Absorbing Boundary Conditions and Perfectly Matched Layer Model for Elastodynamic Analysis

7/22/2019 Absorbing Boundary Conditions and Perfectly Matched Layer Model for Elastodynamic Analysis

1/7

ABSORBING BOUNDARY CONDITIONS AND

PERFECTLY MATCHED LAYER MODEL FOR

ELASTODYNAMIC ANALYSIS

Josif JosifovskiUniversity Ss. Cyril and Methodius, Faculty of Civil Engineering

Skopje, R. [email protected]

Abstract

The solution of linear wave equations for unbounded domains is of special interest in variousfields of science and engineering. In particular, solution of the elastodynamic wave equation

finds application in soil wave propagation analysis which involves continuous geometriesand masses. The definition of such domain requires enforcement of radiational condition inany unbounded direction to ensure that all the energy generated inside the system and carriedout by the soil waves is absorbed.

The paper presents a perfectly matched layer formulation of an absorbing boundarycondition implemented into elastic wave equation. Enforcing the radiation condition in anunphysical layer positioned adjacent to an elastic medium creates a new set of differentialequations that govern the unbounded medium. Basically, the layer represents a perfectlyabsorbing sponge for all impending waves on the artificial boundary independent from theangle or frequency. Although the perfectly matched layer allows to be combined withdifferent numerical methods, here a second-order finite element formulation in frequency

domain has been pursued.

The proposed methodology has been verified by means of well known benchmark exampleswhich unveil the unique properties of perfectly matched layer reported in the literature.Moreover, the calibration of the condition has helped to establish certain guidelines andrecommendations for application in elastodynamics.

The main advantages of perfectly matched layer with respect to other dynamic boundarymethods and formulations have been demonstrated on a representative soil-structureinteraction problem, where two major computational aspects, such as the accuracy and

efficiency have been considered. In fact, the results had shown that the calculation of theproposed model does not only take less computation time but also leads to an improvementof the accuracy with fewer elements. All these features classify the perfectly matched layeras very efficient absorbing boundary condition in solution of not only civil or geotechnicalproblems but also in other fields of engineering.

Keywords- absorbing boundary condition, perfectly matched layer, finite elementmethod, elastic wave propagation, soil-structure interaction.

1 INTRODUCTIONThe investigation of unbounded domains is often of interest in various fields engineering. A

typical example can be found in structural dynamics, where the elastodynamic wave equationfor an unbounded domain needs to be solved in order to describe the dynamic interaction of astructure and its underlying soil. The definition of an unbounded domain [8] requires the


2/7

enforcement of a radiation condition in any unbounded direction. Irregularities in thegeometry of the domain, or in the material, often compel a numerical solution of the problemby using different mathematical formulations, e.g. coupling of the finite element method andboundary element method [7]. Another alternative, however, is an approach that uses adiscrete element method in combination with an artificial boundary taking care of theabsorption of waves travelling to infinity [10]. In the past decade quite a few techniques have

emerged which can solve the problem of the unbounded domains more or less successfully.One of them originally derived by Brenger [3] for application in electromagnetics whichhad turned out to be very promising absorbing boundary condition for linear wave equationsis so-called Perfectly Matched Layer (PML). Moreover, it has been reported in literature [5]that PML is quite accurate for elastodynamic too, independent from the wave type, angle ofincidence or frequency. In general, this approach creates a boundary layer defined with thesame material as the analyzed domain but with additionally introduced attenuation thatdamps the outgoing as well as the reflected wave within the layer. Thus, allowing unbounded

domain to be modelled by a coupled system of bounded domain, usually the region ofinterest, surrounded by a perfectly matched layer. Although the PML allows differentnumerical techniques to be combined with, here a combination with finite element method(FEM) is proposed.

2 GOVERNING EQUATIONS

A linear wave equation of viscoelastic half-plane has been derived in terms of harmonicoscillation. Implementation of the absorbing condition is equivalent to an introduction of arotated system with stretched coordinates, where the stretching parameter is a complexnumber [2]. This procedure creates a new set of differential equations governing the so-called Perfectly Matched Medium (PMM). The medium is then truncated and becomes aperfectly matched layer. Later on, a finite element model as combination of bounded domaini.e. near-field surrounded by an unphysical layer i.e. far-field is presented. In the present

work a second-order finite element formulation of the perfectly matched layer in frequencydomain is derived. Other authors such as, Collino and Tsogka [6] use the coordinatestretching idea to the velocity-stress field in finite element formulation or Hastings et al [10]in finite difference formulation.

