Hi Matt,
Maybe the nonlinear poro-viscoelastic code Jean-Paul and I contributed to
the Code Gallery may help you decide whether adding another phase would be
feasible for your application. Our formulation models a biphasic material
that consists in a nonlinear (finite-strain) viscoelastic solid and a pore
fluid. We solve monolithically for solid displacements and fluid pressure
with a direct solver. The whole code structure is based on that of step-44,
so it should be relatively easy to navigate if you're already familiar with
step-44.
Best,
Ester
El dia dimecres, 5 d’octubre de 2022 a les 15:30:49 UTC+2,
mjri...@gmail.com va escriure:
> This was very helpful!
>
> I am trying to understand the process so I could potentially extend this
> formulation for biphasic materials. The way it handles incompressibility is
> perfect for high water content tissues. My concern revolves around adding
> another phase and explicitly solving for the fluid displacements in the
> system. I was going to follow a similar strategy, but I am worried that
> this approach is very specific.
>
> IMHO the key feature for this approach is the ability to condense out the
> pressure and volumetric ratio variables and recover the displacement only
> formulation but with anti-locking properties baked in as opposed to
> starting with a pure displacement formulation. I would like to preserve
> that if possible. I am just worried I will wind up in a hole I cannot climb
> out of...
>
> These comments were very helpful. One quick question if you just
> proceed without the condensation would that still work? In one of the
> tutorials, If I abandoned the notion of using the CG solver and used
> something like GMRES can I avoid some of the manipulation after I have the
> linearized system?
>
> Thanks for the prompt response and thoughtful answers! This stuff is hard
> and fascinating at the same time.
>
> Matt
>
> On Tue, 4 Oct 2022 at 14:03, Jean-Paul Pelteret <jppel...@gmail.com>
> wrote:
>
>> Hi Matthew,
>>
>> I'm glad that you find step-44 to be a useful tutorial! Let me try to
>> answer your questions directly.
>>
>> *My first question deals with the statement "The Euler-Lagrange equations
>> corresponding to the residual"*
>>
>> *Directly above this sentence is the residual, whose derivation I
>> understand. Where I am lost is that s tact on to equations. I have**
>> only one residual equation. I cannot bridge that disconnect. *
>>
>> This is just another way of saying that the three equations listed there
>> (these E-L equations) are the strong form of the governing equations.
>> Basically, if you take each of these equations (along the way, modifying
>> the definition of the stress in the equilibrium equation), test them with
>> the appropriate test function and sum up the three residual contributions
>> then you recover the (total) residual, or stationary point of the residual,
>> that is listed above. The point is that it not necessarily so straight
>> forward to go from the strong form to the weak form for this mixed
>> formulation, so identifying the conservation equations a-postori is a
>> helpful sanity check here. They seem to align with what we're trying to do
>> here.
>>
>>
>> *I am completely lost here. What is the significance of p and J not
>> having derivatives on them that makes it "easy" to solve for those terms in
>> isolation? *
>>
>> Well, that's a valid point. I can't quite recall what exactly we were
>> trying to identify with this comment. I'll think about it, as that seems to
>> be a point that we could improve on in the documentation. That there is no
>> K_{pp} contribution is significant, because it makes condensing out the p
>> and J fields easier. Maybe we meant to refer to the lack of contribution to
>> K_{pp} (as there is no second derivative involving a variation and
>> linearisation of p.
>>
>> *I am also confused how K_pJ, K_Jp and K_JJ form a block diagonal
>> matrix. I could get there if I ignore the top row but then the equations
>> below do not make sense I think. Some detail on this part of the process
>> would be great. *
>>
>> So this is more easy to explain. We specifically choose discontinuous
>> shape function to discretise these fields. As there are no interface/flux
>> contributions, all local element contributions for these terms will remain
>> local and you therefore end up with an assembled matrix for these
>> contributions that has a block-like structure. The K_{JJ} matrix is
>> evidently block-diagonal, as it is a field that couples with itself. As for
>> the coupling matrices K_{pJ} and K_{Jp}, they are block-diagonal because we
>> chose *exactly* the same discretisation for both fields (i.e. the shape
>> functions and polynomial order match).
>>
>> Does that make sense?
>>
>> Best,
>>
>> Jean-Paul
>> On 2022/09/30 19:16, Matthew Rich wrote:
>>
>> Hi,
>>
>> Step-44 has a lot going on and really sets you up to tackle a variety of
>> real world problems. There is a significant jump in complexity between this
>> tutorial and steps 8,17 & 18.
>>
>> I have been reading the references and I am about there but need few
>> clarifications for why things are the way they are.
>>
>> Any assistance would be much appreciated...
>>
>> My first question deals with the statement "The Euler-Lagrange equations
>> corresponding to the residual"
>>
>> Directly above this sentence is the residual, whose derivation I
>> understand. Where I am lost is that s tact on to equations. I have only one
>> residual equation. I cannot bridge that disconnect.
>>
>> My other question has to deal with the solving procedure for K (see
>> snippet below)
>> [image: Ksolve.png]
>>
>>
>> I am completely lost here. What is the significance of p and J not having
>> derivatives on them that makes it "easy" to solve for those terms in
>> isolation? I am also confused how K_pJ, K_Jp and K_JJ form a block
>> diagonal matrix. I could get there if I ignore the top row but then the
>> equations below do not make sense I think. Some detail on this part of the
>> process would be great.
>>
>> Thanks in advance,
>>
>> Matt
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