Doctoral Thesis No. 49, 2011

Adaptive Reduction of Finite ElementModels in Computational Solid Mechanics

Hakan Jakobsson

Department of Mathematics and Mathematical StatisticsUmea UniversitySE-901 87, Umea, Sweden

Department of Mathematics and Mathematical StatisticsUmea UniversitySE-901 87, Umea, Sweden

Copyright c© 2011 Hakan JakobssonISBN 978-91-7459-227-6ISSN 1102-8300Typeset by the author using LATEX 2εPrinted by: Print & Media, Umea University, Umea, 2011

To Beatrice and Elianne

Abstract

The design of complex structures with high demands on strength, durability,efficiency and performance, require accurate and reliable approximations of thestructural response under typical and extreme loading conditions. The finite ele-ment method is the standard method to compute such approximations. However,in some instances finite element models are too computationally expensive, andreduced finite element models with much fewer degrees of freedom must be used.This thesis deals with the construction of such reduced models. It consists of anintroduction and seven papers.

I. H. Jakobsson and M. G. Larson, Mode Superposition with Submodeling.Submitted.

II. H. Jakobsson, F. Bengzon, and M. G. Larson, Adaptive Component ModeSynthesis in Linear Elasticity, Int. J. Numer. Meth. Engng 86 (2011),829–844.

III. H. Jakobsson and M. G. Larson, A Posteriori Error Analysis of ComponentMode Synthesis for the Elliptic Eigenvalue Problem. Submitted.

IV. H. Jakobsson and M. G. Larson, A Posteriori Error Analysis of ComponentMode Synthesis for the Frequency Response Problem. Submitted.

V. H. Jakobsson, F. Bengzon, and M. G. Larson, Duality Based Adaptive ModelReduction for One-way Coupled Thermoelastic Problems. Submitted.

VI. H. Jakobsson, On Generalized Proper Orthogonal Decompositions. Submit-ted.

VII. H. Jakobsson, M. G. Larson, Model Reduction in Rolling Bearing Simula-tion Based on Static Load Cases and the Proper Orthogonal Decomposition.Technical report.

In the first paper a new approach for local enhancement of mode superpo-sitions that builds on the concept of submodeling is investigated. We impose amultiscale split on the reduced solution into a global part given by the mode su-perposition method, and a local part given by a patchwise finite element problem.The patch problem yields a local correction on the modal approximation.

The second paper deals with a posteriori error estimation for the Craig-Bampton component mode synthesis (CB-CMS) method. We develop the nec-essary framework for a posteriori error estimation in terms of a basic modelproblem. The estimates determine to what degree each CMS subspace influencethe error in the reduced solution. We prove a posteriori error estimates for theerror in a linear goal quantity as well as in the energy and L2 norms.

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In the third paper we apply the methodology developed in the second paperand derive a posteriori error estimates for the error associated with CMS reduc-tion of the elliptic eigenvalue problem. We prove estimates for the approximatedeigenvalues as well as the eigenmodes.

The fourth paper continues the development of a posteriori error estimates forCB-CMS. Here we consider the frequency response problem and prove estimatesin a linear quantity of interest and the energy norm.

In the fifth paper we derive an a posteriori error estimate for a one waycoupled thermoelastic problem where CB-CMS is used in both the thermal andelastic solvers. A main feature with the estimate is that it automatically gives aquantitative measure of the propagation of error between the solvers with respectto a certain computational goal.

In the sixth paper the mathematical foundation of the proper orthogonaldecomposition (POD) in arbitrary separable Hilbert space V is accounted for.It is proved that the solution to the obtained eigenvalue problem is an optimalbasis in V . Further, proofs of a few key properties of the generalized POD basisare provided.

In the seventh paper we present a strategy based on the POD that addressesthe problem of selecting a reduced number of precomputed static modes from alarge set in multibody dynamics simulation.

Keywords: model reduction, reduced order modeling, a posteriori error esti-mation, adaptivity, component mode synthesis, Craig-Bampton method, properorthogonal decomposition

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Acknowledgments

The work presented herein is part of a joint research project between the Mod-elling & Simulation Department at SKF Engineering & Research Centre and theDepartment of Mathematics and Mathematical Statistics at Umea University.Financial support is provided by SKF and the Industrial Graduate School atUmea University.

I would like to thank my supervisor Mats G. Larson for the advice he hasprovided me with during the course of writing this thesis and for sharing hisknowledge in the field of computational mathematics. I would also like to thankmy assistant supervisor Dag Fritzson at SKF together with Iakov Nakhimovskiat SKF and Erik Svensson formerly at SKF for their advice and comments on themanuscripts herein. Many thanks to Jonas Niklasson for his help with varioustechnical issues. My colleagues, past and present, Fredrik Bengzon, August Jo-hansson, Karl Larsson, Robert Soderlund, Tor Troeng, and Per Vesterlund haveall contributed to creating a pleasant and motivating working environment andI thank you all greatly for that. I wish to thank Petter Gustafsson, Director ofthe Industrial Graduate School for the advice he delivered to the students dur-ing our meetings. Finally, I would like to thank Beatrice Soder, our daughterElianne, my parents Per-Martin and Gertrud, together with my mother-in-lawAnn-Kristin Soder for your love, support, and encouragement, without which thisthesis would never have been completed.

