<H1>4. Conclusion</H1>
        Recent literature has emphasized on the relevance of implicit methods for cloth simulation. The implicit Euler method seems effectively a good candidate for most situations involving cloth simulation, because of the robustness resulting from not being prone to numerical instability. This is particularly true when simulating very heterogeneous mechanical systems (elements of various sizes and various mechanical properties) where, using explicit models, the most critical elements would rule the time step size for all the simulation. Contrary the perception of the implicit model iteration being slow because of the linear system resolution it involves, the inverse Euler iteration often proves to be faster than the explicit Runge-Kutta method of higher order, if an adequate approximate linear system resolution is implemented. A limited number of Conjugate Gradient iterations seems suitable for this. Furthermore, while increasing the time step seems not limited by instability with implicit methods, it should be kept in mind that this is still done at the expense of accuracy of the whole simulation. The number of iterations should also be set sensitively to the stiffness of the mechanical problem, for limiting the potential inaccuracies that become particularly visible when an accurate simulation of a dynamical system is wanted. There is an obvious advantage of using implicit methods, and particularly the inverse Euler method, for draping problems where quick convergence to a rest position is required quickly. Our test have shown that the inverse Euler method allow to perform a draping problem almost ten times as fast as with the Runge-Kutta method. While not exactly reproducing real mechanical behavior, the simulation with large time steps provides a quite efficient convergence to equilibrium, and the numerical errors quite often act as extra damping, removing the need of adding them explicitly to the model. For dynamic problems where accurate evolution of the mechanical system along time is needed, the advantage of implicit methods is less obvious. Their stability gives a false sense of efficiency, allowing obtaining quickly a result by “cheating” on the time step size. However, playing back the generated animation, artifacts quickly show up: Excessive damping, wrinkles and folds that fail to disappear, and even objects failing to fall correctly by their own weight. These artifacts are still augmented by the approximations made to the Hessian matrix, possibly in the purpose of reducing instability, while excessive reduction of the Conjugate Gradient iterations produce additional inaccuracy and slow convergence. It seems that there is still some benefit in using the Backward Euler method than any other explicit method for dynamic simulations thanks to the reduced time it takes to compute one iteration, which also only requires one derivation of the particle forces from the state of the system. Our tests have shown a roughly doubled speed for the accuracy corresponding to the limit of stability of the Runge-Kutta method. We got substantial improvements through the implementation of the implicit Midpoint method 
[ VOL 00 ]
, which however had the drawback of increasing the numerical instability problem, forcing additional use of isotropic force gradients, at the expense of accuracy. The explicit methods have still their interest, and should be reserved for simulations requiring high accuracy and particularly those where involving low mechanical damping and where mechanical energy conservation is important. Instability concerns will force parameters and time step size to ensure good accuracy for the simulation of all particles of the discrete mechanical representation, and therefore for the entire mechanical object. This may however require prohibitive computation times for very stiff and discretized models. The 5th-order Runge-Kutta method das proven to be a good solution [ EBE 96 ] [ VOL 97 ]
, because of its high accuracy, and because it furthermore provides integration error evaluation, which is a very good hint to the very sensitive problem of optimal time step size determination. The simpler Midpoint method may have some interest only in very particular cases involving very loose materials with rough discretization, or when numerous fast iterations with small time steps are required for other reasons (high motion sampling, collision detection, very discontinuous models). All these considerations should be carefully taken into account when designing a mechanical simulation engine, as they are the keys to efficient simulation, and therefore complex models that, for garment simulation, express fully visual experience of real fashion models. We intend to pursue our investigations for dealing with damping in a more accurate way. This still remains an important issue to dynamic realism of cloth simulation models, which has to take into account viscosity, the dissipative effect of hysteretic behavior, as well as collision damping and friction. The integration methods have to be tuned to take precisely these effects into account.
      
      
        