research article hybrid stochastic finite element method for...

Research ArticleHybrid Stochastic Finite Element Method forMechanical Vibration Problems

Harri Hakula and Mikael Laaksonen

Department of Mathematics and Systems Analysis School of Science Aalto University PO Box 11100 00076 Aalto Finland

Correspondence should be addressed to Harri Hakula harrihakulaaaltofi and Mikael Laaksonen mikaeljlaaksonenaaltofi

Received 9 April 2015 Revised 23 June 2015 Accepted 28 June 2015

Academic Editor Evgeny Petrov

Copyright copy 2015 H Hakula and M LaaksonenThis is an open access article distributed under theCreative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

We present and analyze a new hybrid stochastic finite element method for solving eigenmodes of structures with random geometryand random elastic modulus The fundamental assumption is that the smallest eigenpair is well defined over the whole stochasticparameter space The geometric uncertainty is resolved using collocation and random material models using Galerkin method ateach collocation point The response statistics expectation and variance of the smallest eigenmode are computed in numericalexperiments The hybrid approach is superior to alternatives in practical cases where the number of random parameters used todescribe geometric uncertainty is much smaller than that of the material models

1 Introduction

In standard engineering models many physical quantitiessuch as material parameters are taken to be constant eventhough their statistical nature is well understood Similarlyassumptions of geometric constants such as thickness ofa structure are not realistic due to manufacturing imper-fections In a detailed report on a state-of-the-art verifica-tion and validation process comparing modern simulationswith the set of experiments performed in the Oak RidgeNational Laboratory in the early 70s Szabo and Muntgesreport discrepancies of over 20 in some quantities ofinterest [1] These discrepancies are attributed to machiningimperfections not accounted for in the computations Also inimportant nonlinear problems such as buckling of a shell it isknown that variation betweenmanufactured specimens has aprofound effect in the actual performance [2] This suggeststhat a stochastic dimension should be added to the models

The modern era of uncertainty quantification starts withthe works of Babuska et al [3 4] and the ETH-group ledby Schwab and Gittelson (eg [5]) with provably fasterconvergence rates than the standard Monte Carlo methodsAlthough the so-called stochastic finite elements had beenstudied for a relative long time before (the classic reference isGhanem and Spanos [6]) their application was thus limitedto highly specific cases

The solution methods can broadly speaking be dividedinto intrusive and nonintrusive ones The same divisionapplies to stochastic eigenvalue problems (SEVPs) as wellSEVPs have attracted a lot of attention recently and variousalgorithms have been suggested for computing approximateeigenpairs [7ndash10] specially the power iteration [11] andinverse iteration [12]

In this paper our focus is on effects ofmaterialmodels andmanufacturing imperfections of geometric nature It shouldbe noted that in the context of this paper it is assumedthat the problems are positive definite and the eigenpair ofinterest is the ground state that is the one with the smallesteigenvalue which in theory can be a double eigenvalue Ourexperimental setup is nonsymmetric and thus a spectralgap exists and the inverse iteration converges to the desiredeigenmode

In stochastic eigenvalue problems one must address twocentral issues that do not arise in stochastic source problemsfirst the eigenmodes are defined only up to a sign andsecond the eigenmodes must be normalized over the wholeparameter space that is every realizationmust be normalizedin the samewayHere our quantity of interest is the eigenvalueand therefore we do not necessarily have to fix the signsThenormalization is handled by solution of a nonlinear system ofequations as in [12]

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 812069 13 pageshttpdxdoiorg1011552015812069

2 Shock and Vibration

R

(a) Nominal domain

R

120579

(b) Perturbed domain

Figure 1 Two realisations of the domain with meshes The nominal domain is the reference The rotation and scaling of the cavity as well asthe direction of the uncertainty in the Youngrsquos modulus is indicated

The main result of this paper is the new hybrid algorithmwhich combines a nonintrusive outer loop (collocation)with an intrusive inner one (Galerkin) The randomness ingeometry is resolved using collocation and that of materialswith Galerkin at every collocation point The model problemis an idealized engine bed vibration problem where a 2Dhollow square is clamped from one side (see Figure 1) Theinner cavity is assumed to be of fixed shape but randomorientation and Youngrsquosmodulus is taken to be random in thenormal direction into the body The choice of the geometricuncertainty models the case of sand casting with moulds offixed shape The hybrid approach combines the geometricflexibility of collocation with much faster computation timesof the Galerkin approach used in the inner loop and thuscombines the best of both worlds

The rest of the paper is organized as follows first theabstract problem is introduced in Section 2 The representa-tion of the random input is given in Section 3 Higher orderfinite elements are described in Section 4 Section 5 is centralto our discussion both solution methods collocation andspectral inverse iteration are presented Numerical experi-ments are discussed in Section 6 and finally conclusions aredrawn in Section 7

2 Problem Statement

We start by presenting the deterministic system of Navierrsquosequations of elasticity and the corresponding weak formWe then extend this to the stochastic setting in order tocover the case of uncertain domain andmodulus of elasticityWe aim to compute statistics of the smallest eigenvalueof the resulting stochastic system The geometry of thecomputational domain is illustrated in Figure 1

21 Deterministic Formulation We use the Navier equationsof elasticity to model the system of interest Find theeigenvalue 120581 the displacement field u = (1199061 1199062) and thesymmetric stress tensor 120590 = (120590

119894119895)2119894119895=1 such that

120590 = 120582div (u) I+ 2120583120598 (u) in 119863

minus div (120590) = 120581u in 119863

u = 0 on 120597119863119863

120590 sdotn = 0 on 120597119863119873

(1)

where 120597119863 = 120597119863119863cup 120597119863

119873is a partitioned boundary of119863 The

Lame constants are

120582 =

119864](1 + ]) (1 minus 2])

120583 =

119864

2 (1 + ])

(2)

with 119864 and ] being Youngrsquos modulus and Poissonrsquos ratiorespectively Further I is the identity tensor n denotes theoutward unit normal to 120597119863

119873 and the strain tensor is

120598 (u) = 12(nablau+nablau119879) (3)

The vector-valued tensor divergence is

div (120590) = (

2sum

119895=1

120597120590119894119895

120597119909119895

)

2

119894=1

(4)

This formulation assumes a constitutive relation correspond-ing to linear isotropic elasticity with stresses and strainsrelated by Hookersquos generalized law

120590V =[

[

[

12059011

12059022

12059012

]

]

]

= D (120582 120583)[

[

[

12059811

12059822

12059812

]

]

]

= D (120582 120583) 120598V (5)

22 Weak Formulation Let us introduce a function space119881119863= V V isin 119867

1(119863) V|

120597119863119863= 0 and define the eigen-

problem find the eigenpair (120581 u) isin R times (119881119863times 119881

119863) such that

119886 (u k) = 120581 (u k)119863 forallk isin 119881119863times 119881

119863 (6)

where the bilinear form

119886 (u k) = int

119863

120590 (u) 120598 (k) 119889x (7)

