Boundary Conditions, Continuum Mechanics

[NeoDevin]

Mods: I wasn't sure if I should put this here or in DE's, if you think it would be better there, feel free to move it.

Everyone:
If I have a bunch of particles, which I'm modelling as a continuum, and I want to put periodic boundary conditions which reflect (completely specular reflection) some fraction of the incoming particles, how would I write this as a boundary condition equation?

Thanks in advance.

 

[Andy Resnick]

Leaving aside the question of how a collection of discrete particles are modeled as a continuum, I would think that a probabilistic equation would do just fine. What happens to the particles that are not reflected?

 

[NeoDevin]

If you have enough particles, you can model them as a continuous density function and velocity/momentum function.

How would you formulate a probabilistic equation in terms of a boundary condition for a differential equation.

The particles not reflected are transmitted.

 

[Andy Resnick]

Are you using some sort of lattice Boltzmann approach?

Shiyi Chen and Daniel Martínez, Renwei Mei
On boundary conditions in lattice Boltzmann methods
Phys. Fluids 8, 2527 (1996); DOI:10.1063/1.869035

Title: Absorbing boundary and free-surface conditions in the phononic lattice solid by interpolation
Authors: Huang, L.-J., Mora, P., & Fehler, M. C.
Journal: Geophysical Journal International, Volume 140, Issue 1, pp. 147-157.

Jeng-Rong Ho, , Chun-Pao Kuo and Wen-Shu Jiaung
Study of heat transfer in multilayered structure within the framework of dual-phase-lag heat conduction model using lattice Boltzmann method
International Journal of Heat and Mass Transfer
Volume 46, Issue 1, January 2003, Pages 55-69

There's plenty more, I'm sure. I'm, not an expert, those are what I was able to quickly find.