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New point-to-face contact algorithm for 3-D contact problems using the augmentedLagrangian method in 3-D DDA
S. Amir Reza Beyabanakia,*, Roozbeh Grayeli Mikolab, S. Omid Reza Biabanakic and Soheil Mohammadid
aPooyesh Rah Mandegar Consulting Engineers, No. 48, Shahr-Tash Alley, North Sohrevardi Avenue, Tehran 1559615311, Iran; bDepartment ofCivil and Environmental Engineering, University of California, Berkeley, CA, USA; cDepartment of Civil Engineering, Sharif University of
Technology, Tehran, Iran; dSchool of Civil Engineering, University College of Engineering, University of Tehran, Tehran, Iran
(Received 13 September 2008; final version received 25 March 2009)
This paper presents a new point-to-face contact algorithm for contacts between two polyhedrons with planar boundaries. A new discrete numericalmethod called three-dimensional discontinuous deformation analysis (3-D DDA) is used and formulations of normal contact submatrices based on theproposed algorithm are derived. The presented algorithm is a simple and efficient method and it can be easily coded into a computer program. Thisapproach does not need to use an iterative algorithm in each time step to obtain the contact plane, unlike the ‘Common-Plane’ method applied in theexisting 3-D DDA. In the present 3-D DDA method, block contact constraints are enforced using the penalty method. This approach is quite simple, butmay lead to inaccuracies that may be large for small values of the penalty number. The penalty method also creates block contact overlap, which violatesthe physical constraints of the problem. These limitations are overcome by using the augmented Lagrangian method that is used for normal contacts in thisresearch. This point-to-face contact model has been programmed and some illustrative examples are provided to demonstrate the new contact rulebetween two blocks. A comparison between results obtained by using the augmented Lagrangian method and the penalty method is presented as well.
Keywords: numerical method; three-dimensional; discontinuous deformation analysis; augmented Lagrangian method; rock mechanics
1. Introduction
Many engineering materials and structures are composed of blocks
in different shapes and sizes. For example, rock masses are divided
into discrete units by joints and faults. Soils are composed of small
particles. Stones and bricks form the fabric of masonry structures.
Contact between blocks may consist of a material, such as mortar
in masonry, or it may be plain interactions of solid objects, such as
joints in rock. The explicit modelling of these contacts, represented
as structural discontinuities, is outside the capability of continuum
idealisations, which generally underlie standard finite element
models. Discrete element models are very appropriate tools to
represent blocky structures. An early numerical approach capable
of modelling the movement and interaction between distinct blocks
was introduced by Cundall under the term ‘distinct element
method’ (DEM) (Cundall 1971). More recently, the ‘discontinuous
deformation analysis’ (DDA) method was developed by Shi (1988,
1993) to model the behaviour of discontinuous media. Similar
methods for simulating blocky rock mass behaviour include the
‘block spring method’ (BSM) (Wang and Garga 1993) and ‘com-
bined DEM/FEM’ formulation (Munjiza 2004), which involves
discretising each distinct element into finite elements (e.g., the
‘discrete finite elements’). Two-dimensional discontinuous defor-
mation analysis (2-D DDA) introduced by Shi (1988, 1993) in the
late 1980s has become a rapidly developing modern numerical
simulation technique and has found wide acceptance by research-
ers and engineers in a variety of mechanical analysis applications.
In the DDA approach, equations governing the equilibrium of
discrete blocks are derived by minimising their total potential
energy. It has been shown that the kinematic interactions between
the blocks can be modelled with great accuracy (Shi 1988, 1993).
MacLaughlin and Doolin (2006) provided a review of more than
100 published and unpublished validation studies on the DDA
approach. Previous DDA studies focused on solving problems in
two dimensions, but in many engineering problems three-dimen-
sional effects have to be considered (Abe et al. 1999, Jones and
Papadopoulos 2001, Duan and Ye 2002) . Up to now, relatively
little work on DDA development in 3-D has been published. Shi
(2001a,b,c) and Wu et al. (2005) provided basic formulations for
matrices for different potential terms. Liu et al. (2004) and Yeung
et al. (2003, 2004) highlighted the application of 3-D DDA. Jiang
and Yeung (2004) developed a point-to-face model for 3-D DDA.
In these researches, contacts between the blocks are detected by
using the ‘‘Common-Plane’’ approach (1988). In this approach, a
plane is located between the blocks, so that the overlap of vertices
or edges on this plane indicates the physical interaction and defines
the contact area. The location and the orientation of the contact
plane are obtained by an iterative algorithm, which may require
many computations. This plane is updated as the blocks move.
In this paper, a new model of 3-D point-to-face contact detec-
tion and mechanics is developed and formulations of normal and
Geomechanics and Geoengineering: An International Journal
Vol. 4, No. 3, September 2009, 221--236
*Corresponding author. Email: a.beyabanaki@gmail.com
ISSN 1748-6025 print=ISSN 1748-6033 online� 2009 Taylor & FrancisDOI: 10.1080=17486020903045370http:==www.informaworld.com
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shear contacts submatrices based on this new model of contact
are presented for 3-D DDA. In the proposed method, a direct
algorithm for contact resolution is provided. The sequence of
steps allows decisions regarding whether the blocks are actually
in contact and the identification of the contact type and geometry
parameters are obtained, depending on the possible block posi-
tions. In this way, unlike the ‘‘Common-Plane’’ approach, an
iterative algorithm in each time step, which may require many
computations, is not required. Also in this new method, the
contact mechanics computation does not need to project vertices
and simply uses only coordinates of block vertices, in opposition
to Jiang and Yeung’s approach(2004).
