model reduction methods with application to frequency

Model reduction methods with application to frequency-dependent viscoelastic finite element model

J. Zhang 1, C. Lein1, M. Beitelschmidt 1

1 TU Dresden, Faculty of Mechanical Science and Engineering,Department of Solid Mechanics, Chair of Dynamics and Mechanism Design01062 Dresden, Germanye-mail: [email protected]

AbstractThis paper investigates the application of model order reduction (MOR) techniques on a laminated beamstructure with viscoelastic core. This beam is created by the Finite Element Method (FEM) based ontraditional shape functions. A second order Golla-Hughes-McTavish (GHM) method takes the frequencydependent property of the viscoelastic core into account, enabling the development of a continuous lineartime-invariant Finite Element model. The GHM-approach introduces auxiliary coordinates, which leads toconsiderably larger models. MOR methods based on Moment-matching and balanced realization in bothstate space and second order form are used to reduce the model. The performance of these methods iscompared at a clamped-free beam structure.

1 Introduction

Viscoelastic materials embedded in mechanical structures are widely used for passive damping to reducevibrations in engineering problems. The most general way to model damping is to consider the viscoelasticityvia convolution integrals over kernel functions. GOLLA [8] and LESIEUTRE [22] developed independentways of augmenting a Finite Element model with internal dissipative coordinates which contain materialdamping information obtained through experiments. In this article, the viscoelastic behavior is expressedthrough the GHM method [8], which introduces auxiliary coordinates to take the frequency-dependentproperty of the materials into account. The main advantage of the method is that the final formulationis expressed in the second order physical coordinate system compared to the state space form via theLESIEUTRE approach. However, auxiliary coordinates lead to system matrices of larger dimension, whichdramatically increase the computation time. To reduce the high dimension, Model Order Reduction (MOR)methods are required.MOR is an efficient mean to reduce the size of the original model while preserving the important propertiesas much as possible. MOR has been widely used in structural dynamics and controller design since GUYAN

[11] proposed the first reduction method. Besides several classifications, reduction methods can be dividedinto two categories. The first category involves methods where the accuracy is highly depending on theadequate selection of master coordinates and with or without damping property (e.g. GUYAN condensation[11], Component Mode Synthesis [5]), the second category contains methods where the accuracy of thereduced model is independent of the selection of master coordinates and the damping property is taken intoaccount at the first place (e.g. Krylov Subspace Method [33, 14], Balanced Truncation [27, 7]).In this article, a laminated beam structure with viscoelastic core is considered. Up to now, reduction methodsused for such viscoelastic models are usually based on the GUYAN condensation in physical configurationspace or Balanced Truncation in state space (e.g. [37, 28]), which lacks information about the other out-standing reduction methods with application to viscoelastic material and their variants in second order form.

1813

The performance of these reduction methods is not yet discovered in detail. This article focuses on theutilization of MOR methods belonging to the second category: the Krylov Subspace Method (KSM) basedon Moment-matching and the Balanced Truncation (BT) method both in state space form (first order) aswell as in second order form. For an adequate comparison, all reduction methods considered are given intheir fundamental form without any optimization process introduced. The methods are then illustrated at theexample of a clamped-free laminated beam model with a viscoelastic core. In the end, the preprocessingtoolbox MORPACK [18] is introduced and sophisticated MOR methods from MORPACK are utilized toreduce the model.Section 2 deals with the principals of the GHM method, which is used for the construction of the FiniteElement model in Section 3. The reduction methods KSM and BT including variants are gathered in Section4. The performance of the reduction methods is demonstrated at a numerical example in Section 5.

2 GHM viscoelastic modelling

Motivated by the need to generate a Finite Element model that is capable of analysing the dynamic behaviorof a structure with viscoelastic material, GOLLA [8] uses mini-oscillators to represent the frequency de-pendent property of viscoelastic materials by introducing internal variables. The Golla-Hughes-McTavish(GHM) method uses a second order physical coordinate formulation, which makes it available for usingmodel reduction methods belonging to both physical space and state space. Equation (1) represents a single-input and single-output (SISO) second order dynamic system with viscous damping

Mq(t) + Dq(t) + Kq(t) = bu(t),

y = cq(t).(1)

Transformation of equation (1) into LAPLACE domain and replacing K by GK leads to(s2M + sD + GK

)q(s) = bu(s), (2)

where G is the elastic modulus from the stress-strain relationship σ = Gε. For a viscoelastic material, thestress-strain relationship is replaced by equation (3). The new equation is composed by two parts [4], inwhich the second part is a convolution integral, representing that the model stress is determined both byinstantaneous strain and deformation history

