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Re: [Getfem-users] Lumped mass matrix

From: Roman Putanowicz
Subject: Re: [Getfem-users] Lumped mass matrix
Date: Thu, 8 Mar 2012 06:14:28 +0100
User-agent: Mutt/1.5.20 (2009-06-14)

> Dear Roman,
Dear Yves,

Thank you for your remarks.
> First a remark : as you built your second version, the loop on all the
> components is very much more expensive than the use of a fourth order
> tensor in the first version. The best way is the second strategy would
> be to iterate on the potential nonzero element, i.e. the "neighbhour"
> degrees of freedom (an exemple exist in the fonction
> asm_generalized_dirichlet_constraints of getfem_assembling.h, I think).

Yes, you are right in the above. In fact, mass lumping can be done
most effectively on the element level. The row summation approach
or HRZ method (as presented in)
require short loops over elements of consistent mass matrix and this is
very local operation. This is equivalent to my first version with the fourth
order tensor. The only reason I have used such tensor is that I haven't seen
the way how to manipulate an element matrix __after__ it is calculated
but __before__ it is assembled into global system. 

The use of a fourth order tensor is the most general but convoluted and 
inefficient way of expressing a linear transformation on element matrix

In other words I was seeking a mechanism expressed in the snippet below:
    generic_assembly assem;
    assem.set("T = sym(comp(Base(#1).Base(#1)));"
              "T = UserMap(T);"
              "M(#1,#1)+= T;");

where UserMap means application of user function passed to assem object
much in the way a nonlinear calculation routines are passed.

Calculation of lumped matrices would benefit from provision of UserMap
as above. However, I am not sure where else it could be useful ?
Maybe in selective/reduced integration techniques? But then, isn't
NonLinear operation sufficient for this? 

> This is more complex to obtain correct
> lumped matrices (i.e. non-singular positive matrices and without to much
> loss of accuracy) for higher order finite element methods.
Yes, you are again right, and I have found references presenting solution
tailored to specific cases.

Currently in GetFEM++ the calculation of mass matrix is hidden for instance
in add_basic_d_on_dt brick. As there are many approaches to mass lumping
it seems to me, that the way to go would be to enable specification of mass
matrix on the time derivative brick level.


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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