# Harmonic analysis using the Krylov method#

## Introduction#

You can use the frequency-sweep Krylov method for a high-performance solution of forced-frequency simulations in acoustic or single-field structural analyses.

Similar to the full harmonic analysis, the frequency-sweep Krylov method uses full system matrices to compute the harmonic response. While the full method solves at every frequency point in the frequency range, the frequency-sweep Krylov method performs the following steps to approximate the response across the frequency range:

• Builds a Krylov subspace set of vectors at the frequency value in the middle of the requested frequency range

• Reduces the system matrices and loading on the entire frequency range

• Solves the reduced system

• Expands the results back to compute the harmonic response

Mechanical APDL provides the following ways to implement a harmonic analysis using the Krylov method:

1. Mechanical APDL commands

2. APDL macros as described in Frequency-Sweep Harmonic Analysis via the Krylov Method in the Structural Analysis guide for Mechanical APDL

PyMAPDL also provides a way to implement a harmonic analysis using the Krylov method. Subsequent sections describe how to use the Krylov method in PyMAPDL.

### Assumptions#

The following assumptions are made when using the Krylov PyMAPDL method to obtain the solution:

• The stiffness, mass, and damping matrices are assumed to be constant (independent of frequency).

• The external load vector is linearly ramped over frequency. Ramping assumes that the frequency at which the Krylov subspace is built is in the middle of the frequency range. If you want to apply stepped loading, there is an option to specify that in the inputs for the `KrylovSolver.solve()` method.

## Krylov method implementation in PyMAPDL#

The PyMAPDL implementation of the Krylov method gives you customization and flexibility because you can access subspace vectors and reduced solutions using the Python programming language for user-defined routines.

If you do not require customization, you can use the Mechanical APDL commands to solve a harmonic analysis with the Krylov method. For more information, including the theory behind this method, see Frequency-Sweep Harmonic Analysis via the Krylov Method in the Structural Analysis guide for Mechanical APDL.

For additional theory information and equations for the Krylov method, see the works of Puri [1] and Eser [2].

The exposure in PyMAPDL follows the same theory as the Mechanical APDL macros and has the following methods:

Warning

These methods must be run consecutively.

## Usage#

This section shows how to implement an analysis identical to that defined by the Mechanical APDL macros.

### Generate the FULL file and FEA model#

Generate the FULL file for the Krylov method and the FEA model using Mechanical APDL:

```  >>> from ansys.mapdl.core import launch_mapdl

>>> mapdl = launch_mapdl()
>>> mapdl.prep7()

# Generate the FEA model (mesh, constraints, loads)
# ...

>>> mapdl.run("/SOLU")
>>> mapdl.antype("HARMIC")  # HARMONIC ANALYSIS
>>> mapdl.hropt("KRYLOV")
>>> mapdl.eqslv("SPARSE")
>>> mapdl.harfrq(0,1000)  # Set beginning and ending frequency
>>> mapdl.nsubst(100)  # Set the number of frequency increments
>>> mapdl.wrfull(1)  # GENERATE .FULL FILE AND STOP
>>> mapdl.solve()
>>> mapdl.finish()
```

### Create an instance of the Krylov class#

```>>> mk = mapdl.krylov
```

Call the `gensubspace` method to create the Krylov subspace and build a subspace of size/dimension 10 at a frequency of 500 Hz:

```>>> Qz = mk.gensubspace(10, 500, True)
```

### Return the Krylov subspace#

Call the `solve` method to reduce the system of equations and solve at each frequency. This code solves from 0 Hz to 1000 Hz with 100 intervals in between, with stepped loading:

```>>> Yz = mk.solve(0, 1000, 100, ramped_load=True)
```

### Return the reduced solution over the frequency range#

Call the `expand` method to expand the reduced solution back to the FE space, output the expanded solution, and calculate the residual:

```>>> result = mk.expand(
...     residual_computation=True, "L-inf", compute_solution_vectors=True, True
... )
```

The preceding code returns a `numpy array` if the kwarg `out_key` is set to `True`. Solution vectors are mapped to user order.

Note

The `numpy array` class returned by the `expand` method contains the node number along with the degrees of freedom (dof) solution for each of the calculated frequencies.

### Get the dof solution at a specific frequency#

This code shows how to get the nodal solution at a specific frequency or step:

```# Get the nodal solution at freq number 3``````
>>> freq = 3
>>> nodal_sol = result[freq - 1]  # Get the nodal solution for each node
```

## Example#

Examples of using the Krylov method in PyMAPDL are available in Harmonic analysis using the frequency-sweep Krylov method.

## Requirements#

To use the Krylov method in PyMAPDL, you must use Mechanical APDL version 2022 R2 or later.

Warning

This feature does not support Distributed Ansys. However, you can still run Mechanical APDL Math commands without specifying the `-smp` flag when launching Mechanical APDL.

## Reference#

For more information on the Krylov method, see Frequency-Sweep Harmonic Analysis via the Krylov Method in the Structural Analysis guide for Mechanical APDL and these resources: