14 Errors, recovery processes and error estimates 14.1 Definition of errors We have stressed from the beginning of this book the approximate nature of the finite element method and on many occasions to show its capabilities we have compared it with exact solutions when these were known. Also on many occasions we have spoken about the ‘accuracy’ of the procedures we suggested and discussed the manner by which this accuracy could be improved. Indeed one of the objectives of this chapter is concerned with the question of accuracy and a possible improvement on it by an a posteriori treatment of the finite element data. We refer to such processes as recovery. We shall also consider the discretization error of the finite element approximation and a posteriori estimates of such error. In particular, we describe two distinct types of error estimators, recovery based error estimators and residual based error estimators. The critical role that the recovery processes play in the computation of these error estimators will be discussed. Before proceeding further it is necessary to define what we mean by error. This we consider to be the difference between the exact solution and the approximate one. This can apply to the basic function, such as displacement which we have called u and can be given as e=u-u
(14.1)
In a similar way, however, we could focus on the error in the strains (i.e., gradients in the solution), such as E or stresses (r and describe an error in those quantities as eE=z--E
(14.2)
e,=a-a
(14.3)
The specification of local error in the manner given in Eqs (14.1)-( 14.3) is generally not convenient and occasionally misleading. For instance, under a point load both errors in displacements and stresses will be locally infinite but the overall solution may well be acceptable. Similar situations will exist near re-entrant corners where, as is well known, stress singularities exist in elastic analysis and gradient singularities develop in field problems. For this reason various ‘norms’ representing some integral scalar quantity are often introduced to measure the error.
366
Errors, recovery processes and error estimates
If, for instance, we are concerned with a general linear equation of the form of Eq. (3.6) (cf. Chapter 3), i.e., Lu+p=O (14.4) we can define an energy norm written for the error as (14.5) This scalar measure corresponds in fact to the square root of the quadratic functional such as we have discussed in Sec. 3.8 of Chapter 3 and where we sought its minimum in the case of a self-adjoint operator L. For elasticity problems the energy norm is identically defined and yields, (14.6) (with symbols as used in Chapter 2). Here e is given by Eq. (14.1) and the operator S defines the strains as
E = S U and C=Sh and D is the elasticity matrix (see Chapter 2), giving the stress as
( 4.7)
DE and &=Dg in which for simplicity we ignore initial stresses and strains. The energy norm of Eq. (14.6) can thus be written alternatively as
( 4.8)
(14.9)
and its relation to strain energy is evident. Other scalar norms can easily be devised. For instance, the L2 norm of displacement and stress error can be written as
( 14.10)
(14.11) Such norms allow us to focus on the particular quantity of interest and indeed it is possible to evaluate 'root mean square' (RMS) values of its error. For instance, the RMS error in displacement, Au,becomes for the domain R (14.12)
Definition of errors 367
Similarly, the RMS error in stress, Acr, becomes for the domain R (14.13) Any of the above norms can be evaluated over the whole domain or over subdomains or even individual elements. We note that m
(14.14) i= I
where i refers to individual elements Ri such that their sum (union) is 0. We note further that the energy norm given in terms of the stresses, the L2 stress norm and the RMS stress error have a very similar structure and that these are similarly approximated. At this stage it is of interest to invoke the discussion of Chapter 2 (Sec. 2.6) concerning the rates of convergence. We noted there that with trial functions in the displacement formulation of degree p , the errors in the stresses were of the order O(hP).This order of error should therefore apply to the energy norm error IJelJ. While the arguments are correct for well-behaved problems with no singularity, it is of interest to see how the above rule is violated when singularities exist. To describe the behaviour of stress analysis problems we define the variation of the relative energy norm error (percentage) as
I le1I x 100% 7 =11~11
(14.15)
where (14.16) is the energy norm of the solution. In Figs 14.1 and 14.2 we consider two similar stress analysis problems, in the first of which a strong singularity is, however, present. In both figures we show the relative energy norm error for an h refinement constructed by uniform subdivision of the initial mesh and of a p refinement in which polynomial order is increased throughout the original mesh. We note two interesting facts. First, the h convergence rates for various polynomial orders of the shape functions are nearly the same in the example with singularity (Fig. 14.1) and are well below the theoretically predicted optimal order O(hP), [or O(NDF)-P/2as the NDF (number of degrees of freedom) is approximately inversely proportional to h2 for a two-dimensional problem]. Secondly, in the case shown in Fig. 14.2, where the singularity is avoided by rounding the corner, the convergence rates improve for elements of higher order, though again the theoretical (asymptotic) rates are not achieved. The reason for this behaviour is clearly the singularity, and in general it can be shown that the rate of convergence for problems with singularity is O(NDF)-[m’”(A>P)l/2
(14.17)
.i . L
m 3
cn .-t VI
3
.-
s
E
LII .-
-5
m Q
x
5 fi
VI ._
0
L
2 m
-s
f
Y
.-6
.5
cn
L
rn 2
VI
c .-
0
e 3
5 ._ 3 m
c .-
s 7Y
m a Q , 1
r
c m
0
rn ._
Lc
2
rn
x m
-2
I&
.-el
370 Errors, recovery processes and error estimates
where X is a number associated with the intensity of the singularity. For elasticity problems X ranges from 0.5 for a nearly closed crack to 0.71 for a 90" corner. The rate of convergence illustrated in Fig. 14.2 approaches the value controlled by the singularity for all values of p used in the elements.
14.2 Superconvergence and optimal sampling points In this section we shall consider the matter of points at which the stresses, or displacements, give their most accurate values in typical problems of a self-adjoint kind. We shall note that on many occasions the displacements, or the function itself, are most accurately sampled at the nodes defining an element and that the gradients or stresses are best sampled at some interior points. Indeed in one dimension at least we shall find that such points often exhibit the quality known as superconvergence (i.e., the values sampled at these points show an error which decreases more rapidly than elsewhere). Obviously, the user of finite element analysis should be encouraged to employ such points but at the same time note that the errors overall may be much larger. To clarify ideas we shall start with a typical problem of second order in one dimension.
