Assemble Vectors and Matrices

Integral (weak) forms are transformed into sparse vectors and matrices by calling the assemble-method of an integral form. The Neo-Hookean material model formulation is used to evaluate both the variation and linerization of its strain energy density function.

import felupe as fem

mesh = fem.Cube(n=3)
region = fem.RegionHexahedron(mesh)
umat = fem.NeoHooke(mu=1, bulk=2)
field = fem.Field(region, dim=3).as_container()

F = field.extract(grad=True, sym=False, add_identity=True)  # deformation-gradient
P = umat.gradient(F)[:1]  # list with first Piola-Kirchhoff stress tensor
A = umat.hessian(F)  # list with fourth-order elasticity tensor

The code for the variation of the total-potential energy, as given in Eq. (1), is very close to the analytical expression.

(1)\[\delta \Pi = \int_V \boldsymbol{P} : \frac{ \partial \delta \boldsymbol{u} }{\partial \boldsymbol{X}} \ dV\]

δΠ = fem.IntegralForm(  # variation of total potential energy
    fun=P,
    v=field,  # container for the test field
    dV=region.dV,  # differential volume
    grad_v=[True],  # use gradient of test field
)
vector = δΠ.assemble()

For the linearization, see Eq. (2).

(2)\[\Delta \delta \Pi = \int_V \frac{ \partial \delta \boldsymbol{u} }{\partial \boldsymbol{X}} : \mathbb{A} : \frac{ \partial \Delta \boldsymbol{u} }{\partial \boldsymbol{X}} \ dV\]

ΔδΠ = fem.IntegralForm(  # linearization of total potential energy
    fun=A,
    v=field,  # container for the test field
    u=field,  # container for the trial field
    dV=region.dV,  # differential volume
    grad_v=[True],  # use gradient of test field
    grad_u=[True],  # use gradient of trial field
)
matrix = ΔδΠ.assemble()