1. Isoparametric Elements
Structural Mechanics
Displacement-based Formulations

2. Fundamental Dilemma
A primary reason engineers go to FEA is complex geometry
But elements give the most accurate results when they have regular shapes (isosceles triangles, squares)
You should always minimize element distortion when you create a mesh (more on this later …)
It is also important to understand how element shape is managed …interpolation (shape) functions

3. Reduced Accuracy
These elements work, but not well …

4. Isoparametric Elements
There are two roles of interpolation in FEA:
Defining the location of interior points within an element in terms of nodal values (geometry interpolation)
Defining the displacement of interior points within an element in terms of nodal values (result interpolation)
There is no fundamental reason why both types of interpolation must be conducted in the same way
But a common class of highly versatile elements does just that
Iso = same; the same basis for geometry and result interpolation

5. Bilinear Quadrilateral (Q4)
Interpolation involves the summation of nodal values multiplied by corresponding shapes functions
geometry interpolation
field variable interpolation
- where -
nodal coordinates
nodal displacements
shape functions

6. Shape Functions
Shape functions have a value of 1.0 at the “corresponding” node and a value of 0.0 at all others (the function “belongs” to a node)
They span a normalized domain, typically [-1,1] over each spatial dimension

7. Element Geometry Interpolation
Edges of adjacent elements match (no overlaps, gaps) as long as common nodes are shared
There are consistent interior point locations defined by the interpolation functions (e.g. you can define the “center” of an element)

8. Example 2D Element N1 = (3,2) N2 = (11,3) N3 = (10,10) N4 = (4,9)Y
N1 = (3,2)
N2 = (11,3)
N3 = (10,10)
N4 = (4,9)
The shape functions establish a geometric equivalence between elements with different node locations…

9. Field Quantity Interpolation
We assume the field quantity (e.g. x-component of displacement) varies within the element as a sum of node-weighted shape functions (visualized here as height above the plane)
If the field quantity actually does vary in this way (or close to it) then the element choice (size, order) is justified X Y
nodal values
u1 = 2
u2 = 3
u3 = 4
u4 = 5
At the element “center” …

10. Multiple Elements N5 = (0,-5) N6 = (9,-3) N3 = (11,3) N4 = (3,2)… X Y
u1 = 2
u2 = 3
u3 = 4
u4 = 5
u5 = 1
u6 = 2.5

11. C0 Continuous A key feature of these elements is their continuity
The value (C0) of an interpolated quantity is continuous across element boundaries
No geometric “gaps” or “overlaps” along the edges of shared elements
Field quantities transition without any “steps” in value
But the slope (C1) (derivative) of a field quantity is not continuous along element edges
As element density increases the transitions become less abrupt
It is possible to construct C1 shape functions, but they require “slope nodes” and are not as computationally efficient as mesh refinement

12. Higher-Order Elements
The overall scheme stays exactly the same
But there are more nodes and different shape function definitions

13. Multiple Elements