Perfectly Matched Layers - An absorbing boundary condition for elastic wave propagation

8/12/2019 Perfectly Matched Layers - An absorbing boundary condition for elastic wave propagation

1/9


2/9

distance. Only then, the PM can be truncated by a fixed (Dirichlet) boundary without any significant

wave reflection, see Figure 1(b).

Outgoing wave

Dirichlet BC

x1

PML

PM BD PM BD

Outgoing wave

Attenuated wave

PMM

x2

x1

x2

Reflected wave

L p (a) (b)

Figure 1 PMM truncation: (a) PMM adjacent to an unbounded domain and (b) PML with a fixed edge

As a consequence, the displacements of the coupled system ( BD PM ) in BD should be almostthe same as those of the unbounded elastic domain . Hence, the Perfectly Matched Layer PM , iscreated from the Perfectly Matched Medium PM

, thus BD PM is considered as a replacement ofthe unbounded elastic domain .

2. GOVERNING EQUATIONS

In the present paper the Perfectly Matched Layer has been derived using a second-order displacement- based finite element formulation. A wave propagation analysis is performed, where the near-field(bounded domain) is discretized with standard isoparametric finite elements, surrounded by a dynamicfar-field representing an unphysical domain (or layer) where the absorbing boundary condition isenforced through special types of functions, which are able to damp the outgoing as well as thereflected wave within the layer.

2.1. Elastic medium

Consider a homogeneous isotropic elastic medium subjected to a time-harmonic excitation. Theoscillation of the elastic medium will be in the form ( ) ( )exp x i t u with as circular frequency. The

governing equations can be summarized as follows:

Equilibrium equation 2ij i j j

u x

=

(no summation on i) (1a)Constitutive relation ij ijkl kl

kl

C = (1b)

Kinematic relation12

iij

j i

uu x x

= +

(1c)

where ijkl C is written in terms of the Kronecker delta ij such that


3/9

( )23ijkl ij kl ik jl il jk

C = + +

. (2)

ij and ij are the components of the stress and strain tensors, respectively, and ijkl C are thecomponents of the material stiffness tensor C ; is the bulk modulus, the shear modulus, and the unit mass density of the medium.

2.2. Perfectly Matched Medium

Consider the governing equations (1) for an elastic medium, where the coordinates i x will be replaced

by stretched coordinates i x~ , defined as

( )0

:i x

i i i x x dx = (3)

This procedure of stretching, in particular, is responsible for the physical mapping of the coordinatesin the dynamic wave equation. The coordinate-stretching formally implies

( )1

i i i i x x x

= (4)

where i x are real coordinates, i are complex stretched-coordinates and ( )i i x are non-zero,continuous, complex-valued coordinate-stretching function. At the same time this procedure creates acomplex formulation for inhomogeneous viscoelastic Perfectly Matched Medium (PMM).A plane-strain elastodynamic motion of a PMM is defined by introducing (4) into the governing

equations (1) as follows:

Equilibrium equation( )

21 iji

j j j j

u x x

=

(no summation on i) (5a)

Constitutive relation,

ij ijkl kl k l

C = (5b)

Kinematic relation( ) ( )

1 1 12

jiij

i i i j j

uu x x x

= +

(5c)

If such two stretched adjacent media have the same i at their interface, then the matching propertywill ensure that any propagating wave will pass through the interface without reflection.

Consider the 1 2 x plane, with two perfectly matched media defined on the

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

The wave propagates from the left medium, upper index ( lm), in x1 direction, through the PMMinterface into right medium, upper index ( rm), as depicted in Figure 2.


4/9

rm

i n t e r f a c e

wavedirection

lm

x 2

BD PM

x 1

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 =

Figure 2 Adjacent PMMs as left ( lm) and right ( rm) media

If ( ) ( )1 10 0lm rm = and

2 2 1lm rm = = , then the two Perfectly Matched Media can be considered as one

PMM, where the continuous 1 is defined piecewise on the two half-planes. (In this case 2 has not to be considered). The homogeneous isotropic elastic medium governed by the equations (1) is a specialcase of a PMM where 1 1 1.

lm rm = = The same arguments hold for the wave propagation in x2 direction.