Shock and Vibration 3

is the integrated tensor contraction

120590 120598 =

2sum

119894119895=1120590119894119895120598119894119895 (8)

and (sdot sdot)119863denotes the mass matrix

(u k)119863 = int

119863

1199061V1 +1199062V2119889x (9)

As usual we define the kinematic relation

120598V (u) =

[

[

[

[

[

[

[

[

120597

12059711990910

0 120597

1205971199092120597

1205971199092

120597

1205971199091

]

]

]

]

]

]

]

]

[

1199061

1199062] (10)

and specify the constitutive matrix

D (120582 120583) =[

[

[

120582 + 2120583 120582 0120582 120582 + 2120583 00 0 120583

]

]

]

(11)

for the purpose of rewriting the bilinear form as

119886 (u k) = int

119863

120598V (u)119879D (120582 120583) 120598V (k) 119889x (12)

23 Stochastic Extension of the Eigenproblem We introducetwo probability spaces (Ω1F1P1) and (Ω2F2P2) tomodel randomness of the domain and the elastic modulusrespectively Here Ω

119894denotes the set of outcomes F

119894is a 120590-

algebra of events and P119894is a probability measure on Ω

119894for

each 119894 = 1 2 Furthermore we set

(ΩFP) =

2⨂

119894=1(Ω

119894F

119894P

119894) (13)

to be the underlying product probability spaceThe geometry of the computational domain 119863 is illus-

trated in Figure 1 We assume that the radius of the cavity119877 Ω1 rarr R and the angle 120579 Ω1 rarr R are randomvariablessatisfying 119877min le 119877(1205961) le 119877max and 120579min le 120579(1205961) le 120579max forsome suitable constants 119877min 119877max 120579min 120579max gt 0 Thus wemay write 119863 = 119863(1205961) = 119863(119877(1205961) 120579(1205961)) In the numericalexamples we set 119877(1205961) and 120579(1205961) to be uniformly distributedrandom variables

Suppose that for each 1205961 isin Ω1 the Young modulus is arandom field on the domain119863(1205961) By this we mean that 119864 =

119864(1205961 sdot sdot) Ω2 times 119863(1205961) rarr R for every 1205961 isin Ω1 We assumethat 119864(1205961 1205962 x) is strictly positive and bounded that is forsome constants 119864min 119864max gt 0 it holds that

119864min le 119864 (1205961 1205962 sdot) le 119864max in 119863(1205961) (14)

The stochastic extension of eigenproblem (6) is now givenby the following find the eigenvalue 120581 Ω rarr R and theeigenfunction u Ω rarr 119881

119863(1205961)times 119881

119863(1205961)such that

119886 (120596 u (1205961 1205962 sdot) k)

= 120581 (1205961 1205962) (u (1205961 1205962 sdot) k)119863(1205961)

forallk isin 119881119863(1205961)

times 119881119863(1205961)

(15)

holds for P-almost every 120596 = (1205961 1205962) isin Ω Here 119886(120596 u k)is the stochastic equivalent of the bilinear form 119886(u k) and isgiven by

119886 (120596 u k) = int

119863(1205961)120590 (120596 u) 120598 (k) 119889x = int

119863(1205961)120598V (u)

119879

sdotD (120582 (1205961 1205962 sdot) 120583 (1205961 1205962 sdot)) 120598V (k) 119889x(16)

The Lame constants 120582(1205961 1205962 x) and 120583(1205961 1205962 x) are simplyrandom fields defined by (2) with 119864 = 119864(1205961 1205962 x)

In most cases we are mainly interested in computingstatistical moments of the solution Suppose that we have arandom variable V(1205961 1205962) The expected value or mean ofV(1205961 1205962) is defined to be

E [V] = int

Ω

V (1205961 1205962) 119889P (120596) (17)

and its variance is

Var [V] = E [(VminusE [V])2] (18)

In this paper our aim is to compute the expected value andvariance of the smallest eigenvalue of system (15)

3 Spectral Representations

For purposes of numerical treatment we need to approximatethe random Young modulus using only a finite numberof random variables A natural way of achieving this isto use the truncated Karhunen-Loeve expansion Once afinite-dimensional approximation has been obtained we mayexpress our solution in a generalized polynomial chaos basisand apply solution schemes based on the stochastic Galerkinfinite element method see [6] for a thorough introductionto this approach Instead of using the Karhunen-Loeveexpansion one could also fix a spatial basis and then estimatethe polynomial chaos coefficients straight frommeasurementdata see for instance [13 14]

31 Karhunen-Loeve Expansion The Karhunen-Loeve ex-pansion is a representation of a random field as a linear com-bination of the eigenfunctions of the associated covarianceoperator Truncating the resulting series we obtain a finite-dimensional approximation of the original random fieldThe Karhunen-Loeve expansion is the optimal choice amonglinear expansions in the sense that it minimizes the meansquare truncation error [6]


For a fixed 1205961 isin Ω1 let 119864(1205962 x) = 119864(1205961 1205962 x) be arandom field on119863 = 119863(1205961) with mean field

119864 (x) = int

Ω2

119864 (1205962 x) 119889P (1205962) (19)

and a covariance function

119862119864(x1 x2) = int

Ω2

(119864 (1205962 x1) minus 119864 (x1))

sdot (119864 (1205962 x2) minus 119864 (x2)) 119889P (1205962)

(20)

that is symmetric and positive definiteThe associated covari-ance operator

(C119864V) (x1) = int

119863

C119864(x1 x2) V (x2) 119889x2 (21)

is compact as an operator C119864 119871

2(119863) rarr 119871

2(119863) and there-

fore admits a countable number of positive eigenvalues120582

119898infin

119898=1 and associated eigenfunctions 120601119898infin

119898=1 The eigen-values 120582

119898infin

119898=1 tend to zero as 119898 tends to infinity and theeigenfunctions 120601

119898infin

119898=1 form an orthonormal basis of 1198712(119863)The eigenpairs may be computed numerically using forexample fast multipole methods [15]

The Karhunen-Loeve expansion of the random field119864(1205962 x) is now given by

119864 (1205962 x) = 119864 (x) +infin

sum

119898=1radic120582

119898120601119898 (

x) 119910119898 (1205962) (22)

where 119910119898infin

119898=1 are centered and orthogonal but not necessar-ily independent random variables

In numerical computations we replace the random field119864(1205962 x)with an approximation obtained by truncating series(22) after 119872 terms The truncation error depends upon thedecay of the Karhunen-Loeve eigenvalues and eigenfunc-tions See [16] for bounds on this decay

Remark 1 It is essential for numerical algorithms that therandom variables 119910

119898infin

119898=1 are mutually independent eventhough this is not in general implied by their orthogonalityA solution to this issue has been proposed in [3] wheresuitable auxiliary density functions have been introduced Inthe numerical examples we assume a representation of therandom field with respect to a set of independent uniformlydistributed random variables Other types of random vari-ables (eg Gaussian) could be considered as well

