# Relaxation for deformable objects

On the deformation side of things, I have investigated nonlinear material models, and looked how optimization methods compare with dynamical systems (`mass-spring' systems, or, more accurately `explicit time integration for nonlinear FEM systems') in these cases. These results have not been published, but will be a part of PhD. thesis.

In the following pictures, you can see a 10 cm long cylinder with tissue-like material parameters under the influence of gravity. The left end of the cylinder is held at a fixed position. Three different elasticity models are applied to this situation. From left to right we have with increasing realism: linear elasticity, linear material with nonlinear geometry and finally compressible neohookean material.

For computing the deformations, we can use a dynamic approach, or an optimization method. For the linear case, the Conjugate Gradient method clearly wins. In the following graph, you see the log of the error of Conjugate Gradient versus damped springs. At time t=0, the gravitational force was applied, and the difference between the final situation was measured.

For nonlinear elasticity, the most striking thing is the overall decrease in speed. In the following graph, the residual force versus computational cost is shown, for a linear dynamic method, and two nonlinear ones (linear and neohookean material). The speed decrease is consistent with the increased complexity of the material model.

When we compare the optimization based approach (a nonlinear CG algorithm) with a damped dynamic system, we see that little is left of speed wise superiority of the optimization approach: it has roughly the same speed as the dynamic system.

This lack of difference between both methods can be explained qualitatively: when there is no linearity, no special structure of the problem can be exploited. CG simply is a relaxation technique where the error decreases exponentially in time, just like a damped movement. However, one advantage of CG over a dynamic system remains: CG does not require tuning of friction and time-step parameters to perform well. It does not react with instability to small elements, merely with a decrease of performance. This is a big advantage for use in interactive systems, as it increases robustness.

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