2.3. Finite element implementation

Consider a two-dimensional homogeneous isotropic elastic continuum undergoing a time-harmonic plane-strain oscillation. This motion will be described in two rectangular Cartesian coordinate:

1) a i x system, with respect to a basis { }ie , and2) a i x system, with respect to a basis { }ie ,

with the two bases related by the rotation matrix Q , with components :ij i jQ e e = . The finite elementformulation implementation of the perfectly matched layer condition is related to a rotated coordinatesystem { }ie through the definition of ( )i i x as coordinate-stretching function globally on thecomputational domain. In this way by replacing i x by i x , the Eqs. (5) are expressed in terms of thecoordinates i x . Afterwards, the equations with the stretched coordinates are transformed back to the

i system with respect to an orthonormal basis { }ie , obtaining

Equilibrium equation ( ) ( ) ( )2 1 1 2 2 x x = u (6a)Constitutive relation = C (6b)

Kinematic relation ( ) ( )12

T T T T = +

u u (6c)

where ( )i i are the coordinate-stretching functions defining the stretch tensors and . Thetransformation of Eq. (6) to unprimed quantities in the basis { }ie is obtained by


5/9

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 (7)

corresponding to the primed quantities via the usual change-of-basis rules for vector and tensorcomponents.The weak form of the governing equations Eq. (6) is derived by taking its inner product with anarbitrary weighting function w and integrating the resultant scalar over the entire computationaldomain using integration-by-parts and the divergence theorem to obtain

2 : d d dm f = w u w n (8)

where = is the boundary of and n is its unit normal; f m is defined by ( ) ( )1 1 2 2:m f x x = .The symmetry of has been used to obtain the first integral on the left hand side, with

( ) ( )12

T T = + w w

. (9)

Assuming element-wise interpolations of u and w in terms of shape functions N , imposing Eq. (6b)and (6c) point-wise in Eq. (7), and restricting the integrals to the element domain e = , gives thestiffness and mass matrices for a PML element. In terms of nodal submatrices, with I and J being thenode numbers, these are expressed as

e

e T ij I J d

= k B DB (10)

e

eij m I J f N N d

= m I (11)

where I is the identity matrix of size (2 x 2), and

4 3 2 3

: 2 3 4 3

+ = +

D

i

i

i i

,

( )

( )

( )

( )

11

12

1 12 1

.

: . I

I I

I I

N

N

N N

=

B ,

( )

( )

( )

( )

21

22

2 22 1

.

: . I

I I

I I

N

N

N N

=

B (12)

with nodal shape functions ( ) j I ij Ii N N ,1 ~:= and ( )2 ,: Ii ij I j N N = , described using T= Q Q

andT= Q Q , known as the left and right stretch tensors , respectively:

( )( )

2 2

1 1

: x

=

i

i and

( )( )

1 1

2 2

1/:

1/

x

=

i

i (13)

hence diagonal with respect to the basis { }ie , i.e. characteristic basis of the PMM. The Eqs. (8) and(9), the functions i in B , B and in f m are defined globally on the computational domain, notelement-wise. Note the evidence of coordinate-stretching in the FE matrices, where the stretch tensors and are incorporated in the nodal compatibility matrices IB and IB , not in the material matrixD . Thus, the system matrices for the are symmetric complex-valued and frequency-dependent


6/9

which have to be computed anew for each frequency.The coordinate-stretching procedure performsmapping of a typical element in the layer such that it extends towards infinity. This is obtained intwo steps illustratively presented in Figure 3.

Outgoing wave

Dirichlet BC

x1

PML

PM BD PM BD

Outgoing wave

Attenuated wave

PMM

x2

x1

x2

Reflected wave

L p

(a)

(b)

Figure 3 Finite element mapping of the unbounded domain

The PML element is first stretched to discretize unbounded domain as in Figure 3(a), then it istruncated (with Dirichlet condition) to a Perfectly Matched Layer element with a depth L p, see Figure3(b). These elements are positioned adjacent to those discretizing the bounded domain.

