Finite Element Method - The patch test, reduced in tegration and non - conforming elements_10

Finite Element Method - The patch test, reduced in tegration and non - conforming elements_10 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.

detail with this test which is applicable to allfinite element forms and will show that (a) it is a necessary condition for assessing the convergence of any finite element

approximation and further that, if properly extended and interpreted, it can provide

(b) a sujicient requirement for convergence,

(c) an assessment of the (asymptotic) convergence rate of the element tested, (d) a check on the robustness of the algorithm, and

(e) a means of developing new and accurate finite element forms which violate

While for elements which a priori satisfy all the continuity requirements, have

correct polynomial expansions, and are exactly integrated such a test is superfluous

in principle, it is nevertheless useful as it gives

(f) a check that correct programming was achieved

For all the reasons cited above the patch test has been, since its inception, and con- tinues to be the most important check for practical finite element codes

The original test was introduced by Irons et al.lP3 in a physical way and could be

interpreted as a check which ascertained whether a patch of elements (Fig 10.1)

subject to a constant strain reproduced exactly the constitutive behaviour of the material and resulted in correct stresses when it became infinitesimally small If it did, it could then be argued that the finite element model represented the real material behaviour and, in the limit, as the size of the elements decreased would therefore reproduce exactly the behaviour of the real structure

Clearly, although this test would only have to be passed when the size of the element patch became infinitesimal, for most elements in which polynomials are used the patch size did not in fact enter the consideration and the requirement that the patch test be passed for any element size became standard

compatibility (continuity) requirements

Convergence requirements 251

Fig 10.1 A patch of element and a volume of continuum subject to constant strain E ~ A physical interpreta-

tion of the constant strain or linear displacement field patch test

Quite obviously a rigid body displacement of the patch would cause no strain, and

if the proper constitutive laws were reproduced no stress changes would result The

patch test thus guarantees that no rigid body motion straining will occur

When curvilinear coordinates are used the patch test is still required to be passed in

the limit but generally will not do so for a finite size of the patch (An exception here is

the isoparametric coordinate system in problems discussed in Chapter 9 since it is

guaranteed to contain linear polynomials in the global coordinates.) Thus for many

problems such as shells, where local curvilinear coordinates are used, this test has

to be restricted to infinitesimal patch sizes and, on physical grounds alone, appears

to be a necessary and suficient condition for convergence

Numerous publications on the theory and practice of the test have followed the

original publications ~ i t e d ~ - ~ and mathematical respectability was added to those

by Strang.7'8 Although some authors have cast doubts on its validity'>" these have

been fully refuted"-l3 and if the test is used as described here it fulfils the require-

ments (a)-(f) stated above

In the present chapter we consider the patch test applied to irreducible forms (see

Chapter 3) but an extension to mixed forms is more important T h s has been studied

in references 13, 14 and 15 and made use of in many subsequent publications The

matter of mixed form patch tests will be fully discussed in the next chapter; however,

the consistency and stability tests developed in the present chapter are always required

One additional use of the patch test was suggested by BabuSka et ~ 1 ' ~ with a

shorter description given by Boroomand and Zienkiewicz l 7 This test can establish

the efficiency of gradient (stress) recovery processes which are so important in error

estimation as will be discussed in Chapter 14

10.2 Convergence requirements

We shall consider in the following the patch test as applied to a finite element solution

of a set of differential equations

252 The patch test, reduced integration, and non-conforming elements

The finite element approximation is given in the form

approaches zero (with some specified subdivision pattern) Stated mathematically

we must find that the error at any point becomes (when h is sufficiently small)

where q > 0 and C is a positive constant, depending on the position

This must also be true for all the derivatives of u defined in the approximation

By the order of convergence in the variable u we mean the value of the index q in the

above definition

To ensure convergence it is necessary that the approximation fulfil both consistency and stability conditions."

