C.L. Bottasso, L. Trainelli
An
Attempt at the Classification of Energy Decaying Schemes for Structural and Multibody Dynamics
Multibody System Dynamics, 12
(2): 173-185 (2004).
also presented at ECCOMAS Thematic Conference “Multibody Dynamics 2003”,
Abstract
Energy decaying schemes represent the most recent
attempt at trying to develop robust algorithms for integrating in time the
semi-discrete equations associated with stiff non-linear finite element
problems, including multibody systems. The basic
motivation behind these schemes is the simple fact that classical algorithms
that are unconditionally stable and high frequency dissipative, two well
understood and appreciated characteristics that are commonly deemed necessary
for many practical engineering applications, do loose their properties in the
non-linear regime.
In practice, all energy decaying schemes are carefully
constructed so that it becomes possible to prove the existence of discrete
bounds on the algorithmic total energy in a typical time step. This discrete
bound implies two fundamental properties: a) unconditional stability in the
non-linear regime; b) damping of the unresolved and spurious high frequencies.
The closely related energy preserving methods are limited to the former
property. The drawback is that one typically has to specialize the scheme to
each new model; for example, the implementation of a basic scheme for a beam or
a shell will reflect the different forms of the two sets of equations governing
these two models. However, what is
gained by this approach seems to amply justify this minor limitation.
Although there appear to be no thorough review works
on energy decaying schemes yet, it is quite clear that each method as applied
to a specific model problem is essentially composed of two main ingredients: a)
a basic underlying time discretization scheme, that
could in principle be applied unchanged to a variety of models; b) a number of
accompanying ``details'' that in general vary from one model to another, but
that are often crucial for the final proof of the energy bound in the specific
case considered. Among these additional ingredients, we can mention for example
the parameterization of finite rotations, if present in the model, or the
details of the spatial discretization scheme, or
specific features of the governing equations. These details can also impact
other conservation properties, as for example the conservation of momentum.
While these are, as said, crucial details, they also tend to hide the nature of
the underlying time discretization scheme, and this
has perhaps not favored so far the development of
unified views on this class of methods.
In this work we discuss the problem of designing
energy decaying schemes, trying to offer a first step towards a unified view on
this topic. In particular, we classify the existing methods based on the
underlying discretizations schemes, and we review the
most interesting aspects related to the accompanying "details".