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structural_mechanics_softening.py
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Sun, Nov 3, 16:35

structural_mechanics_softening.py
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#!/usr/bin/env python
# coding: utf-8
""" structural_mechanics_softening.py: Structural mechanics with softening"""
__author__ = "Nicolas Richart"
__credits__ = [
"Nicolas Richart <nicolas.richart@epfl.ch>",
]
__copyright__ = "Copyright (©) 2018-2021 EPFL (Ecole Polytechnique Fédérale" \
" de Lausanne) Laboratory (LSMS - Laboratoire de Simulation" \
" en Mécanique des Solides)"
__license__ = "LGPLv3"
# # Test of Structural Mechanics
# In this test there is a beam consisting of three parts, all have the same materials.
# The left most node is fixed.
# On the right most node a force is applied in x direction.
#
# After a certain time, the material of the middle _element_ is waekened, lower Young's modulus.
# In each step the modulus is lowered by a coinstant factor.
import akantu as aka
import numpy
import numpy as np
try:
import matplotlib.pyplot as plt
has_matplotlib = True
except ImportError:
has_matplotlib = False
# ### Creating the Mesh
# Create a mesh for the two dimensional case
beam = aka.Mesh(2)
# We now create the connectivity array for the beam.
beam.addConnectivityType(aka._bernoulli_beam_2)
# We need a `MeshAccessor` in order to change the size of the mesh entities.
beamAcc = aka.MeshAccessor(beam)
# Now we create the array to store the nodes and the connectivities and give them their size.
beamAcc.resizeConnectivity(3, aka._bernoulli_beam_2)
beamAcc.resizeNodes(4)
# #### Setting the Nodes
Nodes = beam.getNodes()
Nodes[0, :] = [0., 0.]
Nodes[1, :] = [1., 0.]
Nodes[2, :] = [2., 0.]
Nodes[3, :] = [3., 0.]
# Setting the Connections
Conn = beam.getConnectivity(aka._bernoulli_beam_2)
Conn[0, :] = [0, 1]
Conn[1, :] = [1, 2]
Conn[2, :] = [2, 3]
# Ready
# We have to make the mesh ready.
beamAcc.makeReady()
# Creating the Model
model = aka.StructuralMechanicsModel(beam)
# Setting up the Modell
# Creating and Inserting the Materials
mat1 = aka.StructuralMaterial()
mat1.E = 1e9
mat1.rho = 1.
mat1.I = 1. # noqa: E741
mat1.A = 1.
mat1.GJ = 1.
mat1ID = model.addMaterial(mat1, 'mat1')
mat2 = aka.StructuralMaterial()
mat2.E = 1e9
mat2.rho = 1.
mat2.I = 1. # noqa: E741
mat2.A = 1.
mat2.GJ = 1.
mat2ID = model.addMaterial(mat2, 'mat2')
mat3 = aka.StructuralMaterial()
mat3.E = mat2.E / 100000
mat3.rho = 1.
mat3.I = 1. # noqa: E741
mat3.A = mat2.A / 100
mat3.GJ = 1.
mat3ID = model.addMaterial(mat3, 'mat3')
# ##### Initializing the Model
model.initFull(aka.AnalysisMethod._implicit_dynamic)
# ##### Assigning the Materials
materials = model.getElementMaterial(aka._bernoulli_beam_2)
materials[0][0] = mat1ID
materials[1][0] = mat2ID
materials[2][0] = mat1ID
# ##### Setting Boundaries
# Neumann
# Apply a force of `10` at the last (right most) node.
forces = model.getExternalForce()
forces[:] = 0
forces[2, 0] = 100.
# Dirichlets
# Block all dofs of the first node, since it is fixed.
# All other nodes have no restrictions
boundary = model.getBlockedDOFs()
boundary[0, :] = True
boundary[1, :] = False
boundary[2, :] = False
boundary[3, :] = False
# ### Solving the System
# Set up the system
deltaT = 1e-9
model.setTimeStep(deltaT)
solver = model.getNonLinearSolver()
solver.set("max_iterations", 100)
solver.set("threshold", 1e-8)
solver.set("convergence_type", aka.SolveConvergenceCriteria.solution)
# Perform N time steps.
# At each step records the displacement of all three nodes in x direction.
N = 10000 * 60
disp0 = np.zeros(N)
disp1 = np.zeros(N)
disp2 = np.zeros(N)
disp3 = np.zeros(N)
times = np.zeros(N)
switchT = None
switchEnd = None
softDuration = 1000
SoftStart = (N // 2) - softDuration // 2
SoftEnd = SoftStart + softDuration
if(softDuration > 0):
softFactor = (model.getMaterial('mat3').E
/ model.getMaterial('mat2').E) ** (1.0 / softDuration)
mat2 = model.getMaterial('mat2')
for i in range(N):
times[i] = deltaT * i
if((SoftStart <= i <= SoftEnd) and (softDuration > 0)):
if switchT is None:
switchT = times[i]
elif(i == SoftEnd):
switchEnd = times[i]
#
mat2.E *= softFactor
#
model.solveStep()
disp = model.getDisplacement()
disp0[i] = disp[0, 0]
disp1[i] = disp[1, 0]
disp2[i] = disp[2, 0]
disp3[i] = disp[3, 0]
disps = [disp0, disp1, disp2, disp3]
maxMin = [-1.0, 1.0]
for d in disps:
maxMin[0] = max(np.max(d), maxMin[0])
maxMin[1] = min(np.min(d), maxMin[1])
if has_matplotlib:
# plt.plot(disp0, times, color='k', label = "left node (fix)")
plt.plot(disp1, times, color='g', label="middle, left node")
plt.plot(disp2, times, color='g', linestyle='--',
label="middle, right node")
plt.plot(disp3, times, color='b', label="right node")
if(softDuration > 0):
plt.plot((maxMin[1], maxMin[0]), (switchT, switchT),)
plt.plot((maxMin[1], maxMin[0]), (switchEnd, switchEnd), )
plt.title("Displacement in $x$ of the nodes")
plt.ylabel("Time [S]")
plt.xlabel("displacement [m]")
plt.xlim((maxMin[1] * 1.3, maxMin[0] * 1.1))
plt.legend()
plt.show()
# If the softening is disabled, then the displacement looks wierd.
# Because the displacement first increases and then decreases.
# In this case `softDuration > 0` holds.
#
# However if the softening is enabled, it looks rather good. The left middle
# node will start to vibrate, because it is not pulled in the other direction.

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