The penalty method was originally used by the above-men-
tioned 3-D DDA researchers to enforce contact constraints at the
block interface. The accuracy of the contact solution depends
highly on the choice of the penalty number and the optimal
number cannot be explicitly found beforehand. Obviously, the
penalty number should be very large to achieve zero interpene-
tration distance. However, a very high penalty number leads to
progressive ill-conditioning of the resulting system and thus one
cannot hope to achieve high-accuracy solutions with this
approach. A well-known method to overcome these problems
for equality constrained problems is the augmented Lagrangian
method (Landers and Taylor 1985). The augmented Lagrangian
method has been advocated by Lin et al. (1996) in two-
dimensional discontinuous deformation analysis. In this research,
the same method has been implemented in three-dimensional
discontinuous deformation analysis and some illustrative exam-
ples are presented for demonstrating this new approach.
2. Three-dimensional discontinuous deformation analysis
formulation
DDA calculates the equilibrium equations by minimisation of
the potential energies of single blocks and the contacts between
two blocks. To calculate the simultaneous equilibrium equa-
tions, deformation functions must be defined. The deformation
function calculates the deformation of all the blocks using the
displacement of each block centroid.
Assuming all displacements are small and each block has
constant stress and constant strain throughout, the displacement
(u,v,w) of any point (x,y, z) of a block can be represented by 12
displacement variables. In the 12 variables, (u0,v0, w0) is the rigid
body translation of a specific point (x0,y0,z0) within the block,
(rx,ry, rz) are the rotation angle of the block with a rotation centre
(x0,y0,z0), and "x, "y, "z, �xy, �yz, �zx are the normal and shear
strains in the block. The displacement of any point (x, y, z) in the
block i can be represented by Equation (1).
uvw
� �¼ Ti½ � Di½ � ¼
¼1 0 0 0 ðz� z0Þ �ðy� y0Þ ðx� x0Þ 0 0
ðy�y0Þ2
0ðz�z0Þ
2
0 1 0 �ðz� z0Þ 0 ðx� x0Þ 0 ðy� y0Þ 0ðx�x0Þ
2
ðz�z0Þ2
0
0 0 1 ðy� y0Þ �ðx� x0Þ 0 0 0 ðz� z0Þ 0ðy�y0Þ
2
ðx�x0Þ2
264
375 �
u0v0w0rxryrz"x"y"z�xy�yz�zx
0BBBBBBBBBB@
1CCCCCCCCCCA
ð1Þ
Since 3-D DDA conforms to the minimum total potential
energy principle, the total potential energy is the summation of
all potential energy sources for each block, i.e., the potential
energy contributed by the elastic deformation of the blocks, the
initial stresses, the point load on a block, the volume forces, the
inertia forces, the potential energy when the blocks contact each
other; and the potential energy contributed by the constraint
displacement points. The fixed point is the point where the
prescribed constraint displacement equals zero in DDA.
For a system with N blocks, the total potential energy can be
expressed in matrix form as follows:
� ¼ 1
2fD1gT fD2gT fD3gT : : : fDNgT� �
½K11� ½K12� ½K13� : : : ½K1N �½K21� ½K22� ½K23� : : : ½K2N �½K31� ½K32� ½K33� : : : ½K3N �
..
. ... ..
. . .. ..
.
½KN1� ½KN2� ½KN3� : : : ½KNN �
266666664
377777775
fD1gfD2gfD3g
..
.
fDNg
266666664
377777775
þ fD1gT fD2gT fD3gT : : : fDNgT� �
·
fF1gfF2gfF3g
..
.
fFNg
266666664
377777775þ C
ð2Þ
222 S.A.R. Beyabenaki et al.
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where Dif g represents displacement variables and Fif g indi-
cates loading and moments caused by the external forces and
stress acting on block i. The stiffness submatrices ½Kii�depend
on the material properties of block i, with ½Kij� i�j being defined
by the contacts between blocks i and j; and C is the energy
produced by friction force.
If a first-order displacement function is chosen, there are 12
displacement variables for each block. As a result, Dif g and
Fif g are 12·1 matrices and ½Kij� is a 12·12 matrix.
By minimising the total energy, the simultaneous equations
can be expressed in matrix form as follows:
½K11� ½K12� ½K13� : : : ½K1N �½K21� ½K22� ½K23� : : : ½K2N �½K31� ½K32� ½K33� : : : ½K3N �
..
. ... ..
. . .. ..
.
½KN1� ½KN2� ½KN3�: : :½KNN �
2666664
3777775
fD1gfD2gfD3g
..
.
fDNg
2666664
3777775¼
fF1gfF2gfF3g
..
.
fFNg
2666664
3777775ð3Þ
For only one block, the equilibrium equations for each time
step are derived by minimising the total potential energy, �, in
each variable. For block i, equations:
@�
@u¼ 0;
@�
@v¼ 0;
@�
@w¼ 0 ð4Þ
represent the equilibrium of all loads and contact forces acting
on block i along X; Y and Z directions respectively. The
equations:
@�
@rx
¼ 0;@�
@ry
¼ 0;@�
@rz
¼ 0 ð5Þ
represent the moment equilibrium of all loads and contact
forces acting on block i. The equations:
@�@"x¼ 0; @�
@"y¼ 0; @�
@"z¼ 0
@�@�yz¼ 0; @�
@�zx¼ 0; @�
@�xy¼ 0
8<: ð6Þ
represent the equilibrium of all external forces and stresses on
block i.
The differentiations:
@2�
@dri@dsj
; r; s ¼ 1; 2; ::: ; 12 ð7Þ
form a 12·12 submatrix, which is the submatrix ½Kij� in the
global Equation (3). The differentiations:
� @�ð0Þ@dri
; r; s ¼ 1; 2; ::: ; 12 ð8Þ
are the free terms of the equilibrium equations derived by
minimising the total energy, �. Therefore, all terms of
Equation (8) form a 12·1 submatrix, which is the submatrix
Fif g in Equation (3).
An essential part of any 3-D discrete element method is a
rigorous contact model governing the interaction of many 3-D
discontinuous blocks. The contact model includes two main
steps, named ‘contact detection’ and ‘contact mechanics’. In
the next sections, these two parts of the proposed contact model
are explained.
3. Contact detection scheme
Contact detection is usually performed in two independent
stages. The first stage, referred to as neighbour search, is
merely a rough search that aims to provide a list of all possible
particles in contact. Among available algorithms for neigh-
bour searching, the most recent ones include the sweep and
prone algorithm (Cohen et al. 1995) and the spatial partition-
ing algorithm (Munjiza 2004). A review of neighbour search
methods is available in Bergen (2003). In the second stage,
which is studied in this paper, called geometric resolution,
pairs of contacting particles obtained from the first stage are
examined in more detail to find the contact points and calcu-
late the contact forces.
In the contact theory, it is necessary to determine the type of
contact between any two arbitrary convex-shaped polyhedral
blocks. The type of contact is important because it determines
the mechanical response of the contact. There are six types of
contact for 3-D blocks, i.e., vertex-to-vertex, vertex-to-edge,
vertex-to-face, edge-to-edge, edge-to-face and face-to-face.
Yeung et al. (2003) and Jiang and Yeung (2004) pointed out
that vertex-to-face, edge-to-face and face-to-face contact types
can be converted to the contact of a point to a face. In this paper,
this type of contact will be studied. In the next sections, a new 3-
D point-to-face contact detection algorithm for geometric reso-
lution with new contact force calculation formulas is presented.
This algorithm can be divided into two phases: finding contact
points between two possible blocks in contact, and identifica-
tion of contact type.
3.1 Finding contact points between probable blocksin contact
When determining the distance between two polyhedrons, the
immediate and most natural approach consists in computing
and comparing the distances between their boundary features
(vertices, edges and faces). If the number of vertices and edges
as well as faces is high, searching between all boundary features
of two objects would be costly. To overcome this problem in the
first step, the nearest vertices for two particles are computed
and the search is continued between all faces that share those
vertices in the two polyhedrons. As shown in Figure 1, if R and
Q are the nearest vertex of two polyhedrons A and B respec-
tively, a search for computing the closest point should be
carried out only on the neighbouring faces of R and Q (neigh-
bouring faces of a vertex are those that share that vertex).
Geomechanics and Geoengineering: An International Journal 223
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In order to compute contact points between blocks A and B,
the following steps should be performed:
1.0. Find vertices on the neighbouring faces of R and Q
2.0. Let T denote the plane passing through the face (polygon) f
of block B
3.0. Find vertices of step 1.0 whose distance to plane T falls
within tolerance (in this study tolerance is 2·maximum
displacement in each time step). The following steps
should be done for each of them:
3.1. Calculate projection of the vertex obtained in step 3.0
(e.g., P) on plane T (e.g., point P¢ in Figure 2)
3.2. If P¢ is inside a given polygon f (Figure 2a), this point
and P will be contact points, then Goto 4.0
3.3. Else (If P¢ is outside the polygon f (as shown in Figure
2b)), the following steps should be performed:
3.3.1. Find the nearest point of polygon f to P’ (e.g.,
point P1 in Fig 2b)
3.3.2. Find the nearest point of neighbouring faces of
R to P1 (e.g., P2)
3.3.3. If P2 is exactly P, this point and P1 will be
contact points, then Goto 4.0
3.3.4. Else find the nearest point of polygon f to P2
(e.g., P3)
3.3.4.1. If P3 is exactly P1, these two points
are contact points
3.3.4.2. Else P P2 and P1 P3 then
Goto 3.3.2
3.3.4.3. End If
3.3.5. End If
3.4. End If
4.0. End
A similar approach is proposed by Nezami et al. (2006).
3.2 Identification of contact type
If the distance from contact points to another block falls within
the tolerance, one of the following types of contact is
probable:
� If three or more points of same face fall within the tolerance,
the probability of face-to-face contact type is applicable
(Figure 3).
� If two points of same edge fall in the tolerance, the prob-
ability of edge-to-face contact type is applicable (Figure 4).
� If one vertex falls in the tolerance, the vertex-to-face type is
applicable (Figure 5).
4. Three-dimensional point-to-face contact mechanics
When a point-to-face contact candidate is found in the compu-
tation, as shown in Figure 6, the effects of the contact can be
represented by applying two stiff contact springs in the normal
and tangential directions (Wu et al. 2005). To prevent the
blocks from penetrating each other, 3-D DDA considers the
normal contact forces shown in Figure 6(a) when the blocks
Figure 1. Best candidate for finding the closest points between twopolyhedrons A and B (the neighbouring faces are shaded).
Figure 2. Projection of vertex P of block A on face f of block B (a) projectivepoint P¢ located inside the boundary of f (b) projective point P¢ located outsidethe boundary of f.
Figure 3. Face-to-face contact type.
224 S.A.R. Beyabenaki et al.
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come into contact. However, the forces are disregarded when
the blocks are separated. The usage of the normal contact spring
satisfies the no-penetration and no-tension criteria developed in
the original DDA and the open--close iteration is used to obtain
the converged results at each time step (Shi 2001). In addition, a
shear spring as shown in Figure 6(b) is activated when the shear
force is smaller than the shear resistance of a discontinuity to
diminish the relative displacement of the two blocks in the
form:
Fs < Fn tgð’Þ þ C ð9Þ
where Fs is the shear contact force, ’ is friction angle of the
discontinuity, and Fn is the normal contact force.
The penalty method is used to calculate the potential energy
caused by the contact spring in the existing 3-D DDA. The main
features of this method are (Mohammadi 2003):
� Enforcement of constraints requires no extra equations.
� The solution is easily obtained by simply adding contact
components to the stiffness matrix.
� The constraints are only satisfied in an approximate manner
and the contact solution depends highly on the choice of the
penalty number and the optimal number cannot be explicitly
found beforehand.
� If the penalty number is too low, the constraints are poorly
satisfied, while if it is too large the simultaneous equili-
brium matrix becomes difficult to solve.
In this paper, a more efficient method, named the augmented
Lagrangian method, is used and the results are compared with
the penalty method results. Since (1) shear contact spring is
applied only when the shear force is smaller than the shear
resistance of a discontinuity; (2) its calculation using the aug-
mented Lagrangian method is complicated and not numerically
economical; and (3) a major concern in contact problems is
satisfying normal contact constraints, the augmented
Lagrangian method is used only for normal contacts. The aug-
mented Lagrangian approach uses penalty stiffness but itera-
tively updates the contact traction to impose the contact
constraints with a specified precision. The main features of
this method are:
� No additional equations are required.
� Large penalty values are not required, avoiding the ill con-
ditioning of the stiffness matrices. However, if the initial
penalty number is too small, many iterations are required.
� The constraints are satisfied within a user-defined required
tolerance.
� The algorithm can be used effectively for applications
where the contact pressures become very large in compar-
ison with the material elastic parameters.
4.1 Calculation of normal contact force
When a point-to-face contact occurs, the normal spring with a
stiffness of Pn is introduced into the formulation to return
the point to the surface along the shortest distance. When
using the augmented Lagrangian method, the normal contact
force at the contact point (�n) can be accurately approximated
Figure 4. Edge-to-face contact type.
Figure 5. Vertex-to-face contact type.
Figure 6. Representation of normal and shear contacts in 3-D.
Geomechanics and Geoengineering: An International Journal 225
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by iteratively calculating the Lagrange multiplier ��n. A first-
order updated value for ��n can be written as:
�n � � �n knþ1¼ � �n kn
þ Pndn ð10Þ
where the penalty number, Pn, can be variable and does not
have to be a very large number as in the penalty method. In
Equation (10), � �n kn is the Lagrangian multiplier at the
kthn iteration and ��n knþ1
is the updated Lagrange multiplier
(Mohammadi 2003), and dn is the shortest (normal) distance
from the contact point to the contact face.
As shown in Figure 7, let P1ðx1; y1; z1Þ be a vertex of block i
before the displacement increment, and the polygon P2P3 :::Pm
is the contact face, which is a face of block j. Let ðxi; yi; ziÞ and
ðui; vi;wiÞ be the coordinates and displacement increments,
respectively, of the vertices Pi ði ¼ 1; 2; ::: ;mÞ, and let
P¢i ði ¼ 1; 2; ::: ;mÞ be the respective vertices after the displace-
ment increments are applied. The normal distance between the
P¢1 and the contact face P¢
2P¢3 :::P
¢m, dn, is given by:
dn ¼~n :P¢2P¢
1 ð11Þ
where ~n is the unit vector pointing out of the block that is
normal to the contact face P2P3 :::Pm.
If dn< 0, the point P1 will penetrate the contact face, which
means a penetration takes place.
Let
u1
v1
w1
8<:
9=; ¼ ½Tiðx1; y1; z1Þ� : fDig ð12Þ
and
ul
vl
wl
8<:
9=; ¼ ½Tjðxl; yl; zlÞ� : fDjg; l ¼ 2; 3; 4 ð13Þ
Hence, P¢2P¢
1 can be written with the following form:
P¢2P¢
1 ¼x1 þ u1
y1 þ v1
z1 þ w1
8<:
9=;�
x2 þ u2
y2 þ v2
z2 þ w2
8<:
9=;¼
x1 � x2
y1 � y2
z1 � z2
8<:
9=;þ
u1
v1
w1
8<:
9=;�
u2
v2
w2
8<:
9=;
ð14Þ
Let
fBg ¼x1 � x2
y1 � y2
z1 � z2
8<:
9=; ð15Þ
and using Equations (12) and (13), we have:
P¢2P¢
1 ¼ fBg þ ½Tiðx1; y1; z1Þ� : fDig� ½Tjðx2; y2; z2Þ� : fDjg ð16Þ
and~n can be written as:
~n ¼ P¢2P¢
3 · P¢2P¢
4
P¢2P¢
3 · P¢2P¢
4
�� �� ð17Þ
Assuming displacements of the block in a time step are small,
we have:
P¢2P¢
3 · P¢2P¢
4
�� �� @ P2P3 · P2P4j j
¼~i ~j ~k
x3 � x2 y3 � y2 z3 � z2
x4 � x2 y4 � y2 z4 � z2
������������
������������ ¼ A ð18Þ
and
P2¢P3¢ · P2¢P4¢
¼~i ~j ~k
ðx3 þ u3Þ � ðx2 þ u2Þ ðy3 þ v3Þ � ðy2 þ v2Þ ðz3 þ w3Þ � ðz2 þ w2Þðx4 þ u4Þ � ðx2 þ u2Þ ðy4 þ v4Þ � ðy2 þ v2Þ ðz4 þ w4Þ � ðz2 þ w2Þ
�������
�������¼ðy3 þ v3Þ � ðy2 þ v2Þ ðz3 þ w3Þ � ðz2 þ w2Þðy4 þ v4Þ � ðy2 þ v2Þ ðz4 þ w4Þ � ðz2 þ w2Þ
��������~i
þðz3 þ w3Þ � ðz2 þ w2Þ ðx3 þ u3Þ � ðx2 þ u2Þðz4 þ w4Þ � ðz2 þ w2Þ ðx4 þ u4Þ � ðx2 þ u2Þ
��������~j
þðx3 þ u3Þ � ðx2 þ u2Þ ðy3 þ v3Þ � ðy2 þ v2Þðx4 þ u4Þ � ðx2 þ u2Þ ðy4 þ v4Þ � ðy2 þ v2Þ
��������~k
¼ n1~iþ n2
~jþ n3~k
ð19Þ
that:
1P′
Block i
Block i
Contact face of block j
1P
2P
3P 4P
5P
6P
mP
Figure 7. Point-to-face contact.