32 Legendre Chaos We employ the generalized polynomialchaos framework which essentially means representing oursolution on a basis of orthogonal polynomials In our casewe assume the input random variables to be uniformlydistributed and the polynomial chaos basis is therefore givenby tensorized Legendre polynomials The use of orthogo-nal polynomials as a basis allows us to apply stochasticGalerkin based methods and ensures optimal convergenceof these methods see [17] In this paper we aim to solve

the eigenmodes using a method of spectral inverse iterationintroduced in [12]

Assume that y(1205962) = (1199101(1205962) 119910119872(1205962)) is a vector ofmutually independent random variables that are uniformlydistributed in the interval [minus1 1]The expected value ormeanof a random variable V(y) is now given by

E2 [V] = int

Ω2

V (y (1205962)) 119889P (1205962)

= int

[minus11]119872V (y) 2minus119872119889y

(23)

Here we have used a subscript to indicate that the expectedvalue is taken over the sample space Ω2 only We associateeach multi-index 120572 = (1205721 120572119872) with the multivariateLegendre polynomial

Λ120572(y) =

119872

prod

119898=1119871120572119898(119910

119898) (24)

where 119871119901(119909) denotes the univariate Legendre polynomial of

degree 119901 We assume that the polynomials are normalized sothat E2[Λ

2120572] = 1 for all 120572 isin N119872

0 The system Λ

120572(y) | 120572 isin N119872

0 forms an orthonormalbasis of 1198712([minus1 1]119872) This allows us to express any squareintegrable randomvariable V(y) in a polynomial chaos expan-sion

V (y) = sum

120572isinN1198720

V120572Λ

120572(y) (25)

where the coefficients are given by V120572= E2[VΛ 120572

]For numerical computations we truncate expansion (25)

To this end we choose a finite set of multi-indices A sub N119872

0and approximate V(y) with the series

VA (y) = sum

120572isinA

V120572Λ

120572(y) (26)

Here the choice of the set A is key to obtaining accurateapproximations and is currently a topic of active research Inour numerical examples we consider sets that for a fixed level119871 isin R are given by A

119872119871= 120572 isin N119872

0 | sum119872

119898=1 120572119898radic120582119898 le 119871where 120582

119898infin

119898=1 are the Karhunen-Loeve eigenvalues Moresophisticated adaptive schemes for selecting the multi-indexsets have been presented in [18ndash20]

The most important reason for using the polynomialchaos basis is that it allows fast computation of any expecta-tions involvedThis is because we may evaluate integrals overthe stochastic domain analytically and are thus able to avoidnumerical integration in high dimensions Orthogonality ofthe Legendre polynomials yields

E2 [Λ 120572] = 120575

1205720

E2 [Λ 120572Λ

120573] = 120575

120572120573

(27)

for all multi-indices 120572 120573 isin N119872

0 For notational conveniencewe set

119888119898120573120574

= E2 [119910119898Λ 120572Λ

120573] 119898 isin N 120572 120573 isin N

119872

0

119888120572120573120574

= E2 [Λ 120572Λ

120573Λ

120574] 120572 120573 120574 isin N

119872

0 (28)


These coefficients are easy to evaluate analytically see theappendix in [12] It is also worth noting that most of thecoefficients are evaluated to zero This fact can be exploitedin order to speed up the solution process

4 Higher Order Finite Elements

Here we give a short overview of the ℎ119901-FEM followingclosely the one in [21] In the 119901-version of the FEM thepolynomial degree 119901 is used to control the accuracy ofthe solution while keeping the mesh fixed in contrast tothe ℎ-version or standard finite element method where thepolynomial degree is constant but the mesh size varies Theℎ119901-method simply combines the ℎ- and 119901-refinements

In our setting we can use topologically fixed meshes withhigh-order elements with119901 = 4 and119901 = 8 to ensure sufficientresolution of the deterministic part of the solution

In the following one way to construct a 119901-type quadri-lateral element is given The construction of triangles followssimilar lines First of all the choice of shape functions is notunique We use the so-called hierarchic integrated Legendreshape functions

Legendre polynomials of degree 119899 can be defined by arecursion formula

(119899 + 1) 119875119899+1 (119909) minus (2119899 + 1) 119909119875119899 (119909) + 119899119875119899minus1 (119909) = 0 (29)

where 1198750(119909) = 1 and 1198751(119909) = 119909The derivatives can similarly be computed by using the

recursion

(1minus1199092) 1198751015840119899(119909) = minus 119899119909119875

119899 (119909) + 119899119875119899minus1 (119909) (30)

The integrated Legendre polynomials are defined for 120585 isin[minus1 1] as

120595119899 (120585) = radic

2119899 minus 12

int

120585

minus1119875119899minus1 (119905) 119889119905 119899 = 2 3 (31)

and can be rewritten as linear combinations of Legendrepolynomials

120595119899 (120585) =

1radic2 (2119899 minus 1)

(119875119899 (120585) minus 119875119899minus2 (120585))

119899 = 2 3 (32)

The normalizing coefficients are chosen so that

int

1

minus1

119889120595119894 (120585)

119889120585

119889120595119895 (120585)

119889120585

119889120585 = 120575119894119895 119894 119895 ge 2 (33)

Using these polynomials we can now define the shapefunctions for a quadrilateral reference element over thedomain [minus1 1] times [minus1 1] The shape functions are dividedinto three categories nodal shape functions side modes andinternal modes

There are four nodal shape functions

1198731 (120585 120578) =14(1minus 120585) (1minus 120578)

1198732 (120585 120578) =14(1+ 120585) (1minus 120578)

1198733 (120585 120578) =14(1+ 120585) (1+ 120578)

1198734 (120585 120578) =14(1minus 120585) (1+ 120578)

(34)

which taken alone define the standard four-node quadrilat-eral finite element There are 4(119901 minus 1) side modes associatedwith the sides of a quadrilateral (119901 ge 2) with 119894 = 2 119901

119873(1)119894(120585 120578) =

12(1minus 120578)120595

119894 (120585)

119873(2)119894(120585 120578) =

12(1+ 120585) 120595119894 (120578)

119873(3)119894(120585 120578) =

12(1+ 120578)120595

119894(120578)

119873(4)119894(120585 120578) =

12(1minus 120585) 120595119894 (120585)

(35)

For the internal modes we choose the (119901 minus 1)(119901 minus 1) shapefunctions

1198730119894119895(120585 120578) = 120595

119894 (120585) 120595119895

(120578)

119894 = 2 119901 119895 = 2 119901(36)

The internal shape functions are often referred to as bubble-functions

Note that some additional book-keeping is necessaryThe Legendre polynomials have the property 119875

