Building Spectral Element Dynamic Matrices Using Finite...

Building Spectral Element Dynamic Matrices Using Finite Element Models of Waveguide Slices A. L. Goldstein1, J. R. F. Arruda1, P. B. Silva1,R. Nascimento1 1 DMC, Faculdade de Engenharia Mecânica, UNICAMP Mendeleyev 200, Campinas, Brazil email: [email protected]

Abstract In this work it is shown that it is possible to develop spectral finite elements using a short segment of the waveguide modeled with structural finite elements. The transfer matrix of a short segment of a waveguide is used to derive the wave numbers and wave propagation modes that are use to build a spectral element matrix for the waveguide. The spectral elements can be used to model long homogeneous waveguides with constant cross section over long spans without the need of mesh refinement. Recently there has been great interest in the application of periodic structural configurations like phononic crystals to isolate, attenuate and guide vibrations. The application of the spectral element to model a beam section with periodic inclusions is discussed and the limitations of the method are shown. The dynamic response of a beam with periodic inclusions is computed using a combination of the spectral element method and conventional dynamic matrices obtained processing standard FEA software data. The results obtained with the spectral element formulation are shown to be similar to the results of a finite element analysis of the waveguide, thus validating the proposed method.

1 Introduction

Numerous hybrid waveguide-finite element methods have been developed in recent years. The first approach, proposed a decade ago independently by Gavric [1] and Finnveden [2], was called the Spectral Finite Element Method (SFEM). More recently, a new method was developed where a waveguide model can be derived from a standard FE code used to model a short segment of the waveguide. The wave model can be used to compute the spectral relations, the group and energy velocities, and the forced response [3]. Mace and collaborators [4] have also developed recently a method with a similar approach and have investigated numerical issues based on a previous work by Zhong and Williams [5]. These wave approaches that use a finite element model of a short segment or slice of the waveguide are based on the periodic structures theory developed by Mead [6] in the early seventies. Arruda and Nascimento [8] showed that a spectral element can be derived based on wave propagation methods in the form of a spectral element as defined by Doyle [7] based on a dynamic matrix built from a slice of the waveguide. The method, called the Wave Spectral Element Method (WSEM) was applied to a straight rod problem in [9] and has the potential to be applied to more complex structures such as car tires [10,11] and ducts. In the case of ducts, the method can be applied simulate fault detection methods in a straightforward way. The spectral dynamic matrix is obtained, instead of the standard wave propagation solution (propagation modes and wave numbers). Thus, the proposed method can be easily combined with standard finite elements using a mobility approach. Periodic structure theory has also attracted much interest recently in the investigation of phononic band gap materials, periodic structural configurations that hinder wave propagation for some frequency bands, the so called band gap phenomenon. The fundamentals of wave propagation in elastic media are well established [12] and much of the recent research has focused on theoretical demonstration of band gaps in infinite structures [13,14,15]. More recently, the systematic design of phononic band gap materials in finite structures using topology optimization and FE models was presented in [16]. The WSEM method

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combines conventional finite element analysis (FEA) with periodic structure theory and in this paper we consider its application in the modeling of phononic materials. This paper is organized as follows: initially the formulation of the WSEM method is reviewed, the wave spectral element method is derived using the basic theory for longitudinal rods and used the compute the forced response of a rod subject to uniform longitudinal force excitation. The results are compared with results from a FE analysis, thus validating the approach. Next, the application of the WSEM method to waveguides that include periodic structural configurations is discussed and numerical examples of a rod with periodic inclusions showing band gaps in presented.

2 Finite Element Analysis of Periodic Structures

2.1 Dynamic Stiffness Matrix of a Waveguide Section

We consider a structural wave guide represented as a finite or infinite number of cells and assume that a single cell is cut from the structure and meshed with an equal number of nodes on the left and right sides. The dynamic stiffness matrix of the system can be obtained by taking the Fourier transform of the system of ordinary differential equations such that the equations of motion become:

FqMCiK =−+ ][ 2ωω (1)

where K, M, and C are respectively the stiffness, mass and damping matrices, q is the vector of the displacement degrees of freedom and F the vector of applied external forces. The dynamic stiffness matrix is defined as ][ˆ 2 MCiKD ωω ++= and can be decomposed into right (R) and left (L) boundary degrees of freedom and interior (I) degrees of freedom. Assuming there are no forces applied on the interior nodes, the partitioned matrix can be written as:

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

R

L

R

L

I

RRRLRI

LRLLLI

IRILLL

FF

qqq

DDDDDDDDD

D0

ˆˆˆˆˆˆˆˆˆ

ˆ

(1)

The interior degrees of freedom Iq of the dynamic stiffness matrix can be condensed using the first row of the expression above:

)ˆˆ(ˆ 1RIRLILIII qDqDDq +−= −

(2) Using equation (3), the new condensed dynamic matrix after elimination of the interior DOFs can be written as:

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

R

L

R

L

RRRL

LRLL

FF

qq

DDDD

(3)

Where:

IRIIRIRRRR

ILIIRIRLRL

IRIILILRLR

ILIILILLLL

DDDDD

DDDDD

DDDDD

DDDDD

ˆˆˆˆ

ˆˆˆˆ

ˆˆˆˆ

ˆˆˆˆ

1

1

1

1

−

−

−

−

−=

−=

−=

−=

This matrix depends only on the dynamic stiffness matrix of one section of the waveguide and is used as the basis of the analysis that follows.

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2.2 Transfer Matrix Formulation

The transfer matrix method has been used extensively to solve frame structures and rotor system dynamic problems [17]. Instead of relating the forces and displacements at the left and right faces of an element, the transfer matrix relates forces and displacements at one side to the forces and displacements at the other side. In addition, instead of using direct stiffness assembling of the assembled structure, the state vector (displacements and efforts) is propagated from one end of the assembled structure to the other end and the boundary conditions are applied thus generating the solution to the problem. Referring to Figure 1 and assuming there are no forces applied to the structure:

n

RnL

nR

nL

ff

qq

−=

=+

+

1

1

(4)

Introducing the transfer matrix T that links the displacements and forces in the cross section n and n+1 and using the relations (5) above, it is possible to write:

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+

+

1

1

nL

nL

nR

nR

nL

nL

fq

fq

fqT (5)

Now using the first equation of the condensed dynamic stiffness matrix (4) we get:

)(11 nLLL

nLLR

nL qDfDq −= −+

(6)

Introducing the relation above into the second row of equation (4 ) and using expression (5) gives:

111 )( +−− −=+− n

Ln

LLRRRnLLLLRRRRL ffDDqDDDD (7)

Now using the expressions (7) and (8) finally gives the transfer matrix as:

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+−−−

= −−

−

11

1

LRRRLLLRRRRL

LRLLLR

DDDDDDDDDT (8)

In the general case, with L and R denoting the left and right sections of the waveguide, the eigenvalue problem with the transfer matrix can be written as:

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

L

L

L

L

fq

fq

T λ

(9)

Solving this eigenvalue problem yields the 2n wave numbers Lk nn Δ= /))(ln()( ωλω and the propagation modes, where n is the number of DOFs on each side of the meshed section. The dependence upon frequency is shown to emphasize this feature of the wave numbers and propagation modes. The eigenvalue problem must be solved for each frequency and the pairing of the wave numbers can be done by the correlation between the corresponding eigenvectors. Several numerical issues can arise when taking this approach as errors due to FE discretization, spatial discretization and numerical conditioning as discussed in detail in [18].

3 Numerical Example: Beam Slice Wave Spectral Element

The first example shows a simple application of the method to compute the wave numbers of a uniform beam. For the numerical simulations presented next, a steel beam with height = 9mm, width =6mm and short segment 1mm long was modeled with solid finite elements and is shown in Figure 1. The mass and stiffness matrices of the beam slice obtained from the finite element analysis were post processed to assemble the transfer matrix according to equation (9). The absolute wave numbers computed with the eigen analysis of the numerical transfer matrix are shown in Figure 2. The wave numbers shown

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in the figure correspond to the longitudinal, torsional and bending modes of the beam. These results were showed to agree well with the dispersion curves calculated using Timoshenko beam theory [9,19]. For better clarity the imaginary and real part of the wave numbers are shown in Figure 3 and Figure 4.

Figure 1: Finite element mesh of beam slice using solid structural elements (Solid45) in ANSYS

Figure 2: Absolute value of the eigenvalues obtained of the transfer matrix of the beam slice.

