Page 197 - The Combined Finite-Discrete Element Method

180 TEMPORAL DISCRETISATION

The inertia of the discrete element is defined by the mass of the discrete element, which
is obtained by integration of density over the volume of the discrete element, i.e.

dm = ρdV (5.4)

Discretisation of the discrete element into finite elements also results in discretisation of
the mass. The most convenient way of discretisation of the mass used in the combined
finite-discrete element method is a so-called lumped mass approach. In essence, instead
of considering the mass being distributed over the discrete element, it is assumed that the
mass is lumped into the nodes of the finite element mesh. Thus, the mass associated with
each degree of freedom is given by

 
m 1
 m 2 
 
 m 3 
(5.5)
 
m =  ... 
 
 m i 
...
 
m n
It is worth noting that in the finite element literature, discretisation of mass is done through
the mass matrix, which is in general non-diagonal. However, elimination of non-diagonal
terms leads to a diagonal lumped mass matrix:
 
m 1
m 2
 
 
m 3
 
.. . (5.6)
 
M =  
 
m i
 
...
 
m n
where all non-diagonal terms are zero. This approach, in conjunction with the stiffness
matrix for dynamic problems, is suitable for both implicit and explicit direct integration
in the time domain.
In the context of the combined finite-discrete element method, thousands or even mil-
lions of finite element meshes are present. Deformability together with rigid rotation and
translation is considered, and contact interaction is resolved together with fracture and
fragmentation. Assembling a stiffness matrix and a non-diagonal mass matrix would lead
nowhere, for any available implicit time integration scheme could not be used without
significant modifications.
Thus in the context of the combined finite-discrete element method, no stiffness matrices
are calculated. A time integration scheme is applied on element-by-element, node-by-node
and degree of freedom by degree of freedom bases in an explicit form.
Nodal forces from:
• contact interaction,
• deformation of a discrete element,