14.2.1 A one-dimensional example Here we consider a problem of a second-order equation such as we have frequently discussed in Chapter 3 and which may be typical of either one-dimensional heat conduction or the displacements of an elastic bar with varying cross-section. This equation can readily be written as
ydxk g ) + , B u + Q = o
(14.18)
with the boundary conditions either defining the values of the function u or of its gradients at the ends of the domain. Let us consider a typical problem shown in Fig. 14.3. Here we show an exact solution for u and du/dx for a span of several elements and indicate the type of solution which will result from a finite element calculation using linear elements. We have already noted that on occasions we shall obtain exact solutions for u at nodes (see Fig. 3.4). This will happen when the shape functions contain the exact solution of the homogeneous differential equation (Appendix H) - a situation which happens for Eq. (14.18) when ,B = 0 and polynomial shape functions are used. In all cases, even when ,B is non-zero and linear shape functions are used, the nodal values generally will be much more accurate than those elsewhere, Fig. 14.3(a). For the gradients shown in Fig. 14.3(b) we observe large discrepancies of the finite element solution from the exact solution but we note that somewhere within each element the results are nearly exact. It would be useful to locate such points and indeed we have already remarked in the context of two-dimensional analysis that values obtained within the elements tend to be more accurate for gradients (strains and stresses) than those values calculated at
Superconvergence and optimal sampling points 37 1
("I
Fig. 14.3 Optimal sampling pointsfor the function (a) and its gradient (b) in one dimension (linear elements).
nodes. Clearly, for the problem illustrated in Fig 14.3(b) we should sample somewhere near the centre of each element. Pursuing this problem further in a heuristic manner we shall note that if higher order elements (e.g., quadratic elements) are used the solution still remains exact or nearly exact at the end nodes of an element but may depart from exactness at the interior nodes, as shown in Fig. 14.4(a). The stresses, or gradients, in this case will be optimal at points which correspond to the two Gauss quadrature points for each element as indicated in Fig. 14.4(b). This fact was observed experimentally by Barlow', and such points are frequently referred to as Barlow points. We shall now state in an axiomatic manner that: ( a ) the displacements are best sampled at the nodes of the element, whatever the order of the element is, and (b) the best accuracy is obtainable for gradients or stresses at the Gauss points corresponding, in order, to the polynomial used in the solution. At such points the order of the convergence of the function or its gradients is one order higher than that which would be anticipated from the appropriate polynomial and thus such points are known as superconvergent. The reason for such superconvergence will be shown in the next section where we introduce the reader to a theorem developed by Herrmann.2
372
Errors, recovery processes and error estimates
\-
I
Fig. 14.4 Optimal sampling points for the function (a) and its gradient (b) in one dimension (quadratic
14.2.2 The Herrmann theorem and optimal sampling points The concept of least square fitting has additional justification in self-adjoint problems in which an energy functional is minimized. In such cases, typical of a displacement formulation of elasticity, it can be readily shown that the minimization is equivalent to a least square fit of approximation stresses to the exact ones. Thus quite generally we can start from a theory which states that minimization of an energy functional I3 dejined as
n = -1 2
b
(SII)~ASII dR +
So
uTpdR
(14.19)
Superconvergence and optimal sampling points 373
which at an absolute minimum gives the exact solution u = U this is equivalent to minimization of another functional defined as
n*
n* = I Jn [S(u - U)ITAS(u
-
U) dR
(14.20)
In the above, S is a self-adjoint operator and A and p are prescribed matrices of position. The above quadratic form [Eq. (14.19)] arises in the majority of linear self-adjoint problems. For elasticity problems this theorem is given by Herrmann2 and shows that the approximate solution for Su approaches the exact one SU as a weighted least square approximat ion. The proof of the Herrmann theorem is as follows. The variation of II defined in Eq. (14.19) gives, at u = U (the exact solution),
6II =
I*
4
(SSU)TASiidR + $
+
(SU)~ASSU dR
In
Jn
GUTp dR = 0
(14.21)
or as A is symmetric (SSU)TASUdR
+
J*
SUTpdR
=0
(14.22)
in which Su is any arbitrary variation. Thus we can write
su = u
(14.23)
and (Sii)TASiidR +
In
UTpdR = 0
(14.24)
Subtracting the above from Eq. (14.19) and noting the symmetry of the A matrix, we can write
n=4
6,
[S(U - u)ITAS(U- u) dR -
I
[S(u)lTASudR
(14.25)
where the last term is not subject to variation. Thus
n*= n + constant
( 14.26)
and its stationarity is equivalent to the stationarity of n. It follows directly from the Herrmann theorem that, for one dimension and by a well-known property of the Gauss-Legendre quadrature points, if the approximate gradients are defined by a polynomial of degree p - 1, where p is the degree of the polynomial used for the unknown function u, then stresses taken at these quadrature points must be superconvergent. The single point at the centre of an element integrates precisely all linear functions passing through that point and, hence, if the stresses are exact to the linear form they will be exact at that point of integration. For any higher order polynomial of order p , the Gauss-Legendre points numbering p will provide points of superconvergent sampling. We see this from Fig. 14.5 directly. Here we indicate one, two, and three point Gauss-Legendre quadrature showing why exact results are recovered there for gradients and stresses.
374 Errors, recovery processes and error estimates
I
L
I
Fig. 14.5 The integration property of Gauss points: p = 1, p = 2, and p = 3 which guarantees superconvergence.
For points based on rectangles and products of polynomial functions it is clear that the exact integration points will exist at the product points as shown in Fig. 14.6 for various rectangular elements assuming that the weighting matrix A is diagonal. In the same figure we show, however, some triangles and what appear to be ‘good’ but not necessarily superconvergent sampling points. These are suggested by Moan.3 Though we find that superconvergent points do not exist in triangles, the points shown in Fig. 14.6 are optimal. In Fig. 14.6 we contrast these points with the minimum number of quadrature points necessary for obtaining an accurate (though not always stable) stiffness representation and find these to be almost coincident at all times. In Fig. 14.7 representing an analysis of a cantilever by four rectangular quadratic serendipity elements we see how well the stresses sampled at superconvergent points behave compared to the overall stress pattern computed in each element. It is from results like this that many suggestions have been made to obtain improved nodal values and one method proposed by Hinton and Campbell has proved to be quite widely used.4 However, we shall discuss better recovery procedures later.
