md.manifold¶

Overview

`Manifold`	Base class manifold object.
`Cylinder`	Cylinder manifold.
`Diamond`	Triply periodic diamond manifold.
`Ellipsoid`	Ellipsoid manifold.
`Gyroid`	Triply periodic gyroid manifold.
`Plane`	Plane manifold.
`Primitive`	Triply periodic primitive manifold.
`Sphere`	Sphere manifold.

Details

Manifolds.

A Manifold defines a lower dimensional manifold embedded in 3D space with an implicit function \(F(x,y,z) = 0\). Use Manifold classes to define positional constraints to a given set of particles with:

class hoomd.md.manifold.Cylinder(r, P=(0, 0, 0))¶

Bases: Manifold

Cylinder manifold.

Parameters:

r (float) – radius of the cylinder constraint \([\mathrm{length}]\).
P (tuple [float, float, float]) – point defining position of the cylinder axis (default origin) \([\mathrm{length}]\).

Cylinder defines a right circular cylinder along the z axis:

\[F(x,y,z) = (x - P_x)^{2} + (y - P_y)^{2} - r^{2}\]

Example:

cylinder1 = manifold.Cylinder(r=10)
cylinder2 = manifold.Cylinder(r=5,P=(1,1,1))

class hoomd.md.manifold.Diamond(N, epsilon=0)¶

Bases: Manifold

Triply periodic diamond manifold.

Parameters:

N (tuple [int, int, int] or int) – number of unit cells in all 3 directions. \([N_x, N_y, N_z]\). In case number of unit cells u in all direction the same (\([u, u, u]\)), use N = u.
epsilon (float) – defines CMC companion of the Diamond surface (default 0)

Diamond defines a periodic diamond surface . The diamond (or Schwarz D) belongs to the family of triply periodic minimal surfaces:

\[F(x,y,z) = \cos{\frac{2 \pi}{B_x} x} \cdot \cos{\frac{2 \pi}{B_y} y} \cdot \cos{\frac{2 \pi}{B_z} z} - \sin{\frac{2 \pi}{B_x} x} \cdot \sin{\frac{2 \pi}{B_y} y} \cdot \sin{\frac{2 \pi}{B_z} z} - \epsilon\]

is the nodal approximation of the diamond surface where \([B_x,B_y,B_z]\) is the periodicity length in the x, y and z direction. The periodicity length B is defined by the current box size L and the number of unit cells N \(B_i=\frac{L_i}{N_i}\).

See also

Example:

diamond1 = manifold.Diamond(N=1)
diamond2 = manifold.Diamond(N=(1,2,2))

class hoomd.md.manifold.Ellipsoid(a, b, c, P=(0, 0, 0))¶

Bases: Manifold

Ellipsoid manifold.

Parameters:

a (float) – length of the a-axis of the ellipsoidal constraint \([\mathrm{length}]\).
b (float) – length of the b-axis of the ellipsoidal constraint \([\mathrm{length}]\).
c (float) – length of the c-axis of the ellipsoidal constraint \([\mathrm{length}]\).
P (tuple [float, float, float]) – center of the ellipsoid constraint (default origin) \([\mathrm{length}]\).

Ellipsoid defines an ellipsoid:

Implicit function

\[F(x,y,z) = \frac{(x-P_x)^{2}}{a^{2}} + \frac{(y-P_y)^{2}}{b^{2}} + \frac{(z-P_z)^{2}}{c^{2}} - 1\]

Example:

ellipsoid1 = manifold.Ellipsoid(a=10,b=5,c=5)
ellipsoid2 = manifold.Ellipsoid(a=5,b=10,c=10,P=(1,0.5,1))

class hoomd.md.manifold.Gyroid(N, epsilon=0)¶

Bases: Manifold

Triply periodic gyroid manifold.

Parameters:

N (tuple [int, int, int] or int) – number of unit cells in all 3 directions. \([N_x, N_y, N_z]\). In case number of unit cells u in all direction the same (\([u, u, u]\)), use N = u.
epsilon (float) – defines CMC companion of the Gyroid surface (default 0)

Gyroid defines a periodic gyroid surface. The gyroid belongs to the family of triply periodic minimal surfaces:

\[F(x,y,z) = \sin{\frac{2 \pi}{B_x} x} \cdot \cos{\frac{2 \pi}{B_y} y} + \sin{\frac{2 \pi}{B_y} y} \cdot \cos{\frac{2 \pi}{B_z} z} + \sin{\frac{2 \pi}{B_z} z} \cdot \cos{\frac{2 \pi}{B_x} x} - \epsilon\]

See also

Example:

gyroid1 = manifold.Gyroid(N=1)
gyroid2 = manifold.Gyroid(N=(1,2,2))

class hoomd.md.manifold.Manifold¶

Bases:

Base class manifold object.

Warning

Users should not instantiate Manifold directly, but should instead instantiate one of its subclasses defining a specific manifold geometry.

Warning

Only one manifold can be applied to a given method or active forces.

__eq__(other)¶: Test for equality.

class hoomd.md.manifold.Plane(shift=0)¶

Bases: Manifold

Plane manifold.

Parameters:: shift (float) – z-shift of the xy-plane \([\mathrm{length}]\).

Plane defines an xy-plane at a given value of z:

\[F(x,y,z) = z - \textrm{shift}\]

Example:

plane1 = manifold.Plane()
plane2 = manifold.Plane(shift=0.8)

class hoomd.md.manifold.Primitive(N, epsilon=0)¶

Bases: Manifold

Triply periodic primitive manifold.

Parameters:

N (tuple [int, int, int] or int) – number of unit cells in all 3 directions. \([N_x, N_y, N_z]\). In case number of unit cells u in all direction the same (\([u, u, u]\)), use N = u.
epsilon (float) – defines CMC companion of the Primitive surface (default 0)

Primitive specifies a periodic primitive surface as a constraint. The primitive (or Schwarz P) belongs to the family of triply periodic minimal surfaces:

\[F(x,y,z) = \cos{\frac{2 \pi}{B_x} x} + \cos{\frac{2 \pi}{B_y} y} + \cos{\frac{2 \pi}{B_z} z} - \epsilon\]