Hakan JakobssonUmea, May 2011

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Contents

1 Introduction 11.1 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Mathematical Models of Linear Elasticity 42.1 Continuous Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Weak Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Reduced Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Model Reduction 73.1 Mode Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Component Mode Synthesis . . . . . . . . . . . . . . . . . . . . . . 83.3 Proper Orthogonal Decomposition . . . . . . . . . . . . . . . . . . 93.4 The Reduced Basis Method . . . . . . . . . . . . . . . . . . . . . . 10

4 A Posteriori Error Estimation and Adaptivity 104.1 The Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 A Posteriori Error Estimation . . . . . . . . . . . . . . . . . . . . . 114.4 Adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Multibody Dynamics Simulation 13

6 Summary of Papers 156.1 Paper I. Mode Superposition with Submodeling . . . . . . . . . . . 156.2 Paper II. Adaptive Component Mode Synthesis in Linear Elasticity 156.3 Paper III. A Posteriori Error Analysis of Component Mode Syn-

thesis for the Elliptic Eigenvalue Problem . . . . . . . . . . . . . . 156.4 Paper IV. A Posteriori Error Analysis of Component Mode Syn-

thesis for the Frequency Response Problem . . . . . . . . . . . . . 156.5 Paper V. Duality Based Adaptive Model Reduction for One-way

Coupled Thermoelastic Problems . . . . . . . . . . . . . . . . . . . 166.6 Paper VI. On Generalized Proper Orthogonal Decompositions . . . 166.7 Paper VII. Model Reduction in Rolling Bearing Simulation Based

on Static Load Cases and the Proper Orthogonal Decomposition . 16

Papers I-VII

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1 Introduction

Elasticity is a fundamental physical property inherent to many materials under-going deformation. Accurate and reliable description of elastic behavior is vitalin many scientific and industrial applications. Generally, closed form solutionsto the equations of elasticity cannot be obtained and numerical methods must beused instead, yielding only approximate solutions. In this instance it is importantto have a bound on the error in the approximation, without which its reliabilitymay be difficult to assess.

The finite element method has a long tradition of use in solid mechanics andis commonly the method of choice for solving problems in elasticity. Based ona rigorous mathematical foundation, extensive literature on both its theory andapplication exists, and estimates on its approximation properties are well knownfor many types of problems.

The computational domain in the finite element method is a geometric sub-division of the real domain into tetrahedral or hexahedral elements, the mesh. Afinite dimensional function space of typically piecewise polynomials is defined onthe mesh, and an approximate solution is sought in this space. The accuracy inthe approximation primarily depends on the mesh resolution and properties ofthe approximating function space, e.g. the polynomial order. For complex ge-ometries, possible errors in the geometrical approximation by the finite elementmesh may in addition contribute significantly to the error in the solution, andperhaps especially so when considering derivatives of the solution. Stress, forinstance, depends strongly on geometry. A large number of mesh elements, highpolynomial order, or both, may therefore be required to accurately represent thesolution. Since the unknowns in the finite element method are associated withthe elements in the mesh and the polynomial order of the function space, thenumber of degrees of freedoms in the associated system of equations may besubstantial. Naturally, the computational effort required to solve the problemincreases with the number of degrees of freedom.

Sometimes using the full finite element model is not practical due to the rela-tively high cost of solving the system of equations. In particular this may be thecase when solving certain time dependent problems, large scale eigenvalue prob-lems, or in frequency response analysis of complex structures. In such instancesit may instead be preferable to construct a so called reduced finite element modelwith much fewer degrees of freedom off line, and use this less expensive model inthe on line simulation. This process is commonly referred to as model reduction,reduced order modeling, or model order reduction.

Technically speaking, in model reduction the objective is to find a low di-mensional subspace of the finite element function space that still captures thesolution to a sufficient degree of accuracy. To accomplish this, model reductioncommonly relies on the incorporation of certain a priori knowledge in the model.For instance, if one is interested in the behavior of an elastic body in some specificfrequency range, a low dimensional subspace that captures this behavior may be

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defined by using a basis of elastic eigenmodes vibrating in that same frequencyrange. Likewise, if it is known that forces act on a body in a certain region, basisfunctions designed to capture these forces may be included in the basis. As a re-sult of incorporation of knowledge in the model, its generality is decreased. Thisposes a limitation in instances where for example load patterns are very general,or changing over time, or if the dominating frequency range is very broad. Insuch instances it may be favorable to be able to evaluate a coarse model first,register where for example loads occur, and then update the reduced model toincorporate the a posteriori knowledge obtained. Repeating the argument untilsatisfactory results are obtained yields an iterative method known as an adaptivealgorithm.

One area where model reduction has proved to be both necessary and diffi-cult is rolling bearing simulation. Rolling bearings are rotating machine elementswith demands for precision and load carrying capacity, together with low fric-tion torque, vibration and noise emissions. Rolling bearings may be simulatedusing multibody dynamics simulation technology. Due to the detailed geometrydescription and contact force and deflection calculations required, conventionalsimulation methods are however insufficient.

BEAST, BEAring Simulation Tool, is a simulation software developed by SKFespecially designed for the modeling and simulation of rolling bearings and othermechanical components [35, 36]. Having evolved from a rigid body dynamicssoftware, elasticity is now an integral part of the software. The use of model re-duction is required in BEAST, but the number and size of contacting surfaces ina rolling bearing presents a challenging situation. Conventional reduction meth-ods lead to models that are either inaccurate or too large – where the latter hassignificant negative impact on computation times. This calls for the developmentof new model reduction technology that meets the industrial demands. The chal-lenge is to make transfer between the scale of a FEM model and a reduced modelthat allows for feasible computation time while simultaneously having control ofthe error in the reduced model.

1.1 Thesis Objectives

The main objectives of this thesis have been:

• Develop and analyze efficient and reliable model reduction methods forlarge finite element models of general elastic bodies with large and enclosingcontact surfaces.

• Develop a posteriori error estimates and adaptive algorithms for the studiedmethods.

• Implement and evaluate the performance of the methods in SKF’s multi-body simulation system BEAST.