119899(minus119909) =

(minus1)119899119875119899(119909) This means that every edge must be globally

parameterized in the same way in both elements where itbelongs

41 Curved Boundaries Since we want to use fixed meshtopologies even with perturbed domains it is important torepresent curved boundary segments accurately The linearblending function method of Gordon and Hall [22] is ourchoice for this purpose

In the general case all sides of an element can be curvedbut in our case only one side ismdashas in Figure 1 We assumethat every side is parameterized

119909 = 119909119894 (119905)

119910 = 119910119894 (119905)

minus 1 le 119905 le 1

(37)


where 119894 = 1 number of sides Using capital letters ascoordinates of the corner points (119883

119894 119884

119894) we can write the

mapping for the global 119909-coordinates of a quadrilateral as

119909 =

12(1minus 120578) 1199091 (120585) +

12(1+ 120585) 1199092 (120578)

+

12(1+ 120578) 1199093 (120585) +

12(1minus 120585) 1199094 (120578)

minus

14(1minus 120585) (1minus 120578)1198831 minus

14(1+ 120585) (1+ 120578)1198832

minus

14(1+ 120585) (1+ 120578)1198833 minus

14(1minus 120585) (1+ 120578)1198834

(38)

and symmetrically for the 119910-coordinate Note that if theside parameterizations represent straight edges the mappingsimplifies to the standard bilinear mapping of quadrilaterals

5 The Solution Method

Wewill present a hybrid method of stochastic finite elementsfor solving the stochastic eigenvalue problem (15) More pre-cisely we apply stochastic collocation to resolve the depen-dency on the geometry and the stochastic Galerkin schemefor computing the effects of the random field The methodof stochastic collocation allows for an easy implementationsince the solution statistics are sampled from an ensembleof deterministic solutions This makes the method attractivefrom the point of view of random geometry On the otherhand if a sufficiently accurate parametric representation forthe input random field exists then the use of stochasticGalerkin basedmethods iswellmotivated by their high rate ofconvergencemdashespecially if the number of parameters is largeIn this paper we apply a method of spectral inverse iterationintroduced in [12] to compute the eigenpair of interest at eachcollocation point

51 Discretization Let us consider the stochastic eigenprob-lem (15) for a fixed 1205961 isin Ω1 We replace the Young modulus119864 = 119864(1205961 sdot sdot) with the expression

119864119872(1205962 x) = 119864 (x) +

119872

sum

119898=1radic120582

119898120601119898 (

x) 119910119898 (1205962) (39)

obtained by truncating the Karhunen-Loeve expansion (22)after 119872 terms Plugging (39) into (2) we may write theconstitutive matrix (11) in the form

D (120582 (1205961 1205962 x) 120583 (1205961 1205962 x))

= D0 (1205961 x) +119872

sum

119898=1D119898(1205961 x) 119910119898 (1205962)

(40)

We assume that 119910119898119872

119898=1 is a set of mutually independentand uniformly distributed random variables with supportsscaled to the interval [minus1 1] The sample space Ω2 is thusparameterized by the vector y = (1199101 119910119872) sub [minus1 1]119872With 1205961 isin Ω1 being fixed we may now interpret the

eigenvalue 120581 as a function of y and the eigenfunction u as afunction of y and x

Next we address the spatial discretization of (15) on thefixed domain 119863 = 119863(1205961) Let 120593119896

119873

119896=1 be a set of global basisfunctions for the approximation space of 119881

119863times 119881

119863 Here we

employ the 119901-version of the finite element method so that thecomponents of each 120593

119896are some shape functions (34) (35)

or (36) from Section 4 From (15) we obtain the deterministicGalerkin projection

int

119863

120598V (u)119879D0120598V (120593119896) +

119872

sum

119898=1119910119898120598V (u)

119879D119898120598V (120593119896) 119889x

= 120581int

119863

u sdot120593119896119889x 119896 = 1 119873

(41)

We define the stiffness matrix A(119898) for each119898 = 0 119872 bysetting

A(119898)

119894119895= int

119863

120598V (120593119894)119879D

119898120598V (120593119895) 119889x (42)

and the mass matrixM via

M119894119895= int

119863

120593119894sdot120593

119895119889x (43)

For a fixed 1205961 isin Ω1 we have now reduced (15) to a stochasticmatrix eigenvalue problem find a random variable 120581(1205961 sdot)

[minus1 1]119872 rarr R and a random vector u(1205961 sdot) [minus1 1]119872

rarr

R119873 such that

(A(0)+

119872

sum

119898=1A(119898)

119910119898) u (y) = 120581 (y)Mu (y) (44)

52 Stochastic Collocation In our problem of interest thegeometry of the computational domain depends on therandom variables 119877(1205961) and 120579(1205961) Thus these variables giveus a parametrization of the sample space Ω1 We present ananisotropic collocation operator and apply it for computingthe dependency of our solution on the parameters Imple-mentation of a sparse grid collocation operator (see [4 23])would also be possible but since we only consider collocationin two dimensions we are restricted to full tensor collocationin the following In our case we rely on the Gauss-Legendrequadrature since the random variables 119877(1205961) and 120579(1205961) areuniformly distributed Furthermore for notational simplicitywe assume these to be scaled so that 119877 120579 isin [minus1 1]

Denote by 119911(119899)119896119899

119896=0 the abscissae of the Gauss-Legendrequadrature of order 119899 and by 119908(119899)

119896119899

119896=0 the associated quadra-ture weights The quadrature points 119911(119899)

119896119899

119896=0 are the zerosof the univariate Legendre polynomial of degree 119899 + 1 Theone-dimensional Lagrange interpolation operators I

119899with

respect to the interpolation points 119911(119899)119896119899

119896=0 are defined via

(I119899V) (119911) =

119899

sum

119896=0V (119911(119899)

119896) ℓ

(119899)

119896(119911) for 119899 ge 0 (45)

Here ℓ(119899)119896119899

119896=0 are the Lagrange basis polynomials of degree119899 For information on orthogonal polynomials and computa-tion of the quadrature weights see [24]


The full tensor Lagrange interpolation operator is definedas the tensor product of the univariate operators In our two-dimensional case this is

I119865

120574= I

(119877)

1198991otimesI

(120579)

1198992 (46)

whereI(119877)

1198991andI(120579)

1198992denote the one-dimensional interpola-

tion operators in the variables 119877(1205961) and 120579(1205961) respectivelyand 120574 = 1198991 1198992 are the orders of quadrature Applied on arandom variable V(119877 120579) the full tensor interpolation gives

(I119865

120574V) (119877 120579)

=

1198991

sum

1198961=0

1198992

sum

1198962=0V (119911(1198991)

1198961 119911

(1198992)1198962

) ℓ(1198991)1198961

(119877) ℓ(1198992)1198962

(120579)

(47)

The expected value of a random variable V(119877 120579) is givenby

E1 [V] = int

Ω1

V (119877 (1205961) 120579 (1205961)) 119889P (1205961)

= int

[minus11]2V (119877 120579) 2minus2119889119877119889120579

(48)