Recovery of gradients and stresses 375
Fig. 14.6 Optimal superconvergent sampling and minimum integration points for some Co elements.
The extension of the idea of superconvergent points from one-dimensional elements to two-dimensional rectangles was fairly obvious. However, the full superconvergence is lost when isoparametric distortion occurs. We have shown, however, that results at the pth-order Gauss-Legendre points still remain excellent and we suggest that superconvergent properties of the integration points continue to be used for sampling. In all of the above discussion we have assumed that the weighting matrix A is diagonal, But if such diagonality does not exist then the existence of superconvergent points is questionable. However excellent results are still available through the sampling points defined as above. Finally, we refer readers to references 5-9 for surveys on the superconvergence phenomenon and its detailed analyses.
14.3 Recovery of gradients and stresses In the previous section we have shown that sampling of the gradients and stresses at some particular points is generally optimal and possesses a higher order accuracy when such points are superconvergent. However, we would also like to have similarly accurate quantities elsewhere within each element for general analysis purposes, and in particular we need such highly accurate gradients and stresses when the energy norm or other similar norms have to be evaluated in error estimates. We have already
376
Errors, recovery processes and error estimates
Fig. 14.7 Cantilever beam with four quadratic (Q8)elements. Stress sampling at cubic order (2 x 2) Gauss points with extrapolation to nodes.
shown how with some elements very large errors exist beyond the superconvergent point and attempts have been made from the earliest days to obtain a complete picture of stresses which is more accurate overall. Here attempts are generally made to recover the nodal values of stresses and gradients from those sampled internally and then to assume that throughout the element the recovered stresses CT* are obtained by interpolation in the same manner as the displacements CT* = N,6*
(14.27)
We have already suggested a process used almost from the beginning of finite element calculations for triangular elements, where elements are sampled at the centroid (assuming linear shape functions have been used) and then the stresses are averaged at nodes. We have referred to such recovery in Chapter 4. However this is not the best for triangles and for higher order elements such averaging is inadequate. Here other procedures were necessary, for instance Hinton and Campbell4 suggested a procedure in which stresses at all nodes were calculated by extrapolating the Gauss point values. A further improvement of a similar kind was suggested by Brauchli and Oden" who used the stresses in the manner given by Eq. (14.27) and assumed that these stresses should represent in a least square sense the actual finite element stresses, therefore an L2
Superconvergent patch recovery - SPR
projection. Though t h s has a similarity with the ideas contained in the Herrmann theorem it reverses the order of least square application and has not proved to be always stable and accurate, especially for even order elements. We have already described this procedure in the chapter on mixed elements (see Sec. 11.6) and noted that to obtain results it is necessary to invert a 'mass' type matrix. This can only be achieved without high cost if the mass matrix is diagonal. However, in the following presentation we will show that highly improved results can be obtained by direct polynomial 'smoothing' of the superconvergent values. Here the first method of importance is called superconvergent patch recovery.' '-I3
14.4 Superconvergent patch recovery - SPR 14.4.1 Recovery for gradients and stresses We have already noted that the stresses sampled at certain points in an element possess the superconvergent property (Le., converge at the same rate as displacement) and have errors of order O ( h P + ' ) .A fairly obvious procedure for utilizing such sampled values seems to the authors to be that of involving a smoothing of such values by a polynomial of order p within a patch of elements for which the number of sampling points can be taken as greater than the number of parameters in the polynomial. In Fig. 14.8 we show several such patches each assembled around a central corner node. The first four represent rectangular elements where the superconvergent points are well defined. The last two give patches of triangles where the best sampling points are used which are not superconvergent. If we accept the superconvergence of 6 at certain points s in each element then it is a simple matter (which also turns out computationally much less expensive than the L2 projection) to compute (T* which is superconvergent at all points within the element. The procedure is illustrated for two dimensions in Fig. 14.8, where we shall consider interior patches (assembling all elements at interior nodes) as shown. At the superconvergent point the values of 6 are accurate to order p 1 (not p as is true elsewhere). However, we can easily obtain an approximation given by a polynomial of degree p , with identical order to these occurring in the shape function for displacement, which has superconvergent accuracy everywhere if this polynomial is made to fit the superconvergent points in a least square manner. Thus we proceed for each component &iof 6 as follows: Writing the recovered solution as
+
CT
= p a = [ 1, x , y ,
a = [a],
a2,
"., y P ] a T
" ' 3
(14.28)
am1
we minimize, for an element patch with total n sampling points, n
(14.29)
377
m
c
.U _
h
m 3 0-
-6
L
U m c
m t W
._
M -
v
4-
el u m ._ L D c
U
3
h
m .-U a
m c
U
e
U ' .-c
0-
D 3 m
W
c
c m
W
E
Superconvergent patch recovery - SPR
@ Patch assembly node for boundary interface Recovered boundary and interface values
Fig. 14.9 Recovery of boundary or interface gradients.