226 S.A.R. Beyabenaki et al.
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n1 ¼1 y2 þ v2 z2 þ w2
1 y3 þ v3 z3 þ w3
1 y4 þ v4 z4 þ w4
�������
�������
¼1 y2 z2
1 y3 z3
1 y4 z4
�������
�������þ
1 y2 w2
1 y3 w3
1 y4 w4
�������
�������þ
1 v2 z2
1 v3 z3
1 v4 z4
�������
�������þ
1 v2 w2
1 v3 w3
1 v4 w4
�������
�������¼ n1ð1Þ þ n1ð2Þ þ n1ð3Þ þ n1ð4Þ
ð20Þ
n2 ¼1 z2 þ w2 x2 þ u2
1 z3 þ w3 x3 þ u3
1 z4 þ w4 x4 þ u4
�������
�������¼
1 z2 x2
1 z3 x3
1 z4 x4
�������
�������
þ1 z2 u2
1 z3 u3
1 z4 u4
�������
�������þ
1 w2 x2
1 w3 x3
1 w4 x4
�������
�������þ
1 w2 u2
1 w3 u3
1 w4 u4
�������
�������¼ n2ð1Þ þ n2ð2Þ þ n2ð3Þ þ n2ð4Þ
ð21Þ
n3 ¼1 x2 þ u2 y2 þ v2
1 x3 þ u3 y3 þ v3
1 x4 þ u4 y4 þ v4
�������
�������¼
1 x2 y2
1 x3 y3
1 x4 y4
�������
�������þ
1 x2 v2
1 x3 v3
1 x4 v4
�������
�������
þ1 u2 y2
1 u3 y3
1 u4 y4
�������
�������þ
1 u2 v2
1 u3 v3
1 u4 v4
�������
�������¼ n3ð1Þ þ n3ð2Þ þ n3ð3Þ þ n3ð4Þ
ð22Þ
Assuming the displacements of block j in a time step are small,
the second, third and fourth terms of Equations (20--22) can be
ignored. Therefore:
~n ¼ 1
An1; n2; n3h i @
1
An1ð1Þ; n2ð1Þ; n3ð1Þh i ð23Þ
From Equations (10), (16) and (23):
dn ¼1
An1ð1Þ; n2ð1Þ; n3ð1Þh i : fBg þ ½Tiðx1; y1; z1Þ� : fDigf
�½Tjðx2; y2; z2Þ� : fDjgg ð24ÞLet
G ¼ 1
An1ð1Þ; n2ð1Þ; n3ð1Þh i : fBg ð25Þ
½Hi� ¼1
An1ð1Þ; n2ð1Þ; n3ð1Þh i : ½Tiðx1; y1; z1Þ� ð26Þ
½Qj� ¼1
An1ð1Þ; n2ð1Þ; n3ð1Þh i : ½Tjðx2; y2; z2Þ� ð27Þ
Therefore:
dn ¼ Gþ ½Hi� fDig � ½Qj� fDjg ð28Þ
At the kthn iteration, the potential energy of the normal spring is
given by:
�n ¼ ��nkndn þ
1
2Pnd2
n
¼ ��nknGþ ½Hi� fDig � ½Qj� fDjg� �
þ 1
2Pn Gþ ½Hi� fDig � ½Qj� fDjg� �
: Gþ ½Hi� fDig � ½Qj� fDjg� �
ð29Þ
This equation consists of two components. The first component
is the strain energy resulting from the iteration Lagrange multi-
plier ��nkn, and the penalty constraint creates the second.
Expanding the right side of Equation (29) and minimising �n
by taking derivatives, four 12·12 stiffness submatrices and two
12·1 force submatrices are obtained in the global equilibrium
equation (Equation (3)).
The derivatives of �n:
krs ¼@2�n
@dri @dsi
rs ¼ 1; 2; ::: ; 12 ð30Þ
form a 12 · 12 submatrix which is added to the submatrix ½Kii�in Equation (3):
Pn ½Hi�T ½Hi� ! ½Kii� ð31Þ
The derivatives of �n:
krs ¼@2�n
@dri @dsj
rs ¼ 1; 2; ::: ; 12 ð32Þ
form a 12·12 submatrix which is added to the submatrix ½Kij� inEquation (3):
Pn ½Hi�T ½Qj� ! ½Kij� ð33Þ
The derivatives of �n:
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krs ¼@2�n
@drj @dsi
r; s ¼ 1; 2; ::: ; 12 ð34Þ
form a 12·12 submatrix which is added to the submatrix ½Kji� inEquation (3):
Pn ½Qj�T ½Hi� ! ½Kji� ð35Þ
The derivatives of �n:
krs ¼@2�n
@drj @dsj
r; s ¼ 1; 2; ::: ; 12 ð36Þ
form a 12·12 submatrix which is added to the submatrix ½Kjj� inEquation (3):
Pn ½Qj�T ½Qj� ! ½Kjj� ð37Þ
The derivatives of �n at 0:
fri ¼ �@�nð0Þ@dri
r ¼ 1; 2; ::: ; 12 ð38Þ
form a 12·1 submatrix which is added to the submatrix ½Fi� inEquation (3):
� ð��nknþ Pn :GÞ ½Hi� ! ½Fi� ð39Þ
The derivatives of �n at 0:
frj ¼ �@�nð0Þ@dri
r ¼ 1; 2; ::: ; 12 ð40Þ
form a 12·1 submatrix which is added to the submatrix ½Fj� inEquation (3):
� ð��nknþ Pn :GÞ ½Qj� ! ½Fj� ð41Þ
The final exact contact forces can always be obtained by the
iterative method even with small initial values of the penalty
number.
