Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

716 BAHATTIN KILIC AND ERDOGAN MADENCI
in which c is the damping coefficient and � is the fictitious diagonal density matrix that is expressed in
[Kilic 2008] as
λii ≥ 1
4 �t2 � N Ne � � 18k 1
max abs(ξ)w j e−(|ξ|/l)2
πl 4 |ξ| 2
�
, (11)
e=1 j=1
where max is the function that returns the value of the maximum component of the three-dimensional
vector in its argument. Hence, the densities associated with a particular material point are the same
in every direction of the coordinate frame, making them frame invariant. The function abs(·) returns
a three-dimensional vector whose components are the absolute values of the three components of the
vector in its argument. The vectors X and U represent positions and displacements at the collocation
points, respectively, and they can be expressed as
X T = {x, x2, . . . , xM}, (12)
U T = {u(x1, t), u(x2, t), . . . , u(xM, t)}, (13)
where M is the total number of collocation points. Finally, the vector F is the summation of internal and
external forces, and its i-th component can be written as
Fi = b(xi, t) +
N� Ne �
e=1 j=1
w j f (u(xi, t), u(x ′ k , t), xi, x ′ k , t)). (14)
3.1. Finite element equations. The theory for the development of the finite element method is well
established [Belytschko 1983; Bathe 1982; Zienkiewicz 1977]; however, this section briefly describes
the assembly of finite element equations and the solution of the assembled equations using the adaptive
dynamic relaxation technique. The finite element formulation utilized in this study can be found in
most finite element textbooks such as [Zienkiewicz 1977]. Hence, details will not be given here but the
interested reader can refer to [Kilic 2008]. The present study utilizes direct assembly of finite element
equations without constructing the global stiffness matrix. Hence, the element stiffness vector can be
expressed as
f (e) = k (e) u (e) , (15)
in which k (e) is the element stiffness matrix described in [Zienkiewicz 1977] and u (e) is the vector
representing the nodal displacements of the e-th element. The element stiffness vector, f (e) , includes
internal forces resulting from the deformation of the element. They can be assembled into a global array
of internal forces by using the convention of [Belytschko 1983] as
f int = A e f (e) , (16)
where A is the assembly operator. These operations are strictly performed as additions. Finally, the
equations of motion for adaptive dynamic relaxation can be written as
ü n + cn ˙u n = M −1 F n , (17)
in which M is the mass matrix, c is the damping coefficient, and n indicates the n-th time increment.
The force vector F can be expressed as
F n = f ext (t n ) − f int (u n ) (18)