σ = Gε +

∫ t

0G(t− ς)

dε(ς)

dςdς. (3)

LAPLACE transformation of equation (3) yields

σ(s) = G [1 + h(s)] ε(s). (4)

The kernel function is represented by the sum of rational polynomials

h(s) =∑

k

αk(s2 + 2δkωks)

s2 + 2δkωks + ω2k

, (5)

where αk, δk and ωk are the GHM parameters for the kth mini-oscillator. The second order equation for theviscoelastic model becomes{

s2M + sD + G

[1 +

∑k

αk(s2 + 2δkωks)

s2 + 2δkωks + ω2k

]K

}q(s) = bu(s). (6)

For k = 1, the dissipation coordinates are represented by the following equation

z(s) =ω2

s2 + 2δωs + ω2q(s). (7)

1814 PROCEEDINGS OF ISMA2016 INCLUDING USD2016

Transformation of equation (6) back into the time domain gives[M 00 α

ω2 K

] [qz

]+

[D 0

0 2αδω2 K

] [qz

]+

[K(1 + α) −αK−αK αK

] [qz

]=

[bu(t)

0

]. (8)

The equation (8) only consists of one mini-oscillator and one modulus. To generate better results, equation(8) can be written with N mini-oscillators and multi moduli according to [24]

MG =

M 0 · · · 0

0 α1

ω21Kv 0

...... 0

. . . 00 · · · 0 αN

ω2NKv

, DG =

D 0 · · · 0

0 2α1δ1ω2

1Kv 0

...... 0

. . . 0

0 · · · 0 2αN δN

ω2N

Kv

, (9)

KG =

Ke + Kv(1 +

N∑k=1

αk) −α1Kv · · · −αNKv

−α1KTv α1Kv 0

...... 0

. . . 0−αNKT

v · · · 0 αNKv

. (10)

The matrices MG, DG and KG are the global element matrices for mass, damping and stiffness respectively.Ke represents the elastic part and Kv represents the viscoelastic part of the stiffness matrix.

3 Finite Element model of a laminated beam

In this section, the laminated beam is created by Finite Element discritization. The beam consists of threelayers, where layer 1 and layer 3 are the elastic layers (facesheets) and layer 2 is the viscoelastic core(see Figure 1). The modeling method presented was first proposed by SAINSBURY [32]. MARTIN [23]improved this method and validated the efficiency through experiments. The modeling process is based onthe following assumptions:

• each layer has the same transverse displacement,

• the facesheets behave according to the EULER-BERNOULLI assumptions, meaning that there is nox-z-shear in the facesheets and the thickness of the facesheets in the z-direction does not change,

• the x-z-shear stresses in the core are not neglected.

elastic layer (1)

viscoelastic layer (2)

elastic layer (3)

x

z

Figure 1: Degrees of freedom of the laminated beam element with node i and node j

LIGHTWEIGHT STRUCTURES AND MATERIALS 1815

3.1 Shape functions

The potential energy U for one element is

U =1

2qT

e Kqe, (11)

where qe is the element displacement vector, and K is the element stiffness matrix. The kinetic energy T ofthe element is

T =1

2qT

e Mqe, (12)

where qe is the element velocity vector, and M is the element mass matrix. At each node n, four DoF areintroduced, including transverse displacement wn, rotational degree of freedom θn of the elastic layer andaxial displacements un1 and un3 of the middle plane of the facesheets (see Figure 1). The coordinate vectorof the element reads

qe =

[qi

qj

]=[

wi θi ui1 ui3 wj θj uj1 uj3

]T. (13)

With the Hermite shape function and bar shape function, the displacement field vector can be written aswn

θn

un1

un3

=

Nf

N ′f

N1

N3

. (14)

The shape functions are constructed as following with L being the element length

N1 =[

0 0 1− ζ 0 0 0 ζ 0], (15)

N3 =[

0 0 0 1− ζ 0 0 0 ζ], (16)

Nf =[

1− 3ζ2 + 2ζ3 L(ζ − 2ζ2 + ζ3) 0 0 3ζ2 − 2ζ3 L(−ζ2 + ζ3) 0 0], (17)

N ′f =

[∂N

∂x

]=

[1

L

∂N

∂ζ

], ζ = x/L. (18)

3.2 Element stiffness matrix

The element stiffness matrix of the elastic layers is built through potential energies U1 and U3

U1 =1

2

∫ L

0E1A1

(∂u1

∂x

)2

dx +1

2

∫ L

0E1I1

(∂2w

∂x2

)2

dx, (19)