The consistency requirement ensures that as the size of the elements h tends to zero,

the approximation equation (10.4) will represent the exact differential equation (10.1) and the boundary conditions (10.2) (at least in the weak sense)

The stability condition is simply translated as a requirement that the solution of the

discrete equation system (10.4) be unique and avoid spurious mechanisms which may pollute the solution for all sizes of elements For linear problems in which we solve the system of algebraic equations (10.4) as

this means simply that the matrix K must be non-singular for all possible element

assemblies (subject to imposing minimum stable boundary conditions)

The patch test traditionally has been used as a procedure for verifying the consistency requirement; the stability was checked independently by ensuring non- singularity of matrices." Further, it generally tested only the consistency in satisfac- tion of the differential equation (10.1) but not of its natural boundary conditions In what follows we shall show how all the necessary requirements of convergence can be tested by a properly conceived patch test

A 'weak' singularity of a single element may on occasion be permissible and some elements exhibiting it have been, and still are, successfully used in practice One such case is given by the eight-node isoparametric element with a 2 x 2 Gauss quadrature,

to which we shall refer later here This element is on occasion observed to show peculiar behaviour (though its use has advantages as discussed in Chapter 11) An

element that occasionally fails is termed non-robust and the patch test provides a means of assessing the degree of robustness

The simple patch test (tests A and B) - a necessary condition for convergence 253

10.3 The simple patch test (tests A and 8) - a necessary

We shall first consider the consistency condition which requires that in the limit (as h

tends to zero) the finite element approximation of Eq (10.4) should model exactly the

differential equation (10.1) and the boundary conditions (10.2) If we consider a

‘small’ region of the domain (of size 2h) we can expand the unknown function u

and the essential derivatives entering the weak approximation in a Taylor series

From this we conclude that for convergence of the function and its first derivative

in typical problems of a second-order equation and two dimensions, we require

that around a point i assumed to be at the coordinate origin,

dx j aY i

(10.7)

2 = ( $)i + + O ( h P - ’)

with p 2 2 The finite element approximation should therefore reproduce exactly the

problem posed for any linear forms of u as h tends to zero Similar conditions can

obviously be written for higher order problems This requirement is tested by the

current interpretation of the patch test illustrated in Fig 10.2 We refer to this as

the base solution

In this we compute first an arbitrary linear solution of the differential equation and

the corresponding set of parameters a [see Eq (10.3)] at all ‘nodes’ of a patch which

assembles completely the nodal variable a, (Le., provides all the equation terms

corresponding to it)

In test A we simply insert the exact value of the parameters a into the ith equation

and verify that

(10.8)

K a - f = 0

I J J 1 -

a prescribed on all nodes

a i = K;’ (fi- K a-) V I (j= i)&d

Fig 10.2 Patch test of forms A and B

254 The patch test, reduced integration, and non-conforming elements

where fi is a force which results from any ‘body force’ required to satisfy the base solu-

tion differential equation (10.1) Generally in problems given in Cartesian coordinates the required body force is zero; however, in curvilinear coordinates (e.g., axisym- metric elasticity problems) it can be non-zero

In test B only the values of a corresponding to the boundaries of the ‘patch’ are

inserted and ai is found as

a I - - KT’(f I I I - K ijaj) j # i (10.9) and compared against the exact value

Both patch tests verify only the satisfaction of the basic differential equation and not of the boundary approximations, as these have been explicitly excluded here

We mentioned earlier that the test is, in principle, required only for an infinitesimally small patch of elements; however, for differential equations with constant coefficients and with a mapping involving constant jacobian the size of the patch is immaterial and the test can be carried out on a patch of arbitrary dimensions

Indeed, if the coefficients are not constant the same size independence exists provid- ing that a constant set of such coefficients is used in the formulation of the test (This applies, for instance, in axisymmetric problems where coefficients of the type l/radius enter the equations and when the patch test is here applied, it is simply necessary to enter the computation with such quantities assumed constant.)

If mapped curvilinear elements are used it is not obvious that the patch test posed in global coordinates needs to be satisfied Here, in general, convergence in the mapping coordinates may exist but a finite patch test may not be satisfied However, once again

if we specify the nature of the subdivision without changing the mapping function, in the limit the jacobian becomes locally constant and the previous remarks apply To illustrate this point consider, for instance, a set of elements in which local coordinates are simply the polar coordinates as shown in Fig 10.3 With shape functions using

polynomial expansions in the r, 8 terms the patch test of the kind we have described above will not be satisfied with elements of finite size - nevertheless in the limit as the element size tends to zero it will become true Thus it is evident that patch test

satisfaction is a necessary condition which has always to be achieved providing the

size of the patch is infinitesimal

Fig 10.3 Polar coordinate mapping

Generalized patch test (test C) and the single-element test 255

This proviso which we shall call weakpatch test satisfaction is not always simple to

verify, particularly if the element coding does not easily permit the insertion of

constant coefficients or a jacobian In Sec 10.10 we shall discuss in some detail its

implementation, which, however, is only necessary in very special element forms It

is indeed fortunate that the standard isoparametric element form reproduces exactly

