.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/gallery_examples/01-apdlmath-examples/eigen_solve.py"
.. LINE NUMBERS ARE GIVEN BELOW.
.. only:: html
.. note::
:class: sphx-glr-download-link-note
:ref:`Go to the end <sphx_glr_download_examples_gallery_examples_01-apdlmath-examples_eigen_solve.py>`
to download the full example code
.. rst-class:: sphx-glr-example-title
.. _sphx_glr_examples_gallery_examples_01-apdlmath-examples_eigen_solve.py:
.. _ref_mapdl_math_eigen_solve:
Using APDLMath to solve Eigenproblems
-------------------------------------
Use APDLMath to solve eigenproblems.
This example uses a verification manual input file, but you can use
your own sparse or dense matrices and solve those.
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.. code-block:: default
import time
import matplotlib.pylab as plt
import numpy as np
from ansys.mapdl.core import launch_mapdl
from ansys.mapdl.core.examples import vmfiles
# Start MAPDL as a service and create an APDLMath object
mapdl = launch_mapdl(loglevel="ERROR")
mm = mapdl.math
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First we get the `STIFF` and `MASS` matrices from the full file
after running the input file from Verification Manual 153
.. GENERATED FROM PYTHON SOURCE LINES 30-36
.. code-block:: default
out = mapdl.input(vmfiles["vm153"])
k = mm.stiff(fname="PRSMEMB.full")
m = mm.mass(fname="PRSMEMB.full")
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Display size of the M and K matrices
.. GENERATED FROM PYTHON SOURCE LINES 38-41
.. code-block:: default
print(m.shape)
print(k.shape)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
(126, 126)
(126, 126)
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Allocate an array to store the eigenshapes.
where `nev` is the number of eigenvalues requested
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.. code-block:: default
nev = 10
a = mm.mat(k.nrow, nev)
a
.. rst-class:: sphx-glr-script-out
.. code-block:: none
Dense APDLMath Matrix (126, 10)
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Perform the the modal analysis.
The algorithm is automatically chosen with respect to the matrices
properties (e.g. scalar, storage, symmetry...)
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.. code-block:: default
print("Calling MAPDL to solve the eigenproblem...")
t1 = time.time()
ev = mm.eigs(nev, k, m, phi=a)
print(f"Elapsed time to solve this problem: {time.time() - t1}")
.. rst-class:: sphx-glr-script-out
.. code-block:: none
Calling MAPDL to solve the eigenproblem...
Elapsed time to solve this problem: 0.028902530670166016
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This is the vector of eigenfrequencies.
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.. code-block:: default
print(ev)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
TGOEAE :
Size : 10
3.381e+02 3.381e+02 6.266e+02 6.266e+02 9.283e+02 < 5
9.283e+02 1.250e+03 1.250e+03 1.424e+03 1.424e+03 < 10
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Verify the accuracy of eigenresults
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Check the residual error for the first eigenresult
:math:`R_1=||(K-\lambda_1.M).\phi_1||_2`
First, we compute :math:`\lambda_1 = \omega_1^2 = (2.\pi.f_1)^2`
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.. code-block:: default
# Eigenfrequency (Hz)
i = 0
f = ev[0]
omega = 2 * np.pi * f
lam = omega * omega
.. GENERATED FROM PYTHON SOURCE LINES 82-84
Then we get the 1st Eigenshape :math:`\phi_1`, and compute
:math:`K.\phi_1` and :math:`M.\phi_1`
.. GENERATED FROM PYTHON SOURCE LINES 84-95
.. code-block:: default
# shape
phi = a[0]
# APDL Command: *MULT,K,,Phi,,KPhi
kphi = k.dot(phi)
# APDL Command: *MULT,M,,Phi,,MPhi
mphi = m.dot(phi)
.. GENERATED FROM PYTHON SOURCE LINES 96-98
Next, compute the :math:`||K.\phi_1||_2` quantity and normalize the
residual value.
