1.43 Conforming and nonconforming shape functions

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Elements are called conforming if the functions ϕi—more accurately the displacement

terms of the ϕi—are continuous across interelement boundaries.

What counts as a displacement term depends on the order of the differential

equation. In a second-order equation such as −EAu__, the zero-th

order derivative u is a displacement and the first-order derivative N = EAu_

is a force. In a fourth-order equation like EI wIV , the deflection w and the

slope w_ are displacement terms and the second- and third-order derivatives

210 1 What are finite elements?

1.154. Point

oshenko beam

M = −EI w__ and V = −EIw___ are force terms. Accordingly, conforming

C0 − elements : for second-order equations

C1 − elements : for fourth-order equations

Because discontinuous displacements imply infinite energy, it might seem that

nonconforming shape functions could not be a proper choice for an energy

method, because only the finite part of the energy can be considered in the

analysis. But a whole range of nonconforming elements are successfully employed

in FE analysis. There are different reasons for this:

• The elements are basically conforming elements, and only by enriching

the shape functions with additional terms do they become nonconforming

(Wilson’s element).

• Hybrid variational principles are used or other modifications are applied.

One member of the first class is Wilson’s plane element, which is based on

a conforming bilinear element to which two internal modes are added; see

Sect. 4.4, p. 338. Because these two internal modes ensure that the deformed

elements overlap, the element is nonconforming. But the element is superior

to a standard bilinear element, and if the element size tends to zero h → 0,

the element becomes conforming.

The second class of nonconforming elements is based on hybrid or modified

variational principles, where the “defect” of the element, i.e., the jump in the

displacements at the interelement boundaries, is built into the functional with

the help of Lagrange multipliers. Instead of the principle of minimum potential

energy

Π(u) =

1

2

_

Ω

E •S dΩ −

_

Ω

p •udΩ , (1.593)

beam and b)a Timload

applied to a)

Fig.

an Euler–Bernoulli

elements are classified as C0 or C1 elements, (see Fig. 1.154),

1.44 Partition of unity 211

a hybrid form of this principle is employed:

Π(u, tσ) =

1

2

_

e

_

Ωe

E •S dΩ −

_

Ω

p •udΩ +

_

i

_

Γi

tσ • (u+ − u

−) ds

(1.594)

where u+ − u− are the jump terms of the displacement field at the interelement

boundaries Γi, and the traction vector tσ plays the role of a Lagrange

multiplier.

Now it is of no concern that the displacement field is discontinuous. Hence,

what is a conforming element and what is not depends on the variational

principle employed. The error committed if nonconforming shape functions

are used in the standard functional (1.593) is that the penalty terms at the

interelement boundaries Γi are neglected.

The message is that the “measuring device” that is the strain energy in

the functional Π(u), must be compatible with the peculiarities of the shape

functions; see Eq. (1.594).

The so-called spurious modes also belong in this context. These are shape

functions ϕi(x)             = 0 with zero strain energy:

δWi(ϕi, ϕi) =

_ l

0

EI (ϕ

__

i )2 dx = 0 but ϕi        = 0. (1.595)

The entries on the main diagonal of the stiffness matrix vanish for such shape

functions:

kii = δWi(ϕi, ϕi) = 0. (1.596)

Spurious modes normally only occur if a program author tinkers with the

basic algorithm, if the author reduces the sensitivity of an FE program by

using, for example, reduced integration.

But in mixed formulations or multi-physics problems spurious modes are

not that seldom observed. They are an indication that either the implementation

is not adequate or that the mathematical model is very sensitive to the

physical parameters as for example in the analysis of a nearly incompressible

fluid.

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