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interaction.py
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interaction.py
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from inspect import getargspec
import taichi as ti
from .consts import *
class HessianNotImplemented(NotImplementedError):
pass
class ForceNotImplemented(NotImplementedError):
pass
class Interaction:
def fill_params(self, *args):
return list(args)
class ExternalPotential(Interaction):
def __call__(self, r):
pass
def force(self, r):
raise ForceNotImplemented
def hessian(self, r):
raise HessianNotImplemented
class QuadraticWell(ExternalPotential):
def __init__(self, k, center):
self.k = k
self.center = ti.Vector(center)
def __call__(self, r):
return self.k * ((r - self.center) ** 2).sum() / 2
def force(self, r):
return -self.k * (r - self.center)
def hessian(self, r):
return self.k * IDENTITY
class InverseSquare(ExternalPotential):
def __init__(self, k, center):
self.k = k
self.center = center
def __call__(self, r):
return self.k / (r - self.center).norm()
def force(self, r):
dr = r - self.center
dr2 = (dr ** 2).sum()
return -self.k * dr / dr.norm() / dr2
'''
Pair interactions
'''
class PairInteraction(Interaction):
def __call__(self, r2, args):
raise NotImplementedError
def derivative(self, r2, args):
raise ForceNotImplemented
def second_derivative(self, r2, args):
raise HessianNotImplemented
@ti.func
def force(self, r, r2, args):
return 2. * self.derivative(r2, args) * r
@ti.func
def hessian(self, r, r2, args):
return -4. * r.outer_product(r) * self.second_derivative(r2, args) \
- 2 * IDENTITY * self.derivative(r2, args)
class LennardJones(PairInteraction):
n_params = 3
def __init__(self, rcut=0):
self.rcut = rcut
self.rc2 = rcut ** 2
self.irc6 = 1 / rcut ** 6
self.irc12 = self.irc6 ** 2
@ti.func
def __call__(self, r2, args):
u = 0.0
if ti.static(self.rcut <= 0) or 0 < r2 < self.rc2:
s12, s6, e = args[0], args[1], args[2]
u = 4. * e * (s12 * (1 / r2 ** 6 - self.irc12)
- s6 * (1 / r2 ** 3 - self.irc6))
return u
@ti.func
def derivative(self, r2, args):
s12, s6, e = args[0], args[1], args[2]
return 12. * e / r2 \
* (-2. * s12 / r2 ** 6 + s6 / r2 ** 3)
@ti.func
def second_derivative(self, r2, args):
s12, s6, e = args[0], args[1], args[2]
return 24. * e / r2 ** 2 \
* (7. * s12 / r2 ** 6 - 2. * s6 / r2 ** 3)
def fill_params(self, sigma, epsilon):
s6 = sigma ** 6
s12 = s6 ** 2
return [s12, s6, epsilon]
def combine(self, v1, v2):
s_i, e_i = v1[0], v1[1]
s_j, e_j = v2[0], v2[1]
return self.fill_params(
(s_i + s_j) / 2., ti.sqrt(e_i * e_j))
class Coulomb(PairInteraction):
n_params = 1
@ti.func
def __call__(self, r2, args):
k = args[0]
return k / ti.sqrt(r2)
@ti.func
def derivative(self, r2, args):
k = args[0]
r = ti.sqrt(r2)
return -k / (2 * r * r2)
@ti.func
def second_derivative(self, r2, args):
k = args[0]
r = ti.sqrt(r2)
return 3. * k / (4 * r * r2 * r2)
class HarmonicPair(PairInteraction):
n_params = 2
@ti.func
def __call__(self, r2, args):
k, r0 = args[0], args[1]
r = ti.sqrt(r2)
return k * (r - r0) ** 2 / 2
@ti.func
def derivative(self, r2, args):
k, r0 = args[0], args[1]
return k * (1 - r0 / ti.sqrt(r2)) / 2
@ti.func
def second_derivative(self, r2, args):
k, r0 = args[0], args[1]
r = ti.sqrt(r2)
return k * r0 / (2 * r * r2)
class ParabolicPotential(PairInteraction):
n_params = 1
@ti.func
def __call__(self, r2, args):
k = args[0]
r = ti.sqrt(r2)
return k * r ** 2 / 2
@ti.func
def derivative(self, r2, args):
k = args[0]
return k / 2
@ti.func
def second_derivative(self, r2, args):
return 0.
'''
harmonic bond bending
true harmonic: e = k * (acos(x) - theta0) ** 2
second order approx: e = (x - x0) ** 2 / (1 - x0 ** 2), x0 = cos(theta0)
third order approx: e = (x - x0) ** 2 / (1 - x0 ** 2) + x0 ** 2 * (x - x0) ** 3 /
(1 - x0 ** 2) ** 2
calculate force:
x = cos<r1-r0, r2-r0> = (r1 - r0)^T.(r2 - r0) / (|r1 - r0|*|r2 - r0|)
d|r|/dr = d(sqrt(r^T.r))/dr = e[r] = r / |r|
let v = (r1 - r0)^T.(r2 - r0) = (r1^T.r2 - r0^T.(r1 + r2) + r0^2)
du/dr_1 = (r2 - r0)
du/dr_2 = (r1 - r0)
du/dr_0 = (2r0 - r1 - r2)
let u = |r1 - r0|*|r2 - r0|
dv/dr_1 = |r2 - r0|/|r1 - r0| * (r1 - r0)
dv/dr_2 = |r1 - r0|/|r2 - r0| * (r2 - r0)
dv/dr_0 = -|r1 - r0|/|r2 - r0| * (r2 - r0) - |r2 - r0|/|r1 - r0| * (r1 - r0)
dx = vdu - udv / u**2
dx/dr_1 = [|r1 - r0|*|r2 - r0|*(r2 - r0) - (r1 - r0)^T.(r2 - r0)*|r2 - r0|/|r1 - r0| * (r1 - r0)] / u**2
= [(r2 - r0) - (r1 - r0)^T.(r2 - r0) / (r1 - r0) ** 2 * (r1 - r0)]/u
dx/dr_2 is the same
dx/dr_0 = -dx/dr_1-dx/dr_2
'''
class BondBending(Interaction):
'''
Bond bending potentials operate on the angle cosine
between two bond vectors [r1<--r0-->r2],
U = f(cosx, args)
f* = dU/dr* = f'(cosx) * d(cosx)/dr*
'''
@ti.func
def __call__(self, cosx, args):
raise NotImplementedError
@ti.func
def derivative(self, cosx, args):
raise NotImplementedError
class HarmonicBending(BondBending):
n_params = 2
@ti.func
def __call__(self, cosx, args):
k, theta0 = args[0], args[1]
return 1/2. * k * (ti.acos(cosx) - theta0) ** 2
@ti.func
def derivative(self, cosx, args):
k, theta0 = args[0], args[1]
return - k * (ti.acos(cosx) - theta0) / ti.sqrt(1 - cosx ** 2)