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APDL Math Example

I read about APDL Math a few years ago with much intrigue. A very interesting write-up about it can be found here. What stood out to me was the possibility of computing modal sensitivity relative to different variables similar to what SOL 200 offers in NASTRAN for optimization.

Web Resources
Ansys Help Document: The commands are documented though the available examples are limited
PADT Blog Post: Eric's article gives a good overview of the capabilities
Ansys Knowledge Resource #2025879: Additional guidance in usage.
PADT Blog Post [Edit: Sept 12, 2017]: Awesome post! Results back in 'User Ordering'!
AnsysTips Blog Post [Edit: Oct 11, 2017]: Export Stiffness Matrix

Unfortunately, that's about all I could find anywhere.

APDL Math Example Overview
DOF ordering is not addressed here. I have not yet figured out a good way to do it yet so please share if you can convert to the 'user ordering' all in one shot.

Please see the script on how the solution was performed. The secret sauce is to first export mass and stiffness matrix before computing the natural frequency and mode shapes.
! Extract mass and stiffness matrix
*smat, k0, D, import, full, model0.full, stiff
*smat, m0, D, import, full, model0.full, mass

 !!! solves modal
/solu
antype, modal
modopt, lanb, 10
*eigen, k0, m0, , eigV0, eigM0

To gain an understanding of it's usage, I wrote up a simple script to compute the natural frequencies of a simple plate. After which, the modal sensitivity is calculated via matrix multiplication to estimate the new natural frequency assuming a 1% increase in modulus of elasticity for all elements. Two methods are used. Richardson & Mannan (RM) has a squared relationship:

In "Modal Analysis Theory and Testing" book by Heylen, Lammens & Sas (HLS), they show it inversely proportional to natural frequency:

In the example script, both sensitivities were used as linear gradients to extrapolate linearly to the 'modified' natural frequency that has a 1% increase in stiffness. The results are shown in the table below.

Natural Frequency Comparison

Discussions 
Computed errors were less than 20% which isn't great by most measures. The derivation of both sensitivities notes accuracy drawbacks due to many approximations. Despite the errors, sensitivities values can be computed quickly and can be extremely useful in structural modifications and aid redesign.

Script/Code
The APDL script can be downloaded here:
sensitivity_v2.inp [link]
FromWB.dat [link]

 sensitivity_v2.inp calls FromWB.dat to create a simple Workbench generated geometry.

Update
Related post on exporting stiffness matrix using APDL Math: Link
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