U3 =1

2

∫ L

0E3A3

(∂u3

∂x

)2

dx +1

2

∫ L

0E3I3

(∂2w

∂x2

)2

dx, (20)

where E1 and E3 are the elastic moduli of layer 1 and layer 3, A1 and A3 are the cross section areas, I1 andI3 are the area moments of inertia. The stiffness matrix of the facesheets Ke is obtained by

U1 + U3 =1

2qT

e Keqe. (21)

The stiffness matrix of the core is developed by the shear-strain method proposed by MEAD [25]. Thelongitudinal strain energy is assumed to be zero and the shear-strain is expressed by

γ2 =

[d

h2

∂w

∂x+

u1 − u3

h2

], d = h2 +

1

2(h1 + h3). (22)

The thickness of the three layers is denoted by h1, h2 and h3. The stiffness matrix of the core is obtainedthrough the potential energy stored in the viscoelastic material

U2 =1

2

∫V

Gγ22dV =

1

2

GA2L

h22

qTe Kvqe. (23)


3.3 Element mass matrix

The element mass matrix is computed through the kinetic energy with the total mass m0 = m1 + m2 + m3

T =1

2

∫ L

0

(m0

(∂w

∂t

)2

+ m1

(∂u1

∂t

)2

+ m3

(∂u3

∂t

)2)

dx =1

2qT

e Mqe. (24)

4 Model reduction methods

The viscoelastic modelling approach leads to internal dissipation coordinates which dramatically increasethe dimension of the original system. MOR is an efficient way to find a low-order model that approximatesthe behavior of the original system while keeping its essential properties as much as possible. The low-ordermodel facilitates both the low cost computation and faster response in controller design.In this section, Moment-matching-based and Gramian-based MOR methods for both first order systems (statespace) and second order systems are briefly introduced and the stability of these methods is discussed.

4.1 Moment-matching-based MOR methods

One of the most efficient MOR method is the Krylov Subspace Method (KSM). VILLEMAGNE [36] firstapplied the KSM into model reduction processes and later this method was extended by several authors[20, 6, 33]. The KSM can be applied to large scale models both with or without damping while a lowcomputation cost is required. The reduced model is obtained by matching the Moments of the original andreduced model, in which the Moments are defined as the coefficients of the Taylor series expansion of thetransfer function about an expansion point. The following algorithms for both first order KSM and secondorder KSM are derived from [15, 33].

4.1.1 First Order Krylov Subspace Method

The original KSM for a SISO system was developed from control theory and thus has a state space form

E ˙q = Aq + gu,

y = lq.(25)

The coefficients of the equivalent formulation of the classical second order equation (1) are:

E =

[F 00 M

], A =

[0 F−K −D

], g =

[0b

], l = gT , q =

[qq

]. (26)

The choice of F is arbitrary (e.g. for keeping symmetry F = −K). In this paper, the choice is F = I forillustration. When A is assumed to be nonsingular, the transfer function of equation (25) is obtained throughLAPLACE transformation H(s) = l(sE − A)−1g. By using Taylor expansion, for an arbitrary expansionpoint s0 the transfer function can be rewritten as

H(s) =∞∑

n=0

(−1)nMn(s0)(s− s0)n, (27)

where the coefficients Mn(s0) are called Moments of the transfer function. The objective is to calculate areduced order model of dimension k with k ≪ n, which has the same Moments as the first k Moments of


the original model Mk(s0) = Mk(s0). The reduced model is calculated via orthogonal projection matricesW and V where WTV = I. The projection q = Vqr is applied to system (25)

WTEV ˙qr = WTAVqr + WTgu,

y = lVqr.(28)

The method using both W and V is usually called two-sided MOR method. When W is identical to V, themethod is called one-sided MOR method. The columns of the matrices W and V span the so-called KRYLOV

subspace Kq(A, r) = span{r,Ar,A2r, . . . ,A

k−1r}. If the expansion point of the transfer function is

chosen as s0 = 0, then the first (q1 + q2) Moments match when V is a basis of Kq1(A−1E,A−1g) and

W is a basis of Kq2(A−TET ,A−T l). The columns of the orthogonal basis are usually calculated by well-

established algorithms such as ARNOLDI [1] and LANCZOS [16]. For the first order KSM with W = V,the Moments of the reduced system match the first q1 Moments of the original system. For generating theprojection matrix V, the first order ARNOLDI algorithm is applied (see Algorithm 1).

