sonia parvanova • georgi vasilev • petia dineva • frank wuttke manolis gd, beskos de (1987)...

Numerical Analysis and Modeling of 2D Soil-Structure Interaction Problems by Hybrid BEM-FEM Technique

Dynamic Analysis, Testing and Design of Infrastructure to

Environmental Loads - November 11-13, 2014, Thessaloniki

1


• Sonia Parvanova

• Georgi Vasilev

• Petia Dineva

• Frank Wuttke




2

OUTLINE:

BEM-FEM coupling in frequency domain Algorithm

Numerical results

BEM-FEM coupling in time domain Standard convolution formula

Lubich convolution quadrature method

Validation tests




3 3

Approaches for seismic wave propagation

Analytical These methods are all restricted to media with simple geometry

Numerical FEM BEM FDM….

Hybrid based on a two-step procedure that combines the travel path effects computed by one method and

local site effects evaluated by other method using the first method’s wave field as input. The main disadvantage of the hybrid two-step techniques is that in subsequent steps past the first, any interaction between the backscattering waves from the local heterogeneity with the incoming wave fields emanating from the deeper layers of the geological profile is neglected.




4 4

Analytical

Babich VM (1956) Ray method for the computation of the intensity of wavefronts. Nauka, Moscow. /in Russian/. Chapman CH. (1978) A New Method for Computing Synthetic Seismograms. Geophysical Journal of the Royal

Astronomical Society 54: 481-518. Pao YH, Gajewski RG (1977) The Generalized Ray Theory and Transient Response of Layered Elastic Solids. Physical

Acoustics (Eds Mason, W. P, R. N. Thurston), New York, Academic Press, Vol. 13: 183-265. Panza GF. (1985) Synthetic Seismograms: The Rayleigh Waves Modal Summation. Journal of Geophysics 58: 125-145. Thomson WT. (1950) Transmission of Elastic Waves through a Stratified Solid. Journal of Applied Physics 21: 89-93. Haskell NA (1953) The dispersion of surface waves on multilayered media. Bulletin Seismological Society of America 43:

17-34. Knopoff L. (1964) A Matrix Method of Elastic Wave Problems. Bulletin Seismological Society of America 54: 431-438. Fuchs K, Müller G. (1971) Computation of synthetic seismograms with the reflectivity method and comparison with

observations. Geophys. J. R. Astr. Soc. 23:417–433. Kind R. (1978) The reflectivity method for a buried source. J. Geophys. Res. 44:603–612. Bouchon M. (1981) A simple method to calculate Green's functions for elastic layered media. Bulletin Seismological Society

of America 71: 959-971.


The known analytical methods are ray theory (Babich (1956), Chapman (1978)), generalized ray technique (Pao and Gajewski (1977)), mode mathicng methods (Panza (1985, 1993), Panza et al. (2000)), matrix propagator method (Thomson (1950), Haskell (1953, 1964), Knopoff (1964)), reflectivity method (Fuchs and Müller (1971), Kind (1978)), wave number integration method (Wuttke (2005)), discrete wavenumber summation method (Bouchon (1981))




5 5

Numerical FEM BEM FDM….

The numerical techniques as finite difference method (Moczo (1989)), finite element method (Gatmiri et al. (2008), Gatmiti and Arson (2008)), boundary integral equation method (Manolis and Beskos (1987), Dominguez (1993), Bouchon and Sánchez-Sesma (2007), Dineva et al. (1996)) are suitable for studying complex structures, but usually they require much CPU time and memory

Moczo P. (1989) Finite-difference technique for SH waves in 2-D media using irregular grids: application to the seismic response problem. Geophys.J. Int. 99: 321-329.

Gatmiri B, Arson C, Nguyen KV. (2008) Seismic site effects by an optimized 2D BE/FE method. I. Theory, numerical optimization and application to topographical irregularities. Soil Dynamics and

Earthquake Engineering 28 (8): 632–645. Manolis GD, Beskos DE (1987) Boundary Element Methods in Elastodynamics. Unwin and Allen,

London. Dominguez J. (1993) Boundary elements in dynamics. Computational Mechanics Publications,

Southampton. Bouchon M, Sánchez-Sesma FJ. (2007) Boundary integral equations and boundary elements

methods in elastodynamics. Adv. Geophys. 48: 157-189. Dineva P, Borejko P, Hadjikov L, Zigler F. (1996) Transient elastic waves in a half-space:

comparison of the DBIE - method with the method of generalized ray. Acta Mechanica 115: 203-211.