2.1 Elastic medium

Consider homogeneous isotropic viscoelastic medium subjected to displacements of the

form ( ) ( )expx i tu in absence of body forces with circular frequency at time t. Such anexcitation produces time-harmonic motion which is governed by the following equations,

Equilibrium equation, 2ij

i

j

ux

=

(no summation over i) (1a)

Constitutive relation, ij ijkl kl C = (1b)

Kinematic relation,1

2

i

ij

j i

uu

x x

= +

(1c)

where ijklC is written in terms of the Kronecker delta ijsuch that


3/7

( )2

3ijkl ij kl ik jl il jk C

= + +

. (2)

ij and ij are the components of stress andstrain tensor, respectively. The ijklC are

components of the material stiffness tensor C , is bulk modulus, is shear modulus and

is unit mass density of the medium. If ( ), , , 1, 2i j k l then Eqs. (1) describe a two-dimensional plane-strain motion.

A viscoelastic medium governed by Eqs. (1) allows propagation of different types of elastic

waves, such as P- wave with solution [1] written in the form,

( ) [ ]expi j j P l l u x q ik x p= (3)

where p pk c= is P- wave number with velocity ( )4 3pc = + . The ip and iq are

unit vectors which denote the direction of propagation and particle motion, respectively. In

case of P- wave these vectors are related through i i iq p q= . The elastic medium also

admits S- waves with solution [1] given as,

( ) [ ]expi j j S l l u x q ik x p= (4)

wheres s

k c= is S- wave number with velocitys

c = . Hence ip and iq are

orthonormal vectors, thus related through 0i iq p = . Beside body waves Eqs. (1) also admits

Rayleigh and Stoneley which are fast decaying surface waves. The viscoelastic behaviour of

the medium is described through the correspondence principle [18], which defines complex-

valued parameters, such as bulk ( )1 2i = + or shear ( )1 2i = + modulus, using the

hysteretic damping coefficient oa = with as a damping ratio.

2.2 Perfectly matched medium

The coordinate stretch [4] creates a perfectly matched medium, simply replacing ix coordinates in governing Eqs. (1) by stretched coordinates ix , defined as

( )0

:ix

i i ix x dx= (5)

The procedure actually performs a physical mapping of the coordinates which formallyimplies

( )1

i i i ix x x

=

(6)

where ix

are real and ix are complex-stretched coordinates. The transformation through a

non-zero continuous complex-valued function ( )i ix introduces uncoupled stretches forboth mediums, bounded and infinite, thus creating a complex formulation of inhomogeneousviscoelastic perfectly matched medium [9]. Substituting Eq. (6) into Eqs. (1) creates a newset of differential equations that govern the PMM,


4/7

( )21 ij

i

jj j

uxx

=

(no summation over i) (7a)

ij ijkl kl C = (7b)

( ) ( )1 1 12

jiij

j i i ij j

uux x xx

= +

(7c)

The stretched medium represents an unbounded domain with solutions [2] of a P- wave, as

( ) [ ]expi j j P l l u x q ik x p= (8)and S- wave, as

( ) [ ]expi j j S l l u x q ik x p= (9)Consider unbounded wave propagation in 1 2x x plane where an outgoing wave spreads in

1x -direction from left medium (lm) through the interface into right medium (rm) as shown in

Figure 1.

rm

inter

face

wavedirection

lm

x2

BD

PM

x1

f(x)

0

2

2

=

2(x)=1

2

surface

f (x ) 01 1 =

1(x ) 11 =

f(x)

0

2

2

=

2(x)

1

2

=

1 1(x ) Complex=

f (x ) +Re1 1 =

Fig. 1 Adjacent PMMs as left and right media

In such case it is necessary that two adjacent perfectly matched media are defined as:

1) left halfplane ( ){ }1 2 1, 0x x x= < with ( ) ( )lmi i i ix x = , and2) right halfplane ( ){ }1 2 1, 0x x x= with ( ) ( )rmi i i ix x =

If both stretched media share the same value of i at their interface, then the matching

property will ensure that a propagating wave will pass through the interface without

reflection. If ( ) ( )1 10 0lm rm = and 2 2lm rm = then both perfectly matched media can be

considered as one with continuous 1 defined piecewise on the two half-planes (in this

case 2 has not to be considered). Besides the viscous damping there is no additional wave

attenuation inside BD , thus attenuation function is ( )1 1 0f x = which implies that the


5/7

coordinate stretch is ( )1 1 1x = . Inside PM the attenuation is defined as ( ) ( )1 1 Re pf x L+=

together with ( )1 1x complex = as a coordinate-stretching function. The same arguments

hold for the wave propagation in 2x -direction. As a special case, when 1 1 = then PMM is

transformed to homogeneous isotropic viscoelastic medium. Consequently, if lmis assumed

such that 1jlm

= then as a viscoelastic medium it borders with rm as perfectly matched

medium. In general, ( )j jx is defined [2] for both, propagating waves with 0 1a and

evanescent waves with 0 1a < , as s single stretching function with two terms,ef and

pf , as

( ) ( ) ( )

