Displaying eigenmodes of vibration using warp_by_vector

This example applies the warp_by_vector filter to a cube whose eigenmodes have been computed using the Ritz method, as outlined in Visscher, William M., Albert Migliori, Thomas M. Bell, et Robert A. Reinert. “On the normal modes of free vibration of inhomogeneous and anisotropic elastic objects”. The Journal of the Acoustical Society of America 90, n.4 (october 1991): 2154-62. https://asa.scitation.org/doi/10.1121/1.401643

First, let’s solve the eigenvalue problem for a vibrating cube. We use a crude approximation (by choosing a low max polynomial order) to get a fast computation.

import numpy as np
from scipy.linalg import eigh
import pyvista as pv


def analytical_integral_rppd(p, q, r, a, b, c):
    """Returns the analytical value of the RPPD integral, i.e. the integral
    of x**p * y**q * z**r for (x, -a, a), (y, -b, b), (z, -c, c)."""
    if p < 0:
        return 0.
    elif q < 0:
        return 0.
    elif r < 0.:
        return 0.
    else:
        return a ** (p + 1) * b ** (q + 1) * c ** (r + 1) * \
               ((-1) ** p + 1) * ((-1) ** q + 1) * ((-1) ** r + 1) \
               / ((p + 1) * (q + 1) * (r + 1))


def make_cijkl_E_nu(E=200, nu=0.3):
    """Makes cijkl from E and nu.
    Default values for steel are: E=200 GPa, nu=0.3."""
    lambd = E * nu / (1 + nu) / (1 - 2 * nu)
    mu = E / 2 / (1 + nu)
    cij = np.zeros((6, 6))
    cij[(0, 1, 2), (0, 1, 2)] = lambd + 2 * mu
    cij[(0, 0, 1, 1, 2, 2), (1, 2, 0, 2, 0, 1)] = lambd
    cij[(3, 4, 5), (3, 4, 5)] = mu
    # check symmetry
    assert np.allclose(cij, cij.T)
    # convert to order 4 tensor
    coord_mapping = {(1, 1): 1,
                     (2, 2): 2,
                     (3, 3): 3,
                     (2, 3): 4,
                     (1, 3): 5,
                     (1, 2): 6,
                     (2, 1): 6,
                     (3, 1): 5,
                     (3, 2): 4}

    cijkl = np.zeros((3, 3, 3, 3))
    for i in range(3):
        for j in range(3):
            for k in range(3):
                for l in range(3):
                    u = coord_mapping[(i + 1, j + 1)]
                    v = coord_mapping[(k + 1, l + 1)]
                    cijkl[i, j, k, l] = cij[u - 1, v - 1]
    return cijkl, cij


def get_first_N_above_thresh(N, freqs, thresh, decimals=3):
    """Returns first N unique frequencies with amplitude above threshold based
    on first decimals."""
    unique_freqs, unique_indices = np.unique(
        np.round(freqs, decimals=decimals), return_index=True)
    nonzero = unique_freqs > thresh
    unique_freqs, unique_indices = unique_freqs[nonzero], unique_indices[
        nonzero]
    return unique_freqs[:N], unique_indices[:N]


def assemble_mass_and_stiffness(N, F, geom_params, cijkl):
    """This routine assembles the mass and stiffness matrix.
    It first builds an index of basis functions as a quadruplet of
    component and polynomial order for (x^p, y^q, z^r) of maximum order N.

    This routine only builds the symmetric part of the matrix to speed
    things up.
    """
    # building coordinates
    triplets = []
    for p in range(N + 1):
        for q in range(N - p + 1):
            for r in range(N - p - q + 1):
                triplets.append((p, q, r))
    assert len(triplets) == (N + 1) * (N + 2) * (N + 3) // 6

    quadruplets = []
    for i in range(3):
        for triplet in triplets:
            quadruplets.append((i, *triplet))
    assert len(quadruplets) == 3 * (N + 1) * (N + 2) * (N + 3) // 6

    # assembling the mass and stiffness matrix in a single loop
    R = len(triplets)
    E = np.zeros((3 * R, 3 * R))  # the mass matrix
    G = np.zeros((3 * R, 3 * R))  # the stiffness matrix
    for index1, quad1 in enumerate(quadruplets):
        I, p1, q1, r1 = quad1
        for index2, quad2 in enumerate(quadruplets[index1:]):
            index2 = index2 + index1
            J, p2, q2, r2 = quad2
            G[index1, index2] = cijkl[I, 1 - 1, J, 1 - 1] * p1 * p2 * F(
                p1 + p2 - 2, q1 + q2, r1 + r2, **geom_params) + \
                                cijkl[I, 1 - 1, J, 2 - 1] * p1 * q2 * F(
                p1 + p2 - 1, q1 + q2 - 1, r1 + r2,
                **geom_params) + \
                                cijkl[I, 1 - 1, J, 3 - 1] * p1 * r2 * F(
                p1 + p2 - 1, q1 + q2, r1 + r2 - 1,
                **geom_params) + \
                                cijkl[I, 2 - 1, J, 1 - 1] * q1 * p2 * F(
                p1 + p2 - 1, q1 + q2 - 1, r1 + r2,
                **geom_params) + \
                                cijkl[I, 2 - 1, J, 2 - 1] * q1 * q2 * F(
                p1 + p2, q1 + q2 - 2, r1 + r2, **geom_params) + \
                                cijkl[I, 2 - 1, J, 3 - 1] * q1 * r2 * F(
                p1 + p2, q1 + q2 - 1, r1 + r2 - 1,
                **geom_params) + \
                                cijkl[I, 3 - 1, J, 1 - 1] * r1 * p2 * F(
                p1 + p2 - 1, q1 + q2, r1 + r2 - 1,
                **geom_params) + \
                                cijkl[I, 3 - 1, J, 2 - 1] * r1 * q2 * F(
                p1 + p2, q1 + q2 - 1, r1 + r2 - 1,
                **geom_params) + \
                                cijkl[I, 3 - 1, J, 3 - 1] * r1 * r2 * F(
                p1 + p2, q1 + q2, r1 + r2 - 2, **geom_params)
            G[index2, index1] = G[
                index1, index2]  # since stiffness matrix is symmetric
            if I == J:
                E[index1, index2] = F(p1 + p2, q1 + q2, r1 + r2, **geom_params)
                E[index2, index1] = E[
                    index1, index2]  # since mass matrix is symmetric
    return E, G, quadruplets


