quadratic vs linear elements 1

[alexzive]

Hello there,

I would like to know the difference in using quadratic elements instead of linear ones in the mesh .
As far as I understood, the integration points inside a single element are considerably increased (for a 2D Square from 4 to 9) and the values of the nodes-variables "quadratic" interpolated. Is it correct?
If non linear interpolation occurs, why only "quadratic" interpolation is available in Abaqus?
Does somebody have some stuff to explain this in the theoretical FE framework?(in the official doc. I didn´t find a good mathematical explanation - maybe my search was incomplete)

Thanks
Alex

 

[xerf]

- Linear elements use a linear approximation of displacement field over the domain of the element.
- Quadratic element use quadratic approximation of the displacement field.

There's plenty of theory in ABAQUS documentation but
I suggest you start reading an introductory textbook on FEM.

For example, you can take advantage of this set of lecture notes:

http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/

 

[alexzive]

thank you for the answer.

1)Actually I can use in Abaqus quadratic/linear elements also for heat transfer problem without displacement. In this case the linear/quadratic approximation field you said refer to the temperature field inside the element, right?

2)Why only linear/quadratic approximation and not a general polinomial approximation has been considered?

Thanks, Alex

 

[corus]

Think of it in 1D, a linear element has 2 nodes and so you can only do linear for the 2 unknowns of the line, a quadratic would have 3 nodes for the 3 unknowns of a quadratic equaion.

corus

 

[xerf]

The interpolation functions are used to approximate the primary dependent variable, this depends on the partial differential equation, FEM is used for.

For example:
- displacements field is the primary dependent variable if FEM is used to solve the equilibrium equations. (stress and strain are secondary dependent variables, as they are computed from the displacement solution)
- temperature field is the primary dependent variable if FEM is used to solve the heat transfer equations.
...and so on.

Accurate solutions are obtained with linear, quadratic interpolation functions. Cubic function are rarely used in commercial software.

The p-version of FEM is based on obtaining more accurate solutions by increasing the order of polynomials used as interpolation functions, instead of refining the mesh (h-version of FEM). However, h-version is more popular amongst commercial FE software.

 

[alexzive]

hi Xerf,
thank you for your explanation.

Regards, Alex