ORIGINAL PAPER

A New Viscous Boundary Condition in the Two-DimensionalDiscontinuous Deformation Analysis Method for WavePropagation Problems

Huirong Bao • Yossef H. Hatzor • Xin Huang

Received: 5 December 2011 / Accepted: 17 March 2012

� Springer-Verlag 2012

Abstract Viscous boundaries are widely used in numer-

ical simulations of wave propagation problems in rock

mechanics and rock engineering. By using such bound-

aries, reflected waves from artificial boundaries can be

eliminated; therefore, an infinite domain can be modeled as

a finite domain more effectively and with a much greater

accuracy. Little progress has been made, thus far, with the

implementation and verification of a viscous boundary in

the numerical, discrete element, discontinuous deformation

analysis (DDA) method. We present in this paper a new

viscous boundary condition for DDA with a higher

absorbing efficiency in comparison to previously published

solutions. The theoretical derivation of the new viscous

boundary condition for DDA is presented in detail, starting

from first principles. The accuracy of the new boundary

condition is verified using a series of numerical benchmark

tests. We show that the new viscous boundary condition

works well with both P waves as well as S waves.

Keywords Discontinuous deformation analysis �Dynamic analysis � Viscous boundary condition �Non-reflecting boundary � Absorbing boundary condition �Wave propagation � P wave � S wave

1 Introduction

Artificial boundaries in numerical analysis may introduce

spurious reflected waves which will inevitably contaminate

the numerical solution. In order to properly simulate the

dynamic response of an infinite domain with a finite model,

it is necessary to eliminate such reflected waves. In theory,

this problem could be overcome by constructing a model

sufficiently large so as to obtain the required solution

before the reflected waves arrive. However, such an

approach is not always feasible because, in most cases, the

numerical model size is limited by the available computer

storage and CPU efficiency. Therefore, the correct imple-

mentation of a viscous boundary with high absorbing

capacity is an important prerequisite for accurate and

realistic numerical simulations of dynamic processes

involving wave propagation.

Stability analyses of underground structures such as

tunnels and caverns typically involve an infinite half-space.

Historically, discontinuous deformation analysis (DDA)

models attempting to address this problem utilized artificial

boundaries, almost without exception. Yet, the original

DDA code only provides free ends and pin points as

boundary conditions. In static analysis, a displacement

boundary condition can be applied realistically at the

artificial boundaries with good accuracy (Yeung 1991). In

dynamic analysis, however, non-reflecting boundary con-

ditions are necessary in order to simulate the infinite

domain with a finite model.

Lysmer and Kuhlemeyer (1969) have formulated a

system of dashpots positioned at an artificial boundary that

are capable of damping out most of the reflections. Since

their model is independent of wave frequency, their sug-

gested viscous boundary can absorb both harmonic and

non-harmonic waves. Smith (1974) presented a mathematical

H. Bao � Y. H. Hatzor

Department of Geological and Environmental Sciences,

Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel

X. Huang (&)

School of Civil Engineering, Tianjin University,

Tianjin 300072, China

e-mail: huangxin.cliff@gmail.com

123

Rock Mech Rock Eng

DOI 10.1007/s00603-012-0245-y

procedure which is independent of both frequency and

incidence angle for completely eliminating boundary reflec-

tions. His procedure involves the superposition of solutions

which satisfy both the Dirichlet and the Neumann boundary

conditions, and is, therefore, much more costly, computa-

tionally, than using a single solution. Berenger (1994)

introduced the perfectly matched layer (PML) method which

provides a non-reflecting interface between the region of

interest and the PML layers at all incident angles. Although

the PML method was firstly introduced for the finite differ-

ence time domain method (FDTD) to solve domain trunca-

tion problems in electromagnetic radiation and scattering

problems, it was also proved to be suitable for elastic wave

propagation problems. Liu and Quek Jerry (2003) provided a

mixed method which employs an artificial boundary section

with increasing damping along the wave propagation path

towards the viscous boundary suggested by Lysmer and

Kuhlemeyer (1969).

