Fundamentals for Finite Element Method Part 2

Properties of the Shape Functions1

Property 1. Reproduction property and consistency

The consistency of the shape function within the element depends upon the complete orders of the monomial Pi (x) used in Eq. (3.12), and hence is also dependent upon the number of nodes of the element. If the complete order of monomial is k, the shape functions is said to possess Ck consistency. To demonstrate, we consider a field given by

where Pj(x) are monomials that are included in Eq. (3.12). Such a given field can always be written using Eq. (3.12) using all the basis terms, including those in Eq. (3.29):

where

Using n nodes in the support domain of x, we can obtain the vector of nodal function value de as:

Substituting Eq. (3.32) into Eq. (3.25), we have the approximation of

which is exactly what is given in Eq. (3.30). This proves that any field given by Eq. (3.29) will be exactly reproduced in the element by the approximation using the shape functions, as long as the given field function is included in the basis functions used for constructing the shape functions. This feature of the shape function is in fact also very easy to understand by intuition: any function given in the form ofcan be produced exactly by lettingThis can always

be done as long as the moment matrix P is invertible so as to ensure the uniqueness of the solution for a.

The proof of the consistency of the shape function implies another important feature of the shape function: that is the reproduction property, which states that any function that appears in the basis can be reproduced exactly. This important property can be used for creating fields of special features. To ensure that the shape functions have linear consistency, all one needs to do is include the constant (unit) and linear monomials into the basis. We can make use of the feature of the shape function to compute accurate results for problems by including terms in the basis functions that are good approximations of the problem solution. The difference between consistency and reproduction is

•    consistency depends upon the complete order of the basis functions; and

•    reproduction depends upon whatever is included in the basis functions.

Property 2. Linear independence

Shape functions are linearly-independent. This is because basis functions are of linear independence and P-1 is assumed to exist. The existence of P-1 implies that the shape functions are equivalent to the basis functions in the function space, as shown in Eq. (3.26). Because the basis functions are linearly-independent, the shape functions are hence linearly-independent. Many FEM users do not pay much attention to this linear independence property; however, it is the foundation for the shape functions to have the delta function property stated below.

Property 3. Delta function properties

where Sij is the delta function. The delta function property implies that the shape function Ni should be unit at its home node i, and vanishes at the remote nodes j = i of the element.

The delta function property can be proven easily as follows: because the shape functions Ni (x) are linearly-independent, any vector of length nd should be uniquely produced by linear combination of these nd shape functions. Assume that the displacement at node i is di and the displacements at other nodes are zero, i.e.

and substitute the above equation into Eq. (3.24), we have at x = xj, that

and when i = j, we must have

which implies that

This proves the first row of Eq. (3.34). When i = j, we must have

which requires

This proves the second row of Eq. (3.34). We can then conclude that the shape functions possess the delta function property, as depicted by Eq. (3.34). Note that there are elements, such as the thin beam and plate elements, whose shape functions may not possess the delta function property (see Section 5.2.1 for details).

Property 4. Partitions of unity property

Shape functions are partitions of unity:

if the constant is included in the basis. This can be proven easily from the reproduction feature of the shape function. Let u(x) = c, where c is a constant; we should have

which implies the same constant displacement for all the nodes. Substituting the above equation into Eq. (3.24), we obtain

which gives Eq. (3.41). This shows that the partitions of unity of the shape functions in the element allows a constant field or rigid body movement to be reproduced. Note that Eq. (3.41) does not require

Property 5. Linear field reproduction

If the first order monomial is included in the basis, the shape functions constructed reproduce the linear field, i.e.

where Xi is the nodal values of the linear field. This can be proven easily from the reproduction feature of the shape function in exactly the same manner for proving Property 4. Let u(x) = x, we should have

Substituting the above equation into Eq. (3.24), we obtain

which is Eq. (3.44).

Lemma 1. Condition for shape functions being partitions of unity. For a set of shape functions in the general form

whereis a set of independent base functions, the sufficient and necessary condition for this set of shape functions being partitions of unity is

where

Proof. Using Eq. (3.47), the summation of the shape functions is

which proofs the sufficient condition. To proof the necessary condition, we argue that, to have the partitions of unity, we have

or

Becauseis    a set of independent base functions. The necessary condition for Eq. (3.52) to be satisfied is Eq. (3.48).

