Chapter 4: Finite Element Method

Shape Function / Interpolation Function

• In finite element analysis, field variables at nodal points are found by rigorous analysis.
• But field variables (especially basic variables) at any point inside the element is the function of values at nodal points of the element.
• This function which relates the field variable at any point within the element to the field variables of nodal points, is called a shape function. The field variables approximated by the shape function are not as exact as obtained by rigorous analysis at the same point, and hence, it is also called an “Interpolation function.”

→ In two dimensional stress analysis, where the basic variable is displacement, it is given by:

where summation is over the number of nodes.

For example, for a three-noded triangular element, displacement at $P(x,y)$ is given by:

In matrix form:

Polynomial Shape Function

Polynomial functions are commonly used as shape functions because of the following benefits:

• They are easy to handle mathematically, i.e., differentiation and integration of polynomials is easy.
• Any function can be approximated well by using a polynomial function. If a function is highly nonlinear, we may have to approximate with a higher-order polynomial.

Approximation with polynomial functions of different order:

One Dimensional Polynomial Shape Function

A general one-dimensional polynomial shape function of $n^{th}$ order (with $n+1$ terms) is given by:

In matrix form:

where $[G] = \begin{bmatrix} 1 & x & x^2 & \dots & x^n \end{bmatrix}$ and $\{\alpha\}^T = \begin{bmatrix} \alpha_1 & \alpha_2 & \alpha_3 & \dots & \alpha_{n+1} \end{bmatrix}$

Two Dimensional Polynomial Shape Function

A general form of two-dimensional polynomial shape function of $n^{th}$ order is given by:

In matrix form:

where $G = \begin{bmatrix} 1 & x & y & x^2 & xy & y^2 & x^3 & \dots & y^n \end{bmatrix}$ and $\{\alpha\}^T = \begin{bmatrix} \alpha_1 & \alpha_2 & \dots & \alpha_m & \alpha_{m+1} & \dots & \alpha_{2m} \end{bmatrix}$

The total number of terms ‘$m$’ in an $n^{th}$ degree polynomial is:

• For 1st order polynomial ($n = 1$): $m = \frac{(1+1)(1+2)}{2} = 3$ — Terms: $\alpha_1 + \alpha_2 x + \alpha_3 y$
• For 2nd order polynomial ($n = 2$): $m = \frac{(2+1)(2+2)}{2} = 6$ — Terms: $\alpha_1 + \alpha_2 x + \alpha_3 y + \alpha_4 x^2 + \alpha_5 xy + \alpha_6 y^2$

Pascal’s Triangle

A convenient way to remember complete two-dimensional polynomial terms:

Convergence Requirements of Shape Functions

→ The displacements obtained by finite element analysis are approximate solutions.
→ As the number of finite elements (mesh or no. of nodes) increases, the solution approaches the exact solutions.

In order to ensure convergence criteria, the shape functions should satisfy the following requirements:

i) Compatibility Requirement:

• The displacement models must be continuous within the elements and displacements must be compatible between the adjacent elements.
• The adjacent elements must deform without causing opening, overlaps or discontinuities between the elements.

ii) Rigid Body Displacements:

• In a displacement model there should be a term which permits all points on the element to experience the same displacement.
• If such a term does not exist, shifting of the origin of the coordinate system will cause additional stresses and strains, which should not occur.
• In the displacement model $u = \alpha_1 + \alpha_2 x + \alpha_3 y$, the term $\alpha_1$ provides for the rigid body displacement.
• Hence, to satisfy the requirement of rigid body displacement, there should be a constant term in the shape function selected.

iii) Constant Strain State:

• The displacement model should be such that all points in the element can experience the same strain.
• If we imagine the refinement of mesh, i.e., if elements approach infinitesimal size, the strains within the element approach constant values.
• Unless the shape function term includes these constant strain terms, we cannot hope to converge to a correct solution.
• In displacement model $u = \alpha_1 + \alpha_2 x + \alpha_3 y + \dots$, $\alpha_2$ and $\alpha_3$ provide for uniform strain $\epsilon_x$, and $\alpha_{m+3}$ provides for uniform strain $\epsilon_y$.

iv) Geometric Isotropy / Spatial Isotropy / Geometric Invariance:

• The selection of polynomial shape function for the displacement model should be such that the pattern should be independent of the orientation of the local coordinate system.

Shape Functions Using Polynomials (Cartesian Coordinates)

Shape Function in terms of Natural Coordinate System

L₁=1 P(L₁,L₂) L₁=0
L₂=0 L₂=1
 (1)-------------------X-------------------(2)

ξ = -1 P(ξ) ξ = +1
 (1)-----------------------X------------------------(2)
 N₁ N₂

ξ = -1 ξ = 0 ξ = +1
 (1)------------------------(3)------------------------(2)
 N₁ N₃ N₂

Shape Function using Lagrange Polynomials

If only continuity of basic unknown (displacement) is to be satisfied, Lagrange polynomials can be used to derive shape functions. Lagrange polynomial of one dimension is defined by:

Thus, if $n = 5$ and $k = 3$:

Lagrange Polynomial Approach for 2D/3D Elements

We may extend the concept to two and three dimensions by forming the products of the functions which hold good for the individual one-dimensional coordinate directions. For 2D element:

Shape Function for Serendipity Family Elements

The shape functions are found from consideration that $N_i = 1$ for $i^{th}$ node and zero when referred to any other node.

Shape Functions for Isoparametric Triangular Elements