Jiao et al. (2007) were the first to apply a viscous

boundary in the DDA method based on the standard vis-

cous boundary condition provided in the original DDA

formulation [see Eq. (8) below]. Gu and Zhao (2009) also

applied the same viscous boundary condition in their work.

We show in this paper that results obtained with the pre-

viously published viscous boundary formulae do not pro-

vide an ideal absorbing efficiency as would be desired. In

fact, the absorbing efficiency obtained with the imple-

mented viscous boundary in the DDA method by Jiao et al.

(2007) and Gu and Zhao (2009) proves to be lower than

that obtained with the original viscous boundary condition

implemented by Lysmer and Kuhlemeyer (1969) in the

finite element method.

In this paper, a new viscous boundary submatrix with

high absorbing efficiency developed specifically for DDA

is derived, based on the viscous boundary condition orig-

inally introduced by Lysmer and Kuhlemeyer (1969). In

the present derivation, the analytical velocity of a dashpot

is employed instead of using the finite difference method to

obtain the velocity. The complete derivation of the theo-

retical formulae for the proposed viscous boundary con-

dition for DDA is provided below. Verification of the

formulae is obtained by means of comparisons with well-

established analytical solutions. Finally, we show that the

newly developed viscous boundary condition for DDA

works well with both P and S waves in one-dimensional

wave propagation problems.

2 Theory of the Viscous Boundary Condition

The viscous boundary condition provided by Lysmer and

Kuhlemeyer (1969) is a pair of stresses expressed as

follows:

r ¼ aqVPvn

s ¼ bqVSvt

(ð1Þ

where r and s are the normal and shear stress on the

boundary, respectively; vn and vt are the normal and tan-

gential particle velocities of the boundary, respectively; qis the unit mass; VP and VS are the velocities of P waves

and S waves in the boundary material, respectively; and

a and b are dimensionless parameters.

According to Lysmer and Kuhlemeyer (1969), the

standard viscous boundary corresponding to the choice of

a = b = 1 provides maximum wave absorption. However,

the absorption cannot be perfect over the whole range of

incident angles by any choices of a and b.

The viscous boundary condition corresponds to a situ-

ation in which the boundary is supported by infinitesimal

dashpots oriented normal and tangential to the boundary

(Fig. 1). The resistant forces provided by the dashpots on

the edge of a boundary block are:

Fn ¼ �Zl0

0

aqVPvntdl

Ft ¼ �Zl0

0

bqVSvttdl

8>>>>>>><>>>>>>>:

ð2Þ

where l0 is the length of the boundary to which the dash-

pots are attached; t is the thickness of the domain and, in

two-dimensional problems, it is assumed to be a unit

thickness.

If the rotation of the boundary edge is small, it may be

assumed that vn and vt are constants along the block edge.

Then, Eq. (2) can be written as:

vicous boundary

excited zone

Fig. 1 Illustration of the concept of a viscous boundary in numerical

simulations of dynamic problems using a finite domain to represent an

infinite domain

H. Bao et al.

123

Fn ¼ �aqVPvnl0

Ft ¼ �bqVSvtl0

(ð3Þ

which is equivalent to placing a pair of dashpots on the

midpoint of the edge. And the damping coefficients of the

dashpots are:

cn ¼ aql0VP

ct ¼ bql0VS

(ð4Þ

for the normal and shear directions, respectively.

From the theory of elasticity, the P and S wave veloci-

ties for the boundary material are:

VP ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEð1� lÞ

q0ð1þ lÞð1� 2lÞ

sð5Þ

Vs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

E

2q0ð1þ lÞ

sð6Þ

for P wave and S wave, respectively, where q0 is the

density, E is the Young’s modulus, and l is the Poisson’s

ratio of the boundary block material.

