This topic was nascent for a long time and hence had not checked my account for a long time .

First of all , I thank of all of you for commenting, especially Dr. Biswajit Banerjee for your insightful comments and giving a clear picture of time integration techniques .

Although I have not yet made a code for any 2D problem , but I did made a code for solving simple 2 mass spring problem with different time steps .

For this problem I must say that results were comparable to that obtained with any other time stepping techniques (both in time taken to run the code and the displacements obtained ) , there was no apparent benefit of using AVI here ( I will post the results shortly ), may be I had taken a very simple problem , I am planning to implement AVI for a better 1D problem / 2 D problem and then see how it works out .

Thanks

Keep posting ......

Kapil Nandwana

I think the following reference is even more comprehensive and very easy to read (disregarding the geometric asides, of course) http://www.cds.caltech.edu/~marsden/bib/2004/03-LeMaOrWe2004b/LeMaOrWe20... Julian

I used a similar version of the paper suggested by Nachiket when I started reading about AVIs. Although there is a lot of math this paper does provide a complete picture. Unfortunately I am not aware of a simpler version.

My implementation is based on the "Explicit AVI Algorithm" that appears on page 19 of this paper. Referring to this page in the paper:

Here I have only one particle in 1-D so I can drop the "a" subscript and assume x and v are scalar. Then representing the oscillator as a vector (x,v)^T the position update and velocity update steps can be written as matrices. As a start take the friction coefficient gamma to be 0 and as an exercise try to show that the MATLAB code matches this algorithm. Hope this helps.

Is this:

http://www.cds.caltech.edu/~marsden/bib/2003/05-LeMaOrWe2003/LeMaOrWe200...

a good place to start reading about AVIs? Is there a simpler version of the theory anywhere that treats a one-dimensional problem?. It would be great if someone can point me to such a reference -- I can then correlate it to Will Fong's implementation.

-Nachiket

Hello,

I work with AVI in my research so I have included a MATLAB implementation of AVI for the 1-D harmonic oscillator. Please have a look at my blog entry for more information and a MATLAB file.

//m.limpotrade.com//m.limpotrade.com/node/1007/

Hi Biswajit Banerjee

Your answers jibe with what was written in the variational integrators VI papers by Ortiz, Marsden and friends. But I am wondering whether it works well for dynamical systems with strong dissipation or damping, in particular for systems of the form Ma+Cv+Ku=F, where the damping C is arbitrary and not some linear combination of the mass M and stiffness K. It appears that VIs may not offer much in such cases, but correct me if I'm wrong.

Thanks.

I don' know much about variational integrators, but here's what I think is the basic idea:

1) The question of conserving both momentum and energy is fundamental if you are to run dynamic simulations for long real times. For example, you wish to simulate a perfectly elastic ball bouncing inside a rigid box with perfectly elastic contact. Traditional finite difference algorithms for time integration usually have a dissipative property (for a suitable choice of parameters). If you use such algorithms the ball will lose momentum and energy over time. There are good finite difference schemes available nowadays that claim to be energy and momentum conserving (or close to it).

2) An alternative approach is to formulate the problem (after a typical finite element spatial discretization) in such a way that a Galerkin-type method can also be used to discretize the time derivatives. This type of approach can be found in the space-time finite element literature. Then you can use the well-developed theory of Galerkin methods to prove conservation properties.

3) The third approach is to start directly with a Lagrangian or Hamiltonian (the Euler-Lagrange equations of which are the equations for the balance of momentum, for instance) and integrate over time.

The best way of figuring out how useful a particular algorithm is for your purposes is to code it up and compare various subcycling approaches with it. A good test problem is the 1-D wave equation (first-order in time) and Sod's problem of a travelling shock wave. You will then be able to compare how easy the algorithm is to implement vs how good its conservation properties are and how fast it is.

Hi,

I read a little about AVI, but don't know much and don't have the experience to comment on its significance. Have you tried coding it?

Thanks.