[(xk,yk) corresponding to coordinates of superconvergent points] obtaining immediately the coefficient a as
a = A-'b
(14.30)
where n
n
(14.31) The availability of c* allows the superconvergent values of ii* to be determined at all nodes. As some nodes belong to more than one patch, average values of a* are best obtained. The superconvergence of ts* throughout each element is achieved with Eq. (14.27). It should be noted that on external boundaries and indeed on interfaces where stresses are discontinuous the nodal values should be calculated from interior patches in the manner shown in Fig. 14.9. In Fig. 14.10 we show in a one-dimensional example how the superconvergent patch recovery reproduces exactly the stress (gradient) solutions of order p 1 for linear or quadratic elements. Following the arguments of Chapter 10 on the patch test it is evident that superconvergent recovery is now achieved at all points. Indeed, the same figure shows why averaging (or L2 projection) is inferior (particularly on boundaries). Figure 14.11 shows experimentally determined convergence rates for a onedimensional problem (stress distribution in a bar of length L = 1; 0 < x < 1 and prescribed body forces). A uniform subdivision is used here to form the elements, and the convergence rates for the stress error at x = 0.5 are shown using the direct stress approximation 6,the L2 recovery oL and o* obtained by the SPR procedure using linear, quadratic and cubic elements. It is immediately evident that o* is superconvergent with a rate of convergence being at least one order higher than that of 6. However, as anticipated, the L2 recovery gives much inferior answers, showing superconvergence only for odd values of p and almost no improvement for even
+
379
380
Errors, recovery processes and error estimates
Fig. 14.10 Recovery of exact n of degree p by linear elements ( p = 1) and quadratic elements ( p = 2).
values of p , while n* shows a two-order increase of convergence rate for even order elements (tests on higher order polynomials are reported in reference 14). This ultra convergence has been verified mathemati~ally.'~ Although it is not observed when elements of varying size are used, the important tests shown in Figs 14.12 and 14.13 indicate how well the recovery process works. In the first of these, Fig. 14.12, a field problem is solved in two dimensions using a very irregular mesh for which the existence of superconvergent points is only inferred heuristically. The very small error in a: is compared with the error of C ? ~ and the improvement is obvious. Here a, = &/dx where u is the fluid variable. In the second, i.e., Fig. 14.13, a problem of stress analysis, for which an exact solution is known, is solved using three different recovery methods. Once again the recovered solution n* (SPR) shows the much improved values compared with nL and it is clear that the SPR process should be included in all codes ifsimply to present improved stress values.
Superconvergent patch recovery - SPR
Fig. 14.11 Problem of a stressed bar. Rates of convergence (error) of stress, where x = 0.5 (0 G x G 1). ($-;
(JL
. . . :g* - - - - )
The SPR procedure which we have just outlined has proved to be a very powerful tool leading to superconvergent results on regular meshes and much improved results (nearly superconvergent) on irregular meshes. It has been shown numerically that it produces superconvergent recovery even for triangular elements which do not have superconvergent points within the element. A recent mathematical proof confirms t h s capability of SPR.6 The procedure was introduced by Zienkiewicz and Zhu in 1992”-13 and we still recommend it as the best procedure which is simple to use. However, many investigators have modified the procedure by increasing the functional where
Fig. 14.12 Poisson equation in two dimensions solved using arbitrary shaped quadratic quadrilaterals.
381
382
Errors, recovery processes and error estimates
Fig. 14.13 Plane stress analysis of stresses around a circular hole in a uniaxial field.
the least square fit is performed to include satisfaction of discrete equilibrium equations or boundary conditions, etc. Whle the satisfaction of known boundary tractions can on occasion be useful most of these additional constraints introduced have affected the superconvergent properties adversely and in general the modified versions of SPR by Wiberg et ai.” and by Blacker and BelytschkoI8have not proved to be fully effective.
Recovery by equilibration of patches - REP 383
14.4.2 SPR for displacements The superconvergent patch recovery can be extended to produce superconvergent displacements. The procedure for the displacements is quite simple if we assume the superconvergent points to be at nodes of the patch. However, as we have already observed it is always necessary to have more data than the number of coefficients in the particular polynomial to be able to execute a least square minimization. Here of course we occasionally need a patch which extends further than before, particularly since the displacements will be given by a polynomial one order higher than that used for the shape functions. In Fig. 14.8 however we show for most assemblies that a similar patch as given before can be again applied producing a good approximation for u within its interior. Larger element patches have also been suggested in reference 19. The recovered solution u* has on occasion been used in dynamic problems (e.g., Wiberg*9,20),because in dynamic problems the displacements themselves are often important. We shall find such recovery useful in some problems of fluid dynamics in Volume 3. The SPR recovery technique described in this section takes advantage of the superconvergence property of the finite element solutions and the availability of the optimal sampling points. Very recently a new method of recovery which does not need such information has been devised and will be discussed in the next section.
14.5 Recovery by equilibration of patches
- REP
Although SPR has proved to work well generally, the reason behind its capability of producing an accurate recovered solution even when superconvergent points do not in fact exist remains an open question. We have therefore sought to determine viable recovery alternatives. One of these, known by the acronym REP (recovery by equilibrium of patches), will be described next. This procedure was first presented in reference 21 and later improved in reference 22. To some extent the motivation is similar to that of Ladeveze et who sought to establish (for somewhat different reasons) a fully equilibrating stress field which can replace that of the finite element approximation. However we believe that the process derived in reference 21 is simpler though equilibration is only approximate. The starting point is the governing equilibrium equation
+
STa b = 0
(14.32)
In the finite element approximation this becomes (14.33)
where 6 are the stresses from the finite element solution. In the above flp is the domain of the patch and the last term comes from the tractions on the boundary of the patch domain rp.These can, of course, represent the whole of the problem, an element patch or only a single element.
384
Errors, recovery processes and error estimates
As is well known the stresses 6 which result from the finite element analysis will in general be discontinuous and we shall seek to replace them in every element patch by a recovered system which is smooth and continuous. To achieve the recovery we proceed in an exactly analogous way to that used in the SPR procedure, first approximating the stress in each patch by a polynomial of appropriate order a*,second using this approximation to obtain nodal values of 6' and finally interpolating these values by standard shape functions. The stress a is taken as a vector of appropriate components, which for convenience we write as: (14.34) The above notation is general with, for instance, a1= a,, a2 = ay and a3 = T , ~in two-dimensional plane elastic analysis. We shall write each component of the above as a polynomial expansion of the form: a: = [ 1, x, y ,
...I
a1. - ~ ( xy)ai ,
(14.35)
where p is a vector of polynomials and ai is a set of unknown coefficients for the ith component of stress. For equilibrium we shall always attempt to ensure that the total smoothed stress a* satisfies in the least square sense the same patch equilibrium conditions as the finite element solution. Accordingly, (14.36) where (14.37) written here again for the case of three stress components. Obvious modifications are made for more or less components. It has been found in practice that the constraints provided by Eq. (14.36) are not sufficient to always produce non-singular least square minimization. Accordingly, the equilibrium constraints are split into an alternative form in which each component of stress is subjected to equilibrium requirements. This may be achieved by expressing the stress as a* = liaf = af (14.38) i
I
6=
li6i= i
11
=
a^j*
(14.39)
i
[ 1, 0, 0IT
(14.40)
where
l2= 10,
1, 0lT etc.