Algorithm 1 First Order KSM using Arnoldi Algorithm

Input: g = A−1g, A = A−1EOutput: projection matrix V

1: v1 = normalize(g)2: for i = 2, 3, . . . q do3: vi = Avi−1

4: for j = 1 to i− 1 do5: hj,i−1 = vT

i vj

6: vi = vi − hj,i−1vj

7: if vi = 0 then break8: else9: vi = normalize(vi)

10: end if11: end for12: end for

4.1.2 Second Order Krylov Subspace Method

Differing from the first order KSM, the second order KSM [33] is applied to the system in physical domaininstead of transforming the system into state space, thus the second order structure is preserved. The reducedmodel is obtained by applying a projection q = Vqr directly to the second order system (1)

WTMVqr + WTDVqr + WTKVqr = WTbu,

y = cVqr.(29)

The second order KSM is defined by KQ(A1,A2,G1) = span{P0,P1, · · · ,Pq−1}{P0 = G1, P1 = A1P0

Pi = A1Pi−1 + A2Pi−2. i = 2, 3, · · · (30)

If the expansion point of the transfer function is chosen by s0 = 0, then the first (Q1 + Q2) Moments of theoriginal and reduced model match when V is the basis of KQ1(−K−1D,−K−1M,−K−1b) and W is thebasis of KQ2(−K−TDT ,−K−TMT ,−K−TcT ).For the second order KSM with W = V, the Moments of the reduced order model match the first Q1 Mo-ments of the original model. For generating the projection matrix V, the second order ARNOLDI algorithmis applied (see Algorithm 2).


Algorithm 2 Second Order KSM using Arnoldi Algorithm

Input: D = −K−1D, M = −K−1M, b = −K−1bOutput: projection matrix V

1: v1 = normalize(b)2: for i = 2, 3, . . . Q do3: vi = Dvi−1 + Mgi−1, gi = vi−1

4: for j = 1 to i− 1 do5: hj,i−1 = vT

i vj

6: vi = vi − hj,i−1vj , gi = gi − hj,i−1gj

7: if vi = 0 then break8: else9: vi = normalize(vi), gi = normalize(gi)

10: end if11: end for12: end for

4.1.3 Guaranteed stability

Stability is an important property to be preserved from the original system. A time-invariant system isasymptotically stable, if all the eigenvalues of the system matrix A have negative real parts. In system (26),as denoted by SALIMBAHRAMI [33], if A + AT is negative semi-definite and E = ET is positive semi-definite, then the reduced model using a one-sided method with the choice W = V is stable.For an equivalent state space description of a second order system, even using some typical choices of F (e.g.F = I or F = −K), the conditions for keeping stability cannot be fullfilled. Nevertheless, the stability ofthe reduced system can be preserved by choosing F = K or can be recovered by using restarted ARNOLDI

and LANCZOS algorithms [9, 13].For the system (29), if D + DT is positive semi-definite, M = MT is positive semi-definite and K = KT

is positive definite, then the reduced model using a one-sided method with the choice W = V is stable [33].In the presented Finite Element model, the mass matrix M, damping matrix D and stiffness matrix K are allsymmetric and satisfy the above conditions, thus the stability can be preserved by using second order KSM.

4.2 Balanced Truncation

Another important category is the Balanced Truncation (BT) [27] as it provides a priori error bound [7],which gives a direct measure of the quality of the reduced-order model. The reduced-order model is obtainedin two steps. Firstly, a balanced state is found, in which the states are ordered according to the contributionto the input-output behavior. Secondly, a reduced model is obtained by eliminating the states, which havethe smallest contribution to the overall system behavior. The application of the BT method is restricted tosmall to medium scaled systems as solving the two LYAPUNOV equations is quite time-consuming and space-consuming. To extend the usability, several authors proposed methods for solving larger systems [2, 10, 34]and the reduction process for the second order system was also developed [26, 21, 30].

4.2.1 First Order Balanced Truncation

System (1) is converted into the following state space formulation

˙q = Aq + gu,

y = lq,(31)


where

A =

[0 I

−M−1K −M−1D

], g =

[0

M−1b

], l = gT , q =

[qq

]. (32)

Two important Gramians are used to quantify the contribution of each state to the input-out behavior, namedcontrollability Gramian Wc and observability Gramian Wo:

Wc =

∫ ∞

0eAt · g · gT · eAT tdt, Wo =

∫ ∞

0eAT t · lT · l · eAtdt. (33)

They can be obtained by solving the LYAPUNOV equations:

AWc + WcAT + ggT = 0, ATWo + WoA + lT l = 0. (34)

MOORE [27] has shown that there exists a coordinate system, in which these two Gramians are equal anddiagonal. Such a system is then balanced by applying the transformation q = Tq to system (31)

˙q = Aq + gu,

y = lq,(35)

where A = T−1AT, g = T−1g, l = lT. When MOORE first proposed this method, he assumed that fora stable and minimal system, the Gramians Wc and Wo are positive definite and uses CHOLESKY decom-position as the first step for constructing the balancing transformation matrix T. But for the model createdin this article, poles exist which are identical to zeros, thus the system is not minimal and the Gramians arepositive semi-definite. For such stable but non-minimal systems, as noted in [35], the CHOLESKY factors ofWc and Wo can be computed directly from A, g and l using the method of HAMMARLING [12] without theexplicit calculation of Wc and Wo. For generating the balancing transformation matrix T, the square rootmethod proposed by SAFONOV [31] is applied (see Algorithm 3).

