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BDHI FCM

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Raul edited this page Nov 30, 2018 · 2 revisions

BDHI_FCM is a BDHI (Brownian Hydrodynamics) submodule[4]. Source is at Integrator/BDHI/BDHI_FCM.cuh

As this is a BDHI module. BDHI_FCM computes the terms M·F and B·dW in the differential equation:

dR = K·R·dt + M·F·dt + sqrt(2Tdt)· B·dW

Where:
R: Particle positions
K: Shear matrix (FCM does not implement this in PBC)
M: FCM Mobility tensor
T: Temperature
F: Forces acting on the particles.
dW: Collection of Weinner processes
B: B·B^T = M

Using periodic boundary conditions.

The mobility, M, is computed according to the Force Coupling Method (FCM) tensor[1].

USAGE

Use as any other BDHI method. You have an example in examples/FCM.cu.

BDHI::FCM has the following parameters:

...
BDHI::FCM::Parameters par;
par.temperature = 1.0;
par.viscosity = 1.0;
par.hydrodynamicRadius =  1.0;
par.dt = 1e-3;
par.box = box;
par.tolerance = 1e-3;

auto bdhi = make_shared<BDHI::EulerMaruyama<BDHI::FCM>>(pd, pg, sys, par);
...

A brief summary of the algorithm

The algorithm behind FCM is very similar to the PSE one but without the near field part. The particle forcing is spreaded to a grid using Gaussian kernels and then the fluid velocity (including fluctuations) is solved in Fourier space. Finally the fluid velocity is interpolated back to the particles using the same Gaussian kernels.
Using the operator terminology in [2]:

M·F = σ·St·FFTi·B·FFTf·(S·F+f)
-σ: The volume of a grid cell
-S: An operator that spreads each element of a vector to a regular grid using a gaussian kernel.
-F: Forces acting on the particles
-f: Force density acting on the fluid
-FFT: Fast fourier transform operator.
-B: 1/(\eta k^2)·(I-k\otimes k) in Fourier space. Multiplies fluid forcing by the FCM kernel, resulting in fluid velocity, see eq.59 in [1].

    Related functions:  
    FFT: cufftExecR2C (forward), cufftC2R(inverse)  
    S: FCM_ns::particles2Grid (S), FCM_ns::grid2Particles (σ·St)  
    B: FCM_ns::forceFourier2vel

sqrt(M)·dW: The stochastic contribution is computed in fourier space along M·F as:
M·F + sqrt(M)·dW = σ·St·FFTi·B·FFTf·(S·F+f)+ √σ·St·FFTi·√B·dW =
= σ·St·FFTi( B·FFTf·(S·F+f) + 1/√σ·√B·dW)

Only one St·FFTi is needed, the stochastic term is added as a velocity in fourier space. dW is a gaussian random vector of complex numbers, special care must be taken to ensure the correct conjugacy properties needed for the FFT. See FCM_ns::fourierBrownianNoise

References:

[1] Fluctuating force-coupling method for simulations of colloidal suspensions. Eric E. Keaveny. 2014.
[2] Rapid Sampling of Stochastic Displacements in Brownian Dynamics Simulations - https://arxiv.org/pdf/1611.09322.pdf
[3] Spectral accuracy in fast Ewald-based methods for particle simulations
[4] https://github.com/RaulPPelaez/UAMMD/wiki/Brownian-Hydrodynamics

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