Lemma 2. Condition for shape functions being partitions of unity. Any set of nd shape functions will automatically satisfy the partitions of unity property if it satisfies:

•    Condition I : it is given in a linear combination of the same linearly-independent set of nd basis functions that contain the constant basis, and the moment matrix defined by Eq. (3.21) is of full rank;

•    Condition 2: it possesses the delta function property.

Proof. From Eq. (3.26), we can see that all the nd shape functions are formed via a combination of the same basis functionThis feature, together with the delta function property, can ensure the property of partitions of unity. To prove this, we write a set of shape functions in the general form

where we ensured the inclusion of the constant basis of P1 (x) = 1. The other basis function in Eq. (3.53) can be monomials or any other type of basis functions as long as all the basis functions (including P1(X)) are linearly-independent.

From the Condition 2, the shape functions possess a delta function property that leads to

Substituting Eq. (3.53) into the previous equations, we have

Expanding Eq. (3.56) gives

or in the matrix form

Note that the coefficient matrix of Eq. (3.58) is the moment matrix that has a full rank (Condition 1); we then have

The use of Lemma 1 proves the partitions of the unity property of shape functions.

Lemma 3. Condition for shape functions being linear field reproduction. Any set of nd shape functions will automatically satisfy the linear reproduction property, if it satisfies

•    Condition I : it is given in a linear combination of the same linearly-independent set of nd basis functions that contain the linear basis function, and the moment matrix defined by Eq. (3.21) is of full rank;

•    Condition 2: it possesses the delta function property.

To prove this, we write a set of shape functions in the following general form of

where we ensure inclusion of the complete linear basis functions of p2 (x) = x. The other basis function Pi(X) (i = 1, 3,…, nd) in Eq. (3.53) can be monomials or any other type of basis function as long as all the basis functions are linearly-independent.

Consider a linear field of u(x) = x, we should have the nodal vector as follows:

Substituting the above equation into Eq. (3.24), we obtain

At the nd nodes of the element, we have nd equations:

Using the delta function property of the shape functions, we have

Hence, Eq. (3.63) becomes

Or in matrix form,

Note that the coefficient matrix of Eq. (3.66) is the moment matrix that has a full rank. We thus have

Substituting the previous equation back into Eq. (3.62), we obtain

This proves the property of linear field reproduction.

The delta function property (Property 3) ensures convenient imposition of the essential boundary conditions (admissible condition (b) required by Hamilton’s principle), because the nodal displacement at a node is independent of that at any other nodes. The constraints can often be written in the form of a so-called Single Point Constraint (SPC). If the displacement at a node is fixed, all one needs to do is to remove corresponding rows and columns without affecting the other rows and columns.

The proof of Property 4 gives a convenient way to confirm the partitions of unity property of shape functions. As long as the constant (unit) basis is included in the basis functions, the shape functions constructed are partitions of unity. Properties 4 and 5 are essential for the FEM to pass the standard patch test, used for decades in the finite element method for validating the elements. In the standard patch test, the patch is meshed with a number of elements, with at least one interior node. Linear displacements are then enforced on the boundary (edges) of the patch. A successful patch test requires the FEM solution to produce the linear displacement (or constant strain) field at any interior node. Therefore, the property of reproduction of a linear field of shape function provides the foundation for passing the patch test. Note that the property of reproducing the linear field of the shape function does not guarantee successful patch tests, as there could be other sources of numerical error, such as numerical integration, which can cause failure.

Lemma 1 seems to be redundant, since we already have Property 4. However, Lemma 1 is a very convenient property to use for checking the property of partitions of unity of shape functions that are constructed using other shortcut methods, rather than the standard procedure described in Section 3.4.3. Using Lemma 1, one only needs to make sure whether the shape functions satisfy Eq. (3.48).

Lemma 2 is another very convenient property to use for checking the property of partitions of unity of shape functions. Using Lemma 2, we only need to make sure that the constructed nd shape functions are of the delta function property, and they are linear combinations of the same nd basis functions that are linearly-independent and contain the constant basis function. The conformation of full-rank of the moment matrix of the basis functions can sometimes be difficult. In this topic, we usually assume that the rank is full for the normal elements, as long as the basis functions are linearly-independent. In usual situations, one will not be able to obtain the shape functions if the rank of the moment matrix is not full. If we somehow obtained the shape functions successfully, we can usually be sure that the rank of the corresponding moment matrix is full.

Lemma 3 is a very convenient property to use for checking the property of linear field reproduction of shape functions. Using Lemma 3, we only need to make sure that the constructed nd shape functions are of the delta function property, and they are linear combinations of the same nd basis functions that are linearly-independent and contain the linear basis function.