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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 Putanowicz, PhD  < address@hidden  >
Institute for Computational Civil Engng (L-5)
Dept. of Civil Engng, Cracow Univ. of Technology, tel. +48 12 628 2569, fax 2034

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