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1.2 Main Results

• Formulated new method for model reduction based on global eigenmodesand patchwise finite elements and evaluated its accuracy in static modelproblems. It was also shown that a slightly adjusted method may be used asa post processing operation in time dependent problems. The connection ofthe latter to the mode truncation augmentation method was demonstrated.(Paper I)

• Proved duality based a posteriori error estimates for Craig-Bampton com-ponent mode synthesis (CB-CMS) models. The estimates allows for the de-sign of adaptive algorithms that automatically refines the CMS subspacesaccording to error contribution. Estimates were first derived for a basicmodel problem (Paper II), and then for the elliptic eigenvalue problem(Paper III); the frequency response problem (Paper IV); and for a coupledthermoelastic static problem (Paper V). Numerical examples demonstratethe accuracy of the analytical predictions in all papers.

• Proved that the eigenvalue problem in proper orthogonal decomposition(POD) in a general separable Hilbert space gives rise to an optimal or-thonormal basis. This allows for the use of arbitrary inner product whenformulating POD. (Paper VI)

• Implemented POD in SKF’s simulation software BEAST, and used it on aset of precomputed static modes to construct a reduced set of static modesthat captures the strain energy contained in the original set in an efficientmanner. (Paper VII)

1.3 Future Work

Future work and extensions of the results herein, include:

• Implement adaptive CB-CMS method in BEAST and evaluate the methodon real world test cases. Task is ongoing.

• Extend analytical results for CB-CMS to the multilevel case, i.e. Auto-mated Multilevel Substructuring, c.f. [3].

• Develop theory for choice of inner product space in the generalized PODof Paper VI.

• Further testing of the POD based reduction of Paper VII in BEAST.

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2 Mathematical Models of Linear Elasticity

2.1 Continuous Model

Let Ω be a domain in space with boundary ∂Ω occupied by a homogeneousisotropic elastic material. Assume that the boundary consists of two disjointparts ΓN and ΓD, where a traction force gN acts on ΓN , and displacements uDare prescribed on ΓD. Assume further a force f acts in every point of the domain.

If the strains or deformations within Ω are small (infinitesimal), the propa-gation of waves in Ω is described by a partial differential equation of the form

ρu−∇ · σ(u) = f , x ∈ Ω, t > 0, (2.1a)

u = uD, x ∈ ΓD, t > 0, (2.1b)

n · σ(u) = gN , x ∈ ΓN , t > 0, (2.1c)

u = u0, x ∈ Ω, t = 0, (2.1d)

u = v0, x ∈ Ω, t = 0, (2.1e)

where u = u(x, t) are the vector displacements, u0 and v0 are initial displace-ment and velocity respectively, σ(·) is the stress tensor, and ρ = ρ(x) de-notes density. Further, stress is described in terms of the Cauchy strain tensorε(u) = 1/2(∇u+∇uT ) through the constitutive relation known as Hooke’s law,which may be written

σ(u) = 2µε(u) + κ(∇ · u)I, (2.2)

for isotropic materials. Here µ and κ are the so called Lame material parameters,and I is the d× d identity tensor.

Specific models of elastic behavior may be derived from (2.1) by making theappropriate assumptions. Assuming for instance that f and u are time harmonicof the form f = f0exp(iωt) and u = u0exp(iωt), where i is the imaginary unit,and ω is a given frequency, we obtain the frequency response problem

−ρω2u−∇ · σ(u) = f , x ∈ Ω, (2.3a)

u = uD, x ∈ ΓD, (2.3b)

n · σ(u) = gN , x ∈ ΓN , (2.3c)

which is well posed for all frequencies ω such that ω2 6= λ, where√λ is a resonance

frequency of the body. The resonance frequencies are in turn obtained by solvingan eigenvalue problem for the unknown eigenpairs (λ,u)

−ρλu−∇ · σ(u) = 0, x ∈ Ω, (2.4a)

u = 0, x ∈ ΓD, (2.4b)

n · σ(u) = 0, x ∈ ΓN . (2.4c)

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Spectral theory for elliptic operators stipulate that the solution of (2.4) yields abasis in [L2(Ω)]d, a property that we will return to later on.

Finally, assuming that the object is in static equilibrium, the accelerationterm vanishes in (2.1a), and we obtain the equations of elastostatics

−∇ · σ(u) = f , x ∈ Ω, (2.5a)

u = uD, x ∈ ΓD, (2.5b)

n · σ(u) = gN , x ∈ ΓN . (2.5c)

2.2 Weak Form

Assuming for simplicity that uD = 0, a weak form of for instance (2.5) may bederived by multiplying (2.5a) by v ∈ V = w ∈ [H1(Ω)]3 : w|ΓD

= 0, andintegrating by parts. We then get the problem: find u ∈ V , such that

a(u,v) = b(v), ∀v ∈ V, (2.6)

where a(·, ·) is the bounded, coercive, bilinear form

a(w,y) = (σ(w), ε(y)), ∀w,y ∈ V, (2.7)

and b(·) is a linear functional on [L2(Ω)]d defined by

b(w) = (f ,w) + (gN ,w)ΓN. (2.8)

Weak forms of equations (2.1), (2.3), and (2.4), are similarly derived.

2.3 Finite Element Model

The fundamental idea of the finite element method is to seek approximate solu-tions to a given weak equation in a finite dimensional subspace V h of the corre-sponding infinite dimensional object V . To construct such a finite dimensionalsubspace, the domain Ω is partitioned into for instance tetrahedral or hexahedralelements, and basis functions associated with the partition are defined. Thereare several choices of basis functions, but a common choice is to use piecewisepolynomials.

The finite element method reads: find U ∈ V h, such that

a(U ,v) = b(v), ∀v ∈ V h. (2.9)

Given a basis ϕiNi=1 in V h, the ansatz U =∑Nj=1 ξjϕj , yields the equivalent

problem

N∑j=1

ξja(ϕj ,ϕi) = b(ϕi), i = 1, . . . , N, (2.10)

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which is a linear system of equations

Ku = b, (2.11)

where K is a N ×N matrix referred to as the stiffness matrix, u = [ξ1 · · · ξN ]T isthe vector of unknown coefficients, and b = [b(ϕ1) · · · b(ϕN )]T is the load vector.

