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 a Step and within the newtonrhapson() procedure.

import numpy as np
import felupe as fem

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

# prevent the field-values at the center-point to be treated as dof0
mesh.points_without_cells = mesh.points_without_cells[:0]

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

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

mpc = fem.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 = fem.NeoHooke(mu=1.0, bulk=2.0)
body = fem.SolidBody(umat=umat, field=field)

K = body.assemble.matrix() + mpc.assemble.matrix()
r = body.assemble.vector(field) + mpc.assemble.vector(field)

mesh.plot(plotter=mpc.plot()).show()