The normal and tangential particle velocities of the

boundary can be obtained by the following relationships in

Cartesian coordinates:

vn ¼ vx sin a� vy cos a

vt ¼ vx cos aþ vy sin a

(ð7Þ

where a is the direction angle of the boundary edge cor-

responding to the x-axis (see Fig. 2); vx and vy are the

particle velocities in the x and y directions, respectively.

The positive direction of the tangential velocity on the

boundary is defined by the anticlockwise direction along

the boundary of the domain of interest, i.e., the internal

region is always on the left-hand side of the positive tan-

gential velocity. The positive direction of the normal

velocity is defined by the direction rotated 90� clockwise

from the positive tangential direction (see Fig. 2).

3 Viscous Boundary Submatrix in the DDA Method

In the DDA method, the simultaneous equilibrium equa-

tions can be derived from the minimization of the total

potential energy (P) of the system as follows:

M€dþ C _dþKd ¼ f ð8Þ

where M, C, and K are the mass matrix, damping matrix,

and stiffness matrix, respectively; d is the displacement

unknowns; and f is the force vector.

In a two-dimensional DDA model with n blocks, the

basic element is a block with six unknowns:

di ¼ u0 v0 r0 ex ey cxy

� �T

i; ði ¼ 1; 2; � � � ; nÞ ð9Þ

where (u0, v0) are the rigid body translations; r0 is the

rotation angle of the block with respect to the rotation

center at (x0, y0); and ex, ey, and cxy are the normal and

shear strains of the block, respectively.

As shown by Shi (1988), the complete first-order

approximation of displacements at any point (x, y) has the

following form:

ux

uy

� �i

¼ Tidi; ði ¼ 1; 2; � � � ; nÞ ð10Þ

where

Ti ¼1 0 �ðy� y0Þ ðx� x0Þ 0

ðy�y0Þ2

0 1 ðx� x0Þ 0 ðy� y0Þ ðx�x0Þ2

" #i

ð11Þ

By adopting the first-order displacement approximation,

the stresses and strains are constants in a block, thus,

limiting the accuracy of the DDA method when dealing

with wave propagation problems.

Assuming that the velocity at the beginning of the time

interval is _d0; which can be obtained from the last time

step, and the time interval is D, then the acceleration and

velocity of a block can be written as:

x

y

vt

vnx

y

vt

vn

vn

vn

vn

vt

vt

vt

Fig. 2 Sign convention for the

positive direction of velocities

on the boundary

A New Viscous Boundary Condition in the Two-Dimensional DDA

123

€d ¼ 2

D2ðd� t _d0Þ;

_d ¼ 2

Dd� _d0

8>><>>: ð12Þ

By substituting Eq. (12), Eq. (8) can be rewritten as:

K̂d ¼ f ð13Þ

where K̂ is the equivalent global stiffness matrix.

Equation (13) can be written in submatrix form as:

K11 K12 K13 � � � K1n

K21 K22 K23 � � � K2n

K31 K32 K33 � � � K3n

..

. ... ..

. . .. ..

.

Kn1 Kn2 Kn3 � � � Knn

2666664

3777775

d1

d2

d3

..

.

dn

8>>>>><>>>>>:

9>>>>>=>>>>>;¼

f1

f2

f3

..

.

fn

8>>>>><>>>>>:

9>>>>>=>>>>>;

ð14Þ

where Kijði; j ¼ 1; 2; � � � ; nÞ are 6 9 6 submatrices; di and

f iði ¼ 1; 2; � � � ; nÞ are 6 9 1 submatrices corresponding to

block i.

In the DDA method, the global stiffness matrix K̂ is

obtained by assembling different submatrices, such as the

elastic submatrix, the inertial submatrix, and the constraint

submatrix, etc. All submatrices can be derived from their

corresponding potential energy formulae. Below, we derive the

viscous boundary submatrices. Instead of using the finite dif-

ference method to obtain the velocity of the dashpots, as done

by previous researchers, here, we employ an analytical solu-

tion to formulate new viscous boundary submatrices, resulting,

as will be shown later, in higher absorbing efficiency.

