Source code for sfepy.scripts.plot_condition_numbers

#!/usr/bin/env python
"""
Plot conditions numbers w.r.t. polynomial approximation order of reference
element matrices for various FE polynomial spaces (bases).
"""
from __future__ import absolute_import
import sys
sys.path.append('.')
from argparse import ArgumentParser
import os.path as op
from functools import partial
import numpy as nm
import matplotlib.pyplot as plt

from sfepy import data_dir
from sfepy.base.base import output, assert_
from sfepy.base.timing import Timer
from sfepy.discrete import FieldVariable, Material, Integral
from sfepy.discrete.fem import Mesh, FEDomain, Field
from sfepy.terms import Term
from sfepy.solvers import eig
from sfepy.mechanics.matcoefs import stiffness_from_lame

helps = {
    'basis' :
    'name of the FE basis [default: %(default)s]',
    'max_order' :
    'maximum order of polynomials [default: %(default)s]',
    'matrix_type' :
    'matrix type [default: %(default)s]',
    'geometry' :
    'reference element geometry, one of "2_3", "2_4", "3_4", "3_8"'
    ' [default: %(default)s]',
    'output_dir' :
    'output directory',
    'no_show' :
    'do not show the figures',
}

[docs]def main(): parser = ArgumentParser(description=__doc__) parser.add_argument('--version', action='version', version='%(prog)s') parser.add_argument('-b', '--basis', metavar='name', action='store', dest='basis', default='lagrange', help=helps['basis']) parser.add_argument('-n', '--max-order', metavar='order', type=int, action='store', dest='max_order', default=10, help=helps['max_order']) parser.add_argument('-m', '--matrix', action='store', dest='matrix_type', choices=['laplace', 'elasticity', 'smass', 'vmass'], default='laplace', help=helps['matrix_type']) parser.add_argument('-g', '--geometry', metavar='name', action='store', dest='geometry', default='2_4', help=helps['geometry']) parser.add_argument('-o', '--output-dir', metavar='path', action='store', dest='output_dir', default=None, help=helps['output_dir']) parser.add_argument('--no-show', action='store_false', dest='show', default=True, help=helps['no_show']) options = parser.parse_args() dim, n_ep = int(options.geometry[0]), int(options.geometry[2]) output('reference element geometry:') output(' dimension: %d, vertices: %d' % (dim, n_ep)) n_c = {'laplace' : 1, 'elasticity' : dim, 'smass' : 1, 'vmass' : dim}[options.matrix_type] output('matrix type:', options.matrix_type) output('number of variable components:', n_c) output('polynomial space:', options.basis) output('max. order:', options.max_order) mesh = Mesh.from_file(data_dir + '/meshes/elements/%s_1.mesh' % options.geometry) domain = FEDomain('domain', mesh) omega = domain.create_region('Omega', 'all') orders = nm.arange(1, options.max_order + 1, dtype=nm.int32) conds = [] for order in orders: output('order:', order, '...') field = Field.from_args('fu', nm.float64, n_c, omega, approx_order=order, space='H1', poly_space_base=options.basis) quad_order = 2 * field.approx_order output('quadrature order:', quad_order) integral = Integral('i', order=quad_order) qp, _ = integral.get_qp(options.geometry) output('number of quadrature points:', qp.shape[0]) u = FieldVariable('u', 'unknown', field) v = FieldVariable('v', 'test', field, primary_var_name='u') m = Material('m', D=stiffness_from_lame(dim, 1.0, 1.0)) if options.matrix_type == 'laplace': term = Term.new('dw_laplace(v, u)', integral, omega, v=v, u=u) n_zero = 1 elif options.matrix_type == 'elasticity': term = Term.new('dw_lin_elastic(m.D, v, u)', integral, omega, m=m, v=v, u=u) n_zero = (dim + 1) * dim // 2 elif options.matrix_type in ('smass', 'vmass'): term = Term.new('dw_dot(v, u)', integral, omega, v=v, u=u) n_zero = 0 term.setup() output('assembling...') timer = Timer(start=True) mtx, iels = term.evaluate(mode='weak', diff_var='u') output('...done in %.2f s' % timer.stop()) mtx = mtx[0, 0] try: assert_(nm.max(nm.abs(mtx - mtx.T)) < 1e-10) except: from sfepy.base.base import debug; debug() output('matrix shape:', mtx.shape) eigs = eig(mtx, method='eig.sgscipy', eigenvectors=False) eigs.sort() # Zero 'true' zeros. eigs[:n_zero] = 0.0 ii = nm.where(eigs < 0.0)[0] if len(ii): output('matrix is not positive semi-definite!') ii = nm.where(eigs[n_zero:] < 1e-12)[0] if len(ii): output('matrix has more than %d zero eigenvalues!' % n_zero) output('smallest eigs:\n', eigs[:10]) ii = nm.where(eigs > 0.0)[0] emin, emax = eigs[ii[[0, -1]]] output('min:', emin, 'max:', emax) cond = emax / emin conds.append(cond) output('condition number:', cond) output('...done') if options.output_dir is not None: indir = partial(op.join, options.output_dir) else: indir = None plt.rcParams['font.size'] = 12 plt.rcParams['lines.linewidth'] = 3 fig, ax = plt.subplots() ax.semilogy(orders, conds) ax.set_xticks(orders) ax.set_xticklabels(orders) ax.set_xlabel('polynomial order') ax.set_ylabel('condition number') ax.set_title(f'{options.basis.capitalize()} basis') ax.grid() plt.tight_layout() if indir is not None: fig.savefig(indir(f'{options.basis}-{options.matrix_type}-' f'{options.geometry}-{options.max_order}-xlin.png'), bbox_inches='tight') fig, ax = plt.subplots() ax.loglog(orders, conds) ax.set_xticks(orders) ax.set_xticklabels(orders) ax.set_xlabel('polynomial order') ax.set_ylabel('condition number') ax.set_title(f'{options.basis.capitalize()} basis') ax.grid() plt.tight_layout() if indir is not None: fig.savefig(indir(f'{options.basis}-{options.matrix_type}-' f'{options.geometry}-{options.max_order}-xlog.png'), bbox_inches='tight') if options.show: plt.show()
if __name__ == '__main__': main()