Dynamic Thermoelastic problem

[Ehsan]

Hi,

I want to solve a dynamic coupled thermoelastic problem in deal ii.
I studied deal ii tutorials step-23, step-26, and step-31 and also watched related video lectures, but I still have a few question.

I have 3 questions: first 2 question are related to deal ii, but the third one does not relate to deal ii directly.
I would appreciate it if you help me.

1- I don't want to code from the beginning and I want to modify one of the steps to solve my problem.
Which step do you suggest to be considered as the base program?

2- In step-23 (wave equation) and step-31 (Navier–Stokes equations), new variable is defined to transform the system from second order system into a first order system.
Adding a new variable increases the DoF of system. I would like to know if this method is efficient than dealing with second order derivatives?

3- When I watched the video lectures I got a bit confused about time discretization and stability issues.
In order to have a stable convergence and maintaining the order of accuracy in time discretization of displacement and temperature fields, which time discretization do you suggest?
Is below discretization proper for my problem:

\begin{align*}    
    u &= (1-\theta) u ^{(n-1)} + \theta u ^ {(n)} \\
    T &= (1-\theta) T ^{(n-1)} + \theta T ^ {(n)} \\
    \dot{u} &= \frac{u ^{(n)} - u ^{(n-1)}}{\Delta t} \\
    \dot{T} &= \frac{T ^{(n)} - T ^{(n-1)}}{\Delta t} \\
    \ddot{u} &= \frac{u ^{(n+1)} - 2 u ^{(n)} + u ^{(n-1)}}{\Delta t ^2}
\end{align*}

or I should use higher order time stepping schemes like BDF-2 for time derivatives and three point centred approximation for u and T ?

Best regards.
Ehsan

[Uwe Köcher]

Hej Ehsan,

regarding your questions:

1) hard to say, since none of any tutorial programs solves your problems. I would really suggest to you, that you

understand the programming-flow of a deal.II application an set up a clean program on your own.

2) regarding the wave equations: the second-order time derivatives are hard to simulate directly, so the standard

approach in solving goes down to use well-known first-order ODE integrators for problems which are needed if

you firstly discretised your problem space.

Solving the appearing ODE system with second-order derivatives in time is non-standard and the usual approach

is to use the well-known system of first-order in time. For time-domain wave equations, this means that you

get an additional variable for the velocity.

3) I do not know your problem in the details. The time-discretisation might be stable or only conditionally stable.

Since your problem is well established in the literature, you might look firstly on what discretisation pairs

(space + time) are stable before trying to implement them!

Kind regards

  Uwe

[Ehsan]

Dear Uwe,

So many thanks for your reply.

Regarding the second item, I mean in solving second-order PDEs by defining additional velocity variable and reducing the order of the system,  my specific question is:
What is the benefit of dealing with first-order PDEs instead of the second-order ones, which worth increasing DoF and dealing with larger matrices?
Is it more stable or it has other reasons?

Best regards.
Ehsan

[Wolfgang Bangerth]

On 10/26/2015 05:57 AM, Ehsan wrote:
>
> Regarding the second item, I mean in solving second-order PDEs by defining
> additional velocity variable and reducing the order of the system, my
> specific question is:
> What is the benefit of dealing with first-order PDEs instead of the
> second-order ones, which worth increasing DoF and dealing with larger matrices?
> Is it more stable or it has other reasons?

It's not about stability. It's more about the fact that you have many methods
available for first order systems, including higher order ones (such as RK4)
that do not require you to store many past timesteps.

Generally, the effort for the second equation is relatively small since,
taking the wave equation as an example, the primary equation needs to invert a
Laplace or Helmholtz operator and the other one just a mass matrix. The latter
is rather cheap. So breaking one second-order equation into two first-order
equations does not cost very much.

Best
W.

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

[Uwe Köcher]

Dear Ehsan,

additionally to the notes of Wolfgang given above, I may did not understand your question correctly.

To be precise, using a second variable for the velocity in the wave equation does *not* increase the

DoF's in space as long as the displacement and velocity variables lives in the same space, which is

at least for step-23 the case.

The computational afford is outlined by Wolfgang's post.

To treat the second order in time derivatives in an distributional sense directly, you may use

the (finite-difference) approach of the Newmark's schemes, they are stable or conditionally stable

regarding the choice of the Newmark parameters. They are working, but they are unflexible in

some sense of adaptivity and at most of linear-in-time approximation order. There is a great paper:

Bangerth, Geiger, Rannacher: "Adaptive Galerkin finite element methods for the wave equation"

given in Comput. Meth. Appl. Math. 10, 3--48 (2010)

which brings the affords of different time discretisations for second-order in time wave equations

together.

Additionally, for some higher-order in time variational time discretisations, there are some additional

benefits from using the first-order in time systems. Cf. my current research publications on

http://koecher.cc/uk/uk/papers.php

Using the first-order in time system, without block-elimination, gives you systems which are hardly

to precondition for iterative solvers; cf. esp. my thesis given by the link above,

whereas senseful-eliminated systems gives you the approach of Wolfgang (they are well studied

in the literature).

Fortunately, (and at least) low-order time discretisations can be solved by the approach given in

step-23. Using the reference Bangerth/Geiger/Rannacher, Crank-Nicolson, cG(1) and the Newmark

schemes can be linked together for special choices on the discretisation parameters and

integration of the right hand sides.

Best

  Uwe