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Wavelet Galerkin Methods for Elastic Multibody Systems

Diaz, José; Führer, Claus

Published in:European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS

2004

Link to publication

Citation for published version (APA):Diaz, J., & Führer, C. (2004). Wavelet Galerkin Methods for Elastic Multibody Systems. In P. Neittaanmäki, T.Rossi, S. Korotov, E. Oñate, J. Périaux, & D. Knörzer (Eds.), European Congress on Computational Methods inApplied Sciences and Engineering ECCOMAS (Vol. II). P. Neittaanmäki, T. Rossi, S. Korotov, E. Oñate, J.Périaux, and D. Knörzer (eds.),.

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European Congress on Computational Methods in Applied Sciences and EngineeringECCOMAS 2004

P. Neittaanmaki, T. Rossi, S. Korotov, E. Onate, J. Periaux, and D. Knorzer (eds.)Jyvaskyla, 24–28 July 2004

WAVELET GALERKIN METHODS FOR ELASTICMULTIBODY SYSTEMS - A CASE STUDY: ROLLER

BEARINGS.

Jose Dıaz Lopez, Claus Fuhrer

Numerical AnalysisLund University

Box 118, SE-211 00 Lund, Swedene-mail: jose,claus@maths.lth.se, web page: http://www.maths.lth.se/na

Key words: Galerkin, Wavelet, Elasticity, Roller Bearings

Abstract. In the present paper we discuss simulation with a wavelet element method.The model problem is a simplified 2D-roller bearing model consisting of an elastic outerand inner ring and a roller. This model reflects typical properties of elastic multibodysystems, eg. combination of gross motion with elastic deformation and contact problems.The wavelet element method is implemented in a similar way as the widely used finiteelement method. Its inherent advantages concerning model reduction will be demonstratedas well as the influence of model reduction on simulation time.

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Jose Dıaz, Claus Fuhrer

1 Elastic Roller Bearing Model Problem

In order to demonstrate the power of a wavelet element method we consider an elasticextension of a simplified 2D-roller bearing model presented in [11]. The model consistsof two elastic rings and an elastic roller see Fig. 1. The outer ring is assumed to rotate

Figure 1: 2D roller bearing model and rigid body states

with constant speed ωo, while the inner ring is fixed. The rollers may be in contact withone of the rings depending on their state. The inertial system is placed into the centerof the inner ring and the outer ring is described by means of Cartesian coordinates withrespect to that system by the six rigid state variables (yout, θout, yout, θout), where y ∈ R2

expresses the coordinates of the center of the outer ring and θ its orientation. The motionof the roller is given accordingly by the rigid states (yR, θR, yR, θR).

The elastic deformation is taken into account by additional elastic state variables,qR, qR ∈ RNR for the roller and qout, qout ∈ RNout , qin, qin ∈ RNin for the rings, resultingfrom a wavelet semidiscretization of the rollers and the rings.

The total number of elastic degrees of freedom depends on the discretization and isnormally substantially larger than the number of rigid degrees of freedom.

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Jose Dıaz, Claus Fuhrer

The number of elastic degrees of freedom is classically reduced when using a finiteelement discretization by modal reduction techniques. Using modal reduction techniquesto reduce the number of elastic degrees of freedom may yield a poor description of therelevant quantities for multibody simulation, e.g. direction and magnitude of the contactforces.

In contrast, semidiscretization by wavelets offers direct reduction possibilities whilepreserving the description of the deformation field near the contact point which yieldsbetter results and speeds up the simulation time.

The disposition of the paper is the following: In Section 2 we present the equations ofmotion of a single elastic body. In Sec. 3 a brief introduction to wavelets and the waveletelement method (WEM) adapted to our model problem is presented. In Sec. 4 the contactinteraction is discussed and included in the model. Numerical simulations are presentedin Sec. 6.

2 ELASTIC BODY MOTION

The coupled elastic/rigid motion is described here for the outer ring. The descriptionof the inner ring and the roller follows in a similar way. All three elastic bodies are thencoupled together by contact constraints.

