2024-08-08T13:30:38.000-0400

Simcenter Nastran
Simcenter 3D

The example of the use of Modified Generalized Alpha Method for Time Integration Scheme in Transient Dynamic subcase with a specific boundary condition setup

**Introduction**

In some case due to assumption modifications or model simplifications the obtained results are not in line with our expectations. This can reveal itself in many forms like different location of peak stresses, deformations in a different direction, etc. Another of such examples is the oscillation of the displacement/velocity/acceleration vs time in Transient Dynamic simulations. This can be noticed when specific workflow is performed which results in additional oscillation overlayed on the expected function response (shown in the figure below). In below article the effect of use of Time Integration Schemes and Transient Initial Conditions Constraint on this behavior is described.

Figure 1. Velocity on excited node (Node 3) vs velocity on the node on the free end of the beam (Node 1).

**Description of the problem**

Let’s assume that we want to perform a Transient Dynamic simulation of a simple beam with rectangular cross-section in SOL401. One end is free in all directions and on another end an Enforced Displacement constraint is defined, where we would specify an excitation in Z axis and all other degrees of freedom would be fixed. As an excitation a sinusoid function would be used (fig.2). The analyzed model is shown on fig.3.

Figure 2. Excitation function.

Figure 3. Analyzed model.

For the solver settings the default values were selected. Such modeling approach and model definition results in following results for displacement, velocity and time in Z direction:

Displacement [mm] |

Velocity [mm/s] |

Acceleration [mm/s |

Figure 4. Results for default solver settings.

Above pictures show plots extracted from two nodes:

- Node 3 is the node located on the surface where excitation was defined.
- Node 1 is the node located on the other end of the beam, which can move freely.

From presented results we can see huge acceleration oscillation due to induced higher frequency modes that are impacting the results. This also impact the velocity values through time. This happens due to very big change of acceleration in the first iteration of the solution which results from instant change in velocity from 0 m/s to 8 m/s. This effect may be even amplified when the time step is reduced. This behavior is unphysical and unwanted and it is not occurring when, e.g., model is defined using Large Mass method. However, we have some tools that can be used to minimize the effect of induced higher frequency modes on our solution.

**Solution**

To minimize shown effect of higher frequency impact one can use different option for Time Integration Scheme (TINTMTH). In this KBA only the effect of Modified Generalized Alpha will be presented. Other option are Newmark Method, Hilber-Hughes-Taylor (HHT) Method and Generalized Alpha Method. Time Integration Scheme can be changed in solution Case Controls under Nonlinear Control Parameters (NLCNTLG) in Timer Integration section (fig. 5). Each of the method has different control parameters that can control the behavior of integration scheme.

Figure 5. Nonlinear Control Parameters dialog-box.

The major difference between integration scheme is in their numerical damping. Respective control parameters are controlling the amount of numerical damping in analysis. This is needed as any mesh will produce non-physical, high frequency modes based on the element size used and this damping is used to damp those unwanted frequencies. Despite the fact that in our case the impact of high frequency response is causes by applied boundary conditions it should also work well to minimize their impact on results.

In this example the Modified Generalized Alpha option was selected and Modified Generalized Alpha Scheme Parameter (RHOINF) was set to 0.1. This should be enough to bet rid of the higher frequency modes impact on the results. The value of RHOINF parameter can vary in the following range <0,1>, where 1 result in undamped scheme and 0 gives asymptotic annihilation of the high frequency response. With this modification of solution settings we are getting following results:

Displacement [mm] |

Velocity [mm/s] |

Acceleration [mm/s |

Figure 6. Results with Modified Generalized Alpha Scheme defined.

Above plots confirms that the impact of higher frequency response has been dumped in the solution already after one cycle of load. The obtained plots contain only the oscillation which results from excitation signal and the instabilities introduce by rapid acceleration impulse due to enforced motion definition has been removed. What can be still observed is that velocity and acceleration peak in the first cycle of load which cannot be dumped by integration scheme. However, there’s another option that can handle that. Due to that the model is not moving at the beginning of solution we can observe big jump in acceleration in first iteration. To remove or minimize this behavior we can define Transient Initial Conditions Constraint. It should be defined on the whole solid with the excluded face where Enforced Motion is defined. The initial value of velocity can be estimated basing on the previous velocity value at the first iteration. In this case 8 m/s was selected. When rerunning the solution following results were obtained:

Displacement [mm] |

Velocity [mm/s] |

Acceleration [mm/s |

Figure 7. Results from optimized model.

Above results shows that impact from high frequency responses was fully dumped and impact of rapid acceleration of the structure due to the modeling approach was minimized. The obtained results look as expected.

**Summary**

Above was shows that selection of proper Time Integration Scheme and values for respective parameters are crucial in Transient Dynamics subcase for obtaining the expected results. Implemented changes allowed to stabilize the solution and smooth out the response. The selection of Time Integration Scheme and values for Control Parameters depends on the type of analyzed problem and selected modeling approach. Following general recommendations can be proposed:

- When the damping effects are small and we need to keep a large spectrum of frequency it is recommended to used undamped schemes. Otherwise, with limited effect of damping it is suggested to use a Generalized Alpha method with RHOINF = 0.9-0.8.
- For cases with bigger damping effects it is suggested to use Generalized Alpha method with RHOINF = 0.4-0.1.

**Remark!**

As already stated in the article, different Time Integration Scheme and respective Control Parameters are developed to control the level of numerical dumping introduced to minimize the impact of the high frequency modes induced by finite element meshes. This, when defined properly, improves the stability or/and accuracy of the solution. However, I want to emphasize that care should be taken during selection of Time Integration Scheme and respective Control Parameters values, as it is possible to significantly overdamped the solution and some of important higher frequency responses might not be included in the results which can significantly impact the searched values. There is no general rule for that and a lot of caution should be taken when adjusting those settings. Especially, when it is used in a different use cases as the one described in this article.

More information about Time Integration Schemes are available in the online documentation on Transient dynamic subcase page in Time integration methods section under the link:

Simcenter Nastran Multi-Step Nonlinear User's Guide (SOL 401 and SOL 402) (siemens.com)

Additional two examples of the impact of Time Integration Schemes selection and parameters setup are described in the Samcef documentation. They can even better illustrate the importance of this option as they are also referring to the analytical solutions (however please note that the available Time Integration Schemes may differ from available in SOL401). The links to those examples are below:

Comparison of time integration schemes: Examples/NonLinearMechanics/ (siemens.com)

Simple pendulum with initial velocity: Examples/NonLinearMechanics/ (siemens.com)