(14.41)
Error estimates by recovery 385
and imposing the set of constraints (14.42) The imposition of the approximate equation (14.42) allows each set of coefficients ai to be solved independently reducing considerably the solution cost and here repeating a procedure used with success in SPR. A least square minimization of Eq. (14.42) is expressed as (14.43)
lI = (Hiai- ff)T(H,ai- f f ) where
H~ =
6,
(14.44)
BTljpdQ
and (14.45) The minimization condition results in
ai = [HTH,]-*HTff
(14.46)
For patches in some problems Eq. (14.43) may be unstable. Generally, this may be eliminated by modifying the patch requirement to the minimization of
II* = (Hiai - ff)T(Hiai- f f ) +
a(HFai - ff)T(HTa,- fp)
(14.47)
e
where the added terms represent modification on individual elements and a is a parameter. Minimization now gives
HTH,
+ cr
-1
e
HI.'HI]
[H:ff
+a
e
H14:]
(14.48)
The REP procedure follows precisely the details of SPR near boundaries and gives overall an approximation which does not require knowledge of any superconvergent points. The accuracy of both processes is comparable and we are of the opinion that many other alternative recovery procedures are still possible.
14.6 Error estimates by recovery One of the most important applications of the recovery methods is its use in the computation of the a posteriori error estimators. With the recovered solutions available, we can now evaluate errors simply by replacing the exact values of quantities such as u, 6, etc., which are in general unknown, in Eqs (14.1)-(14.3), by the recovered values which are much more accurate than the direct finite element solution. We write the error estimators in various norms such as llell
Ilell = lb*-41
(14.49)
386
Errors, recovery processes and error estimates
(14.50) (14.51) For example, the energy norm error estimator for elasticity problems has the form of lli5ll =
[/
1;
(o* - ii)TD-'(o*- 6 )d o
R
(14.52)
Similarly, estimates of the RMS error in displacement and stress can be obtained through Eqs (14.12) and (14.13). Error estimators formulated by replacing the exact solution with the recovered solution are sometimes called recovery based error estimators. This type of error estimator was first introduced by Zienkiewicz and Zhu.25 The accuracy or the quality of the error estimator is measured by the effectivity index 0, which is defined as (14.53)
A theorem proposed by Zienkiewicz and Zhu12 shows that for all estimators based on recovery we can establish the following bounds for the effectivity index: 1 --lle*Il < $ < 1 + -
I le1I
lle*11 I le1I
(14.54)
where e is the actual error and e* is the error of the recovered solution, e.g.
Ile*ll = IIU - U*II The proof of the above theorem is straightforward if we write Eq. (14.52) as llell
=
11u* - ull
=
II(u - u) - (u - u*)ll = ]le - e*ll
(14.55)
Using now the triangle inequality we have
llell - Ile*ll =G llell < llell + lle*II
(14.56)
from which the inequality (14.54) follows after division by []ell. Obviously, the theorem is also true for error estimators of other norms. Two important conclusions follow:
1. any recovery process which results 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 estimation. To prove the second point we consider a typical finite element solution with shape functions of order p where we know that the error (in the energy norm) is:
llell = O W ) If the recovered solution gives an error of a higher order, e.g.,
(14.57)
Other error estimators - residual based methods 387
then the bounds of the effectivity index are:
1 - O(P) G
e G 1 + O(P)
(14.59)
and the error estimator is asymptotically exact, that is 6-1
h+O
as
(14.60)
This means that the error estimator converges to the true error. This is a very important property of error estimators based on recovery not generally shared by residual based estimators which we shall discuss in the next section.
14.7 Other error estimators
- residual based methods
Other methods to obtain error estimators have been proposed by many investigators working in the Most of these make use of the residuals of the finite element approximation, either explicitly or implicitly. Error estimators based on these methods are often called residual error estimators. Those using residuals explicitly are termed explicit residual error estimators; the others are called implicit residual error estimators. In this section we are mainly concerned with implicit residual error estimators, in particular, the equilibrated element residual estimator which has been shown to be the most robust among all the residual error estimators.35p37 Here we consider the heat conduction problem in a two-dimensional domain as an example. The differential equation is given by
-vT(kv4) =Q
in R
(14.61)
with boundary conditions
4=4 qTn = q n = q
onr4 onr,
In the above q = -kV4
is the heat flux, n is the outward normal to the boundary r and qn is the flux normal to the boundary (see Chapters 3 and 7). The error of the finite element solution is
e=4-# and for element i the energy norm error is written as (14.62) In what follows we shall construct the equilibrated residual error estimator for this problem. The procedure of constructing an estimator for other problems, such as elasticity problems, is analogous. We start by considering an interior element i. Substitute the finite element solution into Eq. (14.61). Subtracting the resulting equation from Eq. (14.61) gives an
4
388 Errors, recovery processes and error estimates
element boundary value problem for error e given by - V T ( k V e )= ri
in R,
(14.63)
with boundary condition onrj
- ( k V e ) Tn = q , - q ,
Here
+
ri = v T ( k v 4 ) Q is the residual in the finite element and
4,
= ijTn
is the finite element normal flux. We notice immediately that Eq. (14.63) is not solvable because the exact normal flux on the element boundary is in general unknown. A natural strategy to overcome this difficulty is to replace the exact normal flux by a recovered solution qi which can be computed from the finite element flux in element i and its surrounding elements. We can now write the boundary value problem of the element error as - v T ( kv e > = ri