2.4 Reduced Model

As discussed in the introduction, sometimes using the full finite element modelis not practical due to its dimension N , which may be large. It is then possibleto use model reduction to obtain a more low dimensional model of dimensionm N . The construction of the reduced model from the finite element modelis analogous to the construction of the finite element model from the continuousmodel. Consider for simplicity the model problem

a(U ,v) + τ(U ,v) = b(v), ∀v ∈ V h, (2.12)

where τ ∈ R+ is a parameter. The objective in model reduction is to find alow dimensional subspace V h,m ⊂ V h that sufficiently well captures the finiteelement solution U . We then seek the reduced displacements Um ∈ V h,m, suchthat

a(Um,v) + τ(Um,v) = b(v), ∀v ∈ V h,m. (2.13)

Given a basis χimi=1 in V h.m, the ansatz U =∑mj=1 αjχj yields the m × m

linear system of equations

m∑j=1

αja(χj ,χi) + τ

m∑j=1

αj(χj ,χi) = b(χi), i = 1, . . . ,m. (2.14)

Since each χi may be written χi =∑Nk=1 ci,kϕk, for some coefficients ci,k ∈ R,

k = 1, . . . , N , equation (2.14) takes the form

m∑j=1

αj

N∑k=1

cj,k

N∑l=1

ci,la(ϕk,ϕl)

+ τ

m∑j=1

αj

N∑k=1

cj,k

N∑l=1

ci,l(ϕk,ϕl)

=

N∑l=1

ci,lb(ϕl), i = 1, . . . ,m, (2.15)

which may be written

VTKVa + τVTMVa = VTb, (2.16)

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where

V =

c1,1 · · · c1,m...

. . ....

cN,1 · · · cN,m

, (2.17)

is the N ×m matrix of coefficients of the reduced basis.

3 Model Reduction

Reduction of finite element models may be used in instances when the originalmodel is too large to be practically useful. This includes large scale eigenvalueproblems [3, 7], parameter studies such as frequency response analysis [4, 20],and time dependent problems [27, 32]. The challenge is to construct the subspaceV h,m ⊂ V h in such a way that it is low dimensional while still maintaining goodapproximation properties for the problem at hand.

There are naturally many ways to construct the reduced space. We point outtwo main classes of methods: those involving globally supported basis functionsand those relying on locally supported basis functions. Below we briefly describea few common methods. For a detailed account of several more model reductionmethods the reader is referred to the book by Qu [30] and the references therein.

3.1 Mode Superposition

A straight forward and indeed classical model reduction technique is to expandthe solution in a truncated series of eigenmodes. This technique is attractive dueto the orthogonality properties of eigenmodes and its ease of implementation.The eigenmodes are obtained from the finite element eigenvalue problem: findthe eigenpairs (Z, λh) such that

a(Z,v) = λh(Z,v), ∀v ∈ V h. (3.1)

The set of eigenmodes ZiNi=1 forms an orthonormal basis in V h, and a reducedsubspace may be defined for instance by V h,m = spanZimi=1, where m < N .From orthogonality follows that the resulting reduced stiffness and mass matricesare diagonal such that

VTMV = I, (3.2)

VTKV = Λ, (3.3)

where Λii = λhi , i = 1, . . . ,m. When Rayleigh damping is used in the model, thedamping matrix αK + βM is also diagonalized, which is very attractive from anumerical point of view.

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The reduced subspace V h,m will typically capture global behavior well usinga relatively small number of eigenmodes but convergence in the higher end of thespectrum may be slow. Several methods have been developed to accelerate theconvergence of mode superposition in the high frequent regime. See for instance[11].

An aspect to consider with mode superposition is that the eigenvalue problemin itself is expensive to solve, even for a small number of eigenpairs. This leadsto the idea of localization when constructing the reduced model.

3.2 Component Mode Synthesis

Component mode synthesis (CMS) [6, 7, 9, 10, 17] is a class of model reductionmethods that build on the concept of domain decomposition. The computationaldomain is partitioned into non overlapping subdomains and basis functions as-sociated with each subdomain are constructed. In the popular Craig-Bamptonmethod [9] constrained eigenvalue problems define locally supported basis func-tions associated with each of the subdomains. The response in the subdomains iscoupled by inclusion of so called static modes that are defined as the structuralresponse to prescribed displacements on the interface between the subdomains.

Consider the domain Ω for simplicity partitioned into two subdomains Ω1 andΩ2 interfacing at Γ. A decomposition of V h associated with the partition maybe constructed by defining subspaces V hi ⊂ V h associated with Ωi, by

V hi = v ∈ V h : v|Ω\Ωi= 0, i = 1, 2, (3.4)

and a subspace V hΓ associated with Γ, by

V hΓ = Eν ∈ V h : ν ∈ V h|Γ, (3.5)

where V h|Γ denotes the restriction of V h to Γ and Eν ∈ V h denotes the harmonicextension of a function ν ∈ V h|Γ to Ω. That is, Eν is defined by the problem:find Eν ∈ V h, such that

a(Eν,v) = 0, ∀v ∈ V hi , i = 1, 2, (3.6)

Eν|Γ = ν. (3.7)

With the V hi , i = 0, 1, 2, defined as above, it follows that an a-orthogonal decom-position of V h is given by

V h = V hΓ ⊕ V h1 ⊕ V h2 . (3.8)

By constructing bases in in each V hi , i = 0, 1, 2 a basis in V h is obtained.For the subspaces V hi , i = 1, 2, we define the eigenvalue problems: find (λi, zi) ∈R× V hi , i = 1, 2 such that

a(zi,v) =λi(zi,v), ∀v ∈ V hi , (3.9)

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Bases for V hi , i = 1, 2 are then given by

V hi = spanzinii=1, (3.10)

where ni is the dimension of V hi . A basis for V h|Γ is given by continuous piecewisepolynomials along Γ. A basis for V hΓ is then readily computed.

