Harmonic analysis using the Krylov method#

Introduction#

在声学或单场结构分析中，您可以使用频率扫描克雷洛夫方法来高性能地解决强迫频率模拟问题。

与全谐波分析类似，频率扫描克雷洛夫方法使用全系统矩阵来计算谐波响应。全频法对频率范围内的每个频率点进行求解，而频率扫描克雷洛夫法则执行以下步骤，对整个频率范围内的响应进行近似求解：

在请求频率范围中间的频率值处建立一个克雷洛夫子空间向量集
减少整个频率范围内的系统矩阵和负载
求解简化系统
将结果向后扩展，计算谐波响应

Mechanical APDL 提供了使用 Krylov 方法实现谐波分析的以下方法：

Mechanical APDL commands
MAPDL 的 结构分析 指南中的通过克雷洛夫法进行频扫谐波分析中所述的 APDL 宏

PyMAPDL 还提供了一种使用克雷洛夫方法实现谐波分析的方法。下文将介绍如何在 PyMAPDL 中使用克雷洛夫方法。

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:

KrylovSolver.gensubspace(): Creates the Krylov subspace.
KrylovSolver.solve(): Solves the reduced system of equations.
KrylovSolver.expand(): Expands the Krylov subspace.

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 使用频率扫描 Krylov 方法进行谐波分析.

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: