Boundary Conditions, Continuum Mechanics

In summary, the conversation revolved around the topic of modeling discrete particles as a continuum and implementing periodic boundary conditions in this model. The preferred method of applying such boundary conditions was through a probabilistic equation, with the remaining particles being transmitted. The use of lattice Boltzmann methods and heat transfer in multilayered structures within this framework were also discussed. Several studies on this subject were mentioned, including those by Shiyi Chen and Daniel Martínez, Renwei Mei, Huang, L.-J., Mora, P., & Fehler, M. C., and Jeng-Rong Ho, Chun-Pao Kuo, and Wen-Shu Jiaung.
  • #1
NeoDevin
334
2
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.
 
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  • #2
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?
 
  • #3
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.
 
  • #4
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.
 

1. What are boundary conditions in continuum mechanics?

Boundary conditions in continuum mechanics refer to the set of conditions that must be satisfied at the interface or boundary between different materials or regions in a continuum. These conditions govern how the continuum behaves at the boundary and are necessary to ensure the continuity of physical quantities such as stress, strain, and velocity.

2. Why are boundary conditions important in continuum mechanics?

Boundary conditions are important in continuum mechanics because they allow us to solve problems involving complex geometries and material properties. They also help us to accurately model real-world situations and make predictions about the behavior of a continuum under different loading conditions.

3. What are the different types of boundary conditions in continuum mechanics?

The different types of boundary conditions in continuum mechanics include:

  • Dirichlet boundary conditions, which specify the value of a physical quantity at the boundary.
  • Neumann boundary conditions, which specify the flux or gradient of a physical quantity at the boundary.
  • Mixed boundary conditions, which combine Dirichlet and Neumann boundary conditions.
  • Periodic boundary conditions, which impose a repeating pattern at the boundaries of a periodic structure.

4. How are boundary conditions applied in continuum mechanics?

Boundary conditions are typically applied by defining them as constraints in the mathematical equations that govern the behavior of a continuum. These equations, such as the Navier-Stokes equations for fluid mechanics or the elasticity equations for solid mechanics, are then solved numerically to determine the values of the physical quantities at the boundaries.

5. What are some challenges in applying boundary conditions in continuum mechanics?

One challenge in applying boundary conditions in continuum mechanics is ensuring that they accurately represent the real-world behavior of the system being studied. This may require taking into account factors such as material anisotropy, non-linear material behavior, and dynamic loading conditions. Another challenge is selecting the appropriate type and number of boundary conditions to accurately capture the behavior of the continuum without introducing unnecessary complexity.

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