The viscous boundary submatrices are derivatives of the

potential energy stored in the boundary dashpots. In a

single time step, the potential energy in a dashpot must

equal the work done by the reacting force of the dashpot.

Furthermore, the viscous force from a dashpot is assumed

to be proportional to the velocity of the dashpot at the

attaching point, which can be expressed by:

FðtÞ ¼ �c _uðtÞ ð15Þ

where c is the damping coefficient.

At equilibrium, the boundary block satisfies:

m€uðtÞ þ c _uðtÞ ¼ 0 ð16Þ

where m is the mass of the boundary block.

It is important to note here that the gravity force term in

DDA is controlled by the density and unit weight of the

block material, as input by the user. Namely, the gravity

force term can be applied in the material matrix for blocks,

if and when necessary. Therefore, the scope of the current

method is not limited to the solution of problems where

gravity is ignored, even though a gravity force term is not

included in Eq. (16).

Equation (16) can be solved by assuming the following

initial conditions:

_uð0Þ ¼ v0; uð0Þ ¼ 0 ð17Þ

then:

_uðtÞ ¼ v0e�ctm ð18Þ

uðtÞ ¼ m

cðv0 � v0e�

ctmÞ ð19Þ

Combining Eqs. (18) and (19), the velocity of a

boundary block is obtained as:

_uðtÞ ¼ v0 �c

muðtÞ ð20Þ

The velocity at the end of each time step can be obtained

by Eq. (18) as:

v1 ¼ _uðDÞ ¼ v0e�cDm ð21Þ

Since the time step interval D is small:

uðDÞ � v0 þ v1

2D ð22Þ

then:

v0 ¼2uðDÞ

Dð1þ e�cDm Þ

ð23Þ

The differential potential energy of a dashpot is:

dP ¼ �FðtÞdu ¼ c _u2ðtÞdt ð24Þ

The integral of Eq.(24) over each time step is:

P ¼ZD

0

c _u2ðtÞdt ¼ c

ZD

0

v0 �c

muðtÞ

� �_uðtÞdt

¼ cv0uðDÞ � c2

2mu2ðDÞ ð25Þ

For simplicity, let u(D) = u in the following content;

therefore, Eq. (25) can be written as:

P ¼ cv0u� c2

2mu2 ð26Þ

The above derivation is valid for both normally and

tangentially oriented dashpots. Applying Eq. (26) to both

normal and shear dashpots, the total potential energy is:

Pc ¼ Pn þPt ¼ cnv0nun �c2

n

2mu2

n þ ctv0tut �c2

t

2mu2

t ð27Þ

where the subscript n means normal and the subscript t

means tangential.

Rewriting Eq. (27) in matrix form:

Pc ¼ v0n v0tf g cn 0

0 ct

un

ut

� �

� 1

2mun utf g c2

n 0

0 c2t

un

ut

� �ð28Þ

H. Bao et al.

123

where

un

ut

� �¼ LTd ð29Þ

L ¼ sin a � cos acos a sin a

ð30Þ

Let:

v0 ¼v0n

v0t

� �ð31Þ

C ¼ cn 0

0 ct

ð32Þ

where cn = ql0VP and ct = ql0VS.

By substituting Eqs. (29)–(32) into Eq. (28), the total

potential energy is now:

Pc ¼ vT0 CLTd� 1

2mdTTTLTCTCLTd ð33Þ

Let:

oPc

od¼ 0 ð34Þ

TTLTCv0 �1

mTTLTCTCLTd ¼ 0 ð35Þ

The initial velocity vector can be obtained from Eq.(23)

as:

v0 ¼1

DDu ¼ 1

DDLTd ð36Þ

where

D ¼ 2ð1þ e�cnDm Þ�1

0

0 2ð1þ e�ctDm Þ�1

" #ð37Þ

Substituting into Eq. (35), then:

1

DTTLTCDLT� 1

mTTLTCTCLT

� �d ¼ 0 ð38Þ

The obtained submatrix of the viscous boundary is given

by Eq. (39), whereas that introduced by Jiao et al. (2007) is

given by Eq. (40):

Kii ¼1

DTTLTCDLT� 1

mTTLTCTCLT ð39Þ

where the subscript i corresponds to the index of each

boundary block.