the linear polynomial global coordinates (see Chapter 9) and for this reason does not

require special treatment unless some other crime (such as selective or reduced

integration) is introduced

10.4 Generalized patch test (test C) and the singke-

eleftrerrt t@St

The patch test described in the preceding section was shown to be a necessary condition

for convergence of the formulation but did not establish sufficient conditions for it In

particular, it omitted the testing of the boundary ‘load’ approximation for the case

when the ‘natural’ (e.g ‘traction of elasticity’) conditions are specified Further it did

not verify the stability of the approximation A test including a check on the above con-

ditions is easily constructed We show t h s in Fig 10.4 for a two-dimensional plane

problem as test C In this the patch of elements is assembled as before but subject to

prescribed natural boundary conditions (or tractions around its perimeter) correspond-

ing to the base function The assembled matrix of the whole patch is written as

K a = f

Fixing only the minimum number of parameters a necessary to obtain a physically

valid solution (e.g., eliminating the rigid body motion in an elasticity example or a

single value of temperature in a heat conduction problem) a solution is sought for

remaining a values and compared with the exact base solution assumed

Now any singularity of the K matrix will be immediately observed and, as the

vector f includes all necessary source and boundary traction terms, the formulation

will be completely tested (providing of course a sufficient number of test states is

used) The test described is now not only necessary but suficient for convergence

256 The patch test, reduced integration, and non-conforming elements

Fig 10.5 (a) Zero energy (singular) modes for eight- and nine-noded quadratic elements and (b) for a patch

of bilinear elements with single integration points

With boundary traction included it is of course possible to reduce the size of the patch to a single element and an alternative form of test C is illustrated in Fig

10.4(b), which is termed the single-element test '' This test is indeed one requirement

of a good finite element formulation as, on occasion, a larger patch may not reveal the inherent instabilities of a single element This happens in the well-documented case of the plane strain-stress eight-noded isoparametric element with (reduced) four-point Gauss quadrature i.e., where the singular deformation mode of a single element (see Fig 10.5) disappears when several elements are assemb1ed.t It should be noted, however, that satisfaction of a single element test is not a suficient condition f o r conver- gence For suficiency we require at least one internal element boundary to test that consistency of a patch solution is maintained between elements

tThis figure also shows a similar singularity for a patch of four bilinear elements with single-point

quadrature, and we note the similar shape of zero energy modes (see Chapter 9, Sec 9.1 1.3)

Higher order patch tests 257

In the previous section we have defined in some detail the procedures for conducting a

patch test We have also asserted the fact that such tests if passed guarantee that

convergence will occur However all the tests are numerical and it is impractical to

test all possible combinations

In particular let us consider the base solutions used These will invariably be a set of

polynomials given in two dimensions as

(10.10)

where P , are a suitable set of low order polynomials (e.g., 1, x, y for Galerkin forms

possessing only first-order derivatives) and a, are parameters It is fairly obvious that

if patch tests are conducted on each of these polynomials individually any base func-

tion of the form given in Eq (10.10) can be reproduced and the generality preserved

for the particular combination of elements tested This must always be done and is

almost a standard procedure in engineering tests, necessitating only a limited

number of combinations

However, as various possible patterns of elements can occur and it is possible to

increase the size without limit the reader may well ask whether the test is complete

from the geometrical point of view We believe it is necessary in a numerical test to

consider the possibility of several pathological arrangements of elements but that if

the test is purely limited to a single element and a complete patch around a node

we can be confident about the performance on more general geometric patterns

Indeed even mathematical assessments of convergence are subject to limits often

imposed aposteriori Such limits may arise if for instance a singular mapping is used

The procedures referred to in this section satisfy most readers as to the validity and

generality of the test

On some limited occasions it is possible to perform the test purely algebraically and

then its validity cannot be doubted Some such algebraic tests will be referred to later

in connection with incompatible elements

In this chapter we have only considered linear differential equations and linear

material behaviour In Volume 2 non-linear problems will be fully discussed and

on some occasions the patch test can well be used and extended to cover such areas

While the patch tests discussed in the last three sections ensure (when satisfied) that

convergence will occur, they did not test the order of this convergence, beyond

assuring us that in the case of Eq (10.7) the errors were, at least, of order O ( h 2 ) in

u It is an easy matter to determine the actual highest asymptotic rate of convergence

of a given element by simply imposing, instead of a linear solution, exact higher

order polynomial solutions The highest value of such polynomials for which complete

satisfaction of the patch test is achieved automatically evaluates the corresponding con-

vergence rate It goes without saying that for such exact solutions generally non-zero

source (e.g., body force) terms in the original equation (10.1) will need to be involved