3. INVESTIGATIONS

To estimate the performance of the proposed PML formulation it will be compared to other methods

by means of a foundation vibration analysis. Consider a rigid surface massless strip-foundation over ahalf-space, excited by a vertical harmonic displacement with unit amplitude 0 1u = and excitationfrequency , see Figure 4.

,

x1

B = 2b

u(0,t)=u exp(i t)0

x2

half-space

Figure 4 Surface strip-foundation over a half-space excited by vertical oscillations

The foundation dimensions are: width 3 B m= and height 1.2 f h m= . The soil half-space is defined aslinear isotropic viscoelastic material with a shear wave velocity of 92.2 / sc m s= , a unit mass densityof 32000 kg m = , a Poissons ratio of 0.3 = and damping coefficient of 0.05 = .In the present analyses, two parallel finite element models are discretized, one using viscous-damper

as boundary (VDB) and the other a PML as absorbing condition, both are shown in Figure 5. TheVDB finite element model depicted in Figure 5(a) is discretized by 56x46 isoparametric quadrilateralelements (with dimensions / 6 0.25a b B= = = m.


7/9

H = 4 6 a

PM

BD

L = 28b

h = 2 a

B/2 =3b

h=2a

H= 19a

H+ L= 19a+ a

p

p

L + L = 14b + bx p p

PM

BD

L p

=a p

L = 1 4b L p = b p

PM

BD

B/2 =3b

(a) (b)

Figure 5 Finite element model of strip-foundation over elastic half-space with (a) VDB and (b) PML condition

Hence, the viscous-damper boundary condition is enforced at the edge of the model. The PML finiteelement model differs conceptually because it defines a bounded domain BD discretized by 28 x 19isoparametric quadrilateral elements, adjacent to which an absorbing layer PM is positioned as shownin Figure 5(b). This PML with depth 3.5 p L m= enlarges mesh to a 30x20 with additional 68 pn = elements. It should be noted both finite element models discretized an identical region which positionsthe boundary condition at distance 7 L m= from the source of excitation.The analysis quantitatively compares both methodologies by looking at the dimensionless response

functions related through a compliance matrix ( )0aF as follows:

( )0a=u F P (14)

( )

( ) ( )( ) ( )

12

12 1212

2 22 0 2

1 11 0 1 0 1

1 12 0 0

0 0

00

r

r r r r

u F a P

u F a F a P bu M b F a F a

=

(15)

The conversion is done by normalization of the physical parameters the transformation from dimensionalvariables to their dimensionless equivalent variables ^ through the relation:

ii ii F F = (16)

For pure translation degrees ( i=1,2 ) all dependent from the dimensionless frequency ( )0 sa b c = with b (half of the foundation width) as a characteristic dimension. The complex compliance

( ) ( ) ( )Re Im

22 0 22 0 22 0

F a F a iF a= + of the foundation vertical degree with real ( Re ) and imaginary ( Im ) part is presented in Figure 6, respectively.


8/9

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0 0.5 1 1.5 2 2.5 3

PML

VDB

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 0.5 1 1.5 2 2.5 3

PML

VDB

(a) (b)

Figure 6 Comparison of the compliance function of a surface foundation over half-space

with (a) ( )Re22 0 F a and (b) ( )Im22 0 F a

Although an exceptional matching of the curves can be observed in both parts of the complexfunction, there is also a small deviation especially for lower frequencies. The PML as a rigorouscondition is considered to produce exact solutions, in contrast to the VDB which is an approximatecondition. To complete the picture, a relative amplitude difference with respect to the PML methodis illustratively presented in Figure 7.

