Source code for sfepy.tests.test_poly_spaces

"""
Test continuity of polynomial basis and its gradients along an edge on
:math:`y` line (2D) or on a face in :math:`x`-:math:`y` plane (3D) between two
elements aligned with the coordinate system, stack one on top of the other. The
evaluation occurs in several points shifted by a very small amount from the
boundary between the elements into the top and the bottom element.

For H1 space, the basis should be continuous. The components of its gradient
parallel to the edge/face should be continuous as well, while the perpendicular
component should have the same absolute value, but different sign in the top
and the bottom element.

All connectivity permutations of the two elements are tested.

The serendipity basis implementation is a pure python proof-of-concept. Its
order in continuity tests is limited to 2 on 3_8 elements to decrease the tests
run time.
"""
from itertools import product
import numpy as nm
import pytest

from sfepy.base.base import assert_
import sfepy.base.testing as tst

rsels = {
    '2_3' : 'vertices in (y > -0.1) & (y < 0.1)',
    '2_4' : 'vertices in (y > 0.9) & (y < 1.1)',
    '3_4' : 'vertices in (z > -0.1) & (z < 0.1)',
    '3_8' : 'vertices in (z > 0.9) & (z < 1.1)',
}

eps = 1e-5

shifts = {
    '2_3' : nm.array([[0.0, 0.0], [0.0, eps]], dtype=nm.float64),
    '2_4' : nm.array([[0.0, 1.0], [0.0, eps]], dtype=nm.float64),
    '3_4' : nm.array([[0.0, 0.0, 0.0], [0.0, 0.0, eps]], dtype=nm.float64),
    '3_8' : nm.array([[0.0, 0.0, 1.0], [0.0, 0.0, eps]], dtype=nm.float64),
}

def _permute_quad_face(mesh0):
    from sfepy.discrete.fem import Mesh

    coors0, ngroups0, conns0, mat_ids0, desc0 = mesh0._get_io_data()
    im = nm.arange(4, 8)
    cperms = [[0, 1, 2, 3],
              [0, 2, 1, 3],
              [0, 1, 3, 2]]
    meshes = []
    for ip, cperm in enumerate(cperms):
        tr = nm.arange(12)
        tr[im] = tr[im[cperm]]

        coors = coors0.copy()
        coors[im] = coors0[im[cperm]]
        conn = conns0[0].copy()
        conn = tr[conn]

        mesh = Mesh.from_data(mesh0.name + str(ip), coors, ngroups0,
                              [conn], mat_ids0, desc0)
        meshes.append(mesh)

    return meshes

def _get_possible_oris(geom):
    import sfepy.discrete.fem.facets as facets

    if geom in ('2_3', '2_4'):
        oris = facets.ori_line_to_iter.keys()

    elif geom == '3_4':
        oris = facets.ori_triangle_to_iter.keys()

    elif geom == '3_8':
        oris = facets._quad_ori_groups.keys() # Not ori_square_to_iter!

    else:
        oris = []

    return set(oris)

def _gen_common_data(orders, gels):
    import sfepy
    from sfepy.base.base import Struct
    from sfepy.linalg import combine
    from sfepy.discrete import FieldVariable, Integral
    from sfepy.discrete.fem import Mesh, FEDomain, Field
    from sfepy.discrete.common.global_interp import get_ref_coors

    bases = ([ii for ii in combine([['2_4', '3_8'],
                                    ['lagrange', 'serendipity', 'bernstein',
                                     'lobatto']])]
             + [ii for ii in combine([['2_3', '3_4'],
                                      ['lagrange', 'bernstein']])])
    for geom, poly_space_base in bases:
        order = orders[geom]
        if (geom == '3_8') and (poly_space_base == 'serendipity'):
            order = 2

        tst.report('geometry: %s, base: %s, order: %d'
                   % (geom, poly_space_base, order))

        integral = Integral('i', order=order)

        aux = '' if geom in ['2_4', '3_8'] else 'z'
        mesh0 = Mesh.from_file('meshes/elements/%s_2%s.mesh' % (geom, aux),
                               prefix_dir=sfepy.data_dir)
        if (geom == '3_8'):
            meshes = _permute_quad_face(mesh0)

        else:
            meshes = [mesh0]

        gel = gels[geom]

        perms = gel.get_conn_permutations()

        qps, qp_weights = integral.get_qp(gel.surface_facet.name)
        zz = nm.zeros_like(qps[:, :1])
        qps = nm.hstack(([qps] + [zz]))

        shift = shifts[geom]
        rcoors = nm.ascontiguousarray(qps
                                      + shift[:1, :] - shift[1:, :])
        ccoors = nm.ascontiguousarray(qps
                                      + shift[:1, :] + shift[1:, :])

