Page 54 - The Combined Finite-Discrete Element Method

INTRODUCTION 37

which is zero, providing that

δ (u) = 0; C(u) = 0; hence δC(u) = 0 (2.7)

In the finite element approximation, Lagrangian multipliers are approximated using the
base functions. These result in additional unknown variables, the physical meaning of
which is contact force. Thus, in the Lagrange multiplier method, contact is enforced by
increasing the total number of unknowns. In static and implicit transient dynamics prob-
lems, the Lagrange multiplier method is implemented in a rigorous manner by solving
the system of simultaneous algebraic equations. In the explicit transient dynamics prob-
lems, the system of algebraic equations is not solved and the term Lagrange multiplier
method only implies that the impenetrability conditions are satisfied approximately
(often through an iterative solution of coupled system of equations and sufficiently
small time steps).

• The penalty function method: this is introduced with the aim of eliminating the draw-
backs of the Lagrange multipliers method. To enforce contact constraints on the bound-
ary , the additional functional

T
p C (u)C(u)d (2.8)

is added to the original functional

T
˜
(u, λ) = (u) + p C (u)C(u)d (2.9)

where p is the penalty parameter. As

T
C (u)C(u)d ≥ 0 (2.10)

if is a minimum of the solution, then p must be a positive number. The solu-
tion obtained by minimising the modified functional satisfies the contact constraint
only approximately. The larger the value of penalty, the better the contact constraints
achieved. Only with an infinite penalty are the contact constraints satisfied exactly. The
penalty function method is either used to impose an impenetrability condition in an
iterative manner, or to violate this condition in such a way that the correct response
of the physical system is still recovered. This is achieved by using a sufficiently large
penalty term.

In an implementation of any of the above formulations in the combined finite-discrete
element method, the handling of kinematics of contact is of major importance. In the finite
element method, kinematics of contact is considered by employing slideline algorithms,
where one surface is designated as the master (target) surface, while the other is designated
as the slave (contactor) surface. In early algorithms, discretisation of master and slave
surfaces was often performed in such a way that each slave node had a designated master
node, thus only node-to-node contact was handled.