# Survey of Finite Element Method-Based Real-Time Simulations
## 4 Model Order Reduction Techniques  
FE-models are characterized by sufficiently fine discretization of geometry that leads to a high-dimensional system of equations. Solving the resulting system of equations is actually the most demanding and time-consuming part of the overall FEM solution. This fact was the motivation for researchers to develop ideas for performing model order reduction by finding a low-dimensional but still adequate approximation of the initial system of equations. The overall motion in multi-body systems (MBS) is represented as a superposition of rigid-body motion and small deformational motion that is conveniently described in a local (body-fixed) coordinate system. The consideration of deformation in MBS dynamics together with all types of nonlinearities, multi-physics (coupled-field) effects, etc., can be successfully done by means of MBS-FEM co-simulation [20; 21; 22; 23]. The great fidelity of the approach is certainly its undisputable advantage, but it comes at the price of an extreme computational effort combined with the additionally time-consuming data transfer between the two systems. To reduce the effort and therewith significantly improve the simulation speed, model order reduction is extensively used in MBS dynamics involving elastic structures. It should be emphasized that real-time simulation is usually not a primary requirement in this field. Nevertheless, computational efficiency is of great importance in MBS systems, and since they are mostly used for engineering tasks, the accuracy is a demand of equal or even greater importance. Among different approaches to model order reduction, the one typically used in the MBS systems is based on modal vectors and coordinates. The essence of the idea is to perform a great deal of computation prior to simulation. In a step performed prior to simulation, orthogonal mode shapes are determined and they become the degrees of freedom further used to determine the deformational part of the motion. In this manner, FE models with immensely large numbers of degrees of freedom (\(\sim\)10\({}^{6}\)) could be reduced down to a few dozens of degrees of freedom (DOFs) in modal space. The fact that orthogonal mode shapes are used leads to further reduction of computational effort as the resulting equations describing deformation are decoupled, provided modal damping coefficients are used. In fact, this general idea is already known from FEM, where the modal superposition technique is often used to determine structural transient response [24]. In FEM the so-called normal modes are used, which are determined by performing modal analysis for the known kinematic boundary conditions. They represent the patterns of vibration of a system at specific frequencies (eigenfrequency). Figure 2 shows screen-shots from an interactive simulation of a car rear-axle. The original (full extent) FE model contains 300,000 DOFs, while the model reduced in modal space only 10 DOFs and easily enables real-time simulation with suitable accuracy in the lower frequency range (up to the frequency of the highest mode used in the simulation). For improved versatility, MBS systems typically use the component mode synthesis (CMS) technique, particularly the Craig-Bampton method [25]. The idea is to divide the DOFs of the elastic structure into boundary and interior DOFs. The boundary DOFs belong to those nodes of the FE model that are kept in the simulation model for the purpose of interaction with the surrounding (i.e., for the purpose of kinematic and dynamic boundary conditions). In the next step two sets of modes are determined: constraint modes and fixed boundary normal modes. The constraint modes are static modes and their number corresponds to the number of boundary DOFs. A constraint mode is obtained by assigning a unit displacement to one of the boundary DOFs while all remaining boundary DOFs are fixed. The fixed-boundary normal modes are obtained by classical modal analysis whereby all boundary DOFs are fixed.