Implementation of the Finite Element Method in the …paginas.fe.up.pt/~eurodyn2014/CD/papers/080_MS02_ABS...advantages of both outlined methods is presented. The infinite medium of

ABSTRACT: The dynamic soil-structure interaction has to be modeled in an appropriate way in order to provide a realistic assessment of arising vibrations. Modeling of the complete emission- transmission- immission system by means of the emission-transmission causes a high computational effort. In general appropriate modifications are necessary in order to describe the infinite extension of the soil where artificial reflections at the boundaries have to be avoided. An efficient approach to model the complete system is to describe only the irregular geometries of the emission and immission systems with finite elements and to model the transmission system of the soil by using analytical solutions based on Integral Transform Methods. These solutions are obtained for infinite systems using a Fourier-Transformation from the original domain to the wavenumber-frequency domain. In order to couple both methods an adapted Finite Element Method which can be used in the wavenumber-frequency domain has to be applied. In the scope of this paper the following systems are coupled: a halfspace with a cylindrical cavity and a Finite Element mesh which is placed inside the cavity. Applying a transformation along the longitudinal coordinate of the tunnel, the system is 2.5 dimensional which means that the formulation of two-dimensional Finite Elements contains the complete three-dimensional information of the system after an inverse transformation. For each wavenumber-frequency combination, the computed stiffness matrix for the Finite Element mesh is coupled with the respective wavenumber-frequency stiffness of the analytical solution describing the halfspace with cylindrical cavity using the substructure technique.

KEY WORDS: Soil Dynamics; Soil-Structure-Interaction; Integral Transform Method; Fourier-Transformation; Finite Element Method; Wavenumber-Frequency Domain.

1 INTRODUCTION

Vibrations induced by moving trains in a tunnel are of special importance in urban areas. Due to a growing sensibility of both humans and machines, the importance of the prediction of oscillations due to rail and road traffic is increasing. As measurement-based analyses are often not possible during the design phase, when constructional measures would be easy to implement, a reliable method for the prediction of vibrations is of interest.

In order to model the complete system with emission, transmission and immission elements, different numerical and analytical methods are available. A comprehensive overview over the various numerical models and methods is presented in [1]. As one of the possible methods, the Integral Transform Method is based on the Fourier-Transformation and provides an analytical solution for different fundamental systems. Using Integral Transform Methods closed form solutions for the infinite extension of the halfspace can be derived as well as for several modifications of the halfspace, e.g. for a layered halfspace or a halfspace with cylindrical or spherical cavity [2]. The advantage of this method is to be able to describe infinite systems with a low computational effort without any non-physical reflections at artificial boundaries. On the other hand only simple geometries can be described by the fundamental solutions which prevents the modeling of complex emission or immission systems. A detailed modeling of complex geometries can be achieved using the Finite Element Method. However, the modeling of the infinite transmission medium of the soil with finite elements without

further modifications leads to artificial reflections at the boundaries of the discretized domain. Moreover, a high calculation effort is necessary to model the three-dimensional elements to describe the medium.

In the scope of this paper an approach which combines the advantages of both outlined methods is presented. The infinite medium of the transmission system is described by the Integral Transform Method. As fundamental system the solution for a halfspace with a cylindrical excavation is used. On the surface of the cylindrical cavity the displacements are coupled with the respective degrees of freedom of a Finite Element mesh which is fitted into the cylindrical cavity. Thus, an arbitrary shaped geometry inside the cavity can be modeled in detail (see figure 1). In order to couple both substructures, a reference onto the same basis is essential.

First the transmission system is considered. The solution of the fundamental system halfspace with cylindrical cavity is derived by the superposition of the solution of a halfspace with the solution of a fullspace with cylindrical excavation. These methods require different Fourier-Transformations respectively Fourier series expansions. The time domain is transformed into frequency domain and the longitudinal coordinate of the cavity x is transformed into the wavenumber domain kx. On the surface of the halfspace the transverse spatial coordinate y is transformed into the wavenumber ky. Along the cavity the cylindrical coordinate φ is expressed by a Fourier series expansion.

Implementation of the Finite Element Method in the Fourier-Transformed Domain and Coupling with Analytical Solutions

Manuela Hackenberg1, Martin Dengler1, Gerhard Müller1

1Chair of Structural Mechanics, Technische Universität München, Arcisstr. 21, 80333 München, Germany email: [email protected], [email protected], [email protected]

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

583

Forces inducedby moving trains

Transmissionsystem

Emissionsystem

Figure 1. Model of the emission and transmission system.