K0ii ¼2

DTTLTCLT ð40Þ

Analyzing Eqs. (39) and (40) readily reveals that

K 0ii � 2Kii:

The potential energy stored in the dashpots must be

positive, i.e., the viscous force does a negative work on the

system. Hence:

CD

D� CTC

m[ 0 ð41Þ

2ð1þ e�cnD

m Þ�1

D� cn

m[ 0

2ð1þ e�ctDm Þ�1

D� ct

m[ 0

8>>><>>>:

ð42Þ

Since 2 1þ e�cnD

m

� ��1

[ 1 and 2 1þ e�ctDm

� ��1

[ 1;

Eq. (42) can be replaced by a more restricted inequality:

D\ minm

cn

;m

ct

� �ð43Þ

Because cn [ ct, the above inequality is equivalent to:

D\m

cn

ð44Þ

Hence, when time step size satisfies Eq. (44), the

viscous boundary can be guaranteed to absorb energy, and

the smaller the time step, the higher the efficiency.

Furthermore, a time step size of D[ 2m/ct must be

avoided because, in such a case, the viscous boundary will

become an excitation source which will release, rather than

absorb, energy into the system.

4 Verification

The viscous boundary submatrix expressed by Eqs. (39)

and (40) are programmed into the original DDA code. The

corresponding boundaries are referred to as Viscous

Boundary I (VB I, the new viscous boundary presented in

this study) and Viscous Boundary II (VB II, the viscous

boundary proposed by Jiao et al. 2007). In the following

benchmark tests, the viscous boundaries are used in the

analysis of one-dimensional P and S wave propagation

problems. Since the tests focus on the absorption efficiency

of reflected waves, the analytical curve at each measure-

ment point is not shown.

4.1 One-Dimensional P Wave Propagation

An elastic bar (Fig. 3) is used to illustrate the performance

of the viscous boundary conditions in one-dimensional P

wave propagation problems. The bar consists of 100

blocks, where each block is of dimensions 1 9 1 m2. To

prevent attenuation when waves propagate through the

joints, a high-strength joint material (friction angle 45�,

cohesion 20 MPa, and tensile strength 10 MPa) is used to

stick all blocks together as a stiff bar. The parameters of the

joint material are meaningless once they are high enough to

prevent the block from separating. The mechanical prop-

erties of the block material are: density 2,650 kg/m3,

A New Viscous Boundary Condition in the Two-Dimensional DDA

123

Young’s modulus 50 GPa, and Poisson’s ratio 0.25. The

stiffness of contact spring is 2,500 GN/m, which is 50

times the Young’s modulus value.

A one-cycle sinusoidal longitudinal stress wave is gen-

erated by a horizontal point force applied on the left end of

the bar. The frequency of the applied stress wave is

100 Hz, and its amplitude is 1,000 KN. The time history of

the stress wave is shown in Fig. 4. Since the bar section

area is 1 m2, the amplitude of the stress wave is 1 MPa.

Two measurement points are placed at 50.5 m (M1) and

98.5 m (M2) away from the left end, respectively. At the

right end of the bar, different boundary conditions (free

boundary, VB I, and VB II) are applied in a series of tests.