.. GENERATED FROM PYTHON SOURCE LINES 98-107
.. code-block:: default
# APDL Command: *MULT,K,,Phi,,KPhi
kphi = k.dot(phi)
# APDL Command: *NRM,KPhi,NRM2,KPhiNrm
kphinrm = kphi.norm()
.. GENERATED FROM PYTHON SOURCE LINES 108-111
Then we add these two vectors, using the :math:`\lambda_1` scalar
factor and finally compute the normalized residual value
:math:`\frac{R_1}{||K.\phi_1||_2}`
.. GENERATED FROM PYTHON SOURCE LINES 111-120
.. code-block:: default
# APDL Command: *AXPY,-lambda,,MPhi,1,,KPhi
mphi *= lam
kphi -= mphi
# Compute the residual
res = kphi.norm() / kphinrm
print(res)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
3.503338761293556e-11
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This residual can be computed for all eigenmodes
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.. code-block:: default
def get_res(i):
"""Compute the residual for a given eigenmode"""
# Eigenfrequency (Hz)
f = ev[i]
# omega = 2.pi.Frequency
omega = 2 * np.pi * f
# lambda = omega^2
lam = omega * omega
# i-th eigenshape
phi = a[i]
# K.Phi
kphi = k.dot(phi)
# M.Phi
mphi = m.dot(phi)
# Normalization scalar value
kphinrm = kphi.norm()
# (K-\lambda.M).Phi
mphi *= lam
kphi -= mphi
# return the residual
return kphi.norm() / kphinrm
mapdl_acc = np.zeros(nev)
for i in range(nev):
f = ev[i]
mapdl_acc[i] = get_res(i)
print(f"[{i}] : Freq = {f}\t - Residual = {mapdl_acc[i]}")
.. rst-class:: sphx-glr-script-out
.. code-block:: none
[0] : Freq = 338.0666635506364 - Residual = 3.503338761293556e-11
[1] : Freq = 338.06666355063646 - Residual = 5.095765571157227e-11
[2] : Freq = 626.6450980927036 - Residual = 1.7852148894677666e-11
[3] : Freq = 626.645098092704 - Residual = 2.17372918531887e-11
[4] : Freq = 928.2598500574518 - Residual = 1.72025442075496e-11
[5] : Freq = 928.2598500574528 - Residual = 6.747910207805831e-12
[6] : Freq = 1249.8421074363514 - Residual = 2.815524037478371e-11
[7] : Freq = 1249.8421074363525 - Residual = 7.546271373844357e-12
[8] : Freq = 1423.993890941669 - Residual = 4.5372004789272195e-10
[9] : Freq = 1423.9938909416699 - Residual = 1.2160010951553228e-09
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Plot Accuracy of Eigenresults
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.. code-block:: default
fig = plt.figure(figsize=(12, 10))
ax = plt.axes()
x = np.linspace(1, nev, nev)
plt.title("APDL Math Residual Error (%)")
plt.yscale("log")
plt.ylim([10e-13, 10e-7])
plt.xlabel("Frequency #")
plt.ylabel("Errors (%)")
ax.bar(x, mapdl_acc, label="MAPDL Results")
plt.show()
.. image-sg:: /examples/gallery_examples/01-apdlmath-examples/images/sphx_glr_eigen_solve_001.png
:alt: APDL Math Residual Error (%)
:srcset: /examples/gallery_examples/01-apdlmath-examples/images/sphx_glr_eigen_solve_001.png
:class: sphx-glr-single-img
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Stop mapdl
~~~~~~~~~~
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.. code-block:: default
mapdl.exit()
.. rst-class:: sphx-glr-timing
**Total running time of the script:** ( 0 minutes 1.275 seconds)
.. _sphx_glr_download_examples_gallery_examples_01-apdlmath-examples_eigen_solve.py:
.. only:: html
.. container:: sphx-glr-footer sphx-glr-footer-example
.. container:: sphx-glr-download sphx-glr-download-python
:download:`Download Python source code: eigen_solve.py <eigen_solve.py>`
.. container:: sphx-glr-download sphx-glr-download-jupyter
:download:`Download Jupyter notebook: eigen_solve.ipynb <eigen_solve.ipynb>`
.. only:: html
.. rst-class:: sphx-glr-signature
`Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_