Multi-Point Constraints

This How-To demonstrates the usage of multi-point constraints (also called MPC or RBE2 rigid-body-elements) with an independent centerpoint and one or more dependent points. First, a centerpoint has to be added to the mesh. MPC objects are supported as items of the Newton-Rhapson procedure.

import numpy as np
import felupe as fe

# mesh with one additional rbe2-control point
mesh = fe.Cube(n=11)
mesh.points = np.vstack((mesh.points, [2.0, 0.0, 0.0]))
mesh.update(mesh.cells)

region = felupe.RegionHexahedron(mesh)
displacement = felupe.Field(region, dim=3)
field = felupe.FieldContainer([displacement])

An instance of felupe.MultiPointConstraint defines the multi-point constraint. This instance provides two methods, felupe.MultiPointConstraint.assemble.vector() and felupe.MultiPointConstraint.assemble.matrix().

MPC = fe.MultiPointConstraint(
    field=field,
    points=np.arange(mesh.npoints)[mesh.points[:, 0] == 1],
    centerpoint=mesh.npoints - 1,
    skip=(0,1,1),
)

Finally, add the results of these methods to the internal force vector or the stiffness matrix.

umat = felupe.constitution.NeoHooke(mu=1.0, bulk=2.0)

K = fe.IntegralForm(
    umat.hessian(field.extract()), field, region.dV, field
).assemble() + MPC.assemble.matrix()

r = fe.IntegralForm(
    umat.gradient(field.extract()), field, region.dV
).assemble() + MPC.assemble.vector(field)