Note
Go to the end to download the full example code.
Getting Started#
This tutorial covers the essential high-level parts of creating and solving problems
with FElupe. As an introductory example, a quarter model of a solid cube with
hyperelastic material behaviour is subjected to a uniaxial elongation applied at a
clamped end-face. First, let’s import FElupe and create a meshed cube out of hexahedron
cells with n points per axis. A numeric region, pre-defined for hexahedrons, is
created on the mesh. A vector-valued displacement field is initiated on the region.
Next, a field container is created on top of the displacement field.
import felupe as fem
mesh = fem.Cube(n=6)
region = fem.RegionHexahedron(mesh=mesh)
displacement = fem.Field(region=region, dim=3)
field = fem.FieldContainer(fields=[displacement])
A uniaxial load case is applied on the displacement field stored inside the field
container. This involves setting up symmetry planes as well as the absolute value of
the prescribed displacement at the mesh-points on the right-end face of the cube. The
right-end face is clamped: only displacements in direction x are allowed. The dict
of boundary conditions for this pre-defined load case are returned as boundaries
and the partitioned degrees of freedom as well as the external displacements are
stored within the returned dict loadcase.
boundaries, loadcase = fem.dof.uniaxial(field, move=0.2, right=1, clamped=True)
The material behaviour is defined through a built-in Neo-Hookean material formulation. The constitutive isotropic hyperelastic material formulation is applied on the displacement field by the definition of a solid body.
umat = fem.NeoHooke(mu=1.0, bulk=2.0)
solid = fem.SolidBody(umat=umat, field=field)
The problem is solved by an iterative Newton-Rhapson
procedure. A verbosity level of 2 enables a detailed text-based logging.
res = fem.newtonrhapson(items=[solid], verbose=2, **loadcase)
Newton-Rhapson solver
=====================
| # | norm(fun) | norm(dx) |
|---|-----------|-----------|
| 1 | 7.553e-02 | 1.898e+00 |
| 2 | 1.310e-03 | 5.091e-02 |
| 3 | 3.086e-07 | 6.698e-04 |
| 4 | 2.257e-14 | 1.527e-07 |
Converged in 4 iterations (Assembly: 0.07076 s, Solve: 0.02641 s).
Results may be viewed in an interactive window.
field.plot("Displacement", component=0).show()
Total running time of the script: (0 minutes 0.919 seconds)