in ai
(14.64)
with boundary condition - ( k V e ) Tn = q i - q n
onri
The approximate solution of the above equations 2 in the energy norm, [ICIl, is defined as the element residual error estimator. * 30,31 Various recovery techniques can be used to recover the normal flux qn. However, the Neumann problem of Eq. (14.64) will guarantee to have a solution if 41: is computed such that the residuals satisfy (14.65) where Nj is the shape function for node j of element i. Although Nj can be a shape function of any order, a linear shape function seems to be the most practical in the following computation. The residuals which satisfy Eq. (14.65) are said to be equilibrated, thus the recovered solution qz satisfying Eq. (14.65) is called the equilibrated flux. An error estimator which uses the solution of the element error problem of Eq. (14.64) with the equilibrated flux q: is termed an equilibrated residual error estimator. This type of residual error estimator was first introduced by Bank and Weiser3' and later pursued by Ainsworth and Oden.34 It is apparent that the most important step in the computation of the equilibrated residual error estimator is to achieve the recovered normal flux qi which satisfies Eq. (14.65). Once q i is determined, the error problem Eq. (14.64) can be readily solved, over an element, following the standard finite element procedure. Therefore we shall focus on the recovery process. The technique of recovering normal flux by equilibrated residuals was first proposed by Ladevtze et A different version of this technique was later used by Ainsworth and Oden.34
Other error estimators - residual based methods 389
Integrating by parts, we can write Eq. (14.65) in a computationally more convenient form: (14.66) Let the recovered element boundary normal flux, for each edge of the element, have the form
1 + qklTns + Z,
(14.67)
q: = (qj
where the first term on the right-hand side is the average of the normal flux of the finite element solution from element i and its neighbour element k; n, is the outward normal on the edge s of element i; and Z, is a linear function defined on the edge s, shared by elements i and k, with end nodes I and r and
Z,
= L,a;
+ L,as
(14.68)
with 2 2 L, = -(2Nf - NS) L, = -(2Nj' - N,") (14.69) lhsl lhsl where Nj' and N," are linear shape functions defined over edge s and h, is the length of edge s. The unknown parameters as and US, are to be determined from the residual equilibrium equation (14.66). It is easy to verify that
1,
(14.70)
NAL,, d r =,,S
where ,,S
is the Kronecker delta, is given by:
6.. 11 = 1,
j
6.. 11 = 0,
=j ;
j
#j
(14.71)
Let X , denote a typical interior vertex node. Choose Nj = N, in Eq. (14.66) and consider the element patch associated with the linear shape function N,, as shown in Fig. 14.14. A local numbering for the elements and edges connected to node X,, in the patch is given. The edge normals shown here are the results of a global edge orientation. Assume X,, be the end node 1 of all the edges connected with X,,. For element el in the patch, substituting Eq. (14.67) into Eq. (14.66) for each edge and observing that N, is non-zero only on sI and s2 and at the directions of the edge normals, we have
+
SS2
iNn(iel + q e 2 ) ~ n sd2 r
-
I,,
~ n z sd, r
dr
-
S,
N ~ z . d~ r, = 0
(14.72)
where the boundary integral takes a negative sign if the edge normal shown in Fig. 14.15 is inward for the element. Let f,, denote the first four, computable, terms of the above equation and notice that [using Eq. (14.70)]
390
Errors, recovery processes and error estimates
Fig. 14.14 Typical patch with interior vertex nodex, showing a local numbering of elements e, and edges 5,.
and
ss2
NJS, d r =
ss2
+
Nn(Lxn~:nL p ? ) d r = a?n
(14.74)
Equation (14.71) now becomes -usl X"
+ us2X" = -A1
Fig. 14.15 Element interface for equilibrated flux recovery
(14.75)
Other error estimators - residual based methods 391
Similarly for element e2 to e5 we have
-e" +
U?"
= -fe2
-4"- 4"= -fe3
(14.76)
+en + u:n = -f -e" + lgn= -fe5
e4
or in matrix form (14.77)
Aa = b
where --1 A=
1
0 - 1 0 0 0 0 - 1 0
y=--
The solution gives the nodal value element patch.
0 0 1 0 -1 -1
0 0
0 0 0
(14.78)
1 1 0 - 1 -
bTao bTb
4"for each edge connected to node X ,
(14.84) in the
392
Errors, recovery processes and error estimates
Boundary nodes and their related element patches can be considered in the same fashion except that we can take 41: = qn, the known flux, for the element edge being we let the first term on the right-hand side part of r4.For edges coincident with of Eq. (14.67) be zero. By considering each vertex node of the mesh and its associated element patch, we will be able to determine as and a: for every edge, thus the recovered normal flux 41: on the element boundary is achieved. The procedure described above for recovering the normal flux is a recovery by element residual. We note that the non-uniqueness of the solution of Eq. (14.77) represents the nonuniqueness of the equilibrium status of the element residuals. The choice of the arbitrary constant in solving Eq. (14.77) will certainly affect the accuracy of the recovered solution q i , and therefore the accuracy of the error estimator. The local error problem Eq. (14.64) is usually solved by a higher order (e.g., p 1 or even p 2) approximation. The solution of the problem is then employed in the element equilibrated error estimator 1121l i . The global error estimator ll2ll is obtained through Eq. (14.15). The global error estimator has been shown to be an upper bound of the exact error,34 although it is not a trivial task to prove its convergence. We have shown here that the recovery method is the key to the computation of implicit residual error estimators. It can be shown that using a properly designed recovery method some of the explicit residual error estimators or their equivalent can, in fact, be directly derived from recovery based error estimator^.^^'^' Numerical performance of residual based error estimators was tested by BabuSka et a1.35-37and compared with that of recovery based error estimators.