4 Numerically Derived Spectral Element Matrix

4.1 Elementary Rod Theory

Elementary theory considers the rod to be long and slender and assumes it supports only 1-D axial stress. In addition, lateral contraction (or the Poisson’s effect) is neglected. Assuming longitudinal deformation u(x), the axial strain can be written as:

xu

xx ∂∂

=ε (10)

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x 105

0

100

200

300

400

500

600

Frequency, [Hz]

Wav

enum

ber,

[1/m

]


Figure 3: Imaginary part of the eigenvalues of

the transfer matrix of the beam slice.

Figure 4: Real part of the eigenvalues of the

transfer matrix of the beam slice.

The corresponding axial stress is:

xuEE xxxx ∂∂

== εσ (11)

The axial stress gives rise to the resultant axial force:

xuEAdAF

Sxx ∂

∂== ∫σ (12)

Where E is the elasticity modulus and A is the cross-sectional area.

4.2 Derivation of the Spectral Element Matrix

It was shown previously by the authors [8] that is it possible to obtain the spectral element matrix of a rod using the transfer matrix obtained from mass and stiffness matrices obtained from the Finite Element model of the slice and to reduce the displacement field so that the resulting number of nodes is the same of the theoretical spectral matrix. In addition, in [9] it was shown possible to keep all the nodal DOF of the finite element model of the slice and build new numerical spectral elements of a rod with arbitrary cross section and length. It is assumed that the longitudinal displacements can be written as:

∑ −Φ=

r

xrikxrrx eAu

(13)

Where and Φ are the wave numbers and propagation modes for the longitudinal wave modes that are obtained from the eigenvalue decomposition of the transfer matrix obtained from the finite element analysis, respectively. The axial strain for longitudinal deformation only is given by:

∑ −Φ−=

∂∂

=r

xrikxrr

xxx eik

xu

ε

(14)

The expression for the axial force is given by equation (13). Substituting the expression for the axial strain into the expression of the force results in:

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0

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400

600

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Wav

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[1/m

]

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xeSEF

xrik

rxrrx ∂∂

Φ=−

∑

(15)

Using equation (15) the wave numbers and corresponding propagation modes can be written in matrix form for an element of arbitrary length L as:

[ ]

[ ]⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡ΦΦ

⎥⎦

⎤⎢⎣

⎡ΦΦ

=⎭⎬⎫

⎩⎨⎧

Δ

Δ

4

3

2

1

0

00.

1001

.

AAAA

eeu

u

LRk

LLk

RL

RL

L

(16)

Where Rnk and Lnk are the n wave numbers for the left and right faces of the cross section element.

In the same way, using equation (16) we get:

[ ]

[ ]⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

ΦΦ

⎥⎦

⎤⎢⎣

⎡−

−ΦΦ

=⎭⎬⎫

⎩⎨⎧

Δ

Δ

4

3

2

1

0

00.

00

.

AAAA

eikeik

ikik

FF

LRkR

LLkL

RL

R

LRL

L

(17)

For the sake of simplicity, the matrix expressions (18) and (20) can be written as:

{ } [ ]{ }Au Ψ= (18)

and:

{ } [ ]{ }AF Ψ= (19)

Substituting (27) into (28) and after further manipulation we arrive at:

{ } [ ][ ] [ ]( ) [ ] { }uF TT ΨΨΨΨ=−1

(20)

Where:

[ ][ ] [ ]( ) [ ]TTK ΨΨΨΨ=−1

)(ω (21)

Equation (30) gives the numerical spectral element matrix of the beam including all the nodal displacements. K is square and if only a few propagation modes are kept, it is close to singular. Therefore the solution to equation (29) is obtained by a pseudo-inversion using the singular value decomposition (SVD) technique [18]. Solving equation (29) for the displacements and rotations gives:

{ } [ ] { }FKu += )(ω (22)

Where [ ]+)(ωK is the pseudo inverse of a matrix obtained via SVD.