Here we have again used a subscript to designate that theexpectation is taken over the sample space Ω1 From (47) weobtain

E1 [I119865

120574V] =

1198991

sum

1198961=0

1198992

sum

1198962=0V (119911(1198991)

1198961 119911

(1198992)1198962

)119908(1198991)1198961

119908(1198992)1198962

(49)

Hence we may compute the expected value of the solutiononce we have solved the underlying problem at each colloca-tion point

53 Spectral Inverse Iteration We choose to solve (44) at eachcollocation point using amethod of spectral inverse iterationThismethod has been introduced in [12] and there its conver-gence towards the eigenpair with the smallest eigenvalue hasbeen verified in the case of an elliptic operator with randomcoefficients However assuming that the eigenpair of interestis of multiplicity one the method is in fact applicable for amuch wider range of problems

We may consider the spectral inverse iteration as astochastic extension of the deterministic inverse iterationwhich has been given in Algorithm 2 for reference For thegeneralized eigenvalue problem of a stiffness matrix A and amass matrixM the deterministic inverse iteration convergesto the eigenpair forwhich the eigenvalue is closest to the givenparameter 120581

Algorithm 2 (deterministic inverse iteration) Fix 120581 isin R

and tol gt 0 Let u(0) be a normalized initial guess for theeigenvector and set 120581(0) = (u(0))119879Au(0) For 119896 = 1 2 dothe following

(1) Solve (A minus 120581M)k = Mu(119896minus1) for k(2) Set u(119896) = kk

(3) Set 120581(119896) = (u(119896))119879Au(119896)(4) Stop if u(119896) minus u(119896minus1) lt tol and return (120581 u) =

(120581(119896) u(119896))

The idea in the spectral inverse iteration is to interpreteach equation involved in Algorithm 2 in a stochastic senseusing Galerkin projections with respect to a given spectralbasis To this end we start by fixing a finite polynomial chaosbasis Λ

120572(y) | 120572 isin A sub N119872

0 and proceed by formulatingthe respective Galerkin projections Here we only recap theessentials of the process as it has already been thoroughlyexplained in [12]

For some 120581 isin R let us consider the equation

(A (y) minus 120581M) k (y) = Mu (y) (50)

that arises from the first step of Algorithm 2 Here we assumethe matrix A to take the form

A = A(0)+

119872

sum

119898=1A(119898)

119910119898

(51)

as in (44) We replace the random vectors u(y) and k(y) withthe truncated polynomial chaos expansions

uA (y) = sum

120573isinA

u120573Λ

120573(y)

kA (y) = sum

120573isinA

k120573Λ

120573(y)

(52)

The Galerkin projection of (50) on the basis Λ120572120572isinA is now

defined as

E2 [(Aminus 120581M) kAΛ 120572] = E2 [MuAΛ 120572

] 120572 isin A (53)

Using the orthogonality relations (27) this reduces to thelinear system

Ψ120578 = 120577 (54)

where 120577 120578 and Ψ are block vectors and matrices character-ized by [120577

120572] = Mu

120572 [120578

120573] = k

120573and

[Ψ120572120573] = (A(0)

minus 120581M) 120575120572120573+

119872

sum

119898=1A(119898)

119888119898120572120573

(55)

Let us next consider the normalization step inAlgorithm 2 For a vector k(y) we aim to compute w(y)such that

w (y) =k (y)1003817100381710038171003817k (y)100381710038171003817

1003817

(56)

If we set 119899(y) = k(y) then the respectiveGalerkin projectionon the basis Λ

120572120572isinA is defined as

E2 [1198992AΛ 120572

] = E2 [1003817100381710038171003817kA1003817100381710038171003817

2Λ

120572] 120572 isin A (57)


and reduces to

sum

120573isinA

sum

120574isinA

119899120573119899120574119888120572120573120574

= sum

120573isinA

sum

120574isinA

k119879120573Mk

120574119888120572120573120574

120572 isin A (58)

This is a nonlinear system of equations for the coefficients119899

120572120572isinA and can be solved using for example Newtonrsquos

method with the initial guess 119899120572= k

120572(y) The Jacobian is

given explicitly by

J120575120598= 2119899

120598sum

120574isinA

119888120575120598120574 (59)

The Galerkin projection of (56) is now defined as

E2 [wA119899AΛ 120572] = E2 [kAΛ 120572

] 120572 isin A (60)

and reduces to the block linear system

Φ120579 = 120578 (61)

where [120578120572] = k

120572 [120579

120573] = w

120573 and

[Φ120572120573]119894119895= 120575

119894119895sum

120574isinA

119899120574119888120572120573120574

(62)

For computing the eigenvalue we need to be able toevaluate the Rayleigh quotient

119903 (y) = u (y)119879A (y)u (y) (63)

for a vector u(y) As explained in [12] we may approximate119903(y) in the Galerkin sense to obtain the formula

119903120572

= sum

120573isinA

sum

120574isinA

(u119879120573A(0)u

120574+ sum

120575isinA

119872

sum

119898=1u119879120573A(119898)u

120575119888119898120574120575

)119888120572120573120574

(64)

for the Galerkin coefficients 119903120572120572isinA

Algorithm 3 now formulates the method of spectralinverse iteration Similar to the deterministic case the choiceof the parameter 120581 affects the speed of convergence Howeverif 120581 is chosen too close to the actual eigenvalue the iterationmight not converge as (50) becomes singular for some y isin

[minus1 1]119872 We refer to [12] for more information In ournumerical examples we set 120581 = 0 which ensures convergenceto the eigenpair with the smallest eigenvalue as long as thematrix A(y) is positive definite for all y isin [minus1 1]119872

Algorithm 3 (spectral inverse iteration for stochastic eigen-value problems) Fix 120581 isin R and tol gt 0 Let u(0) = u(0)

120572120572isinA

be a normalized initial guess for the eigenvector and set 120581(0) =119903120572(u(0))

120572isinA For 119896 = 1 2 do the following

(1) With u = u(119896minus1)120572

120572isinA solve system (54) for k =

k120572120572isinA

(2) Solve 119899 = 119899120572120572isinA from the nonlinear system (58)

Solve system (61) and set u(119896) = w120572120572isinA

(3) Set 120581(119896) = 119903120572(u(119896))

120572isinA(4) Stop if sum

120572isinA u(119896)120572

minus u(119896minus1)120572

lt tol and return (120581 u) =(120581

(119896) u(119896))

54 Response Statistics We may easily calculate statisticalmoments of a random variable V(119877 120579 y) once we know itspolynomial chaos expansion

VA (119877 120579 y) = sum

120572isinA

V120572 (119877 120579) Λ 120572

(y) (65)

at each collocation point By using the orthogonality relations(27) we obtain

E2 [VA (119877 120579 sdot)] = V0 (119877 120579)

E2 [(VA (119877 120579 sdot))2] = sum

120572isinA

(V120572 (119877 120579))