+
+
14.8 Asymptotic behaviour and robustness of error estimators the BabuJka patch test
-
It is well known that elements in which polynomials of order p are used to represent the unknown u will reproduce exactly any problem for which the exact solution is also defined by such a polynomial. Indeed the verification of this behaviour is an essential part of the ‘patch test’ which has to be satisfied by all elements to ensure convergence, as we have discussed in Chapter 10. Thus if we are attempting to determine the error in a general smooth solution we will find that this error is dominated by terms of order p 1. The response of any patch to an exact solution of order p + 1 will therefore determine the asymptotic behaviour when both the size of the patch and of all the elements tends to zero. If the patch is assumed to be one of a repeatable kind, its behaviour when subjected to an exact solution of order p + 1 will give the exact asymptotic error of the finite element solution. Thus, any estimator can be compared with this exact value and the asymptotic effectivity index can be established. Figure 14.16 shows such a repeatable patch of quadrilateral elements which evaluate the performance for quite irregular meshes. We have indeed shown how true superconvergent behaviour reproduces exactly such higher order solutions and thus leads to an effectivity index of unity in the
+
Asymptotic behaviour and robustness of error estimators - the Babuika patch test
Fig. 14.16 Repeating patch of irregular and quadrilateral elements.
asymptotic limit. In the papers presented by BabuSka et a1.35-37.4'the procedure of dealing with such repeatable patches for various patterns of two-dimensional elements is developed. Thus, if we are interested in solving the differential equation L(u) +f = 0
(14.85)
where L is a linear differential operator of order 2p, we consider exact solutions (harmonic solutions) to the homogeneous equation ( f = 0) of the form u,, =
C U , X ~P(x,y)a; Y"
n = p + 1- m
=
(14.86)
m
The boundary conditions are taken as u e x I x + ~ , = UexIx
and
uexly
Uexly+L,=
(14.87)
where Lx and Ly are periods in the x and y directions, respectively (viz. repeatability Section 9.18). In general, the individual terms of Eq. (14.86) do not satisfy the differential equation and it is necessary to consider linear combinations in terms of the parameters in L as a' = Ta
(14.88)
This solution serves as the basis for conducting a patch test in which the boundary conditions are assigned to be periodic and to prevent constant changes to u.t The correct constant value may be computed from
.r
patch
NU^ + C) dR = J
ueXdR
(14.89)
patch
To compute upper and lower bounds (0, and 0,) on the possible effectivity indices, all possible combinations of the harmonic solution must be considered. This may be achieved by constructing an error norm of the solutions, for example the L2 norm of the flux (or stress)
J
I I ~ ~ I=I ~patch , (qex-qh)T(qex-qh)dR=
i T
T
( a > T EexTa'
t For elasticity type problems the periodic boundary conditions prevent rigid rotations.
(14.90)
393
394 Errors, recovery processes and error estimates Table 14.1 Robustness index for the equilibrated residuals (ERpB) and SPR (ZZ-discrete) estimators for a variety of anisotropic situations and element patterns, p = 2 Estimator
Robustness index
ERpB SPR (ZZ-discrete)
10.21 0.02
and
and solving the eigenproblem
TTEr,Tal= 82TTEe,Ta1
(14.92)
to determine the minimum (lower bound) and maximum (upper bound) effectivity indices. Further details of the process summarized here are given in Boroomand and Zienkiewicz21122 and by Zienkiewicz et These bounds on the effectivity index are very useful for comparing various error estimators and their behaviour for different mesh and element patterns. However, a single parameter called the robustness index has also been devised3’ and is useful as a guide to the robustness of any particular estimator (14.93)
A large value of this index obviously indicates a poor performance. Conversely the best behaviour is that in which 0 L = 0u =
1
(14.94)
and this gives R=O
(14.95)
In the series of tests reported in references 35-41 various estimators have been compared. Table 14.1 shows the highest robustness index value of an equilibrating residual based error estimator and the SPR recovery error estimator for a set of particular patches of triangular elements.37 This performance comparison is quite remarkable and it seems that in all the tests quoted by BabuSka et al.35-41the SPR recovery estimator performs best. Indeed we shall observe that in many cases of regular subdivision, when full superconvergence occurs the ideal, asymptotically exact solution characterized by R = 0 will be obtained. In Table 14.2 we show some results obtained for regular meshes of triangles and rectangles with linear and quadratic elements. In the rectangular elements used for problems of heat conduction type, superconvergent points are exact and the ideal result is obtained for both linear and quadratic elements. It is surprising that this
Asymptotic behaviour and robustness of error estimators - the Babuika patch test 395 Table 14.2 Effectivity bounds and robustness of SPR and REP recovery estimator for regular meshes of triangles and rectangles with linear and quadratic shape function (applied to heat conduction and elasticity problems). Aspect ratio = length(L)/height(H) of elements in patch tested Linear triangles and rectangles (heat conduction/elasticity) SPR Aspect ratio L I H 111 II 2 114 118 1/16 1/32 1/64
REP
@L
0,
R
OL
OIJ
R
1.oooo 1.oooo 1.oooo 1 .oooo
1.oooo 1.oooo 1.oooo 1.oooo 1 .oooo 1.oooo 1.oooo
0.0000 0.0000
1.oooo 1.oooo 1.oooo
0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000
1.oooo 1.oooo 1.oooo
1.oooo 1.oooo 1.oooo 1.oooo 1.oooo 1 .oooo 1.oooo
OL
@U
R
1 .oooo 1 .oooo 1.oooo 1.oooo 1.oooo
1.oooo 1.oooo 1.oooo 1.oooo 1.oooo 1.oooo 1.oooo
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.9991 0.9991 0.9991 0.9991 0.9968 0.9950 0.9945
1.0102 1.0181 1.0136 1.0030 1.0001 1.oooo 1.oooo
0.01 I 1 0.0 189 0.0145 0.0039 0.0033 0.0050 0.0055
I .oooo
1.oooo 1.oooo
0.0000
I .oooo
0.0000 0.0000 0.0000 0.0000
Quadratic rectangles (heat conduction)
Ill 1I 2 114 1I8 1/16 1/32 1/64
@L
0,
R
1 .oooo 1.oooo 1.oooo 1.oooo 1.oooo
1.oooo 1 .oooo 1.oooo 1.oooo 1.oooo 1.oooo 1.oooo
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
1.oooo
I .oooo
I .oooo I .oooo
Quadratic rectangles (elasticity)
111 1I 2 114 1I8 1/16 1/32 1/64
1.oooo
1.oooo 1.oooo 1.oooo
1.oooo
.oooo
1 1.oooo
1.oooo 1.oooo 1 .oooo 1.oooo 1.oooo 1.oooo
I .oooo
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Quadratic triangles (elasticity)
l/l 112 114 118 1/16 1/32 1/64
@L
@IJ
R
@L
@U
R
0.9966 0.9966 0.9967 0.9967 0.9966 0.9966 0.9965
1.0929 1.0931 1.0937 1.0943 1.0946 1.0947 1.0947
0.0963 0.0965 0.0970 0.0976 0.0980 0.0981 0.0982
0.9562 0.9559 0.9535 0.9522 0.9518 0.9517 0.9516
1.0503 1.048 1 1.0455 1.0603 1.0666 1.0684 1.0688
0.0940 0.0923 0.0924 0.1081 0.1148 0.1167 0.1172
also occurs in elasticity where the proof of superconvergent points is lacking (for u > 0). Further, the REP procedure also seems to yield superconvergence except for elasticity with quadratic elements. For regular meshes of quadratic triangles generally superconvergence is not expected and it does not occur for either heat conduction or elasticity problems. However, the robustness index has very small values (R < 0.10 for SPR and R < 0.12 for REP) and these estimators are therefore very good.
