# Survey of Finite Element Method-Based Real-Time Simulations
## 6 Full Extent FE Models in Real-Time Applications
This implies that the system of equations to be resolved remains coupled, which invokes significant numerical effort. The standard solution in commercially available FEA programs is a direct solver. It offers a great stability of solution procedure for various types of structures and engineering accuracy, but lacks options for a trade-off between accuracy and computational speed. Iterative solvers, on the other hand, are rather advantageous with respect to the last aspect, particularly if used with adequately selected preconditioners to improve the convergence rate [61]. One may simply limit the number of iterations and thus obtain a solution that is not necessarily very accurate but still sufficiently good for the purpose of a real-time simulation [62]. In this manner, the several orders of magnitude larger time-step used in an implicit time integration scheme can easily make up for a larger numerical effort needed to resolve the coupled system of equations compared to the numerical effort needed for a time-step within an explicit time integration scheme. More interesting is the innovativeness in proposing nonlinear FE formulations for real-time simulations. In doing that, the aim is to reduce the numerical effort by neglecting certain nonlinear effects and thus simplify the rigorous FE formulations. The developed formulations are supposed to combine the advantages of linear and geometrically nonlinear FE formulations. In other words, they should be stable and as efficient as possible, while covering the geometrically nonlinear effects to a large extent. Probably the most interesting approach is based on the co-rotational FEM formulation [60; 63; 64]. The basic idea is to account for large local rigid-body rotations by means of local coordinate frames that are attached to the structure and perform the same rigid-body rotation as the local area of the structure. The usual simplification implies that the local deformations are considered to be linear with respect to the local coordinate frames. This means that the influence of local deformations and stress state (\"stress stiffening\") onto the tangent stiffness matrix are neglected. The stiffness matrices of the local regions are precomputed and only rotated during the simulation based on the current and initial positions of the local coordinate frames. [PERSON] et al. [65] presented a framework for the skeleton-driven animation of elastically deformable characters. They associated regions of FE meshes with particular bones. Those regions were structural sub-domains whose rigid-body rotations were accounted for by using local coordinate frames, while the FE computations were linearized with respect to them. A similar approach, but extended in order to provide finer resolution of accounting for local rotations was proposed by [PERSON] et al. [66]. They implemented local coordinate frames at the nodes and compute a tensor field at every time-step to describe the local rotations of all the vertices in the mesh. [PERSON] et al. [67] presented development for viscoelastic, highly flexible surfaces. They attached local coordinate frames to triangular elements to cover local rigid-body rotations on the element level. Some researchers extended the approach to 3D elements (tetrahedrons) and also referred to this formulation as the \"warped stiffness\" approach [68]. [PERSON] and [PERSON] analyzed the accuracy in terms of numbers provided by this simplified co-rotational formulation for both solid [69] and shell [70] elements, and showed that if nonlinearities in deformational behavior are dominated by the local rotations, the accuracy for predicted displacements lies within a few percent of the results by rigorous geometrically nonlinear FE formulations. They also demonstrated that FE models of up to several thousand elements are applicable in real-time simulations on conventional hardware components without parallelization and presented this development for the field of surgery simulation [71].