By means of the substructure technique the displacements

of the soil on the cylindrical surface are coupled with the respective degrees of freedom of the Finite Element mesh inside the cavity. As the degrees of freedom of the transmission system are Fourier-transformed along the longitudinal coordinate, the complete three-dimensional system is replaced by a 2.5-dimensional model. This means that the calculations carried out in the transformed domain for every wavenumber kx are two-dimensional but contain the three-dimensional information which is visible after the inverse transformation from the wavenumber into the spatial domain. Therefore, the finite elements that are used to describe the domain inside the tunnel are two-dimensional four-node elements that contain the complete three-dimensional information. Furthermore they have to be adopted to the Fourier-transformed domain.

A second modification of the finite elements is necessary in order to transform the degrees of freedom on the surface of the mesh which shall be coupled to the transmission system from the Cartesian coordinate system into a Fourier series expansion along the peripheral angle. After these modifications, the stiffnesses of both substructures can be coupled.

In this paper the validity of the proposed method is checked by a benchmark test comparing the results of a system where a halfspace with cylindrical cavity is coupled to a finite element mesh that fills the complete cavity and possesses the same material parameters as the surrounding soil with the system of a layered halfspace for which the analytical solution is known.

2 FUNDAMENTAL SOLUTION OF THE HALFSPACE WITH CYLINDRICAL CAVITY

2.1 Basic Equations in Continuum Dynamics

A linear elastic continuum can be described by Lamé’s equation of elastodynamics which consists of three coupled, partial differential equations

0)( iij

jjj

i uuu (1)

with the displacement field u , the Lamé constants of the

material and and the density of the continuum .

Using the Helmholtz decomposition, the partial differential equations are decoupled by replacing the vector field u by the sum of a curl-free field (derivative of a scalar field ) and a

divergence-free field (rotation of a vector field i ).

iklkl

iiu (2)

After this decomposition a system of three partial differential equations is obtained in dependency of the wave-

velocities of the compressional wave pc and the shear wave

sc with 0z [3].

01

01

2

2

i

s

jji

p

jj

c

c

(3)

2.2 Fundamental Solution of a Halfspace

For the system of a halfspace, the system of partial differential equations can be decoupled via a threefold Fourier-Transformation from the Cartesian coordinates x , y into the

wavenumber domain xk , yk and from time domain t into

frequency domain . After these transformations three decoupled ordinary differential equations are obtained

0),,,(

0),,,(

2

2222

2

2222

zkkz

kkk

zkkz

kkk

yxisyx

yxpyx

(4)

that can be solved using an exponential approach for the scalar and vector products

z

iz

ii

zz

eBeB

eAeA22

11

21

21

(5)

with

222

2

2221

syx

pyx

kkk

kkk

(6)

The unknowns 2121 ,,, ii BBAA can be determined with the

boundary conditions on the surface of the halfspace and with Sommerfeld’s radiation conditions [4].

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

584

An advantage of the Integral Transform Method is the possibility to distinguish between the contributions of the solution according to their radiation characteristics. For real

values of i the solutions can physically be interpreted as

surface waves, for imaginary values of i they represent

spatially propagating waves. Using this distinction between near field and far field wave propagation, the computational effort can be reduced [2].

2.3 Fundamental Solution of the Fullspace with Cylindrical Cavity

For the system of a fullspace with cylindrical cavity the Lamé equation is transformed from Cartesian ( zyx ,, ) into

cylindrical coordinates ( ,, rx ). After applying the

Helmholtz decomposition, the vector field i is in a second

step expressed by two independent scalar fields.

iggψ 11

ijj (7)

In order to obtain three ordinary differential equations, again,

a twofold Fourier-Transformation xkx , t and

additionally a Fourier series expansion along the peripheral angle of the cavity are carried out.

0),,,(1

0),,,(1

0),,,(1

2

22

2

2

2

22

2

2

2

22

2

2

nrkr

nk

rrr

nrkr

nk

rrr

nrkr

nk

rrr

x

x

x

(8)

with

22

22

xs

xp

kkk

kkk

(9)

The solution can be expressed by Hankel functions

)()(

)()(

)()(

)2(6

)1(3

)2(5

)1(2

)2(4

)1(1

rkHCrkHC

rkHCrkHC

rkHCrkHC

nnnn

nnnn

nnnn

(10)

The unknowns are determined using the boundary conditions on the surface of the cylindrical cavity as well as the Sommerfeld radiation conditions. As for the halfspace, the solution characteristics of the propagating waves can be used to reduce the computational effort [5].