        all_oris = _get_possible_oris(geom)
        oris = set()

        for (ir, pr), (ic, pc), (im, mesh0) in product(
                enumerate(perms), enumerate(perms), enumerate(meshes),
        ):
            tst.report('im: %d, ir: %d, ic: %d' % (im, ir, ic))
            tst.report('pr: %s, pc: %s' % (pr, pc))

            mesh = mesh0.copy()
            conn = mesh.cmesh.get_conn(mesh0.cmesh.tdim, 0).indices
            conn = conn.reshape((mesh0.n_el, -1))
            conn[0, :] = conn[0, pr]
            conn[1, :] = conn[1, pc]

            conn2 = mesh.get_conn(gel.name)
            assert_((conn == conn2).all())

            cache = Struct(mesh=mesh)

            domain = FEDomain('domain', mesh)
            omega = domain.create_region('Omega', 'all')
            region = domain.create_region('Facet', rsels[geom], 'facet')
            field = Field.from_args('f', nm.float64, shape=1,
                                    region=omega, approx_order=order,
                                    poly_space_base=poly_space_base)
            fis = region.get_facet_indices()
            conn = mesh.cmesh.get_conn_as_graph(region.dim,
                                                region.dim - 1)
            _oris = mesh.cmesh.facet_oris[conn.indptr[fis[:, 0]]
                                          + fis[:, 1]]
            oris |= set(_oris)
            if oris == all_oris:
                break

            var = FieldVariable('u', 'unknown', field)
            tst.report('# dofs: %d' % var.n_dof)

            vec = nm.empty(var.n_dof, dtype=var.dtype)

            ps = field.poly_space

            dofs = field.get_dofs_in_region(region, merge=False)
            edofs, fdofs = nm.unique(dofs[1]), nm.unique(dofs[2])

            rrc, rcells, rstatus = get_ref_coors(field, rcoors,
                                                 cache=cache)
            crc, ccells, cstatus = get_ref_coors(field, ccoors,
                                                 cache=cache)
            assert_((rstatus == 0).all() and (cstatus == 0).all())

            yield (geom, poly_space_base, qp_weights, mesh, im, ir, ic,
                   field, ps, rrc, rcells[0], crc, ccells[0],
                   vec, edofs, fdofs)