4.2 Calculation of shear contact force
Assume the point P0ðx0; y0; z0Þ denotes the projection of P1 on the
contact plane and P¢0 represents this point after the displacement
increment (Figure 8). The shear displacement is calculated as:
ds ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP¢
0P¢1j j2�d2
n
qð42Þ
The potential energy of the shear spring is given by:
�sc ¼1
2Psd
2s ¼
1
2Ps P¢
0P¢1
�� ��2�d2n
�
¼ 1
2Ps
ðx1 þ u1Þ � ðx0 þ u0Þðy1 þ v1Þ � ðy0 þ v0Þðz1 þ w1Þ � ðz0 þ w0Þ
264
375
T
:
ðx1 þ u1Þ � ðx0 þ u0Þðy1 þ v1Þ � ðy0 þ v0Þðz1 þ w1Þ � ðz0 þ w0Þ
264
375� d2
n
0B@
1CA
ð43Þ
WherePs is the stiffness of the shear spring. Let
u1
v1
w1
8<:
9=; ¼ ½Tiðx1; y1; z1Þ� : fDig ð44Þ
and
u0
v0
w0
8<:
9=; ¼ ½Tjðx0; y0; z0Þ� : fDjg ð45Þ
and using Equation (28), we have:
�sc ¼1
2Ps
x1 � x0
y1 � y0
z1 � z0
264
375
T
þfDigT ½Ti�T � fDjgT ½Tj�T
0B@
1CA
·x1 � x0
y1 � y0
z1 � z0
264
375þ ½Ti� fDig � ½Tj� fDjg
0B@
1CA
� 1
2Ps Gþ ½Hi� fDig � ½Qj� fDjg� 2
ð46Þ
By expanding and minimising the potential energy �sc, the
following matrices can be added to the submatrices
½Kii�; ½Kij�; ½Kji�; and ½Kjj� in the global stiffness matrix:
0P′
Block i
1P
Contact face of block j
2P
3P 4P
5P
6PmP
0PL
1P′
Block i
Figure 8. Illustration of 3-D contact.
228 S.A.R. Beyabenaki et al.
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½Kii� ¼ Ps½Ti�T ½Ti� � Ps½Hi�T ½Hi�
½Kij� ¼ �Ps½Ti�T ½Tj� � Ps½Hi�T ½Qj�
½Kji� ¼ �Ps½Tj�T ½Ti� � Ps½Qj�T ½Hi�
½Kjj� ¼ Ps½Tj�T ½Tj� � Ps½Qj�T ½Qj� ð47Þ
And the vectors ½Fi� and ½Fj� are calculated as follows and then
added to the global force vector:
½Fi� ¼ �Ps½Ti�Tx1 � x0
y1 � y0
z1 � z0
24
35þ Ps G ½Hi�T
½Fj� ¼ Ps½Tj�Tx1 � x0
y1 � y0
z1 � z0
24
35þ Ps G ½Qj�T ð48Þ
4.3 Calculation of frictional force
When the state of the contact is sliding, a pair of equal and
opposite frictional forces parallel to the sliding direction is
applied on the contact face at the points P1 and P0. The magni-
tudes and directions of the frictional forces are obtained from a
previous iteration. Coulomb’s law is used to evaluate the dis-
location movements of the block interfaces. When the shear
force F conforms to:
F � N tanð�Þ þ C ð49Þ
in the above formula, N is the normal force on the contact
boundary and � and C are the friction angle and cohesion,
respectively. The frictional force is calculated from the normal
contact compressive force from the previous step (Jiang and
Yeung 2004):
F ¼ �n tanð�Þ ð50Þ
where �n is taken from the previous step. Let L be the direction
of the frictional force (Figure 8) and n̂ be the unit vector
pointing out of the block in a direction normal to the contact
face. We have:
L ¼ P¢0P¢
1 � hP¢0P¢
1; n̂i: n̂ ð51Þ
Therefore, the potential energy of the friction force is given by:
�f ¼ F:ðd:LLj j Þ ð52Þ
where
d ¼ ½u1 v1 w1� � ½u0 v0 w0�¼ fDig T :½Tiðx1; y1; z1Þ� T � fDjg T :½Tjðx0; y0; z0Þ� T
therefore:
�f ¼ F:ð½Di�T ½M� � ½Dj�T ½N�Þ ð53Þ
where
½M� ¼ 1
Lj j ½Tiðx1; y1; z1Þ�T LT ð54Þ
and
½N� ¼ 1
Lj j ½Tjðx1; y1; z1Þ�T LT ð55Þ
By minimising the potential energy, we have:
½Fi� ¼ �F M½ � ð56Þ
½Fj� ¼ F N½ � ð57Þ
which are added to the global force vector.
4.4 Residual forces in augmented Lagrangian method
From physical point of view the Lagrange multiplier, �n, repre-
sents the stiffness of the normal contact between two blocks and
the penalty number, Pn, represents the stiffness of the normal
contact spring, and the final exact contact forces can always be
obtained by the iterative method. As it is mentioned by Lin et al.