2.4 Superposition of the Fundamental Solutions

Superposing the two fundamental systems presented in the preceding chapters, the solution for a halfspace with cylindrical cavity can be derived [6]. As both fundamental

solutions require the Fourier-Transformations xkx and

t the three-dimensional, dynamic problem can be

reduced to a quasi-static, two-dimensional analysis. Therefore the superposition is determined for a two-dimensional case for

every combination of and xk . Furthermore all values at

the surface of the halfspace are transformed into the wavenumber domain using a discrete Fourier-

Transformation yy ksky . The values at the

surface of the cavity are decomposed in a Fourier series expansion regarding the peripheral angle n .

Superposing the two fundamental systems the information is used to fulfill the boundary conditions of the complete system at both surfaces. Therefore fictitious surfaces are introduced, a fictitious cavity surface in the halfspace as well as a fictitious halfspace surface in the fullspace with cavity (see figure 2) [4].

fictitions cavity surface

fictitions halfspace surface

αΓ

βΓ

αΓ

βΓ

Figure 2. Halfspace with fictitious cavity surface (left) and fullspace with cylindrical cavity and fictitious halfspace

surface (right).

Loading the halfspace with unit stress states on the

halfspace surface szi , , for every wavenumber yy ksk

the stresses at the fictitious surface of the cavity ),(,

szinrj can be

calculated. Loading the fullspace with unit stress states on the

cavity surface nrj , , for every Fourier series member n the

stresses on the fictitious halfspace surface ),(,

nrjszi can be

determined (see figure 3). The stresses at both boundaries of the superposed system have to be equal to the external loads

( szip ,

at the halfspace and nrjp ,

at the cavity surface).


585

fictitions cavity surface

fictitions halfspace surface

αΓ

βΓ

αΓ

βΓ

szi,σ̂

nrj,σ̂ ),(,ˆ szinrjσ

),(,ˆ nrjsziσ

Figure 3. Stresses at the fictitious surfaces caused by unit stress states.

nrj

S Inrjnrj

szinrjszi

sziN J

nrjszinrjsziszi

pCC

pCC

,,,),(

,,

,),(

,,,,

ˆˆˆ

ˆˆˆ

(11)

This system of equations can also be written in matrix notation

ITMITM PCS ˆˆ (12)

With the following equation the amplitudes C can be calculated.

ITMITM PSC ˆˆ 1 (13)

As a post processing step the same approach can be used to calculate the displacements at the fictitious cavity caused by the unit stress states at the surface of the halfspace and vice versa. The displacements of these load cases multiplied with the amplitudes C yield the displacements of the halfspace and the cavity in the superposed system

S I

nrjnrjnrj

szinrjszinrj

N J

nrjszinrj

sziszisziszi

uCuCu

uCuCu

),(,,

),(,,,

),(,,

),(,,,

ˆˆˆ

ˆˆˆ

(14)

In analogy to the stresses these equations can be written in matrix notation.

CVu ITMITMˆˆ (15)

Inserting equation (13) in equation (15) the relation between the displacements and the stresses is received.

ITMITMITMITMITMITM PNPSVu ˆˆˆˆˆˆ 1 (16)

with the dynamic flexibility

1ˆˆˆ ITMITMITM SVN (17)

Inverting the dynamic flexibility leads to the dynamic stiffness of a halfspace with cylindrical cavity.

1ˆˆ ITMITM NK (18)

As the calculations are carried out for every combination of

xk and , the results in the transformed domain describe a

two-dimensional, quasi-static system. The three-dimensional, dynamic solution is obtained after an inverse Fourier-Transformation.

3 FINITE ELEMENTS IN THE FOURIER-TRANSFORMED DOMAIN

After the computation of the dynamic stiffness matrix of the halfspace with cylindrical cavity, as a second substructure, the dynamic stiffness of a finite element mesh is computed, as similarly presented in the outlook of [7]. The coupling of both substructures shall be carried out in the transformed domain

for every combination of xk and . Thus, the cylindrical

cavity in the original domain which shall be filled with finite elements, is transformed into a circular cavity in the transformed domain. Therefore, two-dimensional finite elements can be used and thus, the computational effort can be reduced. However, the finite element formulation has to be adopted to the Fourier-transformed domain in dependency on

xk and . Additionally, the degrees of freedom on the

coupling surface i.e. on the circular surface, have to be referring to the same basis. As the solution for the halfspace with finite elements is developed in dependency on the Fourier series members n , also the stiffness matrix of the finite element mesh has to be formulated in dependency on these coordinates.

