Satisfying conditions on the interface for a fluid-structure interaction type of problem

[Artur Safin]

Dear dealii community,

I have a fluid-structure interaction type of problem to solve, where on one domain I have the linearized compressible Navier-Stokes equations, and on the other - equation of motion (elastic deformation). The PDEs that I have are time-harmonic, so there is no time dependence. On the F-S interface, I need to satisfy two continuity conditions:

1) v = ik*u (no voids)

2) τ(v)n = σ(u)n (continuity of stress)

(v - fluid velocity, u - structure displacement; τ,σ - stress tensors; k - constant, i - imaginary number)

So far, what I've done is generated a Robin-Robin type of BCs by adding the two conditions together:

v + a*τ(v)n = ik*u + a*σ(u)n

where a is a parameter that we can choose (one value for the NS-equations, and another for the eqn of motion). In a test-case that I've built, this method seems to work, but I've always wondered about two things:

1) Is this actually an appropriate approach toward solving such a problem?

2) Choosing the correct a-parameters for both the fluid and the structure is always very difficult and unpredictable. Is there a better way to find the two parameters that will satisfy the conditions aside from just plugging in values and seeing what works?

Any help with this would be greatly appreciated.

Many thanks,

Artur

[Wolfgang Bangerth]

Artur,

> I have a fluid-structure interaction type of problem to solve, where on one
> domain I have the linearized compressible Navier-Stokes equations, and on the
> other - equation of motion (elastic deformation). The PDEs that I have are
> time-harmonic, so there is no time dependence. On the F-S interface, I need to
> satisfy two continuity conditions:
>

> 1) *v* = ik**u* (no voids)
> 2) *τ*(*v*)*n* =*σ*(*u*)*n*(continuity of stress)
> (*v* - fluid velocity, *u* - structure displacement; *τ*,*σ* - stress tensors;

> k - constant, i - imaginary number)
>
> So far, what I've done is generated a Robin-Robin type of BCs by adding the
> two conditions together:
>

> *v* + a**τ*(*v*)*n* = ik**u* + a**σ*(*u*)*n*

>
> where a is a parameter that we can choose (one value for the NS-equations, and
> another for the eqn of motion). In a test-case that I've built, this method
> seems to work, but I've always wondered about two things:
>
> 1) Is this actually an appropriate approach toward solving such a problem?

It at least seems non-obvious to me that this is actually correct. The
condition you have does not guarantee that both of these conditions are
satisfied at the same time, and 20 years of experience suggest to me that what
you have is not a formulation that is mathematically correct. One of the
arguments I would have against it is that I can't seem to assign any kind of
physical interpretation to the term
v + a tau n

> 2) Choosing the correct a-parameters for both the fluid and the structure is
> always very difficult and unpredictable. Is there a better way to find the two
> parameters that will satisfy the conditions aside from just plugging in values
> and seeing what works?

The correct starting point would of course be to do a literature search. There
must be hundreds if not thousands of papers on fluid structure interaction.
What do they do?

Another starting point would be to take a look at step-46.

Best
W.

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