By truncating the bases in V hi , i = 1, 2, reduced subspaces V h,mi

i , i = 1, 2,

are obtained. That is, let V h,mi

i be defined by

V h,mi

i = spanzimii=1, (3.11)

where mi < ni. A reduced space V h,m is defined by

V h,m = V hΓ ⊕ Vh,m1

1 ⊕ V h,m2

2 . (3.12)

In a version of the Craig-Bampton method [7], an eigenvalue problem is solved inthe space V hΓ also, yielding a basis of eigenmodes and enabling the use of modetruncation to obtain further reduction.

Several other CMS methods exist as well. See [10] for an extensive overview.See also [8] for a version of CMS with overlapping subdomains.

3.3 Proper Orthogonal Decomposition

The proper orthogonal decomposition (POD) is a multi purpose method for theconstruction of low dimensional representations of high dimensional data. Incontext of model reduction the method has historically been used predominatelyin fluid mechanics, e.g. [5, 15, 34], but interest of using POD in solid mechanicsseems to be increasing, e.g. [19, 23].

The idea in POD is to construct an orthonormal basis that spans a given setof data optimally with respect to mean square error. To make this statementprecise, let V be a Hilbert space and assume that U = uk ⊂ V is an ensembleof functions, for instance samples of some transient process u(x, t). The PODis then concerned with finding a basis ϕj ⊂ V such that n-dimensional rep-resentations of the form un =

∑nj=1(u,ϕj)ϕj in POD basis functions, describe

typical members u ∈ U better than n-dimensional representations in any otherbasis, cf. [16]. The latter statement is formalized by claiming that the meansquare norm 〈‖un‖2〉 of un is maximized, or equivalently that the mean squareerror 〈‖u − un‖2〉 is minimized. Here 〈·〉 denotes an averaging operation, e.g.a time average or an ensemble average. This requirement leads to a variationaleigenvalue problem of the form

〈(u,ϕ)(v,u)〉 = λ(ϕ,v), (3.13)

and its solution yields the sought basis.A typical application of POD for the modeling of a transient process u(x, t),

e.g. as realized by a numerical simulation, would involve first collecting samples

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of the process, by for instance running a short simulation; then solving the PODeigenvalue problem for the basis ϕj; then form the reduced subspace V h,m =spanϕjmj=1; and finally use the reduced space in a full simulation.

For further information of POD the reader is referred to the book [16].

3.4 The Reduced Basis Method

The reduced basis method, see for instance [22, 29], is a model reduction methodconcerned with parameter dependent problems, e.g. of the form: evaluate

s(µ) = `(U), (3.14)

where `(·) is a bounded linear functional, µ is a parameter that varies in someparameter universe D ⊂ Rp, p ∈ N+ and U = U(µ) ∈ V h, satisfies the equation

a(U(µ),v;µ) = f(v), ∀v ∈ V h. (3.15)

Here a(·, ·;µ) is a bilinear form that depends on the parameter µ. The methodrecognizes that due to the parametric dependence, solutions U(µ) ∈ V h likelyreside on a low dimensional manifold M = M(µ). Since solving the problem fora large set of parameter values in the full finite element space V h is expensive,the strategy is to first solve for a representative set of solutions residing on M ,e.g. Ui = U(µi), i = 1, . . . ,m, where m is small compared to the dimension ofV h, then define a reduced subspace V h,m = spanU i, and use this subspace toapproximate solutions for arbitrary µ ∈ D. That is, the reduced problem reads:evaluate

s(µ) = `(Um), (3.16)

for µ ∈ D, where Um ∈ V h,m, satisfies

a(Um(µ),v;µ) = f(v), ∀v ∈ V h,m. (3.17)

The method shares some similarities with POD, which in fact may be viewed asa reduced basis method such that the manifold M is parametrized by time.

For application of the reduced basis method to elasticity, see the thesis [37].

4 A Posteriori Error Estimation and Adaptivity

4.1 The Error

The error e = u − Um = u − U + U − Um in a reduced model can be seento comprise two parts: a discretization error Ed = u−U and a reduction errorEr = U −Um. The discretization error is the error in the finite element modelrelative the continuous model, and the reduction error is the error in the reduced

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model relative the finite element model. The discretization error is controlledthrough mesh refinement, and the reduction error is analogously controlled byrefining the reduced space V h,m appropriately. In the case of for instance modesuperposition or POD, this means adding an appropriate number of mode shapesto the reduced basis. In other methods refinement may become more involved.Naturally it is important to have bounds on the error contribution from both thediscretization and the reduction error. In this thesis we deal with the reductionerror only. It will be referred to as simply the error and is denoted by E.

4.2 Error Estimation

There are two types of error estimates: a priori estimates and a posteriori esti-mates. A priori error estimates bound the error in terms of the exact solutionand some space dependent quantities, e.g. the polynomial order and mesh sizein finite element methods. A posteriori estimates on the other hand bound theerror in terms of the same space dependent quantities and the residual associ-ated with the computed solution. A priori estimates are quite pessimistic andpredominantly used for qualitative estimation, i.e. showing the rate of conver-gence, whereas the a posteriori estimates on the other hand are better suitedfor quantitative error estimation. Since the end result of this thesis concernspractical application, only a posteriori error estimation are considered herein.

4.3 A Posteriori Error Estimation

Much research has been made in the field of a posteriori error estimation forfinite element methods. For detailed information on the subject the reader isreferred to the text book by Bangerth and Rannacher [1] together with articles[12, 13, 18, 21, 28, 31], and the references therein. There are several techniquesbut we mention in particular the dual weighted residual (DWR) method sincethis framework encompasses both much of the original as well as recent results inthe field, and also serves as a basis for the results on a posteriori error estimationobtained in this thesis.

