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tamaas/tests/test_westergaard.py
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test_westergaard.py
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#!/usr/bin/env python
# coding: utf-8
# -----------------------------------------------------------------------------
# @author Lucas Frérot <lucas.frerot@epfl.ch>
#
# @section LICENSE
#
# Copyright (©) 2016 EPFL (Ecole Polytechnique Fédérale de
# Lausanne) Laboratory (LSMS - Laboratoire de Simulation en Mécanique des
# Solides)
#
# Tamaas is free software: you can redistribute it and/or modify it under the
# terms of the GNU Lesser General Public License as published by the Free
# Software Foundation, either version 3 of the License, or (at your option) any
# later version.
#
# Tamaas 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 Lesser General Public License for more
# details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with Tamaas. If not, see <http://www.gnu.org/licenses/>.
# -----------------------------------------------------------------------------
from __future__ import division, print_function
import sys
import numpy as np
from numpy.linalg import norm
from numpy import sin, cos, pi, sqrt, log
import tamaas as tm
def constructSinProfile(size, mode, amplitude):
x = np.linspace(0, 1, size, endpoint=False)
y = np.linspace(0, 1, size, endpoint=False)
x, y = np.meshgrid(x, y)
surface = amplitude * np.sin(2*np.pi*x*mode)
return surface.copy()
def test_westergaard():
tm.initialize()
grid_size = 2187 # Power of 3 for better accuracy
lamda = 1.
delta = 0.1
effective_modulus = 1.
p_star = np.pi * effective_modulus * delta / lamda # Full contact load
load = 0.1 * p_star
surface = constructSinProfile(grid_size, 1 / lamda, delta)
model = tm.ModelFactory.createModel(tm.model_type_basic_2d,
[1., 1.],
[grid_size, grid_size])
model.setElasticity(1, 0)
solver = tm.PolonskyKeerRey(model, surface, 1e-12,
tm.PolonskyKeerRey.pressure,
tm.PolonskyKeerRey.pressure)
solver.setDumpFrequency(1)
solver.solve(load)
tractions = model.getTraction()
displacements = model.getDisplacement()
# Testing contact area against Westergaard solution
contact_area = np.sum(tractions > 0) / grid_size**2
westergaard_contact_size = 2 * lamda / np.pi * np.arcsin(np.sqrt(load / p_star))
area_error = np.abs(contact_area - westergaard_contact_size / lamda) / \
(westergaard_contact_size / lamda)
print("Area error: {}".format(area_error))
# Testing agains pressure from Westergaard solution
x = np.linspace(0, 1, grid_size, endpoint=False)
y = np.linspace(0, 1, grid_size, endpoint=False)
x, y = np.meshgrid(x, y)
x -= 1 / 4
a = westergaard_contact_size / 2
pressure = np.zeros_like(tractions)
pressure = 2 * load * np.cos(np.pi*x/lamda)/np.sin(np.pi*a/lamda)**2 * \
np.sqrt(np.sin(np.pi*a/lamda)**2 - np.sin(np.pi*x/lamda)**2)
pressure[np.abs(x) > a] = 0
pressure_error = norm(pressure - tractions) / norm(pressure)
print("Pressure error: {}".format(pressure_error))
# Energetic error (displacement reference in Johnson)
displacements -= displacements.max()
psi = np.pi * np.abs(x) / lamda
psi_a = np.pi * a / lamda
disp_w = load * lamda / (pi * effective_modulus * sin(psi_a)) * (
cos(2*psi) + 2*sin(psi)*sqrt(sin(psi)**2 - sin(psi_a)**2)
- 2 * sin(psi_a)**2 * log((sin(psi) + sqrt(sin(psi)**2 - sin(psi_a)**2)) / sin(psi_a)))
disp_w[np.abs(x) < a] = load * lamda / (pi * effective_modulus * sin(psi_a)) * \
cos(2*psi[np.abs(x) < a])
disp_w -= disp_w.max()
energy_error = np.sum((tractions-pressure)*(displacements-disp_w)) / \
norm(pressure * disp_w)/grid_size**2
print("Energy error: {}".format(energy_error))
assert area_error < 1e-4 and pressure_error < 5e-4 and energy_error < 2e-8
tm.finalize()
if __name__ == "__main__":
test_westergaard()
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