Algorithm 3 First Order Balancing AlgorithmInput: controllability Gramian Wc, observability Gramian Wo

Output: balancing transformation matrix T1: Wc = UcScVc Wo = UoSoVo ◃ Singular value decomposition2: H = S

1/2o UT

o UcS1/2c

3: H = UhShVh ◃ Singular value decomposition4: T = UoS

−1/2o UhS

1/2h

HANKEL singular values (HSVs) are defined as the square-rooted eigenvalues of the product of the controll-ability Gramian Wc and the observability Gramian Wo

σi =√

λi(WcWo), i = 1, 2, . . . , n. (36)

For Algorithm 3 presented above, the diagonal elements of Sh are the HSVs. The final reduced model{A11, g1, l1} is obtained from the following partition and truncation

[A g

l 0

]−→

A11 A12 g1

A21 A22 g2

l1 l2 0

. (37)


4.2.2 Second Order Balanced Truncation

Similar to the second order KSM, the second order BT also uses the original second order form. In order totransform the system into a balanced state, the free-velocity and zero-velocity Gramians for a second ordersystem are defined at first [26]. Partitioning the Gramians Wc and Wo into block matrices yields

Wc =

[R ×× L

], Wo =

[S ×× Z

]. (38)

The free-velocity Gramians are defined as Wcf = R and Wof = S. The Algorithm 4 for generating thebalancing transformation matrix Tf is similar to Algorithm 3.

Algorithm 4 Free-Velocity Second Order Balancing AlgorithmInput: free-velocity Gramians Wcf and Wof

Output: balancing transformation matrix Tf

1: Wcf = UcScVc Wof = UoSoVo ◃ Singular value decomposition2: H = S

1/2o UT

o UcS1/2c

3: H = UhShVh ◃ Singular value decomposition4: Tf = UoS

−1/2o UhS

1/2h

The balanced state for the second order system (1) is obtained by

TTf MTf qf + TT

f DTf qf + TTf KTfqf = TT

f bu,

y = cTfqf ,(39)

where q = Tfqf . The HANKEL singular values (HSVs) for the second order system are defined as thesquare-rooted eigenvalues of the product of Wcf and Wof

σi =√

λi(WcfWof ), i = 1, 2, . . . , n. (40)

Similar to the first order BT, the reduced model of the second order BT is obtained by partitioning andtruncating system (39).

4.2.3 Guaranteed stability

In case of the first order BT, for a linear time-invariant continuous system in state space form, an asympto-tically, controllable and observable system guarantees an asymptotical stable reduced system [29]. But theconsidered laminated beam model consists of an ill-conditioned Gramian matrix, which reveals that thereexist uncontrollable and unobservable states, thus the general stability cannot be guaranteed. This problemcan be overcome by using alternative methods (e.g. [17] [27]) or by conducting a minimal realization first.For the second order BT, the stability of the reduced model is not guaranteed for general systems [30]. Butfor the one-sided method used in Algorithm 4, the stability is guaranteed similar to the second order KSMdescribed in the last section.

5 Numerical example

A clamped-free beam model is considered, which consists of two elastic layers of T6-6061 aluminum and aviscoelastic core layer of Sorbothane [23]. The dimensions and material properties of each layer are definedin Table 1. The sandwich beam is divided into 18 elements, each element has eight elastic DoF and 16 internal


variables. To ensure a positive definite element mass matrix MG, a spectral decomposition of the stiffnessmatrix related to the viscoelastic part has to be performed. Thus, six internal variables are kept by eliminatingthe null eigenvalues and corresponding eigenvectors. The global matrices are assembled through these elasticDoF and the internal variables are kept in each element (see Figure 4). The final mathematical model has 180DoF in total. As seen from equation (9), the parameters which describe the frequency-dependent property

Input

Output

Output

Figure 2: Clamped-free laminated beam

Layer Length (m) Width (m) Thickness (m) Density (kg/m3) Elastic modulus (Pa)1 0.3 0.038 3.300E-3 2698.79 6.9E+102 0.3 0.038 3.175E-3 1361.57 G3 0.3 0.038 3.300E-3 2698.79 6.9E+10