Parvanova, S., Dineva P., G. Manolis, Fr. Wuttke. (2014a) Seismic response of lined tunnels in the half-plane with surface topography, Bulletin of Earthquake Engineering 12: 981-1005.

Parvanova S, Dineva P, Manolis G, Kochev P. (2014b) Dynamic Response of a Solid with Multiple Inclusions under Anti-plane Strain Conditions by the BEM. Computers and Structures

139:65-83. Parvanova S, Dineva P, Manolis G (2014c) Elastic wave fields in a half-plane with free Surface

relief, tunnels and multiple buried inclusions. Acta Mechanica 225(7): 1843-1865.





6 6


Hybrid based on a two-step procedure that combines the travel path effects computed by one method and

local site effects evaluated by other method using the first method’s wave field as input. The main disadvantage of the hybrid two-step techniques is that in subsequent steps past the first, any interaction between the backscattering waves from the local heterogeneity with the incoming wave fields is neglected.

Zahradnik J, Moczo P. (1996) Hybrid seismic modelling based on discrete wave number and finite difference methods. PAGEOPH 148 (1/2): 21–38.

Moczo P, Bystricky E, Kristek J, Carcione M., Bouchon M. (1997) Hybrid modelling of P-SV seismic motion at inhomogeneous viscoelastic topographic structures. Bulletin Seismological Society of America 87(5): 1305-1323.

Wuttke Fr, Dineva P, Schanz T. (2011) Seismic wave propagation in laterally inhomogeneous geological region via a new hybrid approach. Journal of Sound and Vibration 330: 664–684.

Panza G, Paskaleva I, Dineva P, La Mura Cr. (2009) Earthquake site effects modelling by hybrid MS-BIEM: The case study of Sofia Bulgaria. Rendiconti di Scienze Fisiche by the Accademia dei Lincei 20: 91-116.

Manolis GD, Makra K, Dineva P, Rangelov T. (2013) Seismic motions in a non-homogeneous soil deposit with tunnels by a hybrid computational technique. Earthquakes and Structures 5(2):161-205.

Manolis GD, Parvanova S., Makra K, Dineva P. (2014) Seismic Response of Buried Metro Tunnels by a Hybrid FDM-BEM Approach, Bulletin of Earthquake Engineering, (DOI: 10.1007/s10518-014-9698-6)




7

Complicated tunnel constructions

Is the BEM applicable for such structures?

Will it be easier to model complicated finite domain structures by FEM?




8

Modeling and analysis of the dynamic soil-tunnel interaction system

FEM is weak in modeling and analysis of infinite size domains

BEM is one of the best choices in modeling such domains.

The complexity of the tunnel structure requires detailed modeling of any of the structure components. This is the main reason why here we use the FEM in modeling the tunnel structure. It’s worth mentioning that in many cases the surface relief above the tunnel should also be modeled in details, thus not only the tunnel but also sufficient part of the soil region around it is also modeled with the FEM.

, , ,t t t t

P- or SV- wave

θ 0x

0f02f

01f01x

02x

0 0 0 0, , ,

12

, , ,i i i i

int

free surface1x

2x

relief

BEM-FEM COMBINATION




ALGORITHM 1. Decomposing the system into boundary element domain

(BED) and finite element domain (FED)

2. Modeling the BED through the boundary element method (BEM)

3. Converting the BED model into one macro-finite element (MFE)

4. Importing the MFE in the ANSYS program through the ANSYS user programmable features (UPFs)

5. Modeling the FED through the finite element method

6. Coupling the BED and FED models according to the nodal compatibility and equilibrium conditions

9




10

1. BED-FED DECOMPOSITION BED (BOUNDARY ELEMENT DOMAIN), FED (FINITE ELEMENT DOMAIN)

, , ,t t t t

P- or SV- wave

θ 0x

0f02f

01f01x

02x

0 0 0 0, , ,

12

, , ,i i i i

int

free surface1x

2x

relief

Finite element domain (FED)

Boundary element domain (BED)


Boundary element domain (BED) = +




11


2. MODELING THE BED

1x2x

0 0 0 0, , ,

(n) - unit outward normal

(n) (n)

Direct Boundary Element Method (BEM)

1

2 3 1 12 2u t

1 11 1u t

2 22 2u t 3 3

2 2u t

3 31 1u t

2 21 1u t

1 11 2,x x

2 21 2,x x 3 3

1 2,x x

1x

2x

11 12

N

22 1N

31 12

N

u1=1

u2=1

u3=1

Quadratic BE

Gt Hu Φ




3. CONVERTING BED MODEL INTO MFE MFE (MACRO FINITE ELEMENT)

3.1. DOFs condensation at the BED-FED contact nodes

12

1

2

3 traction free surface traction free surface

contact surface

(n)

(n) (n)

1

2

00

t

t

1 2 0t t 1 2 0t t







Express the interface tractions in terms of interface displacements and load vector:

13

traction free surface traction free surface

contact surface

(n)

(n) (n)

1

2

00

t

t

1 2 0t t 1 2 0t t



1

2

3

(2) (2) t Bu P





Express the interface tractions in terms of interface displacements and load vector:

14

(2) (2) t Bu P

122 21 23 22

1 122 21 23 21 23 2 21 23

3

B G H H A H

tP G H H A H H Ψ Φ G G

t

11 12 13 1 11 12 13 1 1

21 22 23 2 21 22 23 2 2

31 32 33 3 31 32 33 3 3

G G G t H H H u ΦG G G t H H H u ΦG G G t H H H u Φ

111 13 12

31 33 321

11 13 1 11 13 1

31 33 3 31 33 3

H H GA

H H G

H H Φ G G tΨ

H H Φ G G t

1 3

2




3.2. BED to MFE conversion

Gt Hu Φ F Ku F

15

FEM BEM

2t

1t

1u

2u

2F

1F

1u

2u




Assemble the element nodal force vectors:

16

1t

2tBoundary element (e)

(1)2F

(1)1F

(2)2F

(2)1F

(3)2F

(3)1F

Edge of the relevant FE

1x2x

1x2x

0,

0,

e

e T

te T e

s te s

e

e

dsds

L

F N tF N N

t Ntt

tL

Nodal forces for each BE (e) equivalent (replacing) to the element tractions

Standard traction approximation by the nodal tractions and shape functions

Standard FEM expression for derivation of equivalent nodal forces due to distributed loads




Assemble the global nodal force vector:

0,T

e

e s

t

s s

ds FN FM L N Mt

Square (6x6) matrix for k-th element, which gives the relation between nodal forces and tractions for this element

Global (2Nx2N) matrix for the whole model, which gives the relation between nodal forces and tractions for the contact surface

17

M1kL

kL

1kL

2kL




0

(2) (2

,

)

T

e

e s s

t

s

ds

M L N N

t B

F F M

u P

t

(2) , where

K MBK FF Mt MB

Fu

MPMP u

Substitute t(2) in F to obtain the FEM stiffness formulation:

stiffness matrixnodal force vector, equivalent to the seismic excitation in the soil stratum

KF

18

Assemble the global nodal force vector:

1 3

2 BED

FED




4. Implementation of MFE in methodology applied to the steady-state analysis (harmonic analysis)

4.1. Formulation in steady-state (quasistatic state)

19

The BE numerical model of any mechanical system considered in steady state, could be mathematically represented by the following system of linear algebraic equations

ANSYS time-harmonic analysis formulation

Re Im Re Im Re Im

1) SLAE in the frequency domain analysis, where ; ;i i i Ku F K K K u u u F F F

ANSYS ANSYS 2 ANSYS ANSYS ANSYS ANSYS ANSYS

Re Im

2) SLAE in the ANSYS harmonic analysis, wherei i K C M u F F F F

ANSYS ANSYS ANSYS ANSYS ANSYS

Re Im Re Re Im Im

3) Fitting 1) in 2); / ; ; ; K K C K M 0 F F F F

Note: Here the K matrix and the load vector F are generated with an authors’ software program, developed in MATLAB




Importing through ANSYS UPFs

20

Element type MATRIX50

Utility Menu-Preprocessor-Element Type-Add/Edit/Delete and Add Superelement 50, or through ANSYS command line: ET,1,MATRIX50