The field of a posteriori error estimation for model reduction is less maturebut interest seems to be emerging. See for instance [24] for an application of theDWR method to POD. See e.g. [2, 14, 38] for a posteriori error estimation incontext of the reduced basis method.

To introduce the reader to the concepts of a posteriori error estimation inmodel reduction, what follows next is an a posteriori error analysis for the modelproblem (2.12). We begin by observing that subtraction of (2.13) from (2.12)yields the following property

a(E,v) + τ(E,v) = 0, ∀v ∈ V h,m, (4.1)

known as Galerkin orthogonality. It says that if considering the form a(·, ·)+τ(·, ·)

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as an inner product, the error in the approximation is orthogonal to the reducedspace V h,m. In this respect the approximation is the best possible.

Next we introduce the concept of duality and the DWR method. Let J(·)denote a linear functional known as the goal functional, or the goal quantity.As opposed to error estimates in global norms, in the DWR method the error ismeasured in J(·), which may be a localized quantity.

Consider now the following dual problem: find Φ ∈ V h, such that

a(v,Φ) + τ(v,Φ) = J(v), ∀v ∈ V h. (4.2)

Setting v = E, we immediately obtain the error representation formula

J(E) = a(E,Φ) + τ(E,Φ) (4.3)

= (Rh(Um), (I − Pm)Φ), (4.4)

where

(Rh(w),v) = b(v)− τ(w,v)− a(w,v), ∀v ∈ V h, (4.5)

is the residual. Note that we have used the Galerkin orthogonality property (4.1)to subtract a projection PmΦ ∈ V h,m. Note also that the error representationformula does not contain the exact solution U . It does however contain the dualsolution Φ, which must either be computed or estimated for the error represen-tation formula to be useful in practice. It is not meaningful to solve the dualproblem using the full finite element space V h, but in general it is possible toobtain useful information by solving the dual problem in a subspace V h,M suchthat V h,m ⊂ V h,M ⊂ V h. In the mode superposition method V h,M would bechosen as V h,M = spanZiMi=1 with M > m.

Choosing J(·) appropriately we may further derive estimates on the errormeasured in norms such as the L2(Ω) norm ‖ · ‖ =

√(·, ·) or the energy norm

||| · ||| =√a(·, ·). Choosing J(·) = (·,LE) for instance, where L is the linear

operator that satisfies a(w,v) = (Lw,v), we have J(E) = (LE,E) = |||E|||2,and

|||E|||2 = (Rh(Um), (I − Pm)Φ) (4.6)

≤ ‖Rh(Um)‖‖(I − Pm)Φ‖. (4.7)

Depending on the specifics of the reduced space V h,m, it may be possible to obtainan approximation result for the projection PmΦ. For the mode superpositionmethod the following estimate holds for w ∈ Vh:

‖(I − Pm)w‖ ≤ 1√λhm+1

|||w|||. (4.8)

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Using this, we obtain

|||E|||2 ≤ 1√λhm+1

‖Rh(Um)‖|||Φ|||. (4.9)

It can further be shown that the dual solution is stable and bound by the dataE, so that |||Φ||| ≤ |||E|||. Using this we arrive at the a posteriori estimate in theenergy norm

|||E||| ≤ 1√λhm+1

‖Rh(Um)‖. (4.10)

4.4 Adaptivity

The estimate (4.10) may serve as a basis for an adaptive algorithm that auto-matically refines the subspace V h,m to control the error in the reduced model.For our model problem an adaptive algorithm that refines V h,m would read:

1. Solve equation (2.13) using an initial number of modes m.

2. Compute η = ‖Rh(Um)‖/√λhm+1.

3. If η is sufficiently small, stop. Otherwise increase the number of modes mand return to 1.

5 Multibody Dynamics Simulation

The ultimate motivation to the developments herein is multibody dynamics sim-ulation. For completeness follows a brief account of the governing equationstogether with some references where further information on the subject may befound.

Flexible multibody dynamics simulation concerns the modeling and simula-tion of systems of several interconnected flexible bodies undergoing potentiallylarge displacements and rotations. There are several formalisms used to describea multibody system, cf. [39]. The most widely used one [33] is however thefloating frame of reference. Originally developed for the conversion of rigid bodydynamics software to flexible body dynamics, two sets of coordinates are used todescribe the configuration of the body. One set describes the location and orien-tation of a body reference frame, and the second set describe the displacementsin the body reference frame. Under assumption of small displacements, this ap-proach allows the use of the Cauchy strain tensor in the elastic model despitethe presence of large rotations, and hence linear model reduction methods maybe used as well.

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Introducing the partitioned vector of generalized coordinates

q =

RΘqf

, (5.1)

where R and Θ describes the location and orientation of the body referencerespectively, and qf is the elastic coordinates, the equations of motion for abody in a flexible multibody system may be written

Mq + Kq + Cλ = Qe + Qv. (5.2)

Here, M = M(q, t) is the mass matrix

M =

MRR MRθ MRf

Mθθ Mθf

symm. Mff ,

, (5.3)

where MRR is a constant diagonal matrix representing the mass of the body,Mθθ is a time dependent matrix representing the inertia tensor of the body, Mff

is the constant finite element mass matrix, Mθf and MRf are time dependentmatrices representing the inertia coupling between the rigid body motion andelastic deformation, and MRθ is a time dependent matrix representing the inertiacoupling between rigid body translation and rigid body rotation.

Further, K is the constant stiffness matrix

K =

0 0 00 0 00 0 Kff ,

, (5.4)

the matrix C = C(q, t) a constraint Jacobian matrix, λ a vector of Lagrangemultipliers, and Qe = Qe(q, t) together with Qv = Qv(q, t) the vectors of gen-eralized forces and the quadratic velocity vector, respectively. The quadraticvelocity vector includes the Coriolis and Centrifugal forces resulting from therotation of the body reference frame.

Remark 1. In the case of a structural system the body reference is fixed in time,yielding R = Θ = 0 and the system of equations then collapses to the usuallinear system.

