[Dune] question about initial condition

[Christian Engwer]

Dear Rana,

On Mon, Aug 24, 2020 at 11:31:41AM +0500, Rana Abdul Rauf wrote:
> Hello DUNE developers, I am working on phase field modeling. For the
> numerical simulations, I am using Dune PEDlab. Now I am solving the
> Cahn-Hilliard equation to see the spinodal decomposition. For this I want
> to generate random numbers between -1 to 1. Kindly help me how to put
> rand(x,y) as a initial condition, where rand(x,y) is random number between
> -1 to 1.

you can cast any lambda into a discrete function in
PDELab. Interpolation aloows then to use this function as an initial
value.

- use your favorite random number generator, depending on your needs.
- for simpliity I assume a uniform distribution between -1 and 1 is OK
  and I'll use std::uniform_real_distribution (https://en.cppreference.com/w/cpp/numeric/random/uniform_real_distribution)
- you can now create a random scalar function using, working in global coordinates:
  ```
   std::random_device rd;  //Will be used to obtain a seed for the random number engine
   std::mt19937 gen(rd()); //Standard mersenne_twister_engine seeded with rd()
   std::uniform_real_distribution<> dis(-1.0, 1.0);
   auto r = [dis&,gen&](const auto & x){ return dis(gen); };
  ```
  and cast it into a grid function, using infrastruture from dune-functions
  ```
  auto f = Dune::makeAnalyticGridViewFunction(r, gridView);
  ```
- this function `f` an now interpolated into your stating vector using PDELab's `interplate`.

If you know how to interprate your coeffiients you can further
simplify this approach and perhas even use these random values
directly as coefficients.

Ciao
Christian