# Survey of Finite Element Method-Based Real-Time Simulations
## 3 Mass-Spring Systems  
Mass-spring systems represent relatively simple models, in which concentrated masses are interconnected by a set of springs. This heuristic approach actually implies that a 3D (continuum) body is replaced by a truss structure. The idea is relatively old and was proposed by [PERSON] in 1941 [6], who; however, never aimed at real-time simulation but simply at a computational method applicable in the field of structural mechanics. The obvious advantage of the approach resides in the rather simple formulation of the spring (truss) element that transfers the force only in the length direction, thus significantly reducing the number of degrees of freedom. Additionally, if the geometrically nonlinear effects are to be accounted for, it is done in a straightforward manner as the force is always transferred between the current positions of the spring nodes. Figure 1 shows a few illustrative screen-shots from an interactive simulation involving a simple cloth-like structure formed of a mass-spring system and exposed to large, geometrically nonlinear deformations [7]. Hence, the major advantages of the approach are the simplicity of formulation and computational efficiency. Those properties rendered the approach extremely attractive in the 1990s and the beginning of the new millennium as the limited power of available hardware components at that time called for such solutions. In that period, the mass-springs systems were kind of a workhorse for real-time simulations in a number of fields, ranging from entertainment industry up to biomechanics and surgery simulators. Mass-springs systems found their application in numerous medical applications for simulation of soft tissue deformations. This is basically due to the fact that the demand for accuracy in this type of application is reduced to \"plausible physical behavior.\" Hence, in most of such applications, the simulation simply needs to appear realistic to human eye and that kind of behavior can easily be reproduced by mass-spring models even in geometrically nonlinear regimes, despite the significant simplifications implemented. Models with all their specifics were proposed in the field of orthodontics [8], robotic surgery [9], real-time simulation of muscles [10], and various surgery simulations [11; 12; 13], some of which were proposed with GPU accelerated computations [14]. In virtual reality simulators and entertainment products, the use of mass-spring systems is particularly versatile. For instance, they come in handy in modeling very thin structures whose physical behavior is dominated by the membrane stiffness, while the bending stiffness is practically negligible. A typical example would be cloth. [PERSON] and [PERSON] [15] dedicated their work to find proper spring stiffness that enables realistic real-time simulation of collision between a cloth and other objects. In order to achieve enhanced behavior, beside classical springs that provide structural basis of the cloth, they introduced shear and bend springs, which in combination with a specific pattern of interconnections produce more realistic cloth behavior. Another interesting example of mass-spring models application is in the simulation of facial expressions. For this purpose, [PERSON] et al. [16] proposed  
Figure 1: Cloth-like structure as a mass-spring system in an interactive simulation [7]. to model facial skin and a set of facial muscles by means of mass-spring systems. Their models incorporate materially nonlinear springs. Distinctive works in this field addressed the problem of ambiguity of model building. Namely, the major problem with the mass-spring models is how to determine the distribution of point masses and the topology of their connections so as to have acceptable description of the actual physical behavior. There are obviously a large number of possibilities for any considered physical object. [PERSON] et al. [17] proposed the idea to optimize the topology of interconnections by means of genetic algorithms based on the results obtained using a known reference model.