2

(66)

The first and second moments of V(119877 120579 y) are now given by

E [V] = E1 [E2 [V]]

E [V2] = E1 [E2 [V2]]

(67)

These can be computed by applying formula (49) on (66)From these we obtain approximations for the expected valueand variance of V(119877 120579 y)

6 Numerical Experiments

Let us consider a problem with uncertain domain anduncertain field The computational domain is 119863 = [minus2 2] times[minus2 2] 119862 where the cavity 119862 = [minus119877 119877] times [minus119877 119877] cup

annulus with diameter = 14 which is rotated 120579 radians bythe origin Here 119877 = 119880([12 35]) and 120579 = 119880([0 12058712]) arerandom with uniform distribution The bottom edge 119910 = minus2is clamped that is all displacements are inhibited

In Figure 2 the effect of domain perturbation on themodes is illustrated Notice that if one wanted to computethe statistics of the modes it would be necessary to mapevery realisation onto the nominal domain via conformalmappings for instance

The (constant) material parameters are Poisson ratio ] =13 and the mean field of the Young modulus 119864 = 2069 sdot1011 MPa We have used the convention however where thereported results are normalized by the value of 119864

We will proceed in two steps by first letting only thegeometry vary before adding uncertainty to the field as wellThe cases are

(A) uncertain domain with deterministic field solvedwith collocation

(B) uncertain domain with uncertain field solved withthe new hybrid method

In both cases the nominal domain is the same (see Figure 1)In the absence of exact solutions overkill solutions have beenused in convergence graphs unless otherwise specified Wehave also verified the results by comparing to a full MonteCarlo simulation of 620000 draws

61 Case A The values of the smallest eigenvalue over theparameter space are shown in Figure 3 using both contourand surface plots


0

1

2

minus1

minus2

0 1 2minus1minus2

(a) Contour plot 1199061 (119877 = 45)

0 1 2

0

1

2

minus1

minus2

minus1minus2

(b) Contour plot 1199062 (119877 = 45)

0

1

2

minus1

minus2

0 1 2minus1minus2

(c) Contour plot 1199061 (119877 = 45 120579 = 12058712)

0

1

2

minus1

minus2

0 1 2minus1minus2

(d) Contour plot 1199062 (119877 = 45 120579 = 12058712)

Figure 2 Case A effect of perturbation on eigenmodes Contour lines of displacement

We consider a collocation sequence where the collo-cation points are taken to span all 16 cases from 1 times 1to 4 times 4 points over the 119877 times 120579-parameter space Therelative convergence of the smallest eigenvalue over thewholecollocation sequence is shown in Figure 4 We have usedconstant polynomial order 119901 = 8 (1728 degrees of freedom)throughout and 119901 = 16 (6528 degrees of freedom) for thereference computation

The dashed line shows the convergence in the case whenthe deterministic discretization of the reference solution isused also in collocation The exponential convergence inexpectation is due to convergence in parameter space onlyThe poorly performing grids in Figure 4 are the anisotropicones with more points for 120579 This is due to relative effects of 119877

and 120579 not being exactly balanced (see Figure 3)The solid lineindicates that already at rule (2 2) the spatial discretizationused is not sufficient when the reference has been computedwith much higher resolution The convergence stalls com-pletely since the deterministic error dominates Convergencein variance is slower as suggested by theory

62 Case B We proceed to consider a fixed collocation gridwith 4 times 4 collocation points for the parameters 119877 and 120579Assuming normalization of the mean field (119864 = 1) we set thecoefficients in the Karhunen-Loeve expansion of the Youngmodulus (39) to be 120601

119898(x) = cos(119898120587119889(x)) where 119889(x) is the

distance from x to the inner boundary and 120582119898= (119898 + 1)minus3

Thus the series can be considered to converge at an algebraic


12058712

12058724

012 35

120579

R

(a) Contour plot 119877 (119909-axis) versus 120579 (119910-axis)

12058712

0

12

35

120579

R

120582

(b) Surface plot 119877 (119909-axis) versus 120579 (119910-axis)

Figure 3 Case A eigenvalues over the parameter space the standard deviation is 120590 = 00008

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17Grid

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Rela

tive e

rror

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

(a) Expected value

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Grid

Rela

tive e

rror

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance

Figure 4 Case A convergence in eigenvalue relative error versus collocation sequences in log-scale Dashed line fixed deterministicdiscretization in reference Solid line higher resolution reference Plotmarkers uniform collocation grids

rate of 119904 = minus15We set the truncation parameter to be119872 = 3Using 119901 = 4 and 25 multi-indices every linear system has12000 degrees of freedom

At each collocation point we employ the spectral inverseiteration given in Algorithm 3with 120581 = 0 For the polynomialchaos basis we consider multi-index sets of the formA

119872119871=

120572 isin N119872

0 | sum119872

119898=1 120572119898radic120582119898 le 119871 Varying the level 119871 isin

R results in multi-index sets of different size The overkillsolution used as a reference has been computed with 64multi-indices Convergence of the relative error of the small-est eigenvalue towards the overkill solution as a function of

the size of the multi-index set is illustrated in Figure 5 Weobserve algebraic rates of convergence for both the expectedvalue and the variance

The dashed horizontal line in Figure 5 illustrates therelative error between the overkill solution and the MonteCarlo solution The statistics for the eigenvalue are

E [120581] asymp 164286 sdot 10minus2

Var [120581] asymp 919957 sdot 10minus7(68)


1 2 5 10 20Number of multi-indices

1 2 5 10 20Re

lativ

e err

or

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

(a) Expected value

1 2 5 10 20

1 2 5 10 20

Number of multi-indices

Rela

tive e

rror

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance

Figure 5 Case B convergence in eigenvalue in the case of the 4 times 4 collocation grid size of the multi-index sets versus relative error in thehybrid solution in loglog-scale Plotmarkers multi-index sets of different size Dashed line error between the overkill solution and theMonteCarlo solution

for the overkill solution and



for the Monte Carlo solution The central limit theoremsuggests that the standard deviation of theMonte Carlomeanis 121271 sdot 10minus6 which is of the same size range as the errorbetween the overkill solution and the Monte Carlo solutionComparison with standard collocation is more complicatedsince the hybrid method includes a collocation part Foran isotropic grid aligned for the geometric uncertaintydimensions (119877 120579119872111987221198723) = (4 4 8 8 8) that is 8points per every term in the Karhunen-Loeve expansion weget statistics



within expected accuracy as wellAlgorithm 3 also gives us the eigenvector associated with

the smallest eigenvalue More precisely we obtain the spectralcomponents u

120572120572isinA119872119871

at each collocation point In Figure 6we have illustrated themeans of the sorted1198712 and1198671 energiesof these spectral components At each of the 16 collocationpoints we would observe results similar to those presentedin Figure 6Theory suggests that the asymptotic convergencerates for these energies should be 1199031 = 119904 minus 12 and 1199032 = 119904 minus 1for the 1198712 norms and 119867

