Dear Jean-François,
For a vector variable 'u', each line of 'Grad_u' is the gradient of the
ith component of 'u', each of them is tangent to the curve and length
being the derivative with respect to the curvilinear abscissa. The
linearized deformation is a priori Normalized(element_K).(Grad_u *
Normalized(element_K))
The formulas used to compute the gradient and the Hessian can be found
here:
http://getfem.org/project/femdesc.html#geometric-transformations
http://getfem.org/project/appendixA.html#derivative-computation
The hessian of a vector valued variable is also the hessian of each
component.
Best regards,
Yves.
Le 20/11/2017 à 02:10, Jean-François Barthélémy a écrit :
Dear Yves,
Thank you very much for your answer.
It's OK for scalar variables but I do not really understand how Grad_u
is built when u is a displacement vector field of 3 components. I
thought Grad_u would represent the vector du/ds
([dux/ds,duy/ds,duz/ds]) with s the local curvilinear abscissa (so
that Normalized(element_K).Grad_u would give the linearized
longitudinal deformation) but it seems that Grad_u is actually a 3x3
matrix field. Then I do not see how to build the longitudinal
deformation. What would be the best way please? And by the way, what
would be the right syntax to get the second derivative of the
transverse displacement by means of Hermite elements and the Hessian?
Thank you again for your help.
Best regards
Jean-François
2017-11-17 20:52 GMT+01:00 Yves Renard <[email protected]
<mailto:[email protected]>>:
Dear Jean-François,
There is no specific tool yet for that.
You can have access to the tangent with 'element_K' in the generic
assembly language (the unit tangent is then 'Normalized(element_K)')
If you define a scalar quantity "u" on your 1D structure, then
"Grad_u" will be the gradient of the quantity in the sense that it
is a tangent vector whose norm is the derivative of the qunatity
along the curve. So that "Grad_u.Grad_Test_u" is still the
stiffness term for a curvilinear second derivative. For a vector
quantity "u", "Grad_u" is the componentwise gradient.
Best regard,
Yves.
----- Original Message -----
From: "Jean-François Barthélémy" <[email protected]
<mailto:[email protected]>>
To: [email protected] <mailto:[email protected]>
Sent: Friday, November 17, 2017 6:17:13 PM
Subject: [Getfem-users] Curvilinear structures in Getfem
Dear Getfem users,
I wonder whether it is possible to model simple linear elastic
curvilinear
structures submitted to traction, bending, torsion etc... in 2D or
3D in
Getfem. I haven't found a way to have access to the tangential or
normal
parts of vectors in the local basis of a beam and their
derivatives with
respect to the curvilinear abscissa needed to build the
formulation. Does
someone have an answer please?
Thanks in advance
Best regards
Jean-François
--
Yves Renard ([email protected]) tel : (33) 04.72.43.87.08
Pole de Mathematiques, INSA-Lyon fax : (33) 04.72.43.85.29
20, rue Albert Einstein
69621 Villeurbanne Cedex, FRANCE
http://math.univ-lyon1.fr/~renard
---------