(1996) the precision of the solution depends on residual forces
that are produced during the iterative calculations of contact
forces. From a physical point of view, the residual forces are the
unbalanced forces between external and internal forces. The
criterion for convergence is based on the L2 norm of the
residual forces that is (Lin et al. 1996):
½K� ½Dk� � ½K� ½Dk�1�k k½K� ½Dk�1�k k Tol
� �ð58Þ
The tolerance, Tol, is a positive number that is specified by the
user. Lin et al. (1996) reported that a Tol value ranging between
0 (corresponding to no contact) and 0.1 will give excellent
results. Algorithm 1 lists the complete DDA algorithm using
the augmented Lagrangian method.
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Algorithm 1. Discontinuous deformation analysis program
algorithm
1: Initialise data
2: For each ti, i=1, . . ., n time steps do
3: Find nearest point between probable blocks in contact
4: Find vertices falling in the tolerance
5: Find contact points
6: Compute type of contact with regard to number of falling
points in tolerance
7: Assemble stiffness matrix
8: Integrate
9: Repeat {open-close iteration}
10: Initialise �n1 ¼ 0
11: For each iteration kn
12: Solve for displacements (½K� ½D� ¼ ½F�)13: Compute penalty forces (f c
n ¼ ��n ¼ Pn · dnmax)
14: Check for convergence:
14.1: If ��n Tol1ð Þ and½K� ½Dk ��½K� ½Dk�1�k k
½K� ½Dk�1�k k Tol2
�Then
14.2: Goto 18
14.3: End If
15: update Lagrange multipliers �nknþ1¼ �nkn
þ �n
� 16: Goto 11
17: End For
18: Until no-tension, no-penetration
19: Update vertices positions
20: Update blocks stresses
21: End For
5. Examples
The algorithm described in the previous sections has been pro-
grammed in VC++. To investigate it, three examples are pre-
sented and the results are compared with the results obtained by
using the penalty method that is used in the original 3-D DDA.
5.1 Sliding of a block along an inclined plane
This example simulates the sliding of a block along an inclined
plane at an angle � to the horizontal direction with friction
angle � (Figure 9).
Under the action of gravitational force, the displacement s of the
block is determined analytically as a function of time t given as:
sðtÞ ¼ 1
2g sin�� cos� tan�ð Þ t2 ð59Þ
The inclination of the modelled plane is 20 and the density,
Young’s modulus and Poisson’s ratio for both blocks are
2:6 · 103 kg=m3, 5 GPa and 0.25, respectively. The maximum
time increment for each time step is 0.01 s.
Once the problem was solved with � ¼ 0. The accumulated
displacements are calculated up to 5 s. A comparison between
the analytical solution in Equation (59) and 3-D DDA results
using the classic penalty method and the augmented Lagrangian
method for different values of the stiffness of the normal con-
tact spring is shown in Figures 10 and 11, respectively. Figure
12 shows the displacements of a sliding block in X and Z
directions for the penalty method.
The deformation of the block system, using the augmented
Lagrangian method to enforce the contact interface, after 0 s, 3
s, 4 s and 5 s for P = 50 MN/m is shown in Figure 13. It is clear
that no block interpenetration occurs here even though the
penalty number is low. Figure 14 and Figure 15 show the
deformation of the block system, using the classic penalty
method, after 0 s, 3 s, 4 s and 5 s for P = 500 MN/m and P =
50 MN/m, respectively. It indicates that a small penalty number
is unable to enforce the interpenetration constraint.
Again the example was solved with � ¼ 10 and low value of
normal contact stiffness (Pn ¼ 50 MN=m). Figure 16 shows a
comparison between the analytical solution and 3-D DDA
results using the classic penalty method and the augmented
Figure 9. A single block sliding along an inclined plane.
–5
5
15
25
35
45
55
65
0 1 2 3 4 5Time (s)
Dis
pla
cem
ent (m
)
Analytical DDA (P = 50GN/m)
DDA (P = 500MN/m) DDA (P = 50MN/m)
Figure 10. Comparison between the analytical solution and 3-D DDA resultsusing the penalty method.
230 S.A.R. Beyabenaki et al.
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0
5
10
15
20
25
30
35
40
45
0 1 2 3 4 5
Time (s)
Dis
pla
ce
me
nt (m
)Analytical
DDA (P = 50GN/m)
DDA (P = 500MN/m)
DDA (P = 50MN/m)
Figure 11. Comparison between the analytical solution and 3-D DDA resultsusing the augmented Lagrangian method.
–5
5
15
25
35
45
55
65
0 10 20 30 40
X Direction Displacement (m)
Z D
irection D
ispla
cem
ent (m
) DDA (P = 50GN/m)
DDA (P = 500MN/m)
DDA (P = 50MN/m)
Figure 12. Sliding block displacements in X and Z directions using the penaltymethod.
Figure 13. The deformation of the block system, using the augmentedLagrangian method, after (a) 0, (b) 3, (c) 4 and (d) 5 s for P = 50 MN/m.
Figure 14. The deformation of the block system, using the classic penaltymethod, after (a) 0, (b) 3, (c) 4 and (d) 5 s for P = 500 MN/m.
Figure 15. The deformation of the block system, using the classic penaltymethod, after (a) 0, (b) 3, (c) 4 and (d) 5 s for P = 50 MN/m.
0
10
20
30
40
50
60
70
0 1 2 3 4 5
Time (s)
Dis
pla
ce
me
nt
(m)
Analytical
DDA (Penalty Method)
DDA (Augmented Lagrangin Method)
Figure 16. Comparison between the analytical solution and 3-D DDA resultsusing the augmented Lagrangian method.