1 seminorms respectively [25 26]In our example we observe much faster convergence thanthese asymptotic rates would predict However the observed

convergence rates differ by one half which is in accordancewith the theory

The Monte Carlo reference results have been computedfrom 620000 draws using 432 single core CPU hours on IntelXeon (2009 28GHz) The anisotropic collocation took twoand the hybrid algorithm for the highest reported numberof multi-indices (64) exactly one single core CPU hour onthe same machine Naturally in iterative algorithms there aremany tolerances with which to tune the performance butthese reported times can be taken as representative

Remark 4 Notice that in more elaborate models for geo-metric uncertainty with much higher dimensions of theparameter space the asymptotic convergence rates will applyto the collocation method For lower dimensional cases it ispossible to converge at faster rates of course but in real-lifeapplications one should consider the modeling error as wellSimilarly for the Galerkin scheme the size of multi-index setsis too small for the energies of the spectral components toexhibit asymptotic behaviour

7 Conclusions

We have demonstrated the practicality of the new hybridalgorithm in a practical yet idealized application The com-bination of collocation for the geometric uncertainty andGalerkin for that of materials allows us to take advantage ofthe higher computational efficiency of the Galerkin approachwhilst keeping mesh generation simple Indeed in this paperwe have kept the mesh topologically fixed The observedconvergence rates for the smallest eigenvalue are in line withtheoretical predictions


1 2 5 10 20Multi-index (sorted)

Ener

gy1 2 5 10 20

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

(a) 1198712 energies

1 2 5 10 20

1 2 5 10 20

Multi-index (sorted)

Ener

gy

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b) 1198671 energies

Figure 6 Case B energies of the spectral components u120572120572isinA119872119871

of the eigenfunction in the case of the 4 times 4 collocation grid Dashed linealgebraic convergence rates of minus38 and minus33 for the 1198712 norm and 1198671 seminorm respectively Plotmarkers sorted means of the energies inloglog-scale

The main two remaining issues for future research arethe mapping of the eigenmodes to the nominal domain andhandling of double eigenvalues in the general case In 2D weare confident that this extension can be readily done but in3D there are still open mathematical questions on how toconnect the nominal domain and the perturbed ones

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B A Szabo and D E Muntges ldquoProcedures for the verificationand validation of working models for structural shellsrdquo Journalof Applied Mechanics vol 72 no 6 pp 907ndash915 2005

[2] C A Schenk and G I Schueller Uncertainty Assessment ofLarge Finite Element Systems vol 24 of Lecture Notes in Appliedand Computational Mathematics Springer 2005

[3] I Babuska F Nobile and R Tempone ldquoA stochastic collocationmethod for elliptic partial differential equations with randominput datardquo SIAM Journal on Numerical Analysis vol 45 no 3pp 1005ndash1034 2007

[4] F Nobile R Tempone and C G Webster ldquoAn anisotropicsparse grid stochastic collocationmethod for partial differentialequations with random input datardquo SIAM Journal on NumericalAnalysis vol 46 no 5 pp 2411ndash2442 2008

[5] C Schwab and C J Gittelson ldquoSparse tensor discretizationsof high-dimensional parametric and stochastic PDEsrdquo ActaNumerica vol 20 pp 291ndash467 2011

[6] R Ghanem and P Spanos Stochastic Finite Elements A SpectralApproach Dover Publications Mineola NY USA 2003

[7] R Ghanem and D Ghosh ldquoEfficient characterization of therandom eigenvalue problem in a polynomial chaos decomposi-tionrdquo International Journal for Numerical Methods in Engineer-ing vol 72 no 4 pp 486ndash504 2007

[8] C V Verhoosel M A Gutierrez and S J Hulshoff ldquoIterativesolution of the random eigenvalue problem with application tospectral stochastic finite element systemsrdquo International Journalfor Numerical Methods in Engineering vol 68 no 4 pp 401ndash424 2006

[9] R Andreev Implementation of sparse wavelet-Galerkin finiteelement method for stochastic partial differential equations [MSthesis] ETH Zurich Zurich Switzerland 2008

[10] R Andreev and C Schwab ldquoSparse tensor approximationof parametric eigenvalue problemsrdquo in Numerical Analysis ofMultiscale Problems vol 83 of Lecture Notes in ComputationalScience and Engineering pp 203ndash241 Springer Berlin Ger-many 2012

[11] H Meidani and R Ghanem ldquoSpectral power iterations for therandom eigenvalue problemrdquo AIAA Journal vol 52 no 5 pp912ndash925 2014

[12] H Hakula V Kaarnioja and M Laaksonen ldquoApproximatemethods for stochastic eigenvalue problemsrdquo Applied Mathe-matics and Computation 2015

[13] R G Ghanem andADoostan ldquoOn the construction and analy-sis of stochasticmodels characterization and propagation of theerrors associated with limited datardquo Journal of ComputationalPhysics vol 217 no 1 pp 63ndash81 2006

[14] S Das R Ghanem and S Finette ldquoPolynomial chaos repre-sentation of spatio-temporal random fields from experimentalmeasurementsrdquo Journal of Computational Physics vol 228 no23 pp 8726ndash8751 2009

[15] C Schwab and R A Todor ldquoKarhunen-Loeve approximation ofrandomfields by generalized fastmultipolemethodsrdquo Journal ofComputational Physics vol 217 no 1 pp 100ndash122 2006


[16] M Bieri Sparse tensor discretizations of elliptic PDEs with ran-dom input data [PhD thesis] ETHZurich Zurich Switzerland2009

[17] D Xiu and G E Karniadakis ldquoThe Wiener-Askey polynomialchaos for stochastic differential equationsrdquo SIAM Journal onScientific Computing vol 24 no 2 pp 619ndash644 2002

[18] M Bieri R Andreev and C Schwab ldquoSparse tensor discretiza-tion of elliptic SPDEsrdquo SIAM Journal on Scientific Computingvol 31 no 6 pp 4281ndash4304 2009

[19] A Chkifa A Cohen R DeVore and C Schwab ldquoSparseadaptive Taylor approximation algorithms for parametric andstochastic elliptic PDEsrdquo ESAIM Mathematical Modelling andNumerical Analysis vol 47 no 1 pp 253ndash280 2013

[20] A Chkifa A Cohen and C Schwab ldquoHigh-dimensionaladaptive sparse polynomial interpolation and applications toparametric PDEsrdquo Foundations of Computational Mathematicsvol 14 no 4 pp 601ndash633 2014

[21] H Hakula A Rasila and M Vuorinen ldquoOn moduli of ringsand quadrilaterals algorithms and experimentsrdquo SIAM Journalon Scientific Computing vol 33 no 1 pp 279ndash302 2011

[22] W J Gordon and C A Hall ldquoTransfinite element methodsblending-function interpolation over arbitrary curved elementdomainsrdquoNumerischeMathematik vol 21 pp 109ndash129 197374

