Isoparametric Elements Structural Mechanics Displacement-based Formulations

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

Reduced Accuracy These elements work, but not well …

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

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

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

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)

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…

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” …

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

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

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

Multiple Elements