Three different time step sizes (1e–5, 5e-5, and 1e-

4 s) are tested for the three boundary conditions. The

horizontal normal stresses obtained from the original DDA

with the free boundary are shown in Figs. 5 and 6 for

measurement points M1 and M2, respectively. The results

reveal that, in a large time step size test (D = 1e-4 s),

there is an obvious attenuation of the wave energy. This

attenuation is caused by the so called ‘‘algorithmic damp-

ing’’ (Doolin and Sitar 2004) or ‘‘numerical damping’’

(Ohnishi et al. 2005) in the DDA method. The numerical

damping, as implemented in DDA, may be viewed as a

deliberate solution error in numerical methods that results

from the employed time integration scheme. This error

should be reduced by a proper choice of analysis parame-

ters. Generally in the DDA method, a small time step size

will guarantee better accuracy.

The results from VB II are shown in Figs. 7 and 8.

Figure 7 reveals that the reflected wave has the same sign

as that of the incident wave, which means that the prop-

erties of VB II are deflected to properties of a rigid

boundary. In Fig. 7, there is an attenuation of the incident

wave for a time step size of 1e-4 s, and the same phe-

nomenon happens in the reflected wave due to the

numerical damping. In Fig. 8, because of superposition

with the reflected wave, the measured amplitude of the

incident wave is larger than the analytical result.

The results from a time step size of 1e-4 s are closer to the

analytical value (1 MPa) than with the smaller time step

1m

1m

Measure pointM1

Loading point M2

100m

50.5m 48m

Fig. 3 Configuration of the

elastic bar for one-dimensional

P wave propagation

0.000 0.005 0.010 0.015 0.020 0.025 0.030

-1000

-500

0

500

1000

inpu

t for

ce (

KN

)

time (s)

100Hz50hz

Fig. 4 Time history of the input force

0.00 0.01 0.02 0.03 0.04 0.05

-1.0

-0.5

0.0

0.5

1.0

x-no

rmal

str

ess

(MP

a)

time (s)

free-M1(1e-5)free-M1(5e-5)free-M1(1e-4)

Fig. 5 rx at measurement point M1 from the free boundary condition

under different time step sizes

0.00 0.01 0.02 0.03 0.04 0.05

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

x-no

rmal

str

ess

(MP

a)

time (s)

free-M2(1e-5)free-M2(5e-5)free-M2(1e-4)

Fig. 6 rx at measurement point M2 from the free boundary condition

under different time step sizes

H. Bao et al.

123

sizes, because it has a higher numerical damping ratio, but

it does not guarantee a higher absorbing efficiency.

The absorbing efficiency is defined as:

g ¼ Aij j � Arj jAij j

� 100 % ð45Þ

where Ai and Ar are the amplitudes of the incident wave

and the reflected wave, respectively. Accordingly, the

absorbing efficiencies of VB II are 62.3, 62.2, and 62.2 %

for time step sizes 1e-5, 5e-5, and 1e-4 s, respectively.

For VB I, the horizontal normal stresses at M1 and M2

are shown in Figs. 9 and 10, respectively. The absorbing

efficiencies are 96.2, 98.6, and 91.1 % for the same time

steps as before. Our results show that VB I provides a

better absorbing efficiency when a proper time step size is

used [see Eq. (44)]. With decreasing time step size, the

reflected wave varies from a tensional wave to a

compressional wave, i.e., the viscous boundary condition

varies from a free boundary condition to a rigid boundary

condition, but the deflection will never reach the level of

VB II.

It is found that using a time step size of 1e-5 s in this

case provides the least attenuation of the incident waves

among these tests. Accordingly, when this time step size is

used, comparison between VB I and VB II provides the

most meaningful information. These comparison results are

shown in Figs. 11 and 12. Inspection of these figures

reveals that VB I is more efficient than VB II.

