Finite Element Method - Errors, Recovery processes and error estimates _ 14

Finite Element Method - Errors, Recovery processes and error estimates _ 14 The description of the laws of physics for space- and time-dependent problems are usually expressed in terms of partial differential equations (PDEs). For the vast majority of geometries and problems, these PDEs cannot be solved with analytical methods. Instead, an approximation of the equations can be constructed, typically based upon different types of discretizations. These discretization methods approximate the PDEs with numerical model equations, which can be solved using numerical methods. The solution to the numerical model equations are, in turn, an approximation of the real solution to the PDEs. The finite element method (FEM) is used to compute such approximations.

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

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.,

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

and D is the elasticity matrix (see Chapter 2), giving the stress as

in which for simplicity we ignore initial stresses and strains

The energy norm of Eq (14.6) can thus be written alternatively as

4.7)

4.8)

(14.9)

and its relation to strain energy is evident

ment and stress error can be written as

Other scalar norms can easily be devised For instance, the L2 norm of displace-

( 1 4.1 0)

(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 ( h P ) This order of error should therefore apply to the energy norm error I J e l J

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 le1 I

7 = - x 100%

1 1 ~ 1 1 where

(14.15)

(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 ( h P ) , [or

O(NDF)-P/2 as 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 round-

ing 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

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 dis- placements, 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

y k g ) + , B u + Q = o dx (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

( a ) the displacements are best sampled at the nodes of the element, whatever the

( b ) the best accuracy is obtainable for gradients or stresses at the Gauss points

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

We shall now state in an axiomatic manner that:

order of the element is, and

corresponding, in order, to the polynomial used in the solution

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

The concept of least square fitting has additional justification in self-adjoint prob- lems in which an energy functional is minimized In such cases, typical of a displace- ment 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

1

n = - (SII)~ASII dR + So uTp dR (14.19)

2 b

Superconvergence and optimal sampling points 373

which at an absolute minimum gives the exact solution u = U this is equivalent to mini-

mization of another functional n* defined as

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 = 4 I* (SSU)TASii dR + $ I n (SU)~ASSU dR + Jn GUTp dR = 0 (14.21)

or as A is symmetric

(SSU)TASU dR + J* SUTp dR = 0 (14.22)

in which Su is any arbitrary variation Thus we can write

su = u and

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

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 a t 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

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 super-

convergence 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 pro- cedure 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 377

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 poly-

nomial 'smoothing' of the superconvergent values Here the first method of importance

is called superconvergent patch recovery.' ' - I 3

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 supercon-

vergent 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 poly-

nomial 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 &i of 6 as follows: Writing the recovered

Superconvergent patch recovery - SPR 379

@ 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 immedi-

ately the coefficient a as

where

(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 (particu-

larly on boundaries)

Figure 14.1 1 shows experimentally determined convergence rates for a one-

dimensional 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, show-

ing superconvergence only for odd values of p and almost no improvement for even

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, = & / d x 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 381

Fig 14.1 1 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 How-

ever, 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

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 BelytschkoI8 have not proved to be fully effective