User:Tclamb/Journal

=Navier-Stokes (Projection Methods)=

2012/07/20

 * Wikipedia has an article outlining projection methods.
 * Initially, I was interested in Chorin's method (576kB .pdf) due to mention in FEniCS documented demo for this very problem.
 * However, in further research, I've discovered that more modern and effective methods have been developed since 1968.
 * I am currently reading literature on Kim and Moin's method (behind paywall; 897kB .pdf).
 * Another FEniCS solution of this problem is located here in the CBC.Solve project.
 * I can't tell what method they are using here. Does not seem to match Chorin's or Kim and Moin's.
 * Unfortunately, my first, rushed attempt at implementing Kim & Moin's method fails miserably.

2012/07/22

 * Restarting from scratch, I carefully redid my derivation of a weak variational form of Kim and Moin's method.
 * Success! This second attempt has yielded a working implementation of Kim and Moin's method.

=CBC.Pdesys=

2012/07/22

 * I've begun looking into CBC.PDESys.

2012/07/23

 * CBC.PDESys appears to automate implementation Picard iteration to quickly solve systems of PDEs. Before I begin using this library, I am going to implement my own Picard solver for a nonlinear PDE so I can understand the process better. After my initial reading on the topic, Picard iteration seems to be an easy way to explore the effectiveness of various linearizations of nonlinear PDEs.
 * Picard iteration is easy! And since FEniCS solves linear PDEs much faster than its default Newton solver solves nonlinear PDEs, it results in a runtime upwards of 30% faster!

=Readings=

2012/08/01

 * While reading today, I came across the following in Fish & Belytschko (pg. 58):
 * $$\Gamma = \text{boundary}$$
 * $$\Gamma_e = \text{boundaries with essential boundary conditions}$$
 * $$\Gamma_n = \text{boundaries with natural boundary conditions}$$
 * $$\Gamma_e \cap \Gamma_n = 0, \quad \Gamma_e \cup \Gamma_n = \Gamma$$

=References=