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Lagrangian method. It can be seen that unlike the penalty
method, the augmented Lagrangian method is able to solve
the problem very well with low values of penalty numbers.
5.2 Block sliding under two forces
As the second example, a two-block system as shown in Figure 17
is considered. The two blocks are hexahedral, with the fixed
bottom block having dimensions of 5 m · 3:5 m · 2:5 m and
the top block having dimensions of 1 m · 1 m · 1 m. The top
block is subjected to two horizontal forces F1 and F2
ðF1 ¼ 500 N; F2 ¼ 250 NÞ. F1 acts in the x-direction at the
centre of a face parallel to the y--z plane, F2 acts in the negative
y-direction at the centre of a face parallel to the x--z plane. The
density of the blocks is � ¼ 2:0 t=m3
and Young’s modulus and
Poisson’s ratio for both blocks are 100 MPa and 0.3, respectively.
In this example, Pn ¼ 100 MN=m is assumed. The analytical
solution for displacement S as a function of time t is given by:
S ¼ 1
2at2 ¼ 1
2
F
m
� �t2 ð60Þ
where F is the force acted to the block and m is the mass of the
block.
Figure 18 shows time-dependent total displacements up to 4.5 s.
This figure shows that the analytical solutions well agree with the
results computed using the augmented Lagrangian method, but
using the classic penalty method the results are not in agreement
with the theoretical solution. Figures 19 and 20 show the deforma-
tion of the blocks, using the classic penalty method, after 0 s, 2 s,
3.5 s and 4.5 s, respectively. It indicates that a small penalty
number is unable to enforce the interpenetration constraint.
5.3 Block falling
As shown in Figure 21a, this case involves a block falling in
which the inclined plane angle is 20. The block falls freely
initially and then bounces down the slope. The penalty number
Figure 17. Initial configuration of a two-block system.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 1 2 3 4
Time (s)
Dis
pla
cem
ent (m
)
Analytical
Augmented 3D DDA
Penalty 3D DDA
Figure 18. Time-dependent total displacements using the classic penalty andthe augmented Lagrangian methods.
Figure 19. The deformation of the block system, using the augmentedLagrangian method after (a) 0, (b) 2, (c) 3.5 and (d) 4.5 s.
Figure 20. The deformation of the block system, using the penalty methodafter (a) 0, (b) 2, (c) 3.5 and (d) 4.5 s.
232 S.A.R. Beyabenaki et al.
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is assumed low in this example. Figures 21b and 21c show
results of the 3-D DDA using the classic penalty method and
the augmented Lagrangian method, respectively. They indicate
that using the classic penalty method, a small penalty number is
unable to enforce the interpenetration constraint.
Figure 21. (a) Initial configuration of block falling example. (b) Results of3-D DDA using the penalty method. (c) Results of 3-D DDA using theaugmented Lagrangian method.
Figure 22. Initial configuration of a five-block system.
Figure 23. The deformation of the block system, using the augmentedLagrangian method after (a) 2000 steps, (b) 3000 steps, (c) 4000 steps and(d) 5000 steps (Pn = 40 MN/m).
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5.4 A five-block system
This example includes a row of four blocks on one block that fall
over. As the blocks fall, many simultaneous contacts can occur
between them. The values for the elastic modulus, Poisson’s ratio
and mass density for each block are E = 3 GPa, v = 0.2, and � =
2700 kg/m3, respectively. The friction angle is 10 and the max-
imum displacement ratio allowed and the maximum time increment
for each time step are 0.1 and 0.001s, respectively. Initial config-
uration of the example is shown in Figure 22. The example was
solved for 5000 steps with two pairs of low values of contact
stiffness (Pn ¼ 40 MN=m and Pn ¼ 20 MN=m ). The deforma-
tions of the block system using the augmented Lagrangian method
and the penalty method for the first case are shown in Figures 23
and 24, respectively, and for the second case are shown in Figures
25 and 26, respectively. As can be seen in the figures, the penalty
method is unable to enforce the interpenetration constraint, but
using the augmented Lagrangian method the problem can be solved
very well even with small values of penalty numbers. This example
shows the efficiency of the proposed contact algorithm as well.
Figure 24. The deformation of the block system, using the penalty methodafter (a) 2000 steps, (b) 3000 steps, (c) 4000 steps and (d) 5000 steps (Pn = 40MN/m).
Figure 25. The deformation of the block system, using the augmentedLagrangian method after (a) 2000 steps, (b) 3000 steps, (c) 4000 steps and (d)5000 steps (Pn = 20 MN/m).
234 S.A.R. Beyabenaki et al.
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6. Conclusions
In this paper, a new algorithm to detect and calculate the 3-D
point-to-face contacts is presented and the related contact formulas
for the normal spring submatrices are derived in detail, using the
augmented Lagrangian method. The success and accuracy of the
algorithm are demonstrated through several examples involving
two or more blocks. The results presented show that the newly
developed 3-D DDA, involving the 3-D formulations and the
point-to-face contact searching algorithm, correctly simulates the
behaviour of blocks in contact with each other in the 3-D domain
successfully. Several conclusions come out from this study:
� The proposed algorithm is a simple and efficient method
and it can be easily coded into a computer program.
� Unlike the ‘Common-Plane’ method used in the present 3-D
DDA, an iterative algorithm in each time step to obtain the
contact plane is not required in this approach.
� In this new method, the contact mechanics computation does
not need to project vertices and simply uses only coordinates
of block vertices, unlike Jiang and Yeung’s (2004) approach.
� In the existing 3-D DDA, the accuracy of the contact solution
depends highly on the choice of the penalty number and the
optimal number cannot be explicitly found beforehand. These
limitations are overcome by using the augmented Lagrangian
method that is used for normal contacts in this research.
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