The body has the reference configuration Ω, which is the closure of an open convex setΩ ∈ R2 with the boundary Γ. We assume that this boundary is divided into two disjointsections Γ0 and Γ1 and will impose Dirichlet (natural) boundary conditions at Γ0 andNeumann (essential) boundary conditions at Γ1.

The body’s deformation is described in material coordinates, i.e. with respect to thereference configuration of the undeformed body with the displacement field u(x, t), x ∈ Ω,see Fig. 2.

For a fixed time t the displacement field u is a function in the Hilbert space

V =v : vi ∈ H1(Ω), i = 1, 2, 3

with the inner product

〈v, w〉V =2∑

i=1

〈vi, wi〉H1 .

As we will state following the lines of the equations of motion in their weak or variationalform, [10], we have to consider also the subspace of test functions

V0 = v ∈ V : v|Γ0 = 0 .

With the body’s mass m, moment of inertia J(u) =∫

Ωρ(x + u)T (x + u)dx, constant

density ρ, inner volume forces β(x, θ) and surface forces τ(x, θ) on Γ1 the equations of

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Jose Dıaz, Claus Fuhrer

Figure 2: Reference systems in an elastic body

motion of the outer ring take the form:

my + A′(θ)s(u)θ − A(θ)s(u)θ2 + 2A′(θ)θ

∫Ω

ρudx

= A

∫Ω

β(x, θ)dx + A(θ)

∫Γ1

τ(x, θ)ds (1a)

s(u)TA′(θ)Ty + Jθ +

∫Ω

ρ(x + u)TIudx + 2θ

∫Ω

ρ(x + u)Tudx =

=

∫Ω

(x + u)TIβ(x, θ)dx +

∫Γ1

(x + u)TIτ(x, θ)ds (1b)

⟨ρA(θ)Ty, v

⟩+

⟨ρ(θIT − θI)(x + u), v

⟩+

⟨ρ(2ITuθ), v

⟩+ 〈ρu, v〉+ a(u, v)

= 〈l, v〉 ,∀v ∈ V0, (1c)

where A(θ) is the 2D rotation matrix, s(u) =∫

Ωρ(x + u)dx, and I =

(0 −11 0

). a(u, v)

is the bilinear form depending on the constitutive elasticity law and 〈v, l〉 a linear formarising form homogeneisation of boundary conditions at Γo.

For more details, see [10, 8].

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Jose Dıaz, Claus Fuhrer

2.0.1 Galerkin Semidiscretisation

Consider now a finite dimensional subspace Vh0 ⊂ V0 with a basis ζjNq

j=1. Galerkin

semidiscretization seeks for an approximative solution uh ∈ Vh0 . A special case is the

wavelet element method which takes a wavelet basis ζjNq

j=1 for Vh0 .

The usual separation ansatz leads to

uh(x, t) =

Nq∑j=1

qj(t)ζj(x)

with a set of time dependent coefficients qj(t)j, the elastic states of the system. Tocompactify the presentation we write

uh(x, t) = N(x)q(t) =(ζ1(x)...ζj(x)...ζNq(x)

)

q1...qj...

qNq

.

We obtain then N + 3 differential equations to determine y, θ and q:

my + A′ (sr + shq) θ + ACT1 q = A (sr + shq) θ2 − 2A′CT

1 q + fT(θ, q, t) (2a)

(sr + shq) A′Ty + (Jr + Jhq) θ + (C2 + C3q) q = −2 (C4 + Mhq) θq + fR(θ, q, t) (2b)

C1ATy + (C2 + C3(q))θ + Mhq = (C4 + Mhq)θ

2 − 2C3θq −Khq + lh (2c)

where we introduced the following abbreviations:

• Discrete mass, stiffness matrix and forces

Mh =

∫Ω

ρNTNdx

(Kh)j1,j2 = a(ζj1 , ζj2)

