process However, so far no extension of such methodologies to hexahedral elements in space have been made To the knowledge of the authors no efficient hexahedral mesh generators exist for adaptivity, though very many attempts have been reported in the In the more recent mesh generators used for both triangles and tetrahedra the frontal procedure has been largely replaced by Delauney triangulation and the reader is well advised to consult the following references and t e ~ t s ~ ~ , ~ ~ 15.2.2 Predicting the required element size in h adaptivity The error estimators discussed in the previous chapter allow the global energy (or similar) norm of the error to be determined and the errors occurring locally (at the element level) are usually also well represented If these errors are within the limits prescribed by the analyst then clearly the work is completed More frequently these limits are exceeded and refinement is necessary The question which this section addresses is how best to effect this refinement Here obviously many strategies are possible and much depends on the objectives to be achieved In the simplest case we shall seek, for instance, to make the relative energy norm percentage error 77 less than some specified value f j (say % in many engineering applications) Thus Tdij (15.1) is to be achieved In an ‘optimal mesh’ it is desirable that the distribution of energy norm error (i.e., I le1 I k ) should be equal for all elements Thus if the total permissible error is determined (assuming that it is given by the result of the approximate analysis) as Permissible error fj11ull fj(11u112 + llell2 ) ’12 (15.2) here we have used36 lle112 = llU1l2 - 11ull2 (15.3) We could pose a requirement that the error in any element k should be (1 5.4) where m is the number of elements involved 406 Adaptive finite element refinement Elements in which the above is not satisfied are obvious candidates for refinement Thus if we define the ratio (15.5) we shall refine whenevert (1 5.6) can be approximated, of course, by replacing the true error in Eqs (15.4) and (15.5) with the error estimators The refinement could be carried out progressively by refining only a certain number of elements in which E is higher than a certain limit and at each time of refining halve the size of such elements This type of element subdivision process is also known as mesh enrichment This process of refinement though ultimately leading to a satisfactory solution being obtained with a relatively small number of total degrees of freedom, is in general not economical as the total number of trial solutions may be excessive It is more efficient to try to design a completely new mesh which satisfies the requirement that