258 The patch test, reduced integration, and non-conforming elements

In addition, test C in conjunction with a higher order patch test may be used to illustrate any tendency for ‘locking’ to occur (see Chapter 11) Accordingly, element robustness with regard to various parameters (e.g., Poisson’s ratios near one-half for elasticity problems in plane strain) may be established

In such higher order patch tests it will of course first be assumed that the patch is subject to the base expansion solution as described Thus, for higher order terms it will be necessary to start and investigate solutions of the type

10.7 Application of the patch test to plane elasticity elements with ‘standard’ and ‘reduced‘ quadrature

In the next few sections we consider several applications of the patch test in the evaluation of finite element models In each case we consider only one of the necessary tests which need to be implemented For a complete evaluation of a formulation it is necessary to consider all possible independent base polynomial solutions as well as a variety of patch configurations which test the effects of element distortion or alterna- tive meshing interconnections which will be commonly used in analysis As we shall emphasize, it is important that both consistency and stability be evaluated in a properly conducted test

In Chapter 9 (Sec 9.1 1) we have discussed the minimum required order of numerical integration for various finite element problems which results in no loss

of convergence rate However, it was also shown that for some elements such a

minimum integration order results in singular matrices If we define the standard

integration as one which evaluates the stiffness of an element exactly (at least in the

undistorted form) then any lower order of integration is called reduced

Such reduced integration has some merits in certain problems for reasons which we

shall discuss in Chapter 12 (Sec 12.5), but it can cause singularities which should be discovered by a patch test (which supplements and verifies the arguments of Sec

9.1 1.3) Application of the patch test to some typical problems will now be shown

We consider first a plane stress problem on the patch shown in Fig 10.6(a) The

material is linear, isotropic elastic with properties E = 1000 and v = 0.3 The finite

element procedure used is based on the displacement form using four-noded isopara-

metric shape functions and numerical integration Analyses are conducted using the plane element and program described in Chapter 20 Since the stiffness computation

Application of the patch test to plane elasticity elements 259

Fig 10.6 Patch for evaluation of numerically integrated plane stress problems (a) Five-element patch

(b) One-element patch

includes only first derivatives of displacements, the formulation converges provided

that the patch test is satisfied for all linear polynomial solutions of displacements

in the base solution Here we consider only one of the six independent linear poly-

nomial solutions necessary to verify satisfaction of the patch test The solution

The patch test is performed first using 2 x 2 gaussian ‘standard’ quadrature to

compute each element stiffness and resulting reaction forces at nodes For patch

test A all nodes are restrained and nodal displacement values are specified according

to Table 10.1 Stresses are computed at specified Gauss points (1 x 1,2 x 2, and 3 x 3

Gauss points were sampled) and all are exact to within round-off error (double pre-

cision was used which produced round-off errors less than in the quantities

computed) Reactions were also computed at all nodes and again produced the

0.0030

0.0006

0.0 0.0

-0.00186 -0.00120 -0.00024 -0.00036 -0.001 20 -0.00096

260 The patch test, reduced integration, and non-conforming elements

force values shown in Table 10.1 to within round-off limits This approximation satis- fies all conditions required for a finite element procedure (i.e., conforming shape func- tions and normal-order quadrature) Accordingly, the patch test merely verifies that the programming steps used contain no errors Patch test A does not require explicit

use of the stiffness matrix to compute results; consequently the above patch test was repeated using patch test B where only nodes 1 to 4 are restrained with their displacements specified according to Table 10 I This tests the accuracy of the stiffness matrix and, as expected, exact results are once again recovered to within round-off errors Finally, patch test C was performed with node 1 fully restrained and node 4

restrained only in the x-direction Nodal forces were applied to nodes 2 and 3 in

accordance with the values generated through the boundary tractions by a, (i.e.,

nodal forces shown in Table 10.1) This test also produced exact solutions for all other nodal quantities in Table 10.1 and recovered a, of 2 at all Gauss points in each element