(lh)j = 〈l, ζj〉

• Coupling matrices

C1 =

∫Ω

ρNTdx, C2 =

∫Ω

ρ(x1N2 − x2N1)dx

C3 =

∫Ω

ρ(NT2 N1 −NT

1 N2)dx, C4 =

∫Ω

ρ(x1N1 + x2N2)dx

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Jose Dıaz, Claus Fuhrer

• Variation terms

sh =

∫Ω

ρNdx, Jh = 2C4q + qTMhq

• Force terms

fT (θ, t) = A

∫Ω

β(x, θ)dx + A

∫Γ1

τ(x, θ)dx

fR(θ, q, t) =

∫Ω

(x + Nq)TIβ(x, θ)dx +

∫Γ1

(x + Nq)TIτ(x, θ)dx

2.1 Semidiscretized Roller Bearing system

We collect the degrees of freedom in a state vector w =(qin, wR, wout

)Twith the

states for the roller and outer ring wR/out =(yR/out, θR/out, qR/out

)T.

The motion of the uncoupled system is then described the by differential equationsMin

MR

Mout

w =

fin(t, win, win)fR(t, wR, wR)

fout(t, wout, wout)

where the matrices for the roller and outer ring i ∈ R, out are

Mi(wi) =

miI2×2 A′i(sr,i + s∆,iqi) AiC

T1,i

(sr,i + s∆,iqi)TA′

iT Jr,i + J∆,i C2,i + C3,iq

C1AT (C2,i + C3,iqi)

T Mhi

,

fi(t, wi, wi) =

Ai (sr,i + shiq) θ2

i − 2A′iC

T1,iqi + fT,i(θi, t)

−2 (C4,i + Mhiqi) θqi + fR,i(qi, t)

(C4,i + Mhiqi)θ

2i − 2C3,iθiqi −Khi

qi + lhi

,

and those for the inner ring Min = Mhinand fin = Khin

qin + lhin.

The three bodies are coupled by unilateral constraints based on a contact model de-scribed in detail in Sec. 4 below. By introducing constraint forces the final system becomesan index-3 differential algebraic system of the generic form

M(w)w = F (t, w, w)−GT (w)Tµ (3)

0 = gT (w) (4)

with GT = ∂g/∂w.

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Jose Dıaz, Claus Fuhrer

3 Wavelet Element Method (WEM)

In [2] the general setting of the wavelet element method (WEM) is presented. Like inthe finite element case it consists of a Galerkin semidiscretization with locally supportedbasis functions ζj, but unlike finite elements the wavelet basis contains information whichcan be used in an effective way for system reduction. So an extra modal reduction steplike in the FEM case is no longer required.

The discrete space Vh0 ⊂ V0 is constructed by its basis functions ζj in the following

way:

1. An appropriate father wavelet φ ∈ H1(R) is selected, with the property that thetranslated functions φ(·− j), j ∈ Z are linearly independent and span a linear spaceVn0 . . . ⊂ H1(R)

2. We consider father wavelet functions having compact support, so that by truncat-ing the basis, dilating the basis functions and adding additional boundary waveletfunctions a basis for a corresponding finite dimensional subspace of H1([0, 1]) canbe constructed.

3. By forming tensor products of these basis functions a basis for Vn0 ⊂ H1([0, 1]×[0, 1])is obtained.

4. Finally by constructing a continuously differentiable map FΩ : Ω −→ [0, 1]2 thisbasis can be transformed to a basis for Vh

0 ⊂ H1(Ω)

The selection of a father wavelet is based on the smoothness requirements given by theequations of motion (2). We choose the construction presented in [9] based on Daubechieswavelets of order N , [4], denoted as DBN now on. An alternative basis is presented in [3]based on biorthogonal wavelets and can be used for more general geometries.

When scaling and translating father wavelet functions φk,j := φ(2k · −j), j ∈ Z, k ∈ Z+

a basis of a higher dimensional space is constructed and a ladder of spaces is obtained

H1(R) ⊃ ... ⊃ V n ⊃ V n−1... ⊃ Vno

giving a natural way of increasing or decreasing the resolution level n. The lowest possibleresolution level is called no.