5 Numerical Results

5.1 Forced response of an homogeneous rod from a 3D FEM model

Initially, it is shown to be possible to calculate the forced response of a cantilever beam subjected to a homogeneous axial dynamic force. The example considers a steel beam, with E = 2.1e10N/m2 and ρ = 7800kg/m3, height = 9mm, width =6mm and length=100mm fixed at one end and free at the other with a constant axial force applied at all the nodes of the cross section end surface, as shown in Figure 5. The results of axial displacement computed at one of the nodes at the free-end using the proposed method using the 3-D waveguide slice shown in Figure 1 is shown in Figure 6 compared to the result obtained


performing a FE analysis of the entire rod structure modeled with 3-D solid structural elements in a commercial FEA package.

Figure 5: Clamped-free bar with single spectral element subjected to uniform axial dynamic force

Figure 6: Displacement of the beam for a longitudinal uniform force distribution. Circles: FEA;

continuous line: WSEM.

5.2 Forced response of a homogeneous rod from a 2-D FEM model

In this section the WSEM approach is used to model a cantilever rod under uniform longitudinal force based on the analysis of a 2-D FE model of the waveguide segment. The structure material is PVC polymer with E = 3.5e7 N/m2 and ρ = 1200kg/m3 and the dimensions are height = 5mm, width =5mm and length=97.5mm. The rod is fixed at one end and free at the other with a constant axial force applied at all the nodes of the cross section end surface. The spectral element of the rod is calculated using the eigenvalue and eigenvectors of the transfer matrix obtained from the stiffness and damping matrices of a 0.5mm segment of the waveguide created in a commercial FEA package using 2-D structural elements with the plane stress with thickness assumption.

1 2 3 4 5 6 7 8 9 10

x 104

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1

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5x 10-6

Frequency [Hz]

Dis

plac

emen

t rea

l par

t

[m]

h

b

L

F


Figure 7: 2-D FE mesh of waveguide segment with length sliceD

The wavenumber corresponding to the longitudinal wave propagation computed from the eigen analysis of the 2-D FEA model of the rod shown in Figure 7 is presented in Figure 8 for segments of 0.5mm and 2.5mm long. It is important to note the aliasing problems [19] that occur in the WSEM approach determined by the length of the modeled waveguide segment. The limit of the validity of the computed wavelength is determined by sliceD2≥λ , where sliceD is the maximum section dimension of the waveguide segment in the direction of the wave propagation. In this example, the limit for longitudinal wave numbers is for the 0.5mm segment is at sliceDk /limit π= = 6280m-1. The same analysis repeated for a 2.5mm long segment shows the aliasing that occur for k > 3140 m-1 as shown in Figure 8., where this limit is indicated by a horizontal line.

Figure 8: Absolute wave numbers corresponding to longitudinal propagation computed using a

0.5mm waveguide segment (solid line) and a 2.5mm waveguide segment showing aliasing at for a thick (1mm) slice of the rod at k=π/Dslice

The displacement at any point along the rod can be easily representing the finite rod as an assembling of numerical spectral element stiffness matrices. The homogenous rod is modeled as the assembly of five spectral elements 19.5mm long and the displacement at a point along the centerline of the rod and distant

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19.5mm from the clamped end is shown in Figure 9. The response at the same location computed with a 2-D FE model of the entire beam is shown in Figure 10 showing good agreement up to about 15kHz.

Figure 9: Displacement at a position near the rod clamped end computed with the WSEM

method.

Figure 10: Displacement at a position along

the rod near the clamped end computed with ANSYS®

5.3 Model of a rod with a periodic section

In this section the forced response of a rod with a periodic configuration is investigated. The rod material is PVC polymer (E = 3.5e7N/m2 and ρ = 1200kg/m3 ) with steel inclusions (E = 2.1e10N/m2 and ρ = 7800kg/m3). The geometry of the rod is shown schematically in Figure 11.

Figure 11: Geometry of the slender rod with nine periodic inclusions

In the previous section it was shown that the WSEM formulation based on the eigen analysis of the transfer matrix of a waveguide section is limited by the length of the waveguide segment used to compute the transfer matrix. This frequency limit related to the segment length presents problems for the application of the WSEM analysis to model the section of the rod that contains the periodic inclusions, since the waveguide segment would need to be long enough to include an entire cell of the periodic structure. Because of this, in this example the WSEM method is used to obtain dynamic matrices only for the homogeneous sections of the rod. The non-homogeneous section of the rod containing the periodic structures is included in the model using the conventional dynamic stiffness matrices obtained from the dynamic condensation of a 7.5mm long periodic cell of the structure with FE. The spectral-element-based dynamic matrices of the homogeneous section are assembled with the conventional dynamic matrices of the periodic cells in order to compute the forced response of the global structure.