0 0

: 1

e p

j

j j

f x f xx i i

a a

= +

(10)

which attenuates both wave types, independently. In other applications, such aselectromagnetics [15], separate functions are advised, but in the case of elastic waves it is

reasonable [2] to assume same function for both types,p e

i i if f f= = , as

( ) 0m

i

i i

P

xf x f

L

=

(no summation over i) (11)

where ]( ,i Px L L is coordinate in i -th direction residing in perfectly matched medium.The attenuation function parameters: 0f as polynomial amplitude, mas a polynomial order

and PL as depth of the layer are yet to be specified. These three parameters through

attenuation directly control the performance of the condition. If optimal values are assumed

then all waves propagating outwards from bounded domainBD

will be absorbed inside the

PM

as it is shown in Figure 2.

Outgoing wave

Dirichlet BC

x1

PML

PM BD PMBD

Outgoing wave

Attenuated wave

PMM

x2

x1

x2

Reflected wave

Lp(a) (b)

Fig. 2 PMM truncation: (a) PMM adjacent to a bounded domain and (b) PML with afixed edge


6/7

If a wave that propagates out ofBD

enters PM where is absorbed enough in finite

distance PL then PM can be truncated with Dirichlet condition creating a PM , see

Figure 2(b). Thus, it is valid to assume that the displacements insideBD

of such coupled

system BD PM has to be exactly the same as those of the original unbounded medium

.

2.3 Finite element implementation

A plane-strain time-harmonic oscillation of homogeneous isotropic viscoelastic continuumwill be described in two rectangular Cartesian coordinate,

1)i

x system, with respect to a basis { }ie , and

2)i

x system, with respect to a basis { }ie ,

with basis related by the rotation matrix Q with ij i jQ e e= components. The

transformation is necessary due to definition of coordinate-stretching function ( )i ix which

is related to coordinate system { }ie . Introduction of ( )i ix demands that Eqs. (1) arerewritten in terms of

ix coordinates, simply by replacing

ix

with

ix , as

( )21 ij

i

i i j

ux x

=

(no summation ver i) (12a)

ij ijkl kl C = (12b)

( ) ( )1 1 1

2

jiij

j i i ij j

uu

x x xx

= +

(12c)

where

( )i ix

is part of and as stretch tensors which are diagonal in the characteristic

basis { }ie of the PMM. The primed quantities are respective components of the

corresponding tensors in { }ie basis. Afterwards, the governing equations (12) are

transformed back toi

x system with orthonormal basis { }ie , thus obtaining a new set ofdifferential equations that govern the PMM, here presented in tensor notation:

( ) ( ) ( )2 1 1 2 2x x = u (13a)

= C (13b)

( ) ( )1

2

TT T T = +

u u (13c)

The transformation to unprimed quantities, in the basis { }ie , is performed through the

rotation matrix Q , as


7/7

11 12

21 22

T

= =

Q Q ,

11 12

21 22

T

= =

Q Q ,

1

2

u

u

= =

u Qu ,

1

2

x

x

= =

Q (14)

T

= Q Q

andT

= Q Q (15)

The weak form of the governing equations (13) is derived by taking its inner product with anarbitrary weighting function w residing in an admissible space then integrating the resultant

scalar over the entire computational domain using integration by parts and the divergencetheorem to obtain

2 : d d dmf

= w u w n (16)

where = is the boundary of , n is unit normal and ( ) ( )1 1 2 2mf x x = . The

symmetry of has been used to obtain the first integral on the left hand side with

( ) ( )

1

2

TT = + w w

(17)

Assuming element-wise interpolations of u and w in terms of shape functions N , imposing

Eqs. (13b) and (13c) in Eq. (16) and restricting the integrals to the element domaine =

gives the element stiffness and mass matrices expressed in terms of nodal submatrices with

I and Jas

de

e Tij I J

= k B DB (18)

de

eij m I J f N N

= m I (19)

where I is the identity matrix of size 2 x 2 and

4 3 2 3

2 3 4 3

+ = +

D ,

( )

( )

( )

( )

1

11

2

1 1

2 1

.

.

I

I I

I I

N

N

N N

=

B ,

( )

( )

( )

( )

2

12

2

2 2

2 1

.

.

I

I I

I I

N

N

N N

=

B (20)

with nodal shape functions ,I J as

( )1,Ii ij I jN= and

( )2

,Ii ij I jN= (21)

where ij and ij are components of the stretch tensors in Eq. (15) known as the leftand

right stretch tensor. The evidence of the coordinate-stretching is in the nodal compatibility

matrices IB and IB where the stretch tensors

and are incorporated, and not in the

material matrix D . The system matrices are symmetric complex-valued but be aware that i

functions as part of B , B and mf are defined globally not element-wise.

Documents

Absorbing Boundary Conditions and Perfectly Matched Layer Model for Elastodynamic Analysis