Following the steps 1-4 above a ladder of semidiscretization spaces is obtained:

H1(Ω) ⊃ ... ⊃ V n ⊃ V n−1... ⊃ V no

withV n := span

φn,j1,j2 : φn,j1,j2 =

(φn,j1 ⊗ φn,j2

) FΩ

. (5)

For the Galerkin ansatz function uh we set Vho = V n ⊕ V n for some n > no resulting in

N2q = 2 · 22n elastic degrees of freedom qj.In the following sections we formulate the model in this basis.

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Jose Dıaz, Claus Fuhrer

4 ELASTIC CONTACT

We consider now two bodies, the outer ring Ωout and the roller ΩR with boundary∂Ωout = Γout

1 ∪ Γout0 resp. ∂ΩR = ΓR

1 ∪ ΓR0 . Dirichlet boundary conditions are given on Γi

0

and Neumann boundary conditions on Γi1.

The dynamic contact problem is to find at any time point t a state w(t) so that thetwo bodies are either in contact but not penetrating or are free floating. Contact causescontact forces as constraint forces, which are characterized by not being dragging forces.

The part of the Neumann boundary, where contact occurs will be denoted by ΓRc rsp.

Γoutc . These boundary segments depend on the actual state w(t).

4.1 Determination of the contact points

The node-to-element approach by [6] considers the case, where the roller (called theslave body) penetrates the outer ring (called the master body). A penetrating nodal pointis denoted by xs ∈ ΓR

1 .For mathematically describing the contact condition we parameterize a section ΓR

c ⊂ ΓR1

of the roller boundary and a corresponding section of the outer ring boundary Γoutc ⊂ Γout

1 .This parametrization and the corresponding boundary description is based on the orderof the discretization. If Daubechies wavelets DB3 are taken the interpolation error bydescribing these boundary segments by a third degree polynomial are of the same orderas the overall discretization error.

We select the four nearest nodes to xs on both boundaries and denote them byxm1 , xm2 , xm3 , xm4 and xs1 , xs2 , xs3 , xs4 . The corresponding boundary profile sections ofthe undeformed bodies and the corresponding deformations can then be described by fourthird degree polynomials xs(ξs), us(ξs) and xm(ξm), us(ξm).

Let ξs and ξm be the parameter values of the (unknown) contact points, then we canformulate the contact conditions

gcon(w, ξm, ξs) = ys + A(θs)(xs(ξs) + us(ξs))− (ym + A(θm)(xm(ξm) + um(ξm))) = 0

Additionally we need the so-called non penetration condition

gnpe(w, ξm, ξs, λ) = λA(θs)(x′s(ξs) + u′s(ξs))− A(θm)(x′m(ξm) + u′m(ξm)) = 0

stating that the tangents at the contact points are parallel.Together we get four conditions for the four unknown variables µs, ξs, ξm and the tan-

gent scaling λ.The corresponding constraint matrix G is obtained by the requirement that the relative

velocity in the contact points is orthogonal to the boundary normal vector in these points.After having set up the constraints for all contact points we finally get a differential-

8

Jose Dıaz, Claus Fuhrer

algebraic system of the following form

M(w)w = F (t, w, w)−GT (w, ξ, λ)Tµ (6)

0 = gcon(w, ξ, λ) (7)

0 = gnpe(w, ξ, λ) (8)

with ξ ∈ R2nµ being the vector of all contact point parameters and λ ∈ Rnµ the vector ofall tangent scaling variables.

This system has index-3 with λ and xi as index-1 variables, while the Lagrange multi-pliers are index-3 variables. The numerical treatment of these kind of systems is discussedin [7, 1].

5 NUMERICAL SIMULATIONS

5.1 Numerical Algorithms

Before numerically integrating the system (6) its index has to be reduced to a stabilizedindex-2 system, [7, 1]. The resulting system can then be integrated by BDF methods likeDASSL.

To handle switching between contact and no contact a switching function was imple-mented. This function is a vector valued function, with components corresponding toboundary nodal points. A zero of one of its components indicates the transition betweenthe states contact/no contact has to be made. Zeros are detected by a safeguard algorithmbased on inverse interpolation and bisection as described in [7].