5. MODELING THE FED

Finite Element Model PLANE82

ANSYS Release 10.0 Documentation

21




6. Coupling the BED and FED Couple the coincident nodes between the superelement (MFE representing the BED), and

FED model through the Utility Menu-Preprocessor-Coupling/Ceqn-Coincident Nodes

22




Free surface Γff

P-, SV- wave

h x1

x2

φ a

Free surface Γff

P-, SV- wave

h x1

x2

φ a

23

VALIDATION OF THE DEVELOPED BEM - FEM COUPLING

HOLE IN A HOMOGENEOUS HALF-PLANE

BEM model BEM-FEM model

a=7 h=14

b=70

b BEM

FEM




24


P-wave SV-wave

AMPLITUDES OF THE DISPLACEMENT COMPONENTS ALONG THE FREE SURFACE AT FIXED DIMENSIONLESS FREQUENCY

0.25SVa V

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-60 -40 -20 0 20 40 60

U2 BEM-FEMU1 BEM-FEMBEMBEM

0

0.5

1

1.5

2

2.5

3

-60 -40 -20 0 20 40 60


Dis

plac

emen

t am

plitu

des

Dis

plac

emen

t am

plitu

des

x1 x1




0

0.5

1

1.5

2

2.5

3

-60 -40 -20 0 20 40 60


0

0.5

1

1.5

2

2.5

-60 -40 -20 0 20 40 60


25


P-wave SV-wave

AMPLITUDES OF THE DISPLACEMENT COMPONENTS ALONG THE FREE SURFACE AT FIXED DIMENSIONLESS FREQUENCY

Dis

plac

emen

t am

plitu

des

Dis

plac

emen

t am

plitu

des

x1 x1

0.5SVa V




26

NUMERICAL SIMULATION OF A PRACTICAL PROBLEM

7.4 4.77

5 4.

9

1.7

0.4

0.3

0.15

0.3

0.15 0.40

(a)

a=7

x1

x2

70

µ 1, ν1, ρ1, ζ1

µ 2, ν2, ρ2, ζ2

µ 3, ν3, ρ3, ζ3

µ 4, ν4, ρ4, ζ4

µ5, ν5, ρ5, ζ5

(b)

Tunnel construction Soil deposit




27

NUMERICAL SIMULATION OF A PRACTICAL PROBLEM

0 0, exp( )i if f i t x x

x0(x01,x02)

BED E

0, ν0, ρ0, ζ0

FEM (ANSYS) mesh FED

BEM mesh

x01

x 02




0

1

2

3

4

5

6

7

8

9

0.05 0.15 0.25 0.35 0.45

|u2|-point 1

|u2|-point 2

|u2|-point 3

|u2|-point 4

0

0.5

1

1.5

2

2.5

3

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

|u2|-point 1

|u2|-point 2

|u2|-point 3

|u2|-point 4

SVa V

Dis

pla

cem

ent

amp

litu

des

|u

2|

0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4 0.5

|u2|-point 1

|u2|-point 2

|u2|-point 3

|u2|-point 4

Dis

pla

cem

ent

amp

litu

des

|u

2|

SVa V

0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5

|u1|-point 1

|u1|-point 2

|u1|-point 3

|u1|-point 4

Dis

pla

cem

ent

amp

litu

des

|u

1|

SVa V

1 2 3

4

a) b)

c) d)

Dis

pla

cem

ent

amp

litu

des

|u

2|

SVa V

Displacement component amplitudes for fixed representative points along the free surface and at the bottom of the tunnel versus normalized frequency

28

Embedded seismic source

Normal incident SV-wave

Embedded seismic source

Normal incident P-wave




Displacement component amplitudes along the free surface

29

0

0.5

1

1.5

2

2.5

3

3.5

-7 -5 -3 -1 1 3 5 7

|u2| - deep seism. source|u2| - P-wave|u2| - shallow seism. source

0

0.5

1

1.5

2

2.5

3

3.5

4

-7 -5 -3 -1 1 3 5 7

|u1| - SV-wave

|u1| - shallow seism. source

0

1

2

3

4

5

6

-7 -5 -3 -1 1 3 5 7

|u2| - deep seism. source|u2| - P-wave|u2| - shallow seism. source

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-7 -5 -3 -1 1 3 5 7

|u1| - SV-wave|u1| - shallow seism. source

a) b)

c) d)

η=0.25 η=0.25

x1/a x1/a

Ho

rizo

nta

l am

plit

ud

es |

u1|

Ver

tica

l am

plit

ud

es |

u2|

x1/a x1/a

η=0.5 η=0.5

Ho

rizo

nta

l am

plit

ud

es |

u1|

Ver

tica

l am

plit

ud

es |

u2|




Polar distribution of stresses along the soil-tunnel interface

30

(1) The seismic source with its specific geophysical properties; (2) The inhomogeneity and heterogeneity of the wave path

from the source to the local geological region; (3) The geotechnical properties of the near-field local

geological profile (4) The properties of the engineering structure itself




Solutions in time domain

31

P- or SV- wave

θ 0x

0f02f

01f01x 0 0 0 0, , ,

free surface


1x

2x

Validation test 1

If the FED is missing the problem could be solved as follows:

I. BEM solution in the frequency domain followed by IFFT of the corresponding displacement solution and derivation of the displacements in time domain. This original, initial or primary solution is called (0) CANYON

TIME SOLUTION


02x

, , ,t t t t

12

, , ,i i i i

int

relief 1x

2x





32

P- or SV- wave

θ 0x

0f02f

01f01x

02x

0 0 0 0, , ,

free surface


1x

2x

Validation test 1

II. Verification of BEM-FEM coupling in time domain

(2)

F MBu MP u0 u

K FK

K

F

u F

u D F





33

Validation test 1

II. Verification of BEM-FEM coupling in time domain (using COMPLIANCE matrix)

u D FThe inverse FFT of the displacements states that:

. .

. .

1 .2

1 . .2

i t

i t

t e d

t e d t d t t

u u

u u D F D F

Standard discrete convolution formula given in Santamarina, J.C., D. Fratta, DISCRETE SIGNALS AND INVERSE PROBLEMS. An Introduction for Engineers and Scientists, John Wiley & Sons, Ltd (2005)

(1) Compliance convolution solution





34

P- or SV- wave

θ 0x

0f02f

01f01x

02x

0 0 0 0, , ,

free surface


1x

2x

Validation test 1

III. Verification of BEM-FEM coupling in time domain

(2)

F MBu MP u0

K

Fu

u

FK

F

K





35

Validation test 1

III. Verification of BEM-FEM coupling in time domain (using STIFFNESS matrix)

(2) Stiffness convolution solution

11 1 1 1 1 1

11 2 2 1 2 2 1 2 2 1

1

11 1

2

1 3 2 2 3 1 3 3 1 3 2 2 3 1

11

jN

k j j j kk

k jk

j k

K u F u K F

K u K u F u K F K u

K u K u K u F u K F K u K u

K u F u K F K u

1 121

1 1 121

, 2k

k j k jj

t kt

DK K K D K

Derivation of stiffness matrix in time:





36

Validation test 1

SV- wave

θ 0 0 0 0, , ,

free surface1x

2x

0

6

1/ 3

1.10 , 2000, 1r

5% hysteretic damping ratio

-1

-0.5

0

0.5

1

0 2 4 6 8 10

u(t)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 1 2 3 4 5

u(ω)

Re Im Abs

2 0( ) ( ) sin( )tf t f t e t

220

02 )()(

)(

if

f 2(t

)

f 2(ω

)

f, [Hz] t, [s]

-1

-0.5

0

0.5

1

0 2 4 6 8 10

u(t)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 1 2 3 4 5

u(ω)

Re Im Abs

a) b)

2 0( ) ( ) sin( )tf t f t e t

220

02 )()(

)(

if

f 2(t

)

f 2(ω

)

f, [Hz] t, [s]

Time history function





37

-1

-0.5

0

0.5

1

0 1 2 3 4 5 6

(0) canyon time solution

(1) compliance convolution solution

(2) stiffness convolution solution Nyquist frequency=5 Hz

Validation test 1 – RESULTS (standard convolution formula)

x1

x2

SV- wave

Observation point

Time, [s]

Hor

izon

tal d

ispl

acem

ent i

n co

nsis

tent

uni

ts





38

x1

x2

SV- wave

Observation point

-1

-0.5

0

0.5

1

0 5 10 15 20 25





(2) stiffness convolution solution Nyquist frequency=3.333333 Hz

Time, [s]

Hor

izon

tal d

ispl

acem

ent i

n co

nsis

tent

uni

ts





-1

-0.5

0

0.5

1

0 1 2 3 4 5 6




(4) ANSYS


39

x1

x2

SV- wave

Observation point

Time, [s]

Hor

izon

tal d

ispl

acem

ent i

n co

nsis

tent

uni

ts






40

Standard discrete convolution formula

Convolution Quadrature Method developed by Lubich

0

.t

s s s s i is

t d t t

K u F

K u K u

This method numerically approximates the convolution integral by a quadrature rule whose weights are determined by the Laplace transformed function and a linear multistep method.

