Re: [Getfem-users] unexpected non-zero values in a stiffness matrix K: a bar (or truss) problem

[Alex Spring]

Dear Kostas,

Thank you for your quick response. I appreciate your help.
After having your comment, I worked on deriving the weak form language
expression for a bar problem.

The expression:
"A * E * Grad_u(1,1) * Grad_Test_u(1,1)"
was derived for a bar with cross-sectional area A and Young's modulus E.

Attached .py Python3 script validates the expression.

For those who interested, I wrote some documents on the weak form and
its expression in GetFEM++.
Attached .pdf documents describe a bar and a general problem of linear
elasticity.

Attached files are:
- bar_expressions_compared.py: for comparing stiffness matrices come
from different expressions.
- CheatSheetForGradDiv.pdf: notes for gradient and divergence of a
scalar, vector and matrix.
- LinearElasticity.pdf: notes for a general problem of linear elasticity.
- BarProblem.pdf: notes for the weak form of a bar problem.

*Please let me know if anyone finds errors, corrections or has recommendations.
*I hope the files will be of some help.

Best regards,
Alex Spring

2019.9.19(Thu.) 22:18 Konstantinos Poulios <address@hidden>:
>
> Dear Alex,
>
> Thanks for your interest in the framework. I guess it is all about the use of
>
> asm_linear_elasticity
>
> which was probably never meant to be used in 1D. The brick that you have used 
> is equivalent to the generic weak form language expression
>
> (lambda*Trace(Grad_u)*Id(qdim(u)) + mu*(Grad_u+Grad_u')):Grad_Test_u
>
> So basically you need to write the corresponding expression for a bar and use 
> it with the
>
> asm_generic
>
> function instead of asm_linear_elasticity.
>
> Best regards
> Kostas
>
>
> On Thu, Sep 19, 2019 at 2:17 PM Alex Spring <address@hidden> wrote:
>>
>> Dear All,
>>
>> I am handling a bar in GetFEM++ framework. This post is questioning about 
>> unexpected non-zero values in a stiffness matrix K. I believe that thinking 
>> about it helps deeper understanding of the framework.
>>
>> The problem considered is:
>> """
>> Bar:
>>     0----1
>>     (point: 0, 1; convex: 0 (0-1), segment or 1d-simplex))
>>     (2 dof (u, v) on a single point)
>> Coordinate system:
>>     ^ y, v, Fy
>>     |
>>     ----> x, u, Fx
>> Expected for the convex 0 (0-1):
>>              K          U    =    F
>>     [ 1,  0, -1,  0]  [u0]      [Fx0]
>>     [ 0,  0,  0,  0]  [v0]      [Fy0]
>>     [-1,  0,  1,  0]  [u1]      [Fx1]
>>     [ 0,  0,  0,  0]  [v1]      [Fy1]
>>     (K: stiffness matrix, U: displacement vector, F: force vector)
>>     (A bar should have zero stiffness in the transverse direction (i.e. y- 
>> or v-direction here).)
>> """
>>
>> I believe that we can handle a bar using FEM_PK(1, 1) ((1, 1): first 1 for 
>> 1d-simplex (i.e. segment), second 1 for 2-nodes on a segment).
>>
>> However, obtained K has unexpected non-zero values:
>> """
>> Obtained for the convex 0 (0-1) :
>>                       K                  U    =    F
>>     [      1,      0,     -1,      0]  [u0]      [Fx0]
>>     [      0,    0.5,      0,   -0.5]  [v0]      [Fy0]
>>     [     -1,      0,      1,      0]  [u1]      [Fx1]
>>     [      0,   -0.5,      0,    0.5]  [v1]      [Fy1]
>>     (K22, K24, K42, K44 are expected to be 0.)
>> """
>>
>> Can we know why K has 0.5 values like this? This behavior is unexpected for 
>> me (or possibly for people with structural background), but I guess that 
>> this behavior is logically expectable and explainable. I searched 
>> getfem-users-archives and looked into FEM_PK, GT_PK and some implementations 
>> but could not find the reason. The reason would be about the same 
>> interpolation of u and v if |K22|, |K24|, |K42|, |K44| were 1, but actually 
>> they are 0.5.
>>
>> Could you please give your comments? Any ideas are welcome. It really helps 
>> understanding the framework.
>>
>> Attached is the script for python3. The script explicitly obtains stiffness 
>> matrix K using gf.asm_linear_elasticity(...). The other part is similar to 
>> Roman's one:
>> [Getfem-users] Solving truss structure in GetFEM++ (via Python)
>> https://lists.nongnu.org/archive/html/getfem-users/2011-03/msg00009.html
>>
>> *I like GetFEM++ way: separative design (GeoTrans, Mesh, Fem, Im, ...); high 
>> and low layers (Model, asm_...);  point and convex; and so on.and so on.
>>
>> Best regards,
>> Alex Spring
>>

bar_expressions_compared.py
Description: Text document

CheatSheetForGradDiv.pdf
Description: Adobe PDF document

LinearElasticity.pdf
Description: Adobe PDF document

BarProblem.pdf
Description: Adobe PDF document