Table 1: Physical and geometrical properties of the model

of the viscoelastic material are obtained through curve fits. The MATLAB’s fmincon routine is used toestimate these parameters. Figure 3 shows the experimental data of the viscoelastic material Sorbothane from[23] and the curve fitted results. The obtained parameters are shown in Table 2. For the FE-model obtained

101 102 103 104

Frequency (Hz)

105

106

107

Storagemodulus(P

a)

Complex modulus of viscoelastic material

Data from SorbothaneGHM curve fitted results

101 102 103 104

Frequency (Hz)

104

105

106

Loss

modulus(P

a)

Figure 3: Curve fitting for GHM model using two mini-oscillators

in this article, the modes that involve the dissipation coordinates are over-damped, whereas the other modesare under-damped. Within this property, the stability of the model can be validated by examining if all theeigenvalues are located in the left half complex plane. Table 3 shows the natural frequencies and dampingratios for the first four under-damped modes. As pointed out, the DoF are expanded intensively by usingthe GHM method and more than half modes are over-damped. This makes the model reduction methodparticularly valuable, because the size of the model can be reduced and simultaneously, the important under-damped modes can be retained.


0 50 100 150Degrees of Freedom

0

50

100

150

Degrees

ofFreed

om

Global mass matrix


0

50

100

150

Degrees

ofFreed

om

Global damping matrix


0

50

100

150

Degrees

ofFreed

om

Global stiffness matrix

Figure 4: Global system matrices

G (Pa) α1 δ1 ω1 (rad/s) α2 δ2 ω2 (rad/s)176779.74 2.21 5565.7 1447969.28 10.05 2557.12 38616845.2

Table 2: Curve fitted parameters for GHM model with two mini-oscillators

5.1 Reduction using fundamental MOR methods

The KSM and BT for both first order system and second order system are applied to reduce the dimen-sion of the full model. To assess the performance of these four methods, model correlation methods areutilized to compare the full model and reduced model. As indicated by LEIN [19], eigenvector-based meth-ods and frequency-response-based methods are frequently used for model correlation and model updating.Eigenvector-based criteria are initially developed for undamped or under-damped models. Since differentover-damped modes exist in the full model and reduced model, these criteria cannot be applied generally.Hence, frequency-response-based methods are adopted for comparison. The ‘distance’of the transfer func-tions between full model H(s) and reduced model Hr(s) can be described by the relative H2 error norm

ϵH2,rel :=∥H(s)−Hr(s)∥H2

∥H(s)∥H2

. (41)

Figure 2 shows a clamped-free beam with the impulse function applied at the tip of the beam and the output ismeasured at the same location. Figure 5 shows the H2 relative error norm obtained from different reductionmethods with different reduced order. As seen from Figure 5, the H2 relative error norms for the first orderKSM and first order BT method do not appear in each reduced order as the H2 relative error norm tends toinfinity when the reduced system is unstable. However, both reduction methods for second order systemsguarantee stability. A comparison in the frequency domain is shown in Figure 6 and Figure 7. Figure 6shows the frequency response plot for the full model and reduced model using KSM and Figure 7 shows thefrequency response plot for the full model and reduced model using BT. The reduced order is q = 10 (statespace) and Q = 5 (second order) respectively. The bode diagram of the original model is shown as a solidline, the bode diagrams of the reduced models by first order reduction methods and second order reductionmethods are shown as dashed line and dash-dot line respectively. Both the first order KSM and second orderKSM generate a reduced model, where the first two peaks match the full model. The difference is that thesecond order KSM preserves the structure, which can be seen from the behavior at the high frequency level.First and second order BT methods both give promising results as they give a good reduced-order model for

Mode Natural frequency (Hz) Damping ratio (-)1 31.76 0.07642 172.33 0.02473 467.93 0.01764 911.99 0.0119

Table 3: Natural frequencies with damping ratio for the first four under-damped modes


0 5 10 15 20 25 30 35 40Order of the reduced model

10-4

10-3

10-2

10-1

RelativeH

2errornorm

First Order KSM

Second Order KSM

First Order BT

Second Order BT

Figure 5: Relative H2 error norm between the reduced model and full model with different order

-250

-200

-150

-100

-50

Magnitude(dB)

100 101 102 103 104 105-90

0

90

180

Phase

(deg)

Original modelFirst Order KSM (reduced order q = 10)Second Order KSM (reduced order Q = 5)

Frequency (Hz)Figure 6: Frequency response of full model and reduced model using KSM

the frequency range up to 1000Hz. Differences also arise at the high frequency level. It is clear that allthese four methods are able to remove the dissipative coordinates and reduce the size of the full model. TheGramian-based methods yield a better approximation compared to the Moment-matching-based methods forthe same reduced order. Both second order reduction methods guarantee stability when the original modelis stable. However, this article utilizes all the four MOR methods in their most fundamental form, none ofthem is optimized. For different models and purposes, the reduction methods should be chosen accordingly.

