APDL Math Overview¶
APDL Math provides the ability to access and manipulate the large sparse matrices and solve a variety of eigenproblems. PyMAPDL classes and bindings present APDL Math in a similar manner to the popular numpy and scipy libraries. The APDL Math command set is based on tools for manipulating large mathematical matrices and vectors that provide access to standard linear algebra operations, access to the powerful sparse linear solvers of ANSYS Mechanical APDL (MAPDL), and the ability to solve eigenproblems.
Python and MATLAB’s eigensolver is based on the publicly available
LAPACK libraries and provides reasonable solve time for relatively
small DOF (degree of freedom) eigenproblems of perhaps 100,000.
However, ANSYS’s solvers are designed for the scale of 100s of
millions of DOF and there are a variety of situations where users can
directly leverage ANSYS’s high performance solvers on a variety of
eigenproblems. Fortunately, you can leverage this without relearning
an entirely new language as this has been written in a similar manner
as numpy and scipy. For example, here is a comparison between
the scipy linear algebra solver and ANSYS’s solver:
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k_py = k + sparse.triu(k, 1).T
m_py = m + sparse.triu(m, 1).T
n = 10
ev = linalg.eigsh(k_py, k=neqv, M=m_py)
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k = mm.matrix(k_py, triu=True)
m = mm.matrix(m_py, triu=True)
n = 10
ev = mm.eigs(n, k, m)
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What follows is a basic example and a detailed description of the PyMAPDL Math API. For additional PyMAPDL Math examples, visit PyMapdl Math Examples.
MAPDL Matrix Example¶
This example demonstrates how to send an MAPDL Math matrix from MAPDL
to Python and then send it back to be solved. While this example runs
MapdlMath.eigs() on mass
and stiffness matrices generated from MAPDL, you could instead use
mass and stiffness matrices generated from an external FEM tool, or
even modify the mass and stiffness matrices within Python.
First, solve the first 10 modes of a 1 x 1 x 1 steel meter cube in MAPDL.
import re
from ansys.mapdl.core import launch_mapdl
mapdl = launch_mapdl()
# setup the full file
mapdl.prep7()
mapdl.block(0, 1, 0, 1, 0, 1)
mapdl.et(1, 186)
mapdl.esize(0.5)
mapdl.vmesh('all')
# Define a material (nominal steel in SI)
mapdl.mp('EX', 1, 210E9) # Elastic moduli in Pa (kg/(m*s**2))
mapdl.mp('DENS', 1, 7800) # Density in kg/m3
mapdl.mp('NUXY', 1, 0.3) # Poisson's Ratio
# solve first 10 non-trivial modes
out = mapdl.modal_analysis(nmode=10, freqb=1)
# store the first 10 natural frequencies
mapdl.post1()
resp = mapdl.set('LIST')
w_n = np.array(re.findall(r'\s\d*\.\d\s', resp), np.float32)
print(w_n)
We now have solved for the first 10 modes of the cube:
[1475.1 1475.1 2018.8 2018.8 2018.8 2024.8 2024.8 2024.8 2242.2 2274.8]
Next, load the mass and stiffness matrices that are stored by default
at <jobname>.full. First, create an instance of MapdlMath as mm:
mm = mapdl.math
# load by default from file.full
k = mm.stiff()
m = mm.mass()
# convert to numpy
k_py = k.asarray()
m_py = m.asarray()
mapdl.clear()
print(k_py)
These matrices are now solely stored within Python now that we’ve
run Mapdl.clear().
(0, 0) 37019230769.223404
(0, 1) 10283119658.117708
(0, 2) 10283119658.117706
: :
(240, 241) 11217948717.943113
(241, 241) 50854700854.68495
(242, 242) 95726495726.47179
The final step is to send these matrices back to MAPDL to be solved. While we have cleared MAPDL, we could have shutdown MAPDL, or even transferred them to a different MAPDL session to be solved.
my_stiff = mm.matrix(k_py, triu=True)
my_mass = mm.matrix(m_py, triu=True)
# solve for the first 10 modes above 1 Hz
nmode = 10
mapdl_vec = mm.eigs(nmode, my_stiff, my_mass, fmin=1)
eigval = mapdl_vec.asarray()
print(eigval)
As expected, the natural frequencies obtained from
MapdlMath.eigs() is
identical to the result from Mapdl.solve()
within MAPDL.
[1475.1333421 1475.1333426 2018.83737064 2018.83737109 2018.83737237
2024.78684466 2024.78684561 2024.7868466 2242.21532585 2274.82997741]
If you wish to obtain the eigenvectors as well as the eigenvalues,
initialize a matrix eigvec and send that to
MapdlMath.eigs():
nmode = 10
eigvec = mm.zeros(my_stiff.nrow, nmode) # for eigenvectors
val = mm.eigs(nmode, my_stiff, my_mass, fmin=1)
The MAPDL Math matrix eigvec now contains the eigenvectors for the
solution.