The above test was repeated for patch tests A, B, and C but using a 1 x 1 ‘reduced’ Gauss quadrature to compute the element stiffness and nodal force quantities Patch test C indicated that the global stiffness matrix contained two global ‘zero energy modes’ (i.e., the global stiffness matrix was rank deficient by 2), thus producing incorrect nodal displacements whose results depend solely on the round-off errors

in the calculations These in turn produced incorrect stresses except at the 1 x 1 Gauss point used in each element to compute the stiffness and forces Thus, based upon stability considerations, the use of 1 x 1 quadrature on four-noded elements produces a failure in the patch test The element does satisfy consistency require- ments, however, and provided a proper stabilization scheme is employed (e.g., stiff- ness or viscous methods are used in practice) this element may be used for practical calculations .20,2 ’

It should be noted that a one-element patch test may be performed using the mesh shown in Fig 10.6(b) The results are given by nodes 1 to 4 in Table 10.1 For the one- element patch, patch tests A and B coincide and neither evaluates the accuracy or stability of the stiffness matrix On the other hand, patch test C leads to the conclusions reached using the five-element patch: namely, 2 x 2 gaussian quadrature passes a patch test whereas 1 x 1 quadrature fails the stability part of the test (as indeed we would expect by the arguments of Chapter 9, Sec 9.11)

A simple test on cancellation of a diagonal during the triangular decomposition

step is sufficient to warn of rank deficiencies in the stiffness matrix In the profile method, described in Chapter 20, this is easily monitored as compact elimination converts the initial value of a diagonal element to the final value in one step Thus only one extra scalar variable is needed to test the initial and final values

10.7.2 Example 2: Patch test for quadratic elements: quadrature

effects

In Fig 10.7 we show a two-element patch of quadratic isoparametric quadrilaterals Both eight-noded serendipity and nine-noded lagrangian types are considered and a

basic patch test type C is performed for load case 1 For the eight-noded element both

2 x 2 (‘reduced’) and 3 x 3 (‘standard’) gaussian quadrature satisfy the patch test,

Application of the patch test to plane elasticity elements 261

Fig 10.7 Patch test for eight- and nine-noded isoparametric quadrilaterals

whereas for the nine-noded element only 3 x 3 quadrature is satisfactory, with 2 x 2

reduced quadrature leading to failure in rank of the stiffness matrix However, if we

perform a one-element test for the eight-noded and 2 x 2 quadrature element, we

discover the spurious zero-energy mode shown in Fig 10.5 and thus the one-element

test has failed We consider such elements suspect and to be used only with the

greatest of care To illustrate what can happen in practice we consider the simple

problem shown in Fig 10.8(a) In this example the ‘structure’ modelled by a single

element is considered rigid and interest is centred on the ‘foundation’ response

Accordingly only one element is used to model the structure Use of 2 x 2 quadrature

throughout leads to answers shown in Fig 10.8(b) while results for 3 x 3 quadrature

are shown in Fig 10.8(c) It should be noted that no zero-energy mode exists since

more than one element is used There is, however, here a spurious response due to

the large modulus variation between structure and foundation This suggests that

problems in which non-linear response may lead to a large variation in material

parameters could also induce such performance, and thus use of the eight-noded

2 x 2 integrated element should always be closely monitored to detect such anoma-

lous behaviour

Indeed, support or loading conditions may themselves induce very suspect

responses for elements in which near singularity occurs Figure 10.9 shows some

amusing peculiarities which can occur for reduced integration elements and which

disappear entirely if full integration is used.22 In all cases the assembly of elements

is non-singular even though individual elements are rank deficient

In order to demonstrate a higher order patch test we consider the two-element plane

stress problem shown in Fig 10.7 and subjected to bending loading shown as Load 2

As above, two different types of element are considered: (a) an eight-noded serendip-

ity quadrilateral elenent and (b) a nine-noded lagrangian quadrilateral element In our

262 The patch test, reduced integration, and non-conforming elements

Fig 10.8 A propagating spurious mode from a single unsatisfactoty element (a) Problem and mesh (b) 2 x 2

integration (c) 3 x 3 integration

test we wish to demonstrate a feature for nine-noded element mapping discussed in Chapter 9 (see Sec 9.7) and first shown by W a c h ~ p r e s s ~ ~ In particular we restrict

the mapping into the xy plane to be that produced by the four-noded isoparametric

bilinear element, but permit the dependent variable to assume the full range of varia- tions consistent with the eight- or nine-noded shape functions In Chapter 9 we showed that the nine-noded element can approximate a complete quadratic displace-

ment function in x, y whereas the eight-noded element cannot Thus we expect that

the nine-noded element when restricted to the isoparametric mappings of the four- noded element will pass a higher order patch test for all arbitrary quadratic displace- ment fields The pure bending solution in elasticity is composed of polynomial terms