Six Myths of Polynomial Interpolation and Quadrature

I found a profound paper written by Professor Lloyd N. Trefethen entitled Six Myths of Polynomial Interpolation and Quadrature. This paper is profound because it goes against a lot of the mainstream things that I have learned related to polynomials. After watch Professor Trefethen’s lectures , I found that he is right and knows what he is talking about. Read the paper. It is worth your time.

The paper is here if it is ever removed from the link above.
mythspaper

Transmission Line Method

Transmission Line Method (TLM) or Transmission Line Matrix method is an elegant technique used to solve lossless or lossy wave equations.  Here I am gathering together resources on the topic since I have found it is not well study in the US.

 Books

Christopoulos, Christos. “The transmission-line modeling (TLM) method in electromagnetics.” Synthesis Lectures on Computational Electromagnetics 1.1 (2005): 1-132.  ISBN: 1598290509

This book is focused on the point of view of electrical engineering.

De Cogan, Donard. Transmission line matrix (TLM) techniques for diffusion applications. Gordon and Breach Science Publishers, 1998. ISBN: 978-9056991296

De Cogan, Donard, William J. O’Connor, and Susan Pulko. Transmission line matrix (TLM) in computational mechanics. CRC press, 2005. ISBN: 978-0415327176

This is the book I found easiest to read from a Mechanical Engineering perspective.  The book starts by looking at the wave equation in 1D.  It uses the ideas of boundary conditions to explain the scattering effect that occurs in TLM.  It is an easy read for an engineer who has studied the wave equation before.

Papers

Webpages

http://dandadec.com/what-is-tlm/ –  This page is Donard De Cogan’s page.  Donard’s website was lots of information on TLM such as conferences and papers.

Creating a Custom Header in WordPress

I used an image for header that I wanted to cite the photographer with a link to his website. There isn’t a clear way to do this in WordPress tools. Therefore, I searched and found that I could edit the header.php file for the theme I was using to customize the header.  The header.php file is located at ./wp-content/themes/myThemeName where myThemeName was twentytwelve. i.e.  ./wp-content/themes/twentytwelve/header.php.

I added this piece of code to header.php before the ending </header> tag:

Image by <a href=”http://15belowphoto.com/”>Nick Wooley</a>

Like this:

This piece of code has the effect that the header for my WordPress website has a link to the photographer’s website like below.

The downside to modifying header.php is that it gets overwritten when updates to the theme are made. Ugh! Make sure to document the modifications to header.php. Backing it up is undesirable because security updates may change the header.php code and thus restoring an older version may introduce security issues.

See the results of modifying header.php here:
http://www.teamvardo.com/

 

My Introduction to Computational Fluid Dynamics (CFD)

I have been doing a lot of CFD lately. I have quickly learned that the information one can gain from CFD is far more insightful to the physicals then 0D, lumped parameter equations.

One tool I have been using is Agros2d. It can do 2d axis symmetric and planar PDE problems. Agros2d can handle incompressible steady and unsteady CFD.

As an example, orifice problem has been study many times. However, it usually isn’t usually studied from the point of view of an axis symmetric CFD.

Here is the model:
Screenshot from 2016-05-07 00:50:44

The inlet boundary condition is a max velocity of 1.5m/s with a parabolic profile (1-(r/R)^2)*1.5m/s. R is the radius of the pipe 0.003m.

The Agros2d model is attached here: Orifice.a2d. You should be able to open the model with Agros2d and hit the “Solve” button in the lower left hand corner.

Below are some the results from the simulation.

Plot01

Velocity field near the beginning and throat.

 

Plot02

Velocity field just past the throat. Notice the re-circulation.

 

Velocity field at the outlet. Notice the recirculation

Velocity field at the outlet. Notice the recirculation

.

Velocity along the axis of the pipe (velocity in the z direction).

Velocity along the axis of the pipe (velocity in the z direction).

 

Velocity color plot at the throat. Notice that there is a dead space between the wall and the stream. The thinniest part of the stream is known as the vena contracta. See the next picture to see the velocity profile across the center of the throat.

Velocity color plot at the throat. Notice that there is a dead space between the wall and the stream. The thinnest part of the stream is known as the vena contracta. See the next picture to see the velocity profile across the center of the throat.

Plot07

This the velocity profile at the center of the throat. It is interesting to see the velocity is in a step shape. The velocity is constant in the stream and near zero otherwise.

 

Velocity profile at the start of throat where the fluid is accelerating.

Velocity profile at the start of throat where the fluid is accelerating.

Velocity profile at the end of the throat.

Velocity profile at the end of the throat.

Velocity profile at the outlet in the z direction which along the axis of the pipe. Notice the negative velocity part of the profile indicating recirculation.

Velocity profile at the outlet in the z direction which along the axis of the pipe. Notice the negative velocity part of the profile indicating recirculation.

Hopefully you enjoyed this as much as I did making it.

Differential Algebraic Equations (DAE’s)

Often Ordinary Differential Equations (ODE’s) are studied in engineering, especially system dynamics and control theory. However there are many cases where the system is described by a Differential Algebraic Equations (DAE’s) rather than an ODE system.

 

Here are some resources related to the issue:

This book is the most famous book on DAE’s I have com across. When you read articles on DAE’s the author usually references the following book.

Ascher, Uri M., and Linda R. Petzold. Computer methods for ordinary differential equations and differential-algebraic equations. Vol. 61. Siam, 1998.  https://goo.gl/vmpq5l

I came across this book related to control theory and DAE’s

Berger, Thomas. On differential-algebraic control systems. Diss. Technische Universität Berlin, 2013.  http://goo.gl/w4m5ms

 

A Good Nerd Laugh

Donald Knuth wrote the program tex (pronounced tek).  He cares about code correctness and therefore gave out a reward if you ever found a bug in tex.  In 36 years, only 1289 bugs have been found.  In light of tex, Donald joking release iTeX in a 2010 presentation.  It is a joke and you have to know a little of Donald’s quirks to get why the presentation is funny otherwise you may think he was serious.