15.0%

10.2%

6.2% 5.9%4.6%

3.2%3.5% 2.9%2.3% 2.0% 1.6% 1.2%

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

0.5 1 1.5 2 2.5 3

rel.dif f erence

ao

Fzz (Re)Fzz (Im)

Figure 7 Relative differences in compliance functions for a rigid surface foundation on half-space

From the cumulative diagram it is obvious that at the lower excitation frequencies the relativedifference is more pronounced decreasing with the increasing frequency. The qualitative side of theanalysis shows that the VDB model produces 15% error in lower frequency range for 0 0.5a andaround 10% at 0 1.0a = . This could be argued with the fact that at lower frequencies the viscous-damper boundary condition has restrictions with respect to the angle of wave incidence which maylead to some stability problems. Thus, the viscous-damper condition should be positioned at distancesgreater than 7 L m= from the source of the oscillation to obtain the same accuracy as in the case of thePML.Another aspect of the comparative analyses is presented in Table 1. The ratio t/N of the elapsedcalculation time t and the related number of elements N defines the computation time per element.

( ) ( )

( )0 0

0

% 100

PML VDB

PML

F a F a

F a

=

F 2 2

( R e )

F 2 2

I m

^


9/9

Table 1 Comparison of PML and VDB with respect to computation time per element

Method t(min) N(/)

t/N(s)

PML 13.4 600 1.34VDB 24.8 2576 0.58

Since the PML attenuation is defined on the global computational domain, not element-wise, themodel can be discretized with fewer elements which reduce the computational time significantly. Inthis analysis the spatial discretization in the VDB model needs 4.3 times more elements than the PML.In fact, although PML is computationally more expensive due to its complex formulation, it stillremains more efficient because fewer elements are needed for the same level of accuracy. In thecurrent study the time gain was 46 percent.Finally, it can be summarized that the Perfectly Matched Layer formulation not only takes lesscomputation time but also produces better results with fewer elements making it an efficient methodand preferable as a boundary condition in solution of elastodynamic problems.

4. CONCLUSIONS

The formulation and application of a Perfectly Matched Layer in definition of unbounded domains forelastodynamics has been presented. Its adsorption capability was found to be very convincing, even inthe case of a rather small PML thickness and even for relatively low frequencies. In fact, the resultshave shown that the PML absorption remains equally efficient at wavelengths far larger than the PMLthickness. As a consequence, the PML thickness can be kept minimal even for studies involving lowfrequencies, and no rescaling of the model size is required. The recommended value of the PMLthickness is defined through the ratio max 10 p L which will ensure an accurate solution.The performance of the PML condition has been compared with another dynamic boundary method.The results show that the PML does not only take less computational time but also produces betteraccuracy with fewer elements. This suggests that the bounded domain may be restricted to the regionof interest in order to lower the computational cost. This constitutes the major advantage of the PMLwith respect to the other methods classifying it as very efficient and preferable choice in the solutionof elastodynamic problems.

The current contribution brings a formulation of the PML condition derived in the frequency domain.It is recommended that for further research the PML condition should be derived directly in the timedomain, which will eventually enable the investigation of a nonlinear behaviour inside the boundedsoil region. Moreover, the PML offers an efficient modelling alternative for the simulation of wave

propagation in unbounded domains not only for elastodynamic problems but also in other fields ofengineering, such as, for instance, acoustics.

ACKNOWLEDGMENT

The financial support of the first author by the DAAD (Deutscher Akademischer Austausch Dienst)through the SEEFORM (South Eastern European Ph.D. Formation in Engineering) program isgratefully acknowledged.

REFERENCES

Brenger, J.-P. (1994). A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys.114, 185200.

Basu, U. and Chopra, A.K. (2003). Perfectly matched layers for time-harmonic elastodynamics of unboundeddomains. Theory and finite-element implementation, Comput. Methods Appl. Mech. Engrg. 192, 1337 1375.

Harari, I. and Albocher, U. (2006). Studies of FE/PML for exterior problems of time-harmonic elastic waves,Comput. Methods Appl. Mech. Engrg. 195, 38543879.

Documents

Perfectly Matched Layers - An absorbing boundary condition for elastic wave propagation