396 Errors, recovery processes and error estimates
Fig. 14.17.
Asymptotic behaviour and robustness of error estimators - the Babuika patch test 397
In Fig. 14.17 and Table 14.3 very irregular meshes of triangular and quadrilateral elements are analysed in repeatable patterns. It is of course not possible to present here all tests conducted by the effectivity patch test. The results shown are, however, typical - others are given in reference 21. It is interesting to observe that the Table 14.3 Effectivity bounds and robustness of SPR and REP recovery estimator for irregular meshes of triangles (a, b, c, d) and quadrilaterals (e, f, g, h) Linear element (heat conduction) SPR
REP
Mesh pattern
QL
00
R
0L
00
R
a b
0.9626 0.9715 0.9228 0.8341 0.9943 0.9969 0.9987 0.9991
1.0054 1.0156 1.4417 1.2027 1.0175 1.0152 1.0175 1.0068
0.0442 0.0447 0.5189 0.3685 0.0232 0.0183 0.0188 0.0077
0.9709 0.9838 0.8938 0.9463 0.9800 0.9849 0.9987 0.9979
1.0145 1.0167 1.8235 1.9272 1.0589 1.0582 1.0175 1.0062
0.0443 0.0329 0.9297 0.9810 0.0789 0.0733 0.0188 0.0083
C
d e f g
h
Linear elements (elasticity) SPR
REP
0,
00
R
OL
00
R
0.9404 0.8869 0.8550 0.7945 0.9946 1.0038 0.9959 0.9972
1.0109 1.0250 1.6966 1.2734 1.0247 1.0281 1.0300 1.0139
0.0741 0.1520 0.8415 0.4788 0.0301 0.03 18 0.0341 0.0168
0.9468 0.9392 0.8037 0.7576 0.9579 0.9612 0.9960 0.9965
1.0148 1.0275 2.0522 1.9416 1.0508 1.0467 1.0298 1.0122
0.0707 0.0915 1.2486 1.1840 0.0928 0.0855 0.0338 0.0157
Quadratic elements (heat conduction) QL
QU
R
QL
Q L'
R
0.9443 0.8146 0.7640 0.8140 0.9762 0.9691 0.9692 0.9906
1.0295 1.0037 1.0486 1.0141 1.0053 1.0045 1.0004 1.0113
0.0877 0.2313 0.3000 0.2423 0.0296 0.0363 0.0322 0.0207
0.9339 0.9256 0.9559 0.9091 0.9901 0.9901 0.9833 1.0045
1.0098 1.0028 1.2229 1.2808 1.0177 1.0322 1.0024 1.0261
0.0805 0.0832 0.2670 0.3717 0.0276 0.0421 0.0195 0.0307
Quadratic elements (elasticity) QL
0u
R
0L
00
R
0.9144 0.7302 0.7556 0.7624 0.9702 0.965 I 0.9457 0.9852
1.0353 1.0355 1.1024 1.0323 1.0102 1.0085 1.01 15 1.0141
0.1277 0.4038 0.4163 0.3430 0.0408 0.0446 0.0688 0.0290
0.9197 0.8643 0.8387 0.8244 0.9682 0.9749 0.9807 0.9996
1.0244 1.0346 1.2422 1.2632 1.0058 I .0286 1.0125 1.0522
0.1111 0.1905 0.4035 0.4388 0.0386 0.0537 0.0321 0.0526
398
Errors, recovery processes and error estimates
performance measured by the robustness index on quadrilateral elements is always superior to that measured on triangles. The results in a recent paper of BabuSka et show that alternative versions of SPR (such as references 17, 18, 43) generally give much worse robustness index performance than the original version, especially on irregular elements assembled near boundaries.
14.9 Which errors should concern us? In this chapter we have shown how various recovery procedures can accurately estimate the overall error of the finite element approximation and thus provide a very accurate error estimating method. We have also shown how superior are estimators based on SPR recovery to those based on residual computation. The error estimation discussed here concerns however only the original solution and if the user takes advantage of the recovered values a much better solution is already available. In the next chapter we shall be concerned with adaptivity processes aiming at reduction of the original finite element error for which a vast body of literature already exists. Here again we shall show the excellent values of the effectivity index which can be obtained with SPR type methods on examples for which an ‘exact’ solution is available from very fine mesh computations. What perhaps we should also be concerned with are the errors remaining in the recovered solutions, if indeed these are to be made use of. This problem is still unsolved and at the moment all the adaptive methods simply aim at the reduction of various norms of error in the finite element solution directly provided.
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