Variable step-size solvers for coupled DAEs in mechatronic applications

This work addresses a variable step-size formulation of the generalized- time integration scheme for mechanical and mechatronic systems represented by coupled differential-algebraic equations (DAEs). In previous publications, a variant of the generalized- alpha algorithm has been proposed, which is able to deal with a non-constant mass matrix, controller dynamics and kinematic constraints in a consistent way. We have shown that this fixed step-size method can be used to solve efficiently industrial problems. The present work focuses on variable step-size schemes. It is well-known that classical formulations of the generalized-alpha method are no more second-order accurate in this case. We argue that second-order accuracy can be recovered provided a modification of the coefficients of the method. Actually, the value of the coefficients should be modified at each time step according to a simple update formula. This approach can thus be implemented very easily in an existing code. We also report practical experience on the implementation of the method. A strategy for the selection of the step-size is described and the importance of a scaling of the equations and unknowns is highlighted. A number of examples and applications are presented in order to illustrate those results