5.2 Reduction using MORPACK

Sophisticated MOR methods are not yet available in FE-software (e.g. ANSYS). For this purpose, the ModelOrder Reduction Package (MORPACK) [18] is developed at the Institute of Solid Mechanics.The model created in this paper consists of internal dissipative coordinates which are not used for assembling.For importing the model into MORPACK, virtual nodes corresponding to the dissipative coordinates arecreated. The master nodes are defined for GUYAN reduction only. Figure 8 shows the bode diagram for fullmodel and reduced models using three different MOR methods from the MORPACK-toolbox. Performanceof the GUYAN method together with another two sophisticated methods Rational Krylov Subspace Method(RKSM) and Second Order Balanced Truncation with position-velocity balancing (SOBTpv) are compared.


-200

-150

-100

-50Magnitude(dB)

100 101 102 103 104 1050

45

90

135

180

Phase

(deg)

Original modelFirst Order BT (reduced order q = 10)Second Order BT (reduced order Q = 5)

Frequency (Hz)Figure 7: Frequency response of full model and reduced model using BT

Details of the RKSM and SOBTpv are described in [3, 18] and [21] respectively. As expected, the GUYAN

method performs worst, because the damping is not taken into account in the projection matrix. In contrast,both the RKSM and SOBTpv give better results compared to their fundamental form.

-200

-150

-100

-50

Magnitude(dB)

100 101 102 103 104 1050

45

90

135

180

Phase

(deg)

Original modelGuyan (reduced to order 20)RKSM (reduced to order 5)SOBTpv (reduced to order 5)

Frequency (Hz)

Figure 8: Frequency response of full model and reduced model using MOR methods from MORPACK

6 Conclusion

In this paper, the mathematical model of a sandwich structure is obtained by the Finite Element Methodcombined with GHM-approach for viscoelastic material. The frequency dependent viscoelastic dampingis successfully augmented into the global system matrix. The Krylov Subspace Method and BalancedTruncation method are all able to reduce the order of the full system and preserve the dynamic behavior.Regarding the whole frequency range, the Balanced Truncation method performs better compared to theKrylov Subspace Method. Both Krylov Subspace Method and Balanced Truncation method in second orderform preserve the second order structure and stability of the original system, which show superiority to theirreduction algorithm in state space form.


References

[1] W. E. Arnoldi. The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quarterly of Applied Mathematics, 9(1):17–29, 1951.

[2] P. Benner, E. S. Quintana-Ortı, G. Quintana-Ortı. Balanced truncation model reduction of large-scale dense systems on parallel computers. Mathematical and Computer Modelling of Dynamical Systems, 6(4):383–405, 2000.

[3] D. Bernstein. Entwurf einer fehleruberwachten Modellreduktion basierend auf Krylov-Unterraum-verfahren und Anwendung auf ein strukturmechanisches Modell. Master’s thesis, Technische Univer-sitat Dresden, Germany, 2014.

[4] R. Christensen. Theory of viscoelasticity: an introduction. Elsevier, 2012.

[5] R. R. Craig. Coupling of substructures for dynamic analyses: an overview. In Proceedings of AIAA-/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference and exhibit, 1573–1584, 2000.

[6] J. Fehr. Automated and error controlled model reduction in elastic multibody systems. PhD thesis, Technische Universitat Stuttgart, Germany, 2011.

[7] K. Glover. All optimal Hankel-norm approximations of linear multivariable systems and their H∞-error bounds. International journal of control, 39(6):1115–1193, 1984.

[8] D. F. Golla, P. C. Hughes. Dynamics of viscoelastic structures-a time-domain, finite element for-mulation. Journal of applied mechanics, 52(4):897–906, 1985.

[9] E. J. Grimme, D. C. Sorensen, P. V. Dooren. Model reduction of state space systems via an implicitly restarted Lanczos method. Numerical Algorithms, 12(1):1–31, 1996.

[10] S. Gugercin, D. C. Sorensen, A. C. Antoulas. A modified low-rank Smith method for large-scale Lyapunov equations. Numerical Algorithms, 32(1):27–55, 2003.

[11] R. J. Guyan. Reduction of stiffness and mass matrices. AIAA journal, 3(2):380–380, 1965.

[12] S. J. Hammarling. Numerical solution of the stable, non-negative definite lyapunov equation. IMA Journal of Numerical Analysis, 2(3):303–323, 1982.

