Shape function

Equation (2.1) provides an approximate interpolated value for / at position x in terms of its nodal values and two geometrical functions. The geometrical functions in Equation (2.1) are called the shape functions. A simple inspection shows that (a) each function is equal to 1 at its associated node and is 0 at the other node, and (b) the sum of the shape functions is equal to 1. These functions, shown in Figure 2.3, are written according to their associated nodes as Aa and Ab-... 

Inherent in the development of approximations by the described interpolation models is to assign polynomial variations for function expansions over finite elements. Therefore the shape functions in a given finite element correspond to a... 

Shape functions of commonly used finite elements... 

Standard procedures for the derivation of the shape functions of common types of finite elements can be illustrated in the context of two-dimensional triangular and rectangular elements. Let us, first, consider a triangular element having three nodes located at its vertices as is shown in Figure 2.6. 

In the outlined procedure the derivation of the shape functions of a three-noded (linear) triangular element requires the solution of a set of algebraic equations, generally shown as Equation (2.7). 

Shape functions of the described triangular element are hence found on the basis of Equation (2.11) as... 

The described direct derivation of shape functions by the formulation and solution of algebraic equations in terms of nodal coordinates and nodal degrees of freedom is tedious and becomes impractical for higher-order elements. Furthermore, the existence of a solution for these equations (i.e. existence of an inverse for the coefficients matrix in them) is only guaranteed if the elemental interpolations are based on complete polynomials. Important families of useful finite elements do not provide interpolation models that correspond to complete polynomial expansions. Therefore, in practice, indirect methods are employed to derive the shape functions associated with the elements that belong to these families. 

A very convenient indirect procedure for the derivation of shape functions in rectangular elements is to use the tensor products of one-dimensional interpolation functions. This can be readily explained considering the four-node rectangular element shown in Figure 2.8. 

A similar procedure is used to generate tensor-product three-dimensional elements, such as the 27-node tri-quadratic element. The shape functions in two-or three-dimensional tensor product elements are always incomplete polynomials. 

In this element the velocity and pressure fields are approximated using biquadratic and bi-linear shape functions, respectively, this corresponds to a total of 22 degrees of freedom consisting of 18 nodal velocity components (corner, mid-side and centre nodes) and four nodal pressures (corner nodes). 

Using this coordinate system the shape functions for the first two members of the tensor product Lagrange element family are expressed as... 

It can be readily shown that L,J 1,3 satisfy the requirements for shape functions (as stated in Equation 2.3) associated with the triangoilar element. The area of a triangle in tenns of the Cartesian coordinates of its vertices is written as... 

Subparanietric transformations shape functions used in the mapping functions are lower-order polynomials than the shape functions used to obtain finite element approximation of functions. 

Differentiation of locally defined shape functions appearing in Equation (2.34) is a trivial matter, in addition, in isoparametric elements members of the Jacobian matrix are given in terms of locally defined derivatives and known global coordinates of the nodes (Equation 2.27). Consequently, computation of the inverse of the Jacobian matrix shown in Equation (2.34) is usually straightforward. 

As already discussed, variations of a field unknown within a finite element is approximated by the shape functions. Therefore finite element discretization provides a nat ural method for the construction of piecewise approximations for the unknown functions in problems formulated in a global domain. This is readily ascertained considering the mathematical model represented by Equation (2.40). After the discretization of Q into a mesh of finite elements weighted residual statement of Equ tion (2.40), within the space of a finite element T3<, is written as... 

The simplicity gained by choosing identical weight and shape functions has made the standard Galerkin method the most widely used technique in the finite element solution of differential equations. Because of the centrality of this technique in the development of practical schemes for polymer flow problems, the entire procedure of the Galerkin finite element solution of a field problem is further elucidated in the following worked example. 

Within the space of finite elements the unknown function is approximated using shape functions corresponding to the two-noded (linear) Lagrange elements as... 

The weight function used in the Galerkin formulation can be identical to either of the shape functions of a two-node linear element, therefore, for each weight function an equation corresponding to the weak statement (2.53) is derived... 

Although the elemental stiffness Equation (2.55) has a common form for all of the elements in the mesh, its utilization based on the shape functions defined in the global coordinate system is not convenient. Tliis is readily ascertained considering that shape functions defined in the global system have different coefficients in each element. For example... 

Furthermore, in a global syslena limits of definite integrals in the coefficient matrix will be different for each element. This difficulty is readily resolved using a local coordinate system (shown as x) to define the elemental shape functions as... 

Substitution for the shape functions from Equation (2.56) into Equation (2.57) gives... 

Isoparametric mapping described in Section 1.7 for generating curved and distorted elements is not, in general, relevant to one-dimensional problems. However, the problem solved in this section provides a simple example for the illustration of important aspects of this procedure. Consider a master element as is shown in Figure 2.23. The shape functions associated with this element are... 

After the substitution for T from Equation (2.68), dx from Equation (2.70) and global derivatives of shape functions from Equation (2.71) into the elemental stiffness equation (2,.55) we obtain, for the equation corresponding to N[... 

Following the discretization of the solution domain Q (i.e. line AB) into two-node Lagrange elements, and representation of T as T = Ni(x)Ti) in terms of shape functions A, (.v), i = 1,2 within the space of a finite element Q, the elemental Galerkin-weighted residual statement of the differential equation is written as... 

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