Solid rotation of a slotted disk
We describe the benchmark proposed by Zalesak.
Computer codes, used for the acquisition of results, are from Vincent Doyeux.
Problem Description
In order to test our interface propagation method, i.e. the levelset method \phi, we will study the rotation of a slotted disk into a square domain. The geometry can be represented as
We denote \Omega, the square domain [0,1]\times[0,1]. The center of the slotted disk is placed at (0.5,0.75).
To model the rotation, we will apply an angular velocity, centered in (0.5,0.5), as the disk is back to its initial position after t_f=628.
During this test, we observe three different errors to measure the quality of our method. With these values, two kinds of convergence will be studied : the time convergence, with different time step on an imposed grid and the space one, where the space discretization and the time step are linked by a relation. Several stabilization methods are used such as CIP ( Continuous Interior Penalty ) or SUPG ( StreamlineUpwing/PetrovGalerkin ).
Boundary conditions
We set a Neumann boundary condition on the boundary of the domain.
Initial conditions
The velocity is imposed as \boldsymbol{u}=\left( \frac{\pi}{314} (50y),\frac{\pi}{314} (x50) \right)
Here is the velocity look in the square domain
Inputs
The following table displays the various fixed and variables parameters of this testcase.
Name 
Description 
Nominal Value 
Units 
r 
disk radius 
0.15 
m 
l 
slot base 
0.05 
m 
h 
slot height 
0.25 
m 
t_f 
slotted disk rotation period 
628 
s 
Outputs
We observe during this benchmark three different errors.
First at all, the mass error, define by
e_{\text{m}} = \frac{ \left m_{\phi_f}  m_{\phi_0}\right }{m_{\phi_0}} = \frac{ \left \displaystyle \int_{\Omega} \chi( \phi_f < 0 )  \displaystyle \int_{\Omega} \chi( \phi_0 < 0 ) \right }{ \displaystyle \int_{\Omega} \chi( \phi_0 < 0 )}
where \chi is the characteristic function. \chi( f( \phi ) ) = \left\{ \begin{array}{rcl} 1 & \text{ if } & f( \phi ) \neq 0 \\ 0 & \text{ if } & f( \phi ) = 0 \end{array} \right.
However, mass is gain and loose at different emplacements on the mesh, and at the same time, with the level set method.
Secondly, the sign change error e_{\text{sc}} = \sqrt{ \int_\Omega \left( (1H_0)  (1H_f) \right)^2 }
with H_0=H_\epsilon(\phi_0) and H_f=H_\epsilon(\phi_f), H_\epsilon the smoothed Heaviside function of thickness 2ε.
This error is better to define the interface displacement. In fact, we can determine where \phi_0\phi_f<0, in other words where the interface has moved.
Finally, we define the classical L^2 error at the interface, as e_{L^2} = \sqrt{ \frac{1}{\displaystyle \int_\Omega \chi( \delta(\phi_0) > 0 ) } \int_\Omega (\phi_0  \phi_f)^2 \chi( \delta(\phi_0) > 0 ) }.
Discretization
Time convergence
For this case, we set a fixed grid with mesh step size h=0.04, and so 72314 degree of freedom on a \mathbb{P}^1.
Then, after the disk made one round, we measure the errors obtained from two different discretizations ( BDF2 and Euler ) and compared them.
We repeat this with several time step dt\in \{2.14, 1, 0.5, 0.25, 0.20\}.
Only one stabilization method is used : SUPG
Space convergence
We define the following relation, between time step and mesh step size : dt=C\frac{h}{U_{max}}
where C<1 constant and U_{max} the maximum velocity of \Omega.
From the definition of our velocity, U_{max} is reached at the farthest point from the center of \Omega. In this case, we have U_{max}=0.007, and we set C=0.8.
We use the BDF2 method for time discretization. As in time convergence, we wait one round of the disk to measure the errors and we repeat this test for these values of h: 0.32, 0.16, 0.08, 0.04.
We compare the results from different stabilization methods : CIP, SUPG, GLS ( GalerkinLeastSquares ) and SGS ( SubGrid Scale ).
Implementation
Results
Time convergence
dt 
e_{L^2} 
e_{sc} 
e_m 
2.14 
0.0348851 
0.202025 
0.202025 
1.00 
0.0187567 
0.147635 
0.147635 
0.5 
0.0098661 
0.10847 
0.10847 
0.25 
0.008791 
0.0782569 
0.0782569 
0.20 
0.00803373 
0.0670677 
0.0670677 
dt 
e_{L^2} 
e_{sc} 
e_m 
2.14 
0.0118025 
0.0906617 
0.0492775 
1.00 
0.00436957 
0.0445275 
0.0163494 
0.5 
0.00173637 
0.0216359 
0.0100621 
0.25 
0.001003 
0.0125971 
0.00354644 
0.20 
0.000949343 
0.0117449 
0.00317368 
Space convergence
stab 
h 
e_{L^2} 
e_{sc} 
e_m 
CIP 
0.32 
0.0074 
0.072 
0.00029 
0.16 
0.0046 
0.055 
0.00202 

0.08 
0.0025 
0.033 
0.00049 

0.04 
0.0023 
0.020 
0.00110 

SUPG 
0.32 
0.012 
0.065 
0.01632 
0.16 
0.008 
0.049 
0.07052 

0.08 
0.004 
0.030 
0.00073 

0.04 
0.001 
0.018 
0.00831 

GLS 
0.32 
0.013 
0.066 
0.02499 
0.16 
0.008 
0.051 
0.05180 

0.08 
0.004 
0.031 
0.00805 

0.04 
0.001 
0.019 
0.00672 

SGS 
0.32 
0.012 
0.065 
0.01103 
0.16 
0.008 
0.050 
0.07570 

0.08 
0.004 
0.030 
0.00084 

0.04 
0.001 
0.018 
0.00850 
Conclusion
Let’s begin with time convergence results. Tables shows us that sign change error is better to define the quality of the chosen scheme than the mass error. In fact, the loss of mass somewhere can be nullify by a gain of mass elsewhere. Sign change error shows half an order gain from Euler scheme to BDF2 one, as L^2 errors show us a gain of one order. For the slotted disk shape, BDF2 uses the two previous iterations to obtain the current result, while Euler only need the previous iteration. This explain why we can see an asymmetrical tendency in the first one.
As for space convergence, the different stabilization methods we used give us the same convergence rate equals to 0.6, with close error values, for the sign change error. For the L^2 error case, it’s not as evident as the previous one. Aside the CIP stabilization method with a 0.6 convergence rate, the others show us a convergence rate of 0.9.
Bibliography

[Zalesak] Steven T. Zalesak, Fully multidimensional fluxcorrected transport algorithms for fluids, Journal of Computational Physics, 1979.

[Doyeux] Vincent Doyeux, Modelisation et simulation de systemes multifluides. Application aux ecoulements sanguins., Physique Numérique [physics.compph], Université de Grenoble, 2014