For the derivation and further details of equation (5.2) the reader is referredto the book by Shabana [33]. For an overview of different formalisms in multi-body dynamics and extensive list of references, see the review paper [39]. Seethe thesis by Nakhimovski [26] for information on the development of parallelflexible multibody systems software. For information of numerical integration ofmultibody systems see the thesis by Modin [25].

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6 Summary of Papers

6.1 Paper I. Mode Superposition with Submodeling

In this paper a new approach for local enhancement of mode superpositions thatbuilds on the concept of submodeling is investigated. To this end we impose amultiscale split on the reduced solution into a global part given by the modesuperposition method, and a local part given by a patchwise finite element prob-lem. The patch problem yields a local correction on the modal approximation.We evaluate the accuracy of the approximation in elastostatic numerical exam-ples. We also demonstrate how the submodeling technique may be applied asa post processing operation on a set of reduced solutions, e.g. from dynamicssimulation.

6.2 Paper II. Adaptive Component Mode Synthesis in Lin-ear Elasticity

This is the first paper dealing with a posteriori error estimates for the CB-CMSmethod. Here we develop the necessary framework for a posteriori error estima-tion in terms of a basic model problem. The estimates are designed to determineto what degree each CMS subspace influence the error in the reduced solution.We prove a posteriori error estimates for the error in a linear goal quantity aswell as in the energy and L2 norms. We further demonstrate that automatic con-trol of the error in the reduced solution can be accomplished using an adaptivealgorithm that determines suitable dimensions of each CMS subspace.

6.3 Paper III. A Posteriori Error Analysis of ComponentMode Synthesis for the Elliptic Eigenvalue Problem

In this paper we apply the methodology developed in Paper II and derive a pos-teriori error estimates for the error associated with CMS reduction of the ellipticeigenvalue problem. We prove estimates for the approximated eigenvalues as wellas the eigenmodes. The analytical results are accompanied by two numerical ex-amples. The numerical examples comply with the theoretical predictions to ahigh degree of accuracy.

6.4 Paper IV. A Posteriori Error Analysis of ComponentMode Synthesis for the Frequency Response Problem

This paper continues the development of a posteriori error estimates for CB-CMS.Here we consider the frequency response problem and prove estimates in a linearquantity of interest and the energy norm. We illustrate the analytical results inseveral numerical examples. Again it is found that the numerical results agreewith the theoretical predictions to a high degree of accuracy.

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6.5 Paper V. Duality Based Adaptive Model Reduction forOne-way Coupled Thermoelastic Problems

This is the final paper in this thesis concerning a posteriori error estimation forCB-CMS. We derive an a posteriori error estimate for a thermoelastic model prob-lem where CB-CMS is used in both the thermal and elastic solvers. The problemis one-way coupled in the sense that heat transfer affects elastic deformation,but not vice versa. A main feature with the estimate is that it automaticallygives a quantitative measure of the propagation of error between the solvers withrespect to a certain computational goal. The analytical results are accompaniedby a numerical example.

6.6 Paper VI. On Generalized Proper Orthogonal Decom-positions

While the mathematical foundation of the standard POD is well known, thefoundation of the generalized version with an arbitrary inner product is seldomaccounted for. In this paper some details in this direction are provided. Underassumption of separability it is proved that a solution to the generalized problemproblem exists. Further, proofs of a few key properties of the generalized PODbasis are provided. The generalized theory is applied in two analytical examples.

6.7 Paper VII. Model Reduction in Rolling Bearing Sim-ulation Based on Static Load Cases and the ProperOrthogonal Decomposition

The use of mode shapes computed from static load cases is a well known tech-nique to incorporate a priori knowledge and improve accuracy in reduced finiteelement models in flexible multibody dynamics simulation. In certain situationsthe number of static modes required may however become large, leading to aninefficient model. This poses the problem of how to select the most relevant staticload cases to reduce the number required. In this paper we present a strategy thataddresses this problem using the proper orthogonal decomposition. The strategyhas been implemented in the multibody dynamics simulation software BEASTdeveloped by SKF. We further present the results of a preliminary numericalstudy.

References

[1] W. Bangerth and R. Rannacher, Adaptive finite element methods for differ-ential equations, Lectures in Mathematics, Birkhauser, 2003.

[2] M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera, An empirical inter-polation’method: application to efficient reduced-basis discretization of par-

16

tial differential equations, Comptes Rendus Mathematique 339 (2004), no. 9,667–672.

[3] J. K. Bennighof and R. B. Lehoucq, An automated multilevel substructuringmethod for eigenspace computation in linear elastodynamics, SIAM Journalon Scientific Computing 25 (2004), no. 6, 2084–2106.

[4] J.K. Bennighof and M.F. Kaplan, Frequency sweep analysis using multi-level substructuring, global modes and iteration, Proceedings of 39thAIAA/ASME/ASCE/-AHS Structures, Structural Dynamics and MaterialsConference, Citeseer, 1998.

[5] G. Berkooz, P. Holmes, and J. L. Lumley, The proper orthogonal decompo-sition in the analysis of turbulent flows, Annual Review of Fluid Mechanics25 (1993), no. 1, 539–575.

[6] F. Bourquin, Analysis and comparison of several component mode synthesismethods on one dimensional domains, Numerische Mathematik 58 (1990),no. 1, 11–33.

[7] , Component mode synthesis and eigenvalues of second order oper-ators: Discretization and algorithm, Math. Model. Numer. Anal. (1992),no. 26, 385–423.

[8] I. Charpentier, F. De Vuyst, and Y. Maday, A component mode synthesismethod of infinite order of accuracy using subdomain overlapping: numericalanalysis and experiments, Publication du laboratoire d’Analyse NumeriqueR 96002 (1996), 55–65.

[9] R. R. Craig and M. C. C. Bampton, Coupling of substructures for dynamicanalysis, AIAA J (1968), no. 6, 1313–1321.

