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 - where -

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)