In order to show that VB I is independent of wave fre-

quency, a 50-Hz incident wave with the same amplitude is

used (see Fig. 4) and the stress results measured at M1 and

M2 are shown in Figs. 13 and 14, respectively. The results

show that VB I is, indeed, independent of the incident wave

frequency and works well with different wave frequencies.

x-no

rmal

str

ess

(MP

a)

time (s)

Fig. 7 rx at measurement point M1 from VB II under different time

step sizes

x-no

rmal

str

ess

(MP

a)

time (s)

Fig. 8 rx at measurement point M2 from VB II under different time

step sizes

x-no

rmal

str

ess

(MP

a)

time (s)

Fig. 9 rx at measurement point M1 from VB I under different time

step sizes

x-no

rmal

str

ess

(MP

a)

time (s)

Fig. 10 rx at measurement point M2 from VB I under different time

step sizes

A New Viscous Boundary Condition in the Two-Dimensional DDA

123

x-no

rmal

str

ess

(MP

a)

time (s)

Fig. 11 Comparison of rx at measurement point M1 from different

boundary conditions with a time step size of 1e-5 s

x-no

rmal

str

ess

(MP

a)

time (s)

Fig. 12 Comparison of rx at measurement point M2 from different

boundary conditions with a time step size of 1e-5 s

x-no

rmal

str

ess

(MP

a)

time (s)

Fig. 13 rx at measurement point M1 obtained by different boundary

conditions for the 50-Hz incident wave case

x-no

rmal

str

ess

(MP

a)

time (s)

Fig. 14 rx at measurement point M2 obtained by different boundary

conditions for the 50-Hz incident wave case

M0

M1

M2

50m

1x50

=50

m

30.5

m18

.5m

shaking base

Fig. 15 Configuration of the model for one-dimensional S wave

propagation

0.000 0.005 0.010 0.015 0.020

-0.02

-0.01

0.00

0.01

0.02

x-di

spla

cem

ent (

m)

time (s)

Fig. 16 Time history of the horizontal displacement of the shaking

base

H. Bao et al.

123

4.2 One-dimensional S Wave Propagation

A structure of horizontal layers (see Fig. 15) is modeled to

illustrate the performance of VB I in the context of one-

dimensional S wave propagation problems. The model

consists of 50 thin blocks, each of which is 50 m wide and

1 m thick, resting on a rigid shaking base. The incident

wave is generated by horizontal movement of the rigid

base. The time history of the horizontal displacement is

shown in Fig. 16. The block width is very large compared

with its thickness, so as to approach a vertical one-

dimensional S wave propagation.

A high-strength joint material (friction angle 45�,

cohesion 20 MPa, and tensile strength 10 MPa) is used

again to prevent joints dilation during vertical shear wave

propagation. The mechanical properties of the block

material are: unit mass 2,650 kg/m3, Young’s modulus

50 GPa, and Poisson’s ratio 0.25. The time step size is set

at 1e-5 s, which is small enough to guarantee accuracy of

the solution in this case. The stiffness of contact spring is

2,500 GN/m.

Three measurement points are located in the layered

structure as follows: M0 is placed at the center of the

shaking base to monitor the input displacements; M1 and

M2 are placed at 30.5 and 49 m above M0, respectively.

Three different boundary conditions, (1) free boundary

condition, (2) VB I, and (3) VB II are applied at the surface

of the top block in different tests.

The structural responses with the three boundary con-

ditions are shown in Figs. 17 and 18. The graphical outputs

clearly indicate that VB I provides a better absorbing

efficiency than VB II. The absorbing efficiencies of VB I

and VB II are 87.8 and 54.7 %, respectively. The absorbing

efficiency of VB I for S waves is lower than for P waves,

due to the way the DDA program treats shear and normal

contact stiffness. Normally, it is recommended to use a

spring stiffness of 40 times the Young’s modulus of the

rock elements in standard DDA simulations. This recom-

mendation works well for the normal contact spring. The

stiffness of the shear contact spring, however, is automat-

ically scaled down to 40 % of the assigned normal spring

stiffness. This numerical issue may lead to some inaccu-

racies in the DDA simulations of S wave propagation

problems.

To demonstrate this issue, a test with higher contact

stiffness is carried out. The new contact stiffness of the

normal contact spring is set to 12,500 GN/m, which is 250

times the rock’s Young’s modulus, and the shear contact

stiffness is now 100 times the Young’s modulus, accord-

ingly. The measured results are shown in Figs. 19 and 20.