[10] D. De Klerk, DJ Rixen, and SN Voormeeren, General framework for dynamicsubstructuring: history, review, and classification of techniques, AIAA jour-nal 46 (2008), no. 5, 1169.

[11] J. M. Dickens, J. M. Nakagawa, and M. J. Wittbrodt, A critique of modeacceleration and modal truncation augmentation methods for modal responseanalysis, Computers & Structures 62 (1997), no. 6, 985–998.

[12] K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptivemethods for differential equations, Acta Numerica (1995), no. 4, 105–158.

[13] M. B. Giles and E. Suli, Adjoint methods for PDEs: a posteriori error anal-ysis and postprocessing by duality, Acta Numerica 11 (2003), 145–236.

[14] M.A. Grepl, Reduced-basis approximation and a posteriori error estimationfor parabolic partial differential equations, Ph.D. thesis, Massachusetts In-stitute of Technology, 2005.

17

[15] P. Holmes, Low-dimensional models of coherent structures in turbulence,Physics Reports 287 (1997), no. 4, 337–384.

[16] P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, coherent structures,dynamical systems and symmetry, Cambridge Univ. Pr., 1998.

[17] W. C. Hurty, Dynamic analysis of structural systems using componentmodes, AIAA J. (1965), no. 4, 678–685.

[18] S. Irimie and P. Bouillard, A residual a posteriori error estimator for thefinite element solution of the Helmholtz equation, Computer Methods in Ap-plied Mechanics and Engineering 190 (2001), no. 31, 4027–4042.

[19] G. Kerschen, J.-C. Golinval, A. F. Vakakis, and L. A. Bergman, The Methodof Proper Orthogonal Decomposition for Dynamical Characterization andOrder Reduction of Mechanical Systems: An Overview, Nonlinear Dynamics41 (2005), no. 1-3, 147–169.

[20] J.H. Ko and Z. Bai, High-frequency response analysis via algebraic substruc-turing, Int. J. Numer. Meth. Engng 76 (2008), 295–313.

[21] M. G. Larson, A posteriori and a priori error analysis for finite elementapproximations of self-adjoint elliptic eigenvalue problems, SIAM Journalon Numerical Analysis 38 (2001), no. 2, 608–625.

[22] Y. Maday and E.M. Rønquist, A reduced-basis element method, Journal ofscientific computing 17 (2002), no. 1, 447–459.

[23] M. Meyer and H. G. Matthies, Efficient model reduction in non-lineardynamics using the Karhunen-Loeve expansion and dual-weighted-residualmethods, Computational Mechanics 31 (2003), no. 1-2, 179–191.

[24] M. Meyer and H.G. Matthies, Efficient model reduction in non-linear dynam-ics using the Karhunen-Loeve expansion and dual-weighted-residual methods,Computational Mechanics 31 (2003), no. 1, 179–191.

[25] K. Modin, Adaptive geometric numerical integration of mechanical systems,Ph.D. thesis, Centre for Mathematical Sciences, Numerical Analysis, LundUniversity, Sweden, 2009.

[26] I. Nakhimovski, Contributions to the modeling and simulation of mechanicalsystems with detailed contact analyses, Ph.D. thesis, Department of Com-puter and Information Science, Linkopings universitet, Sweden, 2006.

[27] B. Nour-Omid and R.W. Clough, Dynamic analysis of structures using lanc-zos co-ordinates, Earthquake engineering & structural dynamics 12 (1984),no. 4, 565–577.

18

[28] J. T. Oden, S. Prudhomme, and L. Demkowicz, A posteriori error estimationfor acoustic wave propagation problems, Archives of Computational Methodsin Engineering 12 (2005), no. 4, 343–389.

[29] J.S. Peterson, The reduced basis method for incompressible viscous flow cal-culations, SIAM Journal on Scientific and Statistical Computing 10 (1989),777.

[30] Z.-Q. Qu, Model order reduction techniques with applications in finite ele-ment analysis, Graduate Texts in Mathematics, Springer, London, Berlin,Heidelberg, 2004.

[31] R. Rannacher and F. T. Suttmeier, A posteriori error control and mesh adap-tation for FE models in elasticity and elasto-plasticity, Studies in AppliedMechanics (1998), 275–292.

[32] C.W. Rowley, T. Colonius, and R.M. Murray, Model reduction for com-pressible flows using POD and Galerkin projection, Physica D: NonlinearPhenomena 189 (2004), no. 1-2, 115–129.

[33] A.A. Shabana, Dynamics of multibody systems, Cambridge Univ Pr, 2005.

[34] L. Sirovich, Analysis of turbulent flows by means of the empirical eigenfunc-tions, Fluid Dynamics Research 8 (1991), no. 1-4, 85–100.

[35] L.E. Stacke and D. Fritzson, Dynamic behaviour of rolling bearings: simu-lations and experiments, Proceedings of the Institution of Mechanical Engi-neers, Part J: Journal of Engineering Tribology 215 (2001), no. 6, 499–508.

[36] L.E. Stacke, D. Fritzson, and P. Nordling, BEAST–a rolling bearing simu-lation tool, Proceedings of the Institution of Mechanical Engineers, Part K:Journal of Multi-body Dynamics 213 (1999), no. 2, 63–71.

[37] K. Veroy, Reduced-basis methods applied to problems in elasticity: Analysisand applications, Ph.D. thesis, Massachusetts Institute of Technology, 2003.

[38] K. Veroy, C. Prud’homme, and A.T. Patera, Reduced-basis approximation ofthe viscous Burgers equation: rigorous a posteriori error bounds, ComptesRendus Mathematique 337 (2003), no. 9, 619–624.

[39] T.M. Wasfy and A.K. Noor, Computational strategies for flexible multibodysystems, Applied Mechanics Reviews 56 (2003), 553.

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