Iso Parametric Coordination System Of Elements Example

Apart from local and global coordination systems for elements, there is another system based on isoparametric formulation/representation, which is used extensively in finite element method. In BFE also in many places instead of local coordinate system, the iso parametric coordination is used.

For one dimentional objects, usually the greek alphabet ξ is used to represent iso coord of element. it pronounces as ksi or simply will call it xi next. note that xi is different with x where we usually xi for isoparametric coord, but will use x for local coord of bar element.

TODO: for 2 and 3 D models.

Iso Parametric Coordination system for BarElement with two nodes

../../_images/bar-iso-coord1.png

We will define isoparametric coord, as this:

At the beginning point of the element, where x=0 the iso parametric coordinate must be ξ=-1

At the central point of the element, where x=L/2, and L is length of elements, the iso parametric coordinate is taken as ξ=0

At the end point of the element, where x=L, and L is length of elements, the iso parametric coordinate is ξ=1

In bar element with two nodes the relation between isoparamtric ξ coordinate and local x coordinate is:

x = (ξ + 1)*L/2

and subsequently

ξ = (2*x-L)/L

So regardless of length of a beam L, the points xi=-1 xi=0 and xi=1 respectively are pointing to start, middle and end of the beam. for example xi=-0.5 will point to x=l/4.

From code you can convert iso to local coords like this:

var x = 1.0 + 0.0000001;//need to find internal force at a little after x = 1.0 m
var xi = e0.LocalCoordsToIsoCoords(x);

and for iso coord to local coord one could use e0.IsoCoordToLocalCoord()