In this post, we are going to understand how FEA works, but before that, we need to get familiar with some common terms.
Shape Function:
Have you wondered what goes in the background when the solver solves for solutions and how it does so? The solver solves the loads at the nodes and then interpolates the solution through the elements. Now how is the solution interpolated through the elements? Now there are some things called shape functions. The Shape functions are functions that interpolate the solution through the elements. The Shape functions are approximations of the partial differential equations used to solve the problems; these approximations are done to make the calculation easier.
Stiffness:
Stiffness by definition is the quality of an object of being firm and resist deformation. It depends on the geometry as well as the material.
For example, you might be able to bend a rod made of aluminum but won’t be able to bend a rod of the same dimensions made of steel that easily.
In the older posts, we have seen that it is preferred to use quad elements instead of Tria elements as the Tria elements are stiffer than the quad elements. Now let’s try to understand this in lame terms.
Read about use of 1D, 2D & 3D elements here.
We know that a plane can be defined using three points and Tria elements are made of three nodes, therefore while meshing all the nodes of the Tria elements are formed in one plane only, and it is hard to move it out of the plane.
But for Quad elements it is composed of four nodes, thus the fourth node can move out of the plane of the other three nodes while meshing or when the load is applied.
This is one of the reasons why quad elements are preferred in structural and fatigue analysis and Tria elements are preferred for casting simulations.
Degrees of Freedom:
The degrees of freedom of a rigid body refers to the minimum number of independent variables required to define the position of a rigid body in space or the number of independent movements that it has. In FEA it is the freedom of the number of translations and rotations the node has.
Isoparametric Elements:
For irregular geometries, while meshing the quadrilateral and triangle elements will not necessarily follow their original shapes, i.e., a quadrilateral element will not necessarily be square or rectangle, or the triangle element will not necessarily be equilateral.
So how are the interpolation calculations performed?
The answer is Isoparametric elements. Isoparametric elements use mathematical mapping from one coordinate system into another coordinate system, with the first being called the physical – and the second one natural coordinate system. The Shape functions are defined in terms of the natural coordinate systems. Thus, mapping is between the two coordinate systems. The Mapping functions between coordinate systems are a function of the shape function.
Let me know if you want to know more about Isoparametric elements and shape functions.
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