
From:  Konstantinos Poulios 
Subject:  Re: [Getfemusers] unexpected nonzero values in a stiffness matrix K: a bar (or truss) problem 
Date:  Fri, 27 Sep 2019 20:40:28 +0200 
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 crosssectional 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 nonzero values in a stiffness matrix K. I believe that thinking about it helps deeper understanding of the framework.
>>
>> The problem considered is:
>> """
>> Bar:
>> 01
>> (point: 0, 1; convex: 0 (01), segment or 1dsimplex))
>> (2 dof (u, v) on a single point)
>> Coordinate system:
>> ^ y, v, Fy
>> 
>> > x, u, Fx
>> Expected for the convex 0 (01):
>> 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 vdirection here).)
>> """
>>
>> I believe that we can handle a bar using FEM_PK(1, 1) ((1, 1): first 1 for 1dsimplex (i.e. segment), second 1 for 2nodes on a segment).
>>
>> However, obtained K has unexpected nonzero values:
>> """
>> Obtained for the convex 0 (01) :
>> 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 getfemusersarchives 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:
>> [Getfemusers] Solving truss structure in GetFEM++ (via Python)
>> https://lists.nongnu.org/archive/html/getfemusers/201103/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
>>
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