## Recursive Domain Normal Modes in Structural Dynamics

### Intro

Dynamic sub-structuring is a method used to analyze the dynamics of mechanical systems by partitioning the system into smaller subsystems/subcomponents. The problem is solved into these smaller subcomponents and then the global solution is a combination of the sub-components solution.

Dynamic sub-structuring is widely used for the simulation of mechanical vibrations.

Sub-structuring and domain decomposition methods have the following routines:

- A decomposition strategy to split the problem into sub-domains
- Local solvers to find the solution for the sub-domains
- Interface conditions to force compatibility and equilibrium
- Solution strategy for assembling the global solution from local solutions

### Recursive Domain Normal Modes

One of the methods that is based on sub-structuring technology is the Recursive Domain Normal Modes Analysis. The method reduces the dimensionality of the problem by partitioning the matrices. Hence, the eigenvalue problem is solved for these small partitions and the global solution is assembled from the solution of the small eigenvalue problems. The recursive domain normal modes analysis can also be used with parallel computation techniques. Since this is a mathematical reduction technique, the solution is an approximate solution. The nice thing here is that the relative difference in comparison to the classical Lanczos method is really very small. On the other hand, the simulation run time is super fast, and for very large models the solution with recursive domain normal modes can be 100x faster than the classical Lanczos.

The method can be used on the following structural dynamic problems:

- Modal analysis
- Frequency Response
- Shock Simulations
- Random Vibration Simulations

Recursive domain normal modes is implemented in various CAE software packages. For example in Nastran the algorithm is called RDMODES. Figure below shows the sub-structuring technique for the stiffness matrix in Nastran (source: Parallel Processing NX Nastran):

### Case Study

A modal analysis of an e-motor on system level (e-Axle incl. GB) is used here. There are more than 30 parts that are meshed in this model and it has approximately 25 million DOFs.

Below is a graph that illustrates the simulation run-time of the modal analysis of the system without and with RDMODES.