To test the performance and accuracy of the wavelet ansatz we compare it with a corre-sponding FEM semidiscretization. The simulation, contact point detection and switchingevent handling is performed in both cases by identical algorithms.

5.2 Finite elements

FEMlab is used in plane stress mode to describe each individual body and computemass and stiffness matrices Mh,in, Mh,R, Mh,out, Kh,in, Kh,R and Kh,out with linear elements.Typical triangularizations for the outer ring and the roller are depicted in Fig. 3. Theboundary of the rolling elements are under Neumann boundary conditions. The outerboundary of the inner ring is under Neumann boundary conditions, as the inner boundaryof the outer ring. Dirichlet boundary conditions are imposed at the outer boundary ofthe outer ring and inner boundary of the inner ring.

5.3 Wavelets

Own routines in Matlab were written for WEM. Typical wavelet triangularisations ofthe roller and rings are presented in Fig. 4. The Schwarz-Christoffel map is used to mapthe unit square to the circle, see [5]. The exponential map is used to map the unit squareto the rings. The corresponding boundary conditions are periodic at two opposite sides of

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Jose Dıaz, Claus Fuhrer

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2021 22 23

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3132

3334

3536 37 38

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4344

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4748

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45

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78

910

1112

13 141516

171819 20

21 22

2324

25

Figure 3: FEM triangularization of rolling elements and rings

the unit square and Dirichlet / Neumann conditions can be set at the other two. WaveletDB3 was used in the discretization.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Figure 4: Wavelet triangularization of rolling elements and rings

5.4 Comparison WEM-FEM

We present in Fig. 5 the sparsity structure of Kh,R of both approaches, FEM andWEM . The WEM stiffness matrix is always banded, with a bandwidth depending on sizeof the the wavelet support. This feature can be exploited to save computational workusing specially adapted algorithms for banded matrices. FEM generates a sparse butunstructured stiffness matrix.

If ρ is constant The mass matrix is the identity in the WEM case. This means that thespectrum is determined by the stiffness matrix Kh only. This is not the case for a FEM

10

Jose Dıaz, Claus Fuhrer

discretization.Model reduction techniques reduce the number of elastic coordinates q to qred = Chq

with some matrix Ch ∈ RNred×Nq . The mass and stiffness matrices are reduced by

Mh,red = CTh MhCh

Kh,red = CTh KhCh.

This matrix Ch is in the FEM case obtained form the eigenvectors of the generalizedeigenvalue problem

−ω2Mhq = Khq.

We perform model reduction by selecting eigenvectors corresponding to the s lowest fre-quencies ω1, ω2, ..., ωs and assembling them in Ch = (q1, q2, ..., qs).

In the WEM case the situation is different. We construct the matrix Ch by setting toone the diagonal element corresponding to those wavelets that have the contact node intheir support.

In Table 6.4 is the dimensions of the semidiscretisations with FEM, WEM and usingWEM model reduction. Notice that the model reduction depends on the state and possiblecontact contraints applied at a certain time. We give the maximum number of equationsselected by the algorithm for the time interval [0, 0.1].

FEM WEM red WEMRoller 108 128 max 40

Ex. Ring 976 2048 max 128In. Ring 480 512 max 128

Table 1: Dimensions of the discretisations for FEM and WEM

In Fig. 7 we present the reference solution with FEM without model reduction andcompare the step size sequence generated by DB3 and WEM model reduction. We observethat the WEM time step sizes are larger than those required by the FEM. Notice thatthe time points when the step size decreases drastically are those when the roller bouncesagainst one of the rings. Although the number of equations in the reduced WEM modelis considerably lower, the bounces occur at nearly the same instances. This property wasnot observed when using modal reduction.

In Fig. 6 the spectra of the roller element for FEM and WEM with and without modelreduction are depicted. In the reduced WEM we observe that non determinant highfrequency modes are removed, causing a significant speed-up of integration time.

6 CONCLUSIONS

We present a WEM discretisation of a simple 2D roller bearing model with possiblityto model reduction using the properties of wavelet decomposition.

