# Survey of Finite Element Method-Based Real-Time Simulations
## 4 Model Order Reduction Techniques
Such a combination of modes does not make an orthogonal set of modes. In order to keep the numerical advantages of a decoupled system, the so-obtained set of modes is orthogonalized prior to simulation. However, the disadvantage of the orthogonalization is in the fact that the obtained orthogonal modes lack physical interpretation, which, for instance, prevents assignment of (experimentally determined) modal damping coefficients. Regardless of the approach used to obtain the mode shapes, they always characterize the structure in the undeformed (i.e., initial configuration). Hence, the method is intrinsically meant for linear analysis and it is; therefore, suitable for small deformations. However, different approaches were proposed recently in order to extent the approach at least into the realm of moderate geometric nonlinearities. The approach based on the introduction of the geometric stiffness matrix was first introduced in the commercial software package SIMPACK [26]. In that approach it is essential to differentiate between the forces which easily produce relatively large displacements but small strains (i.e., the structure is rather flexible with respect to them) and the forces that give rise to relatively small deformations but large stresses in the structure [27]. Furthermore, the geometric stiffness matrix is computed for each force belonging to the latter group and assuming its magnitude equal to one. Over the course of simulation, the geometric stiffness matrix for each of the forces from the second group is scaled by using the current magnitude of the force. The overall stiffness matrix is obtained by superposition of the linear stiffness matrix and the current values of all geometric stiffness matrices. Thus, the introduced geometric stiffness matrix is force-scaled. In this manner, the exquisite numerical efficiency is retained, while the stress stiffening effects are included for a carefully selected group of forces. [PERSON] and [PERSON] [28; 29] proposed to introduce a geometric stiffness for each mode shape. It is computed prior to simulation using geometrically nonlinear analysis in which the structure is deformed according to the mode shape. In doing that, the intensity of deformation corresponds to the modal coefficient equal to one. Hence, during the simulation, the modal geometric stiffness coefficients are updated by using the current values of the modal coefficients. In this case, the geometric stiffness matrix is deformation-scaled. Figure 2: Modal space reduced finite element (FE) model: Screen-shots from an interactive simulation of a car rear-axle. Frequently the geometric nonlinearities are caused by finite local rotations of structural subdomains. Techniques denoted as modal warping were proposed as a remedy that would again extend the applicability of modal superposition to moderate geometrically nonlinear deformations. [PERSON] and [PERSON] [30] keep the track of local rotations that occur during the deformation computed by the linear model. In the next step, they warp the computed modal basis according to the local rotations of the FE nodes. Similarly, [PERSON] and [PERSON] [29; 31] first identify substructures that can perform relatively large local rotations compared to the rest of the structure. They compute the overall structural response by means of linear mode shapes. Based on the deformed state, the average local rotations of substructures are determined and used to rotate (\"warp\") the nodal displacements previously computed in the classical manner using linear mode shapes. The improvement in accuracy of determining the deformed shape was demonstrated in term of numbers for a simple cantilever beam and a car rear-axle. It should be emphasized that the procedure of \"warping\" the displacements introduces a rather small additional numerical effort so that the numerical efficiency of the reduced model based on modal space remains extremely high. Another approach can be applied in cases where the nonlinear behavior is restricted to a relatively small part of a structure.