N = 8  # maximum order of x^p y^q z^r polynomials
rho = 8.0  # g/cm^3
l1, l2, l3 = .2, .2, .2  # all in cm
geometry_parameters = {'a': l1 / 2., 'b': l2 / 2., 'c': l3 / 2.}
cijkl, cij = make_cijkl_E_nu(200, 0.3)  # Gpa, without unit
E, G, quadruplets = assemble_mass_and_stiffness(N, analytical_integral_rppd,
                                                geometry_parameters, cijkl)

# solving the eigenvalue problem using symmetric solver
w, vr = eigh(a=G, b=E)
omegas = np.sqrt(np.abs(w) / rho) * 1e5  # convert back to Hz
freqs = omegas / (2 * np.pi)
# expected values from (Bernard 2014, p.14),
# error depends on polynomial order ``N``
expected_freqs_kHz = np.array(
    [704.8, 949., 965.2, 1096.3, 1128.4, 1182.8, 1338.9, 1360.9])
computed_freqs_kHz, mode_indices = get_first_N_above_thresh(8, freqs / 1e3,
                                                            thresh=1,
                                                            decimals=1)
print('found the following first unique eigenfrequencies:')
for ind, (freq1, freq2) in enumerate(
        zip(computed_freqs_kHz, expected_freqs_kHz)):
    error = np.abs(freq2 - freq1) / freq1 * 100.
    print(
        f"freq. {ind + 1:1}: {freq1:8.1f} kHz," + \
        f" expected: {freq2:8.1f} kHz, error: {error:.2f} %")

Out:

found the following first unique eigenfrequencies:
freq. 1:    705.1 kHz, expected:    704.8 kHz, error: 0.04 %
freq. 2:    949.1 kHz, expected:    949.0 kHz, error: 0.01 %
freq. 3:    965.7 kHz, expected:    965.2 kHz, error: 0.05 %
freq. 4:   1096.3 kHz, expected:   1096.3 kHz, error: 0.00 %
freq. 5:   1128.6 kHz, expected:   1128.4 kHz, error: 0.02 %
freq. 6:   1183.9 kHz, expected:   1182.8 kHz, error: 0.09 %
freq. 7:   1339.0 kHz, expected:   1338.9 kHz, error: 0.01 %
freq. 8:   1361.8 kHz, expected:   1360.9 kHz, error: 0.07 %

Now, let’s display a mode on a mesh of the cube.

# Create the 3D NumPy array of spatially referenced data
#   (nx by ny by nz)
nx, ny, nz = 30, 31, 32

x = np.linspace(-l1 / 2., l1 / 2., nx)
y = np.linspace(-l2 / 2., l2 / 2., ny)
x, y = np.meshgrid(x, y)
z = np.zeros_like(x) + l3 / 2.
grid = pv.StructuredGrid(x, y, z)

slices = []
for zz in np.linspace(-l3 / 2., l3 / 2., nz)[::-1]:
    slice = grid.points.copy()
    slice[:, -1] = zz
    slices.append(slice)

vol = pv.StructuredGrid()
vol.points = np.vstack(slices)
vol.dimensions = [*grid.dimensions[0:2], nz]

for i, mode_index in enumerate(mode_indices):
    eigenvector = vr[:, mode_index]
    displacement_points = np.zeros_like(vol.points)
    for weight, (component, p, q, r) in zip(eigenvector, quadruplets):
        displacement_points[:, component] += weight * vol.points[:, 0] ** p * \
                                             vol.points[:, 1] ** q * \
                                             vol.points[:, 2] ** r
    if displacement_points.max() > 0.:
        displacement_points /= displacement_points.max()
    vol[f'eigenmode_{i:02}'] = displacement_points

warpby = 'eigenmode_00'
warped = vol.warp_by_vector(warpby, factor=0.04)
warped.translate([-1.5 * l1, 0., 0.])
p = pv.Plotter()
p.add_mesh(vol, style='wireframe', scalars=warpby)
p.add_mesh(warped, scalars=warpby)
p.show()
warp by vector eigenmodes

Finally, let’s make a gallery of the first 8 unique eigenmodes.

p = pv.Plotter(shape=(2, 4))
for i in range(2):
    for j in range(4):
        p.subplot(i, j)
        current_index = 4 * i + j
        vector = f"eigenmode_{current_index:02}"
        p.add_text(
            f"mode {current_index}," + \
            f" freq. {computed_freqs_kHz[current_index]:.1f} kHz",
            font_size=10)
        p.add_mesh(vol.warp_by_vector(vector, factor=0.03), scalars=vector)
p.show()
warp by vector eigenmodes

Total running time of the script: ( 0 minutes 8.017 seconds)

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