Four-node elements with linear shape functions are chosen for the modeling of the emission system. Additional to the two degrees of freedom in the plane of the two-dimensional element, a third degree of freedom is introduced in order to describe the displacements in the longitudinal direction of the tunnel (see figure 4).

η

ζ

xu

yu

zu x

Figure 4. Four node finite element with linear shape function and three degrees of freedom.

The stiffness matrix of an element is calculated using the

weak formulation of the principle of virtual work

dVxxWV

i ),,(),,()(

(19)

with the local coordinates and . Both the virtual strains

and the stresses depend on the x -coordinate in the original domain. According to the Parseval identity, the integral of the virtual work formulation can also be formulated

with respect to the wavenumber xk [8].


586

xxx dkktkudxxtxu )()(2

1)()(

(20)

Therefore, the element stiffness matrix eK is computed for

every combination of xk and with a Gauß point

integration with m Gauß points as

kkk

m

kkk

Te w)det(),(),(

1

JBDBK

(21)

Due to the Fourier-Transformation of the longitudinal coordinate, the matrix B contains the multiplication of the

shape functions 1N to 4N with xik instead of the derivative

of the shape function with respect to x .

202101

220110

02201

1

200100

020010

002001

Nxikz

NNxik

z

Ny

N

z

N

y

N

z

N

Nxiky

NNxik

y

Nz

N

z

Ny

N

y

N

NxikNxik

B (22)

The mass matrix M is computed analogously to the stiffness matrix.

4 COUPLING OF THE SUBSTRUCTURES

The coupling of the two substructures halfspace with cylindrical cavity and finite element mesh takes places at the two-dimensional circular coupling surface in the transformed domain. As the stiffness of the halfspace with cylindrical cavity is described with respect to polar coordinates on the cylindrical surface and developed into a Fourier series, also the stiffness of the finite element mesh has to be adopted.

In a first step, the degrees of freedom are rearranged according to their position inside the mesh (see figure 5).

FE

FE

FE u

uu (23)

FE

uΓ

FE

uΩ

Figure 5. Degrees of freedom at the boundary and inside the domain of the finite element mesh.

Thus, the system of equations for the finite element mesh

can be formulated as

FE

FE

FE

FE

p

p

u

u

KK

KK (24)

As shown in figure 6, the degrees of freedom at the surface of the finite element mesh are transformed from Cartesian coordinates into a polar basis using the transformation matrix

1T .

)sin()cos(

)cos()sin(0

)1sin()1cos(

)1cos()1sin(0

22

22

1

0000000

0010000000000000001

T (25)

FE

uΓ

cylFEu

,Γ

FE

uΩ

FEuΩ

Figure 6. Transformation of the degrees of freedom at the boundary of the finite element mesh from Cartesian to polar

coordinates.

Afterwards, the degrees of freedom at the surface are

developed into a Fourier series regarding the peripheral angle

with a second transformation matrix 2T (figure 7).

2221

11

11

1211

00

000

000

00

2

inin

in

in

inin

ee

e

e

ee

T (26)


587

FE

uΩ

cylFE

u,Γ

FE

uΩ

ITMuΓ

Figure 7. Development of the degrees of freedom at the boundary of the finite element mesh from polar coordinates to

a Fourier series.

Applying both transformations one obtains the following

system of equations describing the finite elements.

FE

FE

FE

ITM

p

pT

u

u

KTK

KTTKT 111

(27)

with FEITM

uTu 1 and 21 TTT

Coupling both substructures which are described by the Integral Transform Method respectively the Finite Element Method according to the substructure technique leads to the model of the complete system halfspace with cylindrical cavity filled with finite elements described by

FE

ITM

FE

ITM

ITM

FEFE

FEFEITMITM

ITMITM

p

p

p

u

u

u

KK

KKKK

KK

0

0

(28)

where marks the surface of the halfspace.

5 VALIDATION

The presented theoretical derivations are checked comparing the displacements of the following depicted systems (figure 8 and figure 11).

αΓ

βΓ

Centerline

Surface of theHalfspace Λp Λp

Figure 8. Comparison of a halfspace with cylindrical cavity filled with finite elements (left) and a layered halfspace (right)

loaded at the surface of the halfspace.

A halfspace with a cylindrical cavity that is filled with finite

elements, that have the same material parameters as the surrounding medium of the halfspace, is excited by a vertical dynamic load of the surface of the halfspace. The resulting displacements on the surface of the halfspace (figure 9) as well as on the centerline of the mesh are computed (figure 10). As a benchmark a layered halfspace is used. The results

of the coupled system show a good accordance with the benchmark solution.

