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Re: [Getfem-users] Stiffness matrix for linear elasticity
From: |
Roman Putanowicz |
Subject: |
Re: [Getfem-users] Stiffness matrix for linear elasticity |
Date: |
Wed, 29 Dec 2010 02:16:36 +0100 |
User-agent: |
Mutt/1.5.13 (2006-08-11) |
Dear Yves,
> This formula implements in fact a weak term. This represents
>
> lambda div(u).div(v) + mu grad(u):grad(v) + mu grad(u)^T:grad(v)
> y
> where A:B is the Frobenius (scalar) product of matrices.
>
Thank you for your comment, it was much helpful. Somehow I couldn't
see the forest for the trees.
Now I can link the implementation and the formal derivations.
Below I include the explanation of this link I have built for myself.
It is rather informal so please correct me if I to brutally violate
some math rules :-)
For linearised elasticity (assuming epsilon = 1/2 (grad(u) + grad^T(u))
we have:
Cauchy stress:
sigma = lambda div(u) I + 2 mu (1/2 (grad(u) + grad^T(u)))
sigma = lambda div(u) I + mu grad(u) + mu grad^T(u)
The force balance equation (neglecting inertia term and body forces)
div(sigma) = 0
Weak form of weighted residual equation
int{div(sigma):v} -> int{sigma:grad(v)} + boundary term
Considering the term under the integral sign
sigma:grad(v) ->
lambda div(u) I:grad(v) + mu grad(u):grad(v) + mu grad^T(u):grad(v)
but as I:grad(v) = div(v)
Then we get:
lambda div(u).div(v) + mu grad(u):grad(v) + mu grad^T(u): grad(v)
Well now I see the correspondence with
lambda.t(:,i,i,:,j,j) + mu.t(:,i,j,:,i,j) + mu.t(:,j,i,:,i,j)
where: t = comp(vGrad(#1).vGrad(#1))
Somehow I couldn't get this using the index notation (maybe was not patient
enough :).
I decided to write this (now obvious :) elaborate derivation because
the above way the to calculate the stiffness matrix for linearised elasticity
is a bit different from what one can find in the elementary books on FEM.
Is it true that this particular way of calculation is driven
a) by the availability of tensor t = comp(vGrad(#1).vGrad(#1))
b) avoidance of problems with incompressible materials ?
Thanks again and best regards,
Roman
--
Roman Putanowicz, PhD < address@hidden >
Institute for Computational Civil Engng (L-5)
Dept. of Civil Engng, Cracow Univ. of Technology
www.l5.pk.edu.pl, tel. +48 12 628 2569, fax 2034