felupe 0.0.4

Finite Element Analysis

FElupe - Finite Element Analysis

FElupe is an open-source finite element package focussing on the formulation and numerical solution of nonlinear problems in continuum mechanics of solid bodies. Its name is a combination of FE (finite element) and the german word Lupe (magnifying glass) as a synonym for getting a little insight how a finite element analysis code looks like under the hood. The user code for defining the integral form of equilibrium equations as well as their linearizations over a region are kept as close as possible to the analytical expressions. FElupe is both written in Python and fast in execution times thanks to NumPy (and optional Numba). No complicated installation, just pure Python. Another key feature is the easy and straightforward definition of mixed field formulations for nearly incompressible material behavior. Several useful utilities are available, i.e. an incremental approach for the application of boundary conditions and subsequent solution of the nonlinear equilibrium equations or the calculation of forces and moments on boundaries. Finally, results are ready to be exported to VTK or XDMF files using meshio.

Question: Okay, why not use existing projects like Fenics or scikit-fem? Fenics is great but way too complicated to install on Windows. Scikit-fem is close to perfection but slow (too slow) for my problems of interest in combination with hyperelastic material formulations. In fact the utilities of FElupe could also be wrapped around the core code of scikit-fem but in the end I decided to create a code base which works the way I think it should.

Installation

Install Python, fire up a terminal an run pip install felupe; import FElupe as follows in your script.

import felupe as fe

Getting started

Start setting up a problem in FElupe by the creation of a Region with a geometry (Mesh), a finite Element and a Quadrature rule.

Region

A region essentially pre-calculates basis functions and derivatives evaluated at every quadrature point of every element. An array containing products of quadrature weights multiplied by the geometrical jacobians is stored as the differential volume.

mesh = fe.mesh.Cube(n=9)
element = fe.element.Hex1()
quadrature = fe.quadrature.Linear(dim=3)

region = fe.Region(mesh, element, quadrature)

dV = region.dV
V = dV.sum()

Field

In a second step fields may be added to the Region. These may be either scalar or vector-valued fields. The nodal values are obtained with the attribute values. Interpolated field values at quadrature points are calculated with the interpolate() method.

displacement = fe.Field(region, dim=3)

u  = displacement.values
ui = displacement.interpolate()

Additionally, the displacement gradient w.r.t. the undeformed coordinates is calculated for every quadrature point of every element in the region with the field method grad(). interpolate and grad methods are also callable from functions of the math module. The deformation gradient is obtained by a sum of the identity and the displacement gradient.

dudX = displacement.grad()

# use math function for better readability
from felupe.math import grad, identity

dudX = grad(displacement)

F = identity(dudX) + dudX

Constitution

The material behavior has to be provided by the first Piola-Kirchhoff stress tensor as a function of the deformation gradient. FElupe provides a constitution library which contains a definition for the Neo-Hookean material model and the associated fourth-order elasticity tensor.

$$\boldsymbol{P} = \mu J^{-2/3} \left(\boldsymbol{F} - \frac{\boldsymbol{F} : \boldsymbol{F}}{3} \boldsymbol{F}^{-T} \right) + K (J-1) J \boldsymbol{F}^{-T} $$

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

P = umat.f_u(F)
A = umat.A_uu(F)

Boundary Conditions

Next we introduce boundary conditions on the displacement field. Boundary conditions are stored inside a dictionary of multiple boundary instances. First, we fix the left surface of the cube. Displacements on the right surface are fixed in directions y and z whereas displacements in direction x are prescribed with a value. A boundary instance hold useful attributes like nodes or dof.

import numpy as np

f0 = lambda x: np.isclose(x, 0)
f1 = lambda x: np.isclose(x, 1)

boundaries = {}
boundaries["left"]  = fe.Boundary(displacement, fx=f0)
boundaries["right"] = fe.Boundary(displacement, fx=f1, skip=(1,0,0))
boundaries["move"]  = fe.Boundary(displacement, fx=f1, skip=(0,1,1), value=0.5)

Partition of deegrees of freedom

The separation of active and inactive degrees of freedom is performed by a so-called partition. External values of prescribed displacement degrees of freedom are obtained by the application of the boundary values to the displacement field.

dof0, dof1, _ = fe.doftools.partition(displacement, boundaries)
u0ext = fe.doftools.apply(displacement, boundaries, dof0)

Integral forms of equilibrium equations

The integral (or weak) forms of equilibrium equations are defined by the IntegralForm class. The function of interest has to be passed as the fun argument whereas the virtual field as the v argument. Setting grad_v=True FElupe passes the gradient of the virtual field to the integral form. FElupe assumes a linear form if u=None (default) or create a bilinear form if a field is passed to the argument u.