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disp

lace

men

t mag

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F


The free-end uniform section of rod is modeled with a single spectral-element-based matrix and the uniform section at the fixed end is modeled with the assembly of two spectral element stiffness matrices representing 7.5mm long segments. The section of the rod with periodic inclusions is modeled by assembling nine identical conventional dynamic stiffness matrices one for each of the periodic cells. The displacement of the rod at a point after the periodic inclusions can be divided by the displacement at the applied force location to show the effect of the periodic inclusions in hindering wave propagation. This ratio computed for a point along the rod centerline and distant 7.5mm from the clamped end is shown in Figure 12. The response at the same location computed with a 2-D FE model of the 97.5mm rod is shown in Figure 13 showing good agreement up to about 20 kHz. The response shows a band gap starting at about 6 kHz. Note that the band gap shows some resonant peaks due to the finite dimensions of the rod and the lack of structural damping. Structural damping can be easily added to the spectral elements that model the ends of the rod by using a complex elasticity modulus. The ratio of the response of the rod at a location 7.5 mm from the clamped end divided by the displacement at the free end, where the force is applied, with the addition of structural damping is shown in Figure 14.


the rod near the clamped end after the periodic section, computed with WSEM


the rod near the clamped end after the periodic section, computed with ANSYS®

Figure 14: Displacement at a position along the rod near the clamped end after the periodic section divided by the displacement at the free end where the force is applied (transmissibility) computed

with WSEM with structural damping added to the spectral elements at the rod ends

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Next, a rod periodic structural configuration with the same materials as before and including nine smaller inclusions, resulting in a smaller filling fraction, as shown in Figure 18 is modeled using the same steps for the example above. The displacement of the rod with periodic inclusions at a point along its centerline and distant 7.5 mm from the clamped end divided by the displacement at a point at the free end where the force is applied is shown in Figure 19. The equivalent response computed with a 2-D FE model of the entire beam is shown in Figure 20 showing good agreement. The rod with thinner inclusions still shows band gaps starting at around 6 kHz, but the gaps are not as wide or deep as the band gaps shown in Figure 12 and Figure 13.

Figure 15: Geometry of the slender rod with nine smaller periodic inclusions



Figure 17 Displacement at a position along the

rod near the clamped end after the periodic section, computed with WSEM

For many structural problems it would be interest to have band gaps at frequencies lower than showed above that would required wider spacing between the periodic inclusions. The rod with periodic structural configuration with five inclusions as shown in Figure 18 is modeled as the assembly of spectral element stiffness matrices modeling the homogeneous segments of the rod and conventional dynamic stiffness matrices representing each of periodic cells. The displacement of the rod with periodic inclusions at a point along its centerline and distant 7.5 mm from the clamped end divided by the displacement at a point at the free end where the force is applied is shown in Figure 19. The equivalent response computed with a 2-D FE model of the entire beam is shown in Figure 20 showing good agreement. The rod with larger spacing between the inclusions but with only five inclusions shows partial band gaps starting at around 3 kHz, but that are not as wide or deep as the band gaps found in the rod with nine inclusions.

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Figure 18: Geometry of the slender rod with periodic inclusions with larger spacing





6 Conclusion

In this work it was shown that using the wavenumber and the longitudinal wave mode obtained from the 2-D or 3-D segment of a waveguide it is possible to obtain a numerical spectral element matrix of the waveguide using the so called WSEM approach. This approach was used to compute the forced response of a uniform rod that was validated against the FEA model of the structured. The validity of the results at higher frequencies due to aliasing effects determined by the segment length were also discussed and shown to restrain the application of the method to thin waveguide segments. It was also shown that the WSEM method can be combined with conventional dynamic matrices obtained from the FEA of a periodic structure cell to obtain the forced response of a rod with homogeneous and periodic sections.

Acknowledgements

The authors are grateful to the government funding agency Fundação de Amparo à Pesquisa de São Paulo – FAPESP.

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F 7.5 mm


References

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