Homogeneous halfspaceHalfspace with Finite Elements

y-axis

Re(u

z )Im

(uz )

-50 -40 -30 -20 -10 0 10 20 30 40 50-202468

x 10-4

-50 -40 -30 -20 -10 0 10 20 30 40 50

x 10

05

1015

Figure 9. Comparison of real and imaginary part of the vertical displacements at the surface of the halfspace.


Re(u

z )Im

(uz )

x 10-4

x 10-5

-20 -10 0 10 20-5-15 5 15

-20 -10 0 10 20-5-15 5 15y-axis

-2

-1

0

1

-0.6-0.4-0.2

0

-0.8

Figure 10. Comparison of real and imaginary part of the vertical displacements at the centerline of the finite element

mesh.

In a second computation the systems are excited by a load inside the domain. For the finite element calculation the nodes on the centerline are loaded and compared to a layered halfspace where both layers have the same material parameters that is loaded analogously.

αΓ

βΓ

Centerline

Surface of theHalfspace

FEΩpFEΩp

Figure 11. Comparison of a halfspace with cylindrical cavity filled with finite elements (left) and a layered halfspace (right)

loaded at the centerline of the finite element mesh.


588

The results of the comparison are presented in figures 12 and 13. Also for the load inside the domain a good match between the two systems is obtained.


y-axis

Re(u

z )Im

(uz )

-1-0.5

00.5

11.5

x 10-4

-4-202

-50 -40 -30 -20 -10 0 10 20 30 40 50

-50 -40 -30 -20 -10 0 10 20 30 40 50

x 10-4

-6

Figure 12. Comparison of real and imaginary part of the vertical displacements at the surface of the halfspace.


Re(u

z )Im

(uz )

x 10-4

x 10-4

-20 -10 0 10 20-5-15 5 15

-20 -10 0 10 20-5-15 5 15y-axis

-2

-1

0

1

8

10

12

6

Figure 13. Comparison of real and imaginary part of the vertical displacements at the centerline of the finite element

mesh.

6 CONCLUSION

By a coupling of the Integral Transform Method with the Finite Element Method, the advantages of both approaches can be combined. Thus, an emission-transmission-system can be modeled with both an infinite extension of the transmission medium and a detailed geometry of the emission part. As the calculations of the Integral Transform Method are carried out in the Fourier-transformed domain, the formulation of the B-Matrix of the finite elements has to be adopted to the Fourier-transformed domain. For the coupling of the degrees of freedom on the cylindrical coupling surface the finite elements have to be transformed from Cartesian into polar coordinates and developed into a Fourier series. Afterwards, the coupling can be carried out and the results can be checked using a first benchmark test where the finite elements are representing the same material as the surrounding medium.

REFERENCES [1] D. Clouteau, R. Cottereau and G. Lombaert, Dynamics of structures

coupled with elastic media – A review of numerical models and methods, Journal of Sound and Vibration 332, p. 2415-2436, 2013.

[2] G. Frühe, Überlagerung von Grundlösungen in der Elastodynamik zur Behandlung der dynamischen Tunnel-Boden-Bauwerk-Interaktion, Shaker, Technische Universität München, Germany, 2010.

[3] C. F. Long, On the completeness of the Lame potentials, Acta Mechanica 3, p. 371-375, 1967.

[4] K. Müller, Dreidimensionale dynamische Tunnel-Halbraum-Interaktion, Shaker, Technische Universität München, Germany, 2007.

[5] G. Müller, Ein Verfahren zur Erfassung der Fundament-Boden-Wechselwirkungen unter Einwirkung periodischer Lasten, Mitteilungen aus dem Institut für Bauingenieurwesen I, Technische Universität München, Germany, 1989.

[6] G. Frühe and G. Müller, Modelling of the dynamic interaction of tunnel- or longitudinal trench-structures in the half-space by the application of a set of various fundamental solutions, Proceedings of the Institute of Acoustics & Belgium Acoustical Society, Noise in the Built Environment, Ghent, 2010.

[7] G. Müller, G. Frühe and M. Hackenberg. Fundamental solutions in elastodynamics coupled with FEM applied to the dynamic tunnel-soil-building interaction, Eurodyn 2011, Leuven, 2011.

[8] D. W. Kammler, A First Course in Fourier Analysis, Cambridge University Press, New York, United States of America, 2007.


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Implementation of the Finite Element Method in the …paginas.fe.up.pt/~eurodyn2014/CD/papers/080_MS02_ABS...advantages of both outlined methods is presented. The infinite medium of