K̂

Lubich, C.: Convolution Quadrature and Discretized Operational Calculus. I. Numerische Mathematik,

52, 129-145, 1988. Lubich, C.: Convolution Quadrature and Discretized Operational Calculus. II. Numerische Mathematik,

52, 413-425, 1988.

References




-1

-0,5

0

0,5

1

0 5 10 15 20 25



(3) Lubich convolution solution (MATLAB)


41

x1

x2

SV- wave

Observation point

Time, [s]

Hor

izon

tal d

ispl

acem

ent i

n co

nsis

tent

uni

ts

Validation test 1 – RESULTS (Lubich convolution formula)




-1

-0,5

0

0,5

1

0 1 2 3 4 5 6



(3) Lubich convolution solution (MATLAB)

(4) ANSYS (Lubich input data)


42

x1

x2

SV- wave

Observation point

Time, [s]

Hor

izon

tal d

ispl

acem

ent i

n co

nsis

tent

uni

ts

Validation test 1 – RESULTS (Lubich convolution formula)




43

CANYON IN THE HOMOGENEOUS HALF-PLANE

60 0

0 0

1/ 3, 1.10 ,2000, 0%

0 0

60

sin ,

2.10

tF f e t

f

1r d

1x

2x

1x

2x

0 0sintF f e t 6

0 0

0 0

1/ 3, 1.10 ,2000, 0%

61 1

1 1

1/ 3, 1.10 ,2000, 0%

FED

BED

5r

BEM model BEM-FEM model

d

d=2.5 d=10

Solutions in time domain Validation test 2




-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 1 2 3 4 5 6


(4) ANSYS Lubich input

44

Solutions in time domain Validation test 2 – RESULTS

Time, [s]

Verti

cal d

ispl

acem

ent i

n co

nsis

tent

uni

ts

x1

x2

Observation

point

d=2.5 – seismic source in the FED

-1

-0.5

0

0.5

1

0 2 4 6 8 10

u(t)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 1 2 3 4 5

u(ω)

Re Im Abs

2 0( ) ( ) sin( )tf t f t e t

220

02 )()(

)(

if

f 2(t

)

f 2(ω

)

f, [Hz] t, [s]

-1

-0.5

0

0.5

1

0 2 4 6 8 10

u(t)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 1 2 3 4 5

u(ω)

Re Im Abs

a) b)

2 0( ) ( ) sin( )tf t f t e t

220

02 )()(

)(

if

f 2(t

)

f 2(ω

)

f, [Hz] t, [s]

0 10; 1.




-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0 1 2 3 4 5 6



45


Time, [s]

Verti

cal d

ispl

acem

ent i

n co

nsis

tent

uni

ts

d=10 – seismic source in the BED

-1

-0.5

0

0.5

1

0 2 4 6 8 10

u(t)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 1 2 3 4 5

u(ω)

Re Im Abs

2 0( ) ( ) sin( )tf t f t e t

220

02 )()(

)(

if

f 2(t

)

f 2(ω

)

f, [Hz] t, [s]

-1

-0.5

0

0.5

1

0 2 4 6 8 10

u(t)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 1 2 3 4 5

u(ω)

Re Im Abs

a) b)

2 0( ) ( ) sin( )tf t f t e t

220

02 )()(

)(

if

f 2(t

)

f 2(ω

)

f, [Hz] t, [s]

0 10; 1.

x1

x2

Observation

point




46


Time, [s]

Verti

cal d

ispl

acem

ent i

n co

nsis

tent

uni

ts

d=10 – seismic source in the BED

-1

-0.5

0

0.5

1

0 2 4 6 8 10

u(t)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 1 2 3 4 5

u(ω)

Re Im Abs

2 0( ) ( ) sin( )tf t f t e t

220

02 )()(

)(

if

f 2(t

)

f 2(ω

)

f, [Hz] t, [s]

-1

-0.5

0

0.5

1

0 2 4 6 8 10

u(t)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 1 2 3 4 5

u(ω)

Re Im Abs

a) b)

2 0( ) ( ) sin( )tf t f t e t

220

02 )()(

)(

if

f 2(t

)

f 2(ω

)

f, [Hz] t, [s]

0 10; 0.3

x1

x2

Observation

point

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0 2 4 6 8 10 12 14



sonia parvanova • georgi vasilev • petia dineva • frank wuttke manolis gd, beskos de (1987)...

Documents