[23] M Bieri ldquoA sparse composite collocation finite elementmethodfor elliptic SPDEsrdquo SIAM Journal onNumerical Analysis vol 49no 6 pp 2277ndash2301 2011

[24] W Gautschi Orthogonal Polynomials Computation andApproximation NumericalMathematics and Scientific Compu-tation Oxford University Press New York NY USA 2004

[25] A Cohen RDeVore andC Schwab ldquoConvergence rates of best119873-term Galerkin approximations for a class of elliptic sPDEsrdquoFoundations of Computational Mathematics vol 10 no 6 pp615ndash646 2010

[26] A Cohen R Devore and C Schwab ldquoAnalytic regularity andpolynomial approximation of parametric and stochastic ellipticPDErsquosrdquo Analysis and Applications vol 9 no 1 pp 11ndash47 2011

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DistributedSensor Networks



R

(a) Nominal domain

R

120579

(b) Perturbed domain

Figure 1 Two realisations of the domain with meshes The nominal domain is the reference The rotation and scaling of the cavity as well asthe direction of the uncertainty in the Youngrsquos modulus is indicated

The main result of this paper is the new hybrid algorithmwhich combines a nonintrusive outer loop (collocation)with an intrusive inner one (Galerkin) The randomness ingeometry is resolved using collocation and that of materialswith Galerkin at every collocation point The model problemis an idealized engine bed vibration problem where a 2Dhollow square is clamped from one side (see Figure 1) Theinner cavity is assumed to be of fixed shape but randomorientation and Youngrsquosmodulus is taken to be random in thenormal direction into the body The choice of the geometricuncertainty models the case of sand casting with moulds offixed shape The hybrid approach combines the geometricflexibility of collocation with much faster computation timesof the Galerkin approach used in the inner loop and thuscombines the best of both worlds

The rest of the paper is organized as follows first theabstract problem is introduced in Section 2 The representa-tion of the random input is given in Section 3 Higher orderfinite elements are described in Section 4 Section 5 is centralto our discussion both solution methods collocation andspectral inverse iteration are presented Numerical experi-ments are discussed in Section 6 and finally conclusions aredrawn in Section 7

2 Problem Statement

We start by presenting the deterministic system of Navierrsquosequations of elasticity and the corresponding weak formWe then extend this to the stochastic setting in order tocover the case of uncertain domain andmodulus of elasticityWe aim to compute statistics of the smallest eigenvalueof the resulting stochastic system The geometry of thecomputational domain is illustrated in Figure 1

21 Deterministic Formulation We use the Navier equationsof elasticity to model the system of interest Find theeigenvalue 120581 the displacement field u = (1199061 1199062) and thesymmetric stress tensor 120590 = (120590

119894119895)2119894119895=1 such that

120590 = 120582div (u) I+ 2120583120598 (u) in 119863

minus div (120590) = 120581u in 119863

u = 0 on 120597119863119863

120590 sdotn = 0 on 120597119863119873

(1)

where 120597119863 = 120597119863119863cup 120597119863

119873is a partitioned boundary of119863 The

Lame constants are

120582 =

119864](1 + ]) (1 minus 2])

120583 =

119864

2 (1 + ])

(2)

with 119864 and ] being Youngrsquos modulus and Poissonrsquos ratiorespectively Further I is the identity tensor n denotes theoutward unit normal to 120597119863

119873 and the strain tensor is

120598 (u) = 12(nablau+nablau119879) (3)

The vector-valued tensor divergence is

div (120590) = (

2sum

119895=1

120597120590119894119895

120597119909119895

)

2

119894=1

(4)

This formulation assumes a constitutive relation correspond-ing to linear isotropic elasticity with stresses and strainsrelated by Hookersquos generalized law

120590V =[

[

[

12059011

12059022

12059012

]

]

]

= D (120582 120583)[

[

[

12059811

12059822

12059812

]

]

]

= D (120582 120583) 120598V (5)

22 Weak Formulation Let us introduce a function space119881119863= V V isin 119867

1(119863) V|

120597119863119863= 0 and define the eigen-

problem find the eigenpair (120581 u) isin R times (119881119863times 119881

119863) such that

119886 (u k) = 120581 (u k)119863 forallk isin 119881119863times 119881

119863 (6)

where the bilinear form

119886 (u k) = int

119863

120590 (u) 120598 (k) 119889x (7)



120590 120598 =

2sum

119894119895=1120590119894119895120598119894119895 (8)


(u k)119863 = int

119863

1199061V1 +1199062V2119889x (9)


120598V (u) =

[

[

[

[

[

[

[

[

120597

12059711990910

0 120597

1205971199092120597

1205971199092

120597

1205971199091

]

]

]

]

]

]

]

]

[

1199061

1199062] (10)


D (120582 120583) =[

[

[

120582 + 2120583 120582 0120582 120582 + 2120583 00 0 120583

]

]

]

(11)


119886 (u k) = int

119863

120598V (u)119879D (120582 120583) 120598V (k) 119889x (12)



119894is a 120590-


119894for


(ΩFP) =

2⨂

119894=1(Ω

119894F

119894P

119894) (13)







119863(1205961)times 119881

119863(1205961)such that

119886 (120596 u (1205961 1205962 sdot) k)

= 120581 (1205961 1205962) (u (1205961 1205962 sdot) k)119863(1205961)

forallk isin 119881119863(1205961)

times 119881119863(1205961)

(15)


119886 (120596 u k) = int

119863(1205961)120590 (120596 u) 120598 (k) 119889x = int

119863(1205961)120598V (u)

119879




E [V] = int

Ω

V (1205961 1205962) 119889P (120596) (17)

and its variance is

Var [V] = E [(VminusE [V])2] (18)







119864 (x) = int

Ω2

119864 (1205962 x) 119889P (1205962) (19)


119862119864(x1 x2) = int

Ω2

(119864 (1205962 x1) minus 119864 (x1))

sdot (119864 (1205962 x2) minus 119864 (x2)) 119889P (1205962)

(20)


(C119864V) (x1) = int

119863

C119864(x1 x2) V (x2) 119889x2 (21)


2(119863) rarr 119871



119898infin



119898infin


119898infin



119864 (1205962 x) = 119864 (x) +infin

sum

119898=1radic120582

119898120601119898 (

x) 119910119898 (1205962) (22)





119898infin





E2 [V] = int

Ω2

V (y (1205962)) 119889P (1205962)

= int

[minus11]119872V (y) 2minus119872119889y

(23)


Λ120572(y) =

119872

prod

119898=1119871120572119898(119910

119898) (24)



2120572] = 1 for all 120572 isin N119872

0 The system Λ

120572(y) | 120572 isin N119872


V (y) = sum

120572isinN1198720

V120572Λ

120572(y) (25)





VA (y) = sum

120572isinA

V120572Λ

120572(y) (26)


119872119871= 120572 isin N119