# Boundary conditions and density when using smoothed particle hydrodynamics/corrective smoothed particle method for two-temperature models

I am trying to solve the dual-hyperbolic two temperature model (TTM) using a modified smoothed particle hydrodynamics (SPH) scheme known as CSPM (corrective smoothed particle method), as demonstrated in these two papers (can't add links, sorry).

1. DOI: 10.1080/104077801300348842
2. 10.1002/(SICI)1097-0207(19990920)46:2<231::AID-NME672>3.0.CO;2-K This essentially models the ablation of a thin metal film by a femtosecond laser pulse, $T$ represents temperature (electron and lattice) and $q$ is the heat flux.

I coded the CSPM scheme successfully and tested it through some simple examples (including a simple 1 temperature model in paper 2). However, for the TTM, the results are close to what I need (i.e. qualitative shape of curves), but they're off. I think it might be the boundary conditions and/or the way I'm assigning particle mass and density.

The boundary conditions I need to use are: $q_{E, L}(z = 0, t) = q_{E, L}(z = L, t) = 0$ where L is the film length.

My questions are as follows:

1. How do I correctly pick the boundary particle positions? I have tried:

• A: Boundary particles with their centers on the boundary (z = 0, z = L).

• B: Boundary particles which do not lie on the boundary (z = dz, z = L-dz).

For both cases, the particles are equally spaced, with $h = dz$ (2 recommends this value of $h$). Boundary particle B gives me better results (peak temp. is around 750 compared to expected 650, for example). Particle (A) has worse results (peak 850 vs 650). The boundary particle has half the mass of the internal particles, otherwise, the zigzaging bhavior in the picture below prevails. Curves have the correct shape but they're not smooth. Most SPH papers deal with hydrodynamics and I haven't found an in-depth discussion of how to position my boundary particles, especially that CSPM is slightly different from SPH. I am just enforcing my $q$ boundaries to 0 and not updating them as you would in as mesh-based method. 1. The bigger issue, how do I define mass and density in this case? Other SPH applications seem to have a physical meaning for both these quantities (and density seems to be involved in the PDEs themselves in most cases), but in my case I have no idea how to define these. Treat the particles as metal atoms and use those values?

Currently I'm using:

$m = dz\cdot1\times10^9$ for internal particles, and half that for boundary particles. My $dz$ is in nm, if I use such a small value for mass it just doesn't work.

For density I tried two formulations: - Constant rho = 1. Gives best results. - Using the equation $\rho_i = \sum_j^N W_{ij}m_j$. Results are zig-zagging as shown above.

Note that in all cases $\sum_j W_{ij} m_j/\rho_j$ is not equal to 1 for the first few particles near the boundary, but is equal to 1 for internal particles. Is this expected?

I apologize for the long question, I am just completely stuck and I have run out of ideas.

• I'm not sure cross posting is allowed, you might check. – user108787 Jul 26 '16 at 0:42
• It seems it's not, I have deleted it from Comp Sci, thanks for letting me know. – Ash Jul 26 '16 at 0:44
• Spelling out acronyms in the title helps people who should pass on a question to pass on a question. – garyp Jul 26 '16 at 0:57