PhD: Biofilms model with NXFEM
Posted on 20/10/2018, in PhD.This note is for chapter 6 in my thesis. I will use NXFEM coupling with LSM to simulate a biofilm model.
Notes
 Chap6 in reMarkable
 Handwriting notes (main)
 This site.
Models
Very simple biofilm model
 Below model is in chopp 2007 xfem biofilm growth.pdf.
 The same (more specific): xfem moving interface THESIS  Bryan G. Smith.pdf
 Is mentioned in chopp duddu et al 2006 combine nxfem levelset.pdf, page 18 (figure biofilm, layer growth a little bit  Fig 5). Fig in this article is consistent with result in chopp 2007 xfem.
 $u$: substrate
 $v$: biomass
Coefficients:
Three test cases
In chopp duddu et al 2006 combine nxfem levelset.pdf, Chopp listed 4 test cases for biofilms. He also describes the properties of the domain, the condition’s values and so on. Page 16  18.
 $100\times 100$ triangles mesh size.
 $\Omega = 0.5\times 0.5$ mm
 $dt = \frac{dx}{\max{F}}$
 $S = 8.3\times 10^{6}\, \frac{mgO_2}{mm^3}$
 $\nabla_n S=0, \nabla_n \Phi = 0$ on sides of $\Omega$
 First example, only 1 semicircle whose radius is $0.01$ mm, grow in $44$ days. Fig 5 and [49] (chopp 2007 xfem biofilm growth.pdf).
 Second, 3 semicircles, the middle has radius $0.02$ mm whereas others has $0.01$ mm for each. How to implement???. Fig 6
 Third, see in the article.
Chopp06combine –[49]> Chopp07xfem –[4]> Chopp06simulating (splitting) and –[6] kalpperFinger and –[5] dependQuorum (parameters)
Others
 frederic MASTER THESIS 2010 biofilms.pdf page 21, 85.
CFL  time step

If $\Delta x$ is very small in comparison with $\Delta t$, there may be instability and divergence (sometimes). That’s why we need CFL condition (relation between velocity $u$, time step $\Delta t$ and mesh size $\Delta x$)

For the Crank–Nicolson numerical scheme, a low CFL number is not required for stability, however it is required for numerical accuracy (wiki).
Affected components
All changeable components in the code,
 Time:
.Tmax
,.dt
pa.r0
: interface’s radius $\kappa$ (
.kapU
,.kapV
)  Penalty coefficient $\lambda$ (
.lamU
,.lamV
)  (May take an affect)
.bcU
,.bcV
Note that,
 $\kappa, \lambda$ depends on the choice of using or not using ghost penalty!
Modify main_chopp2007.m
 Why we need
cp
instead ofpa
? Because in some functions to find global matrix/load vector, we need the diff coef inside pa, but how about diff foru
andv
? If we usepa.kapU
,pa.kapV
, it’s not good because we need to consider upto 4 parameters!  Why main_sys_linda can use
pa
? No, it didn’t!!!  Form of $\kappa_i$?
 Smallcut, many other functions are not modified to the new version yet!
Tobe coded
 Periodic boundary condition in x direction.
 multi biofilm shape (separated)
 Chopp06stimulating: section 5, there are some results to be compared!
 Time scale??
Don’t forget to do
 Adaptive mesh (start with freefem++ first)
 Run the test in Arnold’s book!
 Model main_sys_linda is not the one in article1 because both of u and v have the interface conditions [u], [v]! Check the file main_article1 instead!