Numerical vibration analysis

In order to accurately compute the dynamics of components and technical systems, contact interactions must be resolved with sufficient spatial and temporal precision. Common simulation methods exhibit artifacts such as numerical oscillations, instability, and/or numerical damping. The numerical methods we have developed enable the simulation of high-fidelity models. For example, we have developed fully coupled methods for fluid-structure interaction. In addition to the nonlinear contact interactions, the inherently nonlinear behavior of the flow is modeled in order to determine the aerodynamic damping and excitation of blades in engines.

Harmonic balance can be used to calculate periodic solutions of nonlinear ordinary and differential algebraic equations. The approach uses a truncated Fourier series and exploits the fact that in many cases only a few Fourier terms are needed to describe the oscillations very accurately. Furthermore, the typically lengthy settling process (transient) associated with light damping does not need to be simulated. We have published a simple tool that also offers options for numerical path continuation; see the NLvib subpage.

In addition to periodic oscillations, non-periodic and transient processes are becoming increasingly important. Even in periodic cases, impact-like processes can lead to slow convergence of Harmonic Balance. We are therefore developing methods for numerical time step integration that are specifically suited to frictionless and frictional contact. Conventional finite element methods associate a finite mass with each node, including those at the contact interface. As illustrated, this leads to artificial oscillations and large errors. Our approaches lead to a massless boundary in order to avoid such errors. At the same time, this reduces the computational effort by several orders of magnitude without compromising the excellent agreement with the measurements.

Numerical path continuation is often used to determine how oscillations evolve with a free parameter (e.g., the excitation frequency). In high-fidelity models of complex nonlinear systems, the sequential nature inherent to path continuation can become an insurmountable obstacle. The broadly applicable concept we have developed is very simple, yet has the potential for a breakthrough: The idea is to combine path continuation and model refinement in a new and creative way. First, numerical path continuation is applied to a low-fidelity model (requiring only low computational effort). Then, a subset of interesting points is selected, and based on these, points are calculated on the solution curve of the high-fidelity model. The iterative correction can be performed independently for each point and is therefore ideal for parallel computation. This concept has proven to be numerically very robust and enables calculations to be accelerated by several orders of magnitude.