### 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.

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.

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

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.

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.

The APDL script can be downloaded here:

sensitivity_v2.inp [link]

FromWB.dat [link]

Related post on exporting stiffness matrix using APDL Math: Link

**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

Good post.

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