Daniel Duque Campayo

Welcome to my personal website // Bienvenido a mi página personal

@dduque at Mastodon

Current links / Enlaces actuales

UPM CFD course 2022/2023

Daniel Duque Campayo

Old links / Enlaces históricos

UPM CFD course 2020/2021

UPM CFD course 2017/2018

UPM undergrad CFD course 2018/2019

UPM undergrad CFD course 2017/2018

UPM CFD course 2016/2017

Seminar on Camley and Brown's work

SPHeric 2018 talk

References

Created with reveal.js, the HTML Presentation Framework

Particle-in-cell idea

“You don't have to move all your particles, you may keep some fixed, as in the old Particle-in-Cell method”

J Monaghan, SPHeric 2015 at Parma

Starting with particles + velocity values

Let's project onto a fixed mesh

Using some triangulation (pFEM style)

The problem is on the mesh only

We compute new velocities (and pressure)

We project back onto the particles ...

Particles are finally moved, Lagrangian-ly

Possible advantages

From particles: linear algebra may be done once !!
From particles: easier boundary conditions and FSI
From mesh: avoidance of false diffusion through particles

pFEM simulations

p-FEM:
particle + FEM

proj-FEM:
particle + ( mesh + FEM )

The problem with projection

Unfortunately, projection may spoil convergence
$\frac{d A}{d t} = Q$ $ \frac{d r }{d t} = u $

$ E= T $ $ (\Delta t)^2 $ $ \left( \ddot{Q} + \frac{Q'' u_0^2}{ \mathrm{Co}^2} \right) $ $ + \underbrace{T \color{red}{ (\Delta t)} A'' \frac{ u_0^2 }{ \mathrm{Co}_h^2} \left( 1+ \frac{1}{ m^2} \right) }_\mathrm{proj} $
But, this can be fixed!

Improving projection

The former holds for linear projection !

$ E= T \color{red}{ (\Delta t)^2 } \left( \ddot{Q} + \frac{Q'' u_0^2}{ \mathrm{Co}^2} \right) + \underbrace{T \color{red}{ (\Delta t) } A'' \frac{ u_0^2 }{ \mathrm{Co}_h^2} \left( 1+ \frac{1}{ m^2} \right) }_\mathrm{proj} $
with a quadratic projection:
$ E= T \color{red}{ (\Delta t)^2 } \left( \ddot{Q} + \frac{Q'' u_0^2}{ \mathrm{Co}^2} \right) + \underbrace{T \color{red}{ (\Delta t)^2 } A''' u_0^3 \left( \frac{1}{ \mathrm{Co}_H^3} + \frac{1}{ \mathrm{Co}_h^3} \right) }_\mathrm{proj} $
... or the offending term can be made small by particle crowding

Zalesak's disk

	Linear FE	Quad mesh	Quad mesh + parts
Equal parts. mesh
Part crowding

Taylor-Green vortex sheet

pFEM	projFEMq
projFEMq	projFEM6

Errors for Taylor-Green

Velocity err vs $\Delta t$

Errors for Taylor-Green

Pressure err vs $\Delta t$

Errors for Taylor-Green

Velocity err vs CPU time

Errors for Taylor-Green

Pressure err vs CPU time

Rayleigh-Taylor instability

pFEM

projFEMq

projFEM6

OpenFOAM

Extra: Kelvin-Helmholtz instability

pFEM

projFEMq

Conclusions

Linear projection is not a good idea ...
but quadratic projection is.
Future: Other boundaries, free surface.
Future: quadratic FEM + better time integrator.

Thank you for your attention

CGAL programming


void areas(Triangulation& T) {

  for(F_f_it fc=T.finite_faces_begin();
      fc!=T.finite_faces_end();
      fc++) {

    Periodic_triangle pt=T.periodic_triangle(fc);

    Triangle t=T.triangle(pt);

    fc->area=t.area();

  }

  return;
}