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## Re: [Getfem-users] unexpected non-zero values in a stiffness matrix K: a

 From: Konstantinos Poulios Subject: Re: [Getfem-users] unexpected non-zero values in a stiffness matrix K: a bar (or truss) problem Date: Thu, 19 Sep 2019 15:18:11 +0200

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_

So basically you need to write the corresponding _expression_ for a bar and use it with the

asm_generic

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.

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