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Jose Dıaz, Claus Fuhrer

0 20 40 60 80 100 120 140 160 180

0

20

40

60

80

100

120

140

160

1800 50 100 150 200 250 300 350 400 450 500

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50

100

150

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350

400

450

500

0 50 100 150 200 250 300 350 400 450 500

0

50

100

150

200

250

300

350

400

450

500

nz = 126624

Figure 5: FEM and Wavelet (single and multiresolution) sparsity structure of stiffness matrices

We show that WEM model reduction techniques are robust in the sense that we donot need to know in advance where possible forces/interactions occur to apply modalreduction or static mode analysis.

An improvement with respect to linear FEM discretisation is shown.We observe that the order of approximation of the boundary is the same, but WEM

model reduction conserves this order after reduction.

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Jose Dıaz, Claus Fuhrer

−2000 −1500 −1000 −500 0

−800

−600

−400

−200

0

200

400

600

800

Real Part

Imag

inar

y Pa

rt

−1600 −1400 −1200 −1000 −800 −600 −400 −200 0

−600

−400

−200

0

200

400

600

Real Part

Imag

inar

y Pa

rt

−400 −350 −300 −250 −200 −150 −100 −50 0

−150

−100

−50

0

50

100

150

Real Part

Imag

inar

y P

art

Figure 6: FEM and Wavelet spectrum of the stiffness matrix for the rolling element

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.110

−12

10−10

10−8

10−6

10−4

10−2

100

time

Ste

p S

ize

FEMDB4−red

Figure 7: Step size sequence of the example

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Jose Dıaz, Claus Fuhrer

REFERENCES

[1] Martin Arnold and Helmut Netter. Ein modifizierter Korrektor fur die stabi-lizierte Integration diffenrential-algebraicher Systeme mit von Hessenberg abweichen-der Strucktur. Robotik und Systemdynamik (FF-DR) IB.Nr 515-93-03, DLR, DLR,Oberpfaffenhofen, 1993.

[2] Claudio Canuto, Anita Tabacco, and Karsten Urban. The wavelet element method.II. Realization and additional features in 2D and 3D. Appl. Comput. Harmon. Anal.,8(2):123–165, 2000.

[3] Wolfgang Dahmen, Angela Kunoth, and Karsten Urban. Biorthogonal spline waveletson the interval—stability and moment conditions. Appl. Comput. Harmon. Anal.,6(2):132–196, 1999.

[4] Ingrid Daubechies. Ten lectures on wavelets. Society for Industrial and AppliedMathematics (SIAM), Philadelphia, PA, 1992.

[5] Tobin A. Driscoll. A MATLAB toolbox for Schwarz-Christoffel mapping. ACMTrans. Math. Soft., (22):168–186, 1996.

[6] Peter Eberhard. Kontaktuntersuchungen durch hybride Mehrkorpersystem/ finiteElemente Simulationen. Berichte aus dem Machinenbau ISBN 3-8265-8107-5, ShakerVerlag GmbH, Aachen Germany, 2000.

[7] Edda Eich-Soellner and Claus Fuhrer. Numerical methods in multibody dynamics.European Consortium for Mathematics in Industry. B. G. Teubner, Stuttgart, 1998.

[8] Jose Dıaz Lopez. A study on wavelet semidiscretization of elastic multibody systems.Licenciate Thesis 2002:4, Lunds Universitet, Numerisk Analys Box 117 SE-211 0Lund Sweden, 2002.

[9] Pascal Monasse and Valerie Perrier. Orthonormal wavelet bases adapted for partialdifferential equations with boundary conditions. SIAM J. Math. Anal., 29(4):1040–1065 (electronic), 1998.

[10] Bernd Simeon. Numerische Simulation gekoppelter Systeme von partiellen unddifferential-algebraische Gleichungen in der Mehrkorperdynamik. Fortschritt-BerichteVDI, 20,325, 2000.

[11] Anders Sjo. Numerical aspects in contact mechanics and rolling bearing simulation.Licenciate Thesis 1996, Numerisk Analys, Lunds Universitet, 1996.

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