[13] I. M. Jaimoukha, E. M. Kasenally. Implicitly restarted Krylov subspace methods for stable partial realizations. SIAM Journal on Matrix Analysis and Applications, 18(3):633–652, 1997.

[14] P. Koutsovasilis. Model order reduction in structural mechanics: coupling the rigid and elastic multi body dynamics. PhD thesis, Technische Universitat Dresden, Germany, 2009.

[15] P. Koutsovasilis, M. Beitelschmidt. Comparison of model reduction techniques for large mechanical systems. Multibody System Dynamics, 20(2):111–128, 2008.

[16] C. Lanczos. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. United States Governm. Press Office Los Angeles, CA, 1950.

[17] A. J. Laub, M. T. Heath, C. C. Paige, R. C. Ward. Computation of system balancing transfor-mations and other applications of simultaneous diagonalization algorithms. Automatic Control, IEEE Transactions on, 32(2):115–122, 1987.

[18] C. Lein and M. Beitelschmidt. MORPACK-Schnittstelle zum Import von FE-Strukturen nach SIM-PACK. at-Automatisierungstechnik Methoden und Anwendungen der Steuerungs-, Regelungs-und In-formationstechnik, 60(9):547–560, 2012.


[19] C. Lein, M. Beitelschmidt. Comparative study of model correlation methods with application to model order reduction. In Proceedings 26th ISMA (International Conference on Noise and Vibration Engineering), 2683–2700, Leuven, Belgium, 2014.

[20] C. Lein, M. Beitelschmidt, D. Bernstein. Improvement of Krylov-Subspace-Reduced Models by Iterative Mode-Truncation. IFAC-PapersOnLine, 48(1):178–183, 2015.

[21] M. Lenz. Error-controlled Model Reduction in Structural Mechanics based on Balanced Truncation. Master’s thesis, Technische Universitat Dresden, Germany, 2014.

[22] G. A. Lesieutre, E. Bianchini. Time domain modeling of linear viscoelasticity using anelastic displacement fields. Journal of Vibration and Acoustics, 117(4):424–430, 1995.

[23] L. A. Martin. A Novel Material Modulus Function for Modeling Viscoelastic Materials. PhD thesis, Virginia Tech, USA, 2011.

[24] D. J. McTavish, P. C. Hughes. Modeling of linear viscoelastic space structures. Journal of Vibration and Acoustics, 115(1):103–110, 1993.

[25] D. J. Mead, S. Markus. The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions. Journal of sound and vibration, 10(2):163–175, 1969.

[26] D. G. Meyer, S. Srinivasan. Balancing and model reduction for second-order form linear systems. Automatic Control, IEEE Transactions on, 41(11):1632–1644, 1996.

[27] B. C. Moore. Principal component analysis in linear systems: Controllability, observability, and model reduction. Automatic Control, IEEE Transactions on, 26(1):17–32, 1981.

[28] C. H. Park, D. J. Inman, M. J. Lam. Model reduction of viscoelastic finite element models. Journal of Sound and Vibration, 219(4):619–637, 1999.

[29] L. Pernebo, L. M. Silverman. Model reduction via balanced state space representations. Automatic Control, IEEE Transactions on, 27(2):382–387, 1982.

[30] T. Reis, T. Stykel. Balanced truncation model reduction of second-order systems. Mathematical and Computer Modelling of Dynamical Systems, 14(5):391–406, 2008.

[31] M. G. Safonov, R. Y. Chiang. A Schur method for balanced-truncation model reduction. Automatic Control, IEEE Transactions on, 34(7):729–733, 1989.

[32] M. G. Sainsbury, Q. J. Zhang. The Galerkin element method applied to the vibration of damped sandwich beams. Computers & structures, 71(3):239–256, 1999.

[33] S. B. Salimbahrami. Structure preserving order reduction of large scale second order models. PhD thesis, Universitat Munchen, Germany, 2005.

[34] V. Simoncini. A new iterative method for solving large-scale Lyapunov matrix equations. SIAM Journal on Scientific Computing, 29(3):1268–1288, 2007.

[35] M. S. Tombs, I. Postlethwaite. Truncated balanced realization of a stable non-minimal state-space system. International Journal of Control, 46(4):1319–1330, 1987.

[36] C. D. Villemagne, R. E. Skelton. Model reductions using a projection formulation. International Journal of Control, 46(6):2141–2169, 1987.

[37] S. Zghal, M. L. Bouazizi, N. Bouhaddi, R. Nasri. Model reduction methods for viscoelastic sandwich structures in frequency and time domains. Finite Elements in Analysis and Design, 93:12–29, 2015.


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