$\int_V P_i^{\ J} : \frac{\partial \delta u^i}{\partial X^J} \ dV \qquad$ and $\qquad \int_V \frac{\partial \delta u^i}{\partial X^J} : \mathbb{A}_{i\ k\ }^{\ J\ L} : \frac{\partial u^k}{\partial X^L} \ dV$

linearform   = fe.IntegralForm(P, displacement, dV, grad_v=True)
bilinearform = fe.IntegralForm(A, displacement, dV, displacement, grad_v = True, grad_u=True)

An assembly of the forms lead to the (nodal) internal forces and the (sparse) stiffness matrix.

r = linearform.assemble(parallel=True).toarray()[:,0]
K = bilinearform.assemble(parallel=True)

Prepare (partition) and solve the linearized equation system

In order to solve the linearized equation system a partition into active and inactive degrees of freedom has to be performed. This system may then be passed to the sparse direct solver. Given a set of nonlinear equilibrium equations $\boldsymbol{g}$ the unknowns $\boldsymbol{u}$ are found by linearization at a valid starting point and an iterative Newton-Rhapson solution prodecure. The incremental values of inactive degrees of freedom are given as the difference of external prescribed and current values of unknowns. The (linear) solution is equal to the first result of a Newton-Rhapson iterative solution procedure. The resulting nodal values du are finally added to the displacement field.

$\boldsymbol{g}_1(\boldsymbol{u}) = -\boldsymbol{r}_1(\boldsymbol{u}) + \boldsymbol{f}_1$

$\boldsymbol{g}_1(\boldsymbol{u} + d\boldsymbol{u}) \approx -\boldsymbol{r}_1 + \boldsymbol{f}_1 - \frac{\partial \boldsymbol{r}_1}{\partial \boldsymbol{u}_1} \ d\boldsymbol{u}_1 - \frac{\partial \boldsymbol{r}_1}{\partial \boldsymbol{u}_0} \ d\boldsymbol{u}_0 = \boldsymbol{0}$

$d\boldsymbol{u}_0 = \boldsymbol{u}_0^{(ext)} - \boldsymbol{u}_0$

solve $\boldsymbol{K}_{11}\ d\boldsymbol{u}1 = \boldsymbol{g}1 - \boldsymbol{K}{10}\ d\boldsymbol{u}{0}$

$\boldsymbol{u}_0 += d\boldsymbol{u}_0$

$\boldsymbol{u}_1 += d\boldsymbol{u}_1$

system = fe.solve.partition(displacement, K, dof1, dof0, r)
du = fe.solve.solve(*system, u0ext).reshape(*u.shape)
# displacement += du

A very simple newton-rhapson code looks like this:

for iteration in range(8):
    dudX = grad(displacement)
    F = identity(dudX) + dudX
    P = umat.f_u(F)
    A = umat.A_uu(F)

    r = fe.IntegralForm(P, displacement, dV, grad_v=True).assemble().toarray()[:,0]
    K = fe.IntegralForm(A, displacement, dV, displacement, True, True).assemble()

    system = fe.solve.partition(displacement, K, dof1, dof0, r)
    du = fe.solve.solve(*system, u0ext).reshape(*u.shape)

    norm = np.linalg.norm(du)
    print(iteration, norm)
    displacement += du

    if norm < 1e-12:
        break

0 8.174180680860697
1 0.2940958778404002
2 0.02083230945148837
3 0.00010289925344210932
4 6.0171532522337204e-09
5 5.490863535572125e-16

Export of results

Results can be exported as VTK or XDMF files using meshio.

fe.utils.save(region, displacement, filename="result")

Utilities

The above code snippets and explanations cover only the essential parts of FElupe. Several utilities are available - please have a look at the scripts folder or the source code itself.

License

FElupe - finite element analysis (C) 2021 Andreas Dutzler, Graz (Austria).

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses/.