Cardiac electro-mechanics problem using deal.ii

[navneet roshan]

Dear deal.ii community,

  I am solving cardiac tissue problem, i.e. a time-dependent electro-mechanical problem. In this problem electrical potential generated from the pacemaker is taken care by a reaction-diffusion equation, the mechanical response is an elastic contraction. A picture is attached below showing how a plane wave of the electrical signal passes the tissue from left to right and how contraction follows.

Till now I have done only the electrical part of the problem i.e. a reaction-diffusion problem using adaptively refined meshes. 

              For the full electro-mechanics problem, I am now coupling the electrical potential to the mechanical response of the tissue.  My doubts are following,

Q1. Is it possible/numerically correct to adaptively refine a deforming mesh?

For reaction-diffusion(electrical) part, a highly refined mesh is needed(due to constraints on time steps) but for solving the mechanical deformation problem coarser mesh will do, because doing mechanics calculations on the highly refined mesh will be expensive, for efficiency I wanted to solve the mechanics problem on a coarser version of the same mesh used for solving only the electrical part at each time step.

Q2. How do I get a mesh and quadrature points of a mesh that is one or two levels less refined than the one used for the solving purely electrical part? how to get the quadrature points of such mesh? How do I interpolate electrical potential solution on this coarser mesh?

Note: I am using "totally Lagrangian approach" for the mechanics part, i.e. the reference configuration is the initial undeformed configuration.

[Wolfgang Bangerth]

Navneet,

>   I am solving cardiac tissue problem, i.e. a time-dependent
> electro-mechanical problem. In this problem electrical potential generated
> from the pacemaker is taken care by a reaction-diffusion equation, the
> mechanical response is an elastic contraction. A picture is attached below
> showing how a plane wave of the electrical signal passes the tissue from left
> to right and how contraction follows.
>
>
> Till now I have done only the electrical part of the problem i.e. a
> reaction-diffusion problem using adaptively refined meshes.
>
>
>  For the full electro-mechanics problem, I am now coupling the electrical
> potential to the mechanical response of the tissue.  My doubts are following,
>
>
> Q1. Is it possible/numerically correct to adaptively refine adeforming mesh?

Yes, you can do that. You have to make sure that you move hanging nodes in
such a way that they continue to be at the midpoint of the longer edge. An
easy way to make sure that you do is if you deform a mesh by a displacement
that is computed using a continuous finite element. (I'll note that the easier
way to do this is to actually use the MappingQEulerian class -- it allows you
to not actually move vertices, but just uses a finite element field as the
displacement vector.)

> For reaction-diffusion(electrical) part, a highly refined mesh is needed(due
> to constraints on time steps) but for solving the mechanical deformation
> problem coarser mesh will do, because doing mechanics calculations on the
> highly refined mesh will be expensive, for efficiency I wanted to solve the
> mechanics problem on a coarser version of the same mesh used for solving only
> the electrical part at each time step.
>
>
> Q2. How do I get a mesh and quadrature points of a mesh that is one or two
> levels less refined than the one used for the solving purely electrical part?
> how to get the quadrature points of such mesh? How do I interpolate electrical
> potential solution on this coarser mesh?

Using different meshes is possible, but awkward. Have you though of just using
a higher-order element for the electrical part, and a lower-order element for
the displacement? This way, both variables live on the same mesh, making data
exchange simpler.

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/