The absorbing efficiencies of VB I and VB II increase to

97.0 and 63.0 %, respectively. In each viscous boundary, a

higher absorbing efficiency is obtained than previously,

Fig. 17 Horizontal displacements at measurement point M1 obtained

by different boundary conditions

Fig. 18 Horizontal displacements at measurement point M2 obtained

by different boundary conditions

Fig. 19 Horizontal displacements at measurement point M1 obtained

by different boundary conditions with contact stiffness at 250E

A New Viscous Boundary Condition in the Two-Dimensional DDA

123

when a low shear contact spring stiffness was used. These

results imply that, clearly, the accuracy of any imple-

mented viscous boundary depends on the accuracy of the

DDA solution.

5 Summary and Conclusions

A new viscous boundary condition is provided for the

discontinuous deformation analysis (DDA) method. The

implementation of a viscous boundary in the artificial

domain boundaries in numerical simulations is necessary

for the accurate representation of an infinite medium using

a finite domain. The submatrix of the viscous boundary

condition is derived from the potential energy of dashpots.

The proposed submatrix is easy to implement in the ori-

ginal DDA code, and proves to provide good absorbing

efficiency in both P wave and S wave propagation prob-

lems. The absorbing efficiency depends on the accuracy of

the DDA solution, which can be optimized by choosing to

use properly conditioned numerical control parameters

such as the time step size and the stiffness of contact

springs.

In the verification section, the new viscous boundary

condition is applied for one-dimensional P wave and S

wave problems. It is shown that the proposed viscous

boundary can absorb wave energy effectively in any

direction in a two-dimensional problem. Therefore, theo-

retically, the new viscous boundary condition can be

applied to two-dimensional problems. Verifications for

two-dimensional problems is planned to be performed in

future research.

Acknowledgments This study was funded by the Israel Science

Foundation through grant ISF-2201, contract no. 556/08.

References

Berenger J-P (1994) A perfectly matched layer for the absorption of

electromagnetic waves. J Comput Phys 114(2):185–200

Doolin DM, Sitar N (2004) Time integration in discontinuous

deformation analysis. J Eng Mech 130(3):249–258

Gu J, Zhao ZY (2009) Considerations of the discontinuous deforma-

tion analysis on wave propagation problems. Int J Numer Anal

Methods Geomech 33(12):1449–1465. doi:10.1002/nag.772

Jiao YY, Zhang XL, Zhao J, Liu QS (2007) Viscous boundary of

DDA for modeling stress wave propagation in jointed rock. Int J

Rock Mech Min Sci 44(7):1070–1076

Liu GR, Quek Jerry SS (2003) A non-reflecting boundary for

analyzing wave propagation using the finite element method.

Finite Elem Anal Des 39(5–6):403–417

Lysmer J, Kuhlemeyer RL (1969) Finite dynamic model for infinite

media. J Eng Mech Div ASCE 95(EM4):859–877

Ohnishi Y, Nishiyama S, Sasaki T, Nakai T (2005) The application of

DDA to particle rock engineering problems: issue and recent

insights. In: Proceedings of the 7th International Conference on

the Analysis of Discontinuous Deformation (ICADD-7), Hono-

lulu, Hawaii, December 2005

Shi G (1988) Discontinuous deformation analysis—a new numerical

model for the statics and dynamics of block systems. Ph.D.

thesis, University of California, Berkeley

Smith WD (1974) A nonreflecting plane boundary for wave

propagation problems. J Comput Phys 15(4):492–503

Yeung MR (1991) Application of Shi’s discontinuous deformation

analysis to the study of rock behavior. Ph.D. thesis, University of

California, Berkeley

Fig. 20 Horizontal displacements at measurement point M2 obtained

by different boundary conditions with contact stiffness at 250E

H. Bao et al.

123