[docs]@pytest.fixture(scope='module') def gels(): from sfepy.discrete.fem.geometry_element import GeometryElement gels = {} for key in ['2_3', '2_4', '3_4', '3_8']: gel = GeometryElement(key) gel.create_surface_facet() gels[key] = gel return gels
[docs]def test_partition_of_unity(gels): from sfepy.linalg import combine from sfepy.discrete import Integral, PolySpace ok = True orders = {'2_3' : 5, '2_4' : 5, '3_4' : 5, '3_8' : 5} bases = ( [ii for ii in combine([['2_4', '3_8'], ['lagrange', 'serendipity', 'bernstein']] )] + [ii for ii in combine([['2_3', '3_4'], ['lagrange', 'bernstein']])] ) for geom, poly_space_base in bases: max_order = orders[geom] for order in range(max_order + 1): if (poly_space_base == 'serendipity') and not (0 < order < 4): continue tst.report('geometry: %s, base: %s, order: %d' % (geom, poly_space_base, order)) integral = Integral('i', order=2 * order) coors, _ = integral.get_qp(geom) ps = PolySpace.any_from_args('ps', gels[geom], order, base=poly_space_base) vals = ps.eval_base(coors) _ok = nm.allclose(vals.sum(axis=-1), 1, atol=1e-14, rtol=0.0) tst.report('partition of unity:', _ok) ok = ok and _ok assert_(ok)
[docs]@pytest.mark.slow def test_continuity(gels): ok = True orders = {'2_3' : 3, '2_4' : 3, '3_4' : 4, '3_8' : 3} bads = [] bad_families = set() for (geom, poly_space_base, qp_weights, mesh, im, ir, ic, field, ps, rrc, rcell, crc, ccell, vec, edofs, fdofs) in _gen_common_data(orders, gels): if poly_space_base in ('lagrange', 'serendipity', 'bernstein'): rbf = ps.eval_base(rrc) cbf = ps.eval_base(crc) else: rbf = ps.eval_base(rrc, ori=field.ori[:1]) cbf = ps.eval_base(crc, ori=field.ori[1:]) dofs = nm.r_[edofs, fdofs] res = nm.zeros((2, dofs.shape[0]), dtype=nm.int32) res[0, :] = dofs for ii, ip in enumerate(dofs): vec.fill(0.0) vec[ip] = 1.0 evec = vec[field.econn] rvals = nm.dot(rbf, evec[rcell]) cvals = nm.dot(cbf, evec[ccell]) _ok = nm.allclose(rvals, cvals, atol=1e-14, rtol=0.0) res[1, ii] = _ok if not _ok: bads.append([geom, poly_space_base, im, ir, ic, ip]) bad_families.add((geom, poly_space_base)) ok = ok and _ok tst.report('results (dofs, status: 1 ok, 0 failure):\n%s' % res) if not ok: tst.report('continuity errors:\n', bads) tst.report('%d in total!' % len(bads)) tst.report('continuity errors occurred in these spaces:\n', bad_families) assert_(ok)
[docs]@pytest.mark.slow def test_gradients(gels): from sfepy.discrete.fem.mappings import VolumeMapping ok = True orders = {'2_3' : 3, '2_4' : 3, '3_4' : 4, '3_8' : 3} bads = [] bad_families = set() for (geom, poly_space_base, qp_weights, mesh, im, ir, ic, field, ps, rrc, rcell, crc, ccell, vec, edofs, fdofs) in _gen_common_data(orders, gels): gel = gels[geom] conn = mesh.get_conn(gel.name) geo_ps = field.gel.poly_space rmapping = VolumeMapping(mesh.coors, conn[rcell:rcell+1], poly_space=geo_ps) rori = field.ori[:1] if field.ori is not None else None rvg = rmapping.get_mapping(rrc, qp_weights, poly_space=ps, ori=rori) rbfg = rvg.bfg cmapping = VolumeMapping(mesh.coors, conn[ccell:ccell+1], poly_space=geo_ps) cori = field.ori[1:] if field.ori is not None else None cvg = cmapping.get_mapping(crc, qp_weights, poly_space=ps, ori=cori) cbfg = cvg.bfg dofs = nm.r_[edofs, fdofs] res = nm.zeros((2, dofs.shape[0]), dtype=nm.int32) res[0, :] = dofs for ii, ip in enumerate(dofs): vec.fill(0.0) vec[ip] = 1.0 evec = vec[field.econn] rvals = nm.dot(rbfg, evec[rcell])[0] cvals = nm.dot(cbfg, evec[ccell])[0] okx = nm.allclose(rvals[:, 0], cvals[:, 0], atol=1e-12, rtol=0.0) if gel.dim == 2: oky = nm.allclose(rvals[:, 1], -cvals[:, 1], atol=1e-12, rtol=0.0) _ok = okx and oky else: oky = nm.allclose(rvals[:, 1], cvals[:, 1], atol=1e-12, rtol=0.0) okz = nm.allclose(rvals[:, 2], -cvals[:, 2], atol=1e-12, rtol=0.0) _ok = okx and oky and okz res[1, ii] = _ok if not _ok: bads.append([geom, poly_space_base, im, ir, ic, ip]) bad_families.add((geom, poly_space_base)) ok = ok and _ok tst.report('results (dofs, status: 1 ok, 0 failure):\n%s' % res) if not ok: tst.report('gradient continuity errors:\n', bads) tst.report('%d in total!' % len(bads)) tst.report('gradient continuity errors occurred in these' ' spaces:\n', bad_families) assert_(ok)
[docs]def test_hessians(gels): """ Test the second partial derivatives of basis functions using finite differences. """ from sfepy.linalg import combine from sfepy.discrete import Integral, PolySpace ok = True orders = {'2_3' : 3, '2_4' : 3, '3_4' : 4, '3_8' : 3} bases = ([ii for ii in combine([['2_3', '2_4', '3_4', '3_8'], ['lagrange']])]) for geom, poly_space_base in bases: tst.report('geometry: %s, base: %s' % (geom, poly_space_base)) order = orders[geom] integral = Integral('i', order=order) coors, _ = integral.get_qp(geom) ps = PolySpace.any_from_args('ps', gels[geom], order, base=poly_space_base) dim = coors.shape[1] h1 = nm.zeros((coors.shape[0], dim, dim, ps.n_nod), nm.float64) eps = 1e-8 for ir in range(dim): cc = coors.copy() cc[:, ir] -= eps aux0 = ps.eval_base(cc, diff=1) cc[:, ir] += 2 * eps aux1 = ps.eval_base(cc, diff=1) h1[:, :, ir, :] = 0.5 * (aux1 - aux0) / eps h2 = ps.eval_base(coors, diff=2) _ok = nm.allclose(h1, h2, rtol=0, atol=50*eps) tst.report('hessians: error: %.2e ok: %s' % (nm.abs(h1 - h2).max(), _ok)) ok = ok and _ok assert_(ok)