Page 228 - The Combined Finite-Discrete Element Method

THE COMBINED FINITE-DISCRETE ELEMENT SIMULATION 211

the complexity of the scheme and the CPU time needed to achieve a given accuracy.
Higher order schemes for the same accuracy are not necessarily faster (i.e. CPU more
efficient) than lower order schemes. The CPU results shown are subject to limitations
of a one degree of freedom system, and may be different when applied to large scale
combined finite-discrete element systems, where apart from contact force evaluation pro-
cedures, contact detection, fracture, fragmentation, fluid coupling, etc. may be involved.
The presence of this considerably changes the CPU cost at each time step. However,
in this context, higher order schemes requiring multiple force evaluation are much less
efficient than lower order schemes.

5.4 THE COMBINED FINITE-DISCRETE ELEMENT SIMULATION OF
THE STATE OF REST

Many combined finite-discrete element problems include transient motion that leads to
the state of rest. Through energy dissipation mechanisms such as fracture, friction and
permanent deformations, the energy of the combined finite-discrete element system is
steadily reduced until all the discrete elements are virtually at a state of rest.
There is also a whole class of combined finite-discrete element problems where the
state of rest is more important than the transient motion sequence preceding it. These
problems are, by nature, static, and require efficient procedures for integration of govern-
ing equations.
It has already been mentioned in this chapter that the combined finite-discrete element
systems may comprise thousands, even millions, of finite element meshes. Assembling
all stiffness matrices and possible problems with inversion of the stiffness matrices
has already resulted in the elimination of implicit time integration schemes as possible
candidates for solving governing equations. Explicit time integration schemes for both
deformable and rigid discrete elements are used instead. In a similar way, for static prob-
lems any solver involving the assembly of a stiffness matrix would require unreasonable
CPU times, and would be coupled with huge algorithmic difficulties.
Thus in the combined finite-discrete element method, the state of rest and static cases
in general are treated as special cases of transient dynamics problems, where energy
dissipation mechanisms are such that in a relatively short time a state of rest is achieved
or the load is applied at a such a slow rate that no dynamic effects are induced. The
method is called dynamic relaxation.
In dynamic relaxation the static system is replaced by an equivalent transient dynamic
system
Kx + M¨x + C˙x = p (5.126)

or less often with
Kx + C˙x = p (5.127)

where K is the stiffness matrix, M is the mass matrix and C is the damping matrix.
The dynamic relaxation is said to converge if the steady state solution of the equiva-
lent dynamic system is identical to the static solution. By definition, dynamic relaxation
assumes an explicit time integration of the equivalent dynamic system. A wide range of