mixed methods 1

[Krzychu]

Hello everyone, I have one problem:
I'm trying to make a FEM algorithm for Euler-Bernoulli cantilever beam based on Hellinger-Reissner principle with Hermite interpolation. Even though I obtain a very good convergence for displacements and quite good for rotations, the convergence for bending moments and shear forces is tragic! Is enyone who can tell me what is the reason for this behaviour?
Thank you in advance and sorry for my english

 

[pja]

What is the constraint ratio of your method compared to the ideal based upon the number of equilibrium equations and shear and moment constraints?

 

[Krzychu]

Stiffness matrix of element is 8x8 K={{Dmm,Dwm},{Dwm^T,0}} Dmm and Dwm and 0 is 4x4 and the rank of K is 6 but after enforce kinematic boundary conditions (w1=fi1=0)the rank of K is of course 8. Vector d=[Mi,Qi,Mj,Qj,wi,(fi)i,wj,(fi)j]^T and f=[0,0,0,0,f1,f2,f3,f4]^T and K.d=f. I don't enforce any shear and miment boundaty conditions becouse they are weak formulated. The cantilever is upon uniform load q and on right hand end of it is upon concentrated force P.

 

[pja]

You are using Hermite shape functios(third order) for the w but only linear shape functions for the Moment and Shear so your convergence with respect to number of elements will be worse for Moment and Shear..if you plot the error versus element size it should be linear..is it?

Also why are you using a mixed method for a thin beam formulation..you might as well add in shear stiffness and make the beam a Mindlin type if you are going to use a mixed method.

 

[pja]

I also forgot to mention that usually in mixed formulations, the mixed variables are internal(generalized dof) to the element so that they can be condensed out at the element level.

 

[Krzychu]

I use Hermite interpolation also for Moment and Shear. I use mixed methods for thin beam formulation becouse I'm only learning about FEM and I'm very curious why it isn't working well. Thank you for your replies!

 

[pja]

So you get a cubic type convergence rate for w. What are you getting for moment and shear?

 

[corus]

There is an exact formulation for beams in finite elements taking into account shear deformation. From the displacements and rotations at the ends you can approximate the deflection inbetween points/nodes by a cubic.

corus

 

[Krzychu]

Maybe I have to big requirements to convergence. w and fi are exact even for one element at the end of cantilever. I though that for this simple problem also M and Q will have such a good convergence in fixed end, but for exemple when I have 10 elements there should be 17 kN but I have only 15.75 kN, for 60 elements 16,79 kN. Maybe it isn't bad convergence?

 

[pja]

What reference(book or paper) are you using to generate your element? Your interpolation functions used for moment and shear don't seem to be correct.

 

[Krzychu]

I am learning from "Advanced Finite Element Methods (ASEN 5367)" Carlos A. Felippa

http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/Home.html

, but I discretize Hellinger-Reissner functional by myself so I can't be sure that it is good. Why can't I use cubic functions to approximate the Moment and Shear? Should I try linear functions for the Moment and Shear and cubic for the deflection and slope? What do you think about convergence that I have just mentioned above?

 

[Krzychu]

Sorry I mean linear functions only for moment. In this case the Shear should have been omitted.

Just a few second ago I have tried my algorithm without uniform load q (only with concentrated force). The solution was exact for all quantities, even for M and Q!!! I think this is because of strong formulated constitutive eqations. If w and fi are exact so M and q are. On the other hand if q is constant and differs from 0 the solution for w and fi are exact only at nodes and and functional "want" to map M and Q from bad daflections and slopes. Maybe this is the reason of this slow convergence? Maybe it is also bad idea to use mixed method to thin beam problem .

with kindest regards
Krzychu

 

[pja]

The convergence doesn't seem correct...also you can't be interpolating BOTH shear and moment with hermites if you only have 4 dof to describe them..perhaps you are writing the moment as a hermite with moments and shears at both nodes as the dof?

 

[Krzychu]

No, I dont approximate shear with hermites. Sher is derivative of moment so it isn't approximated with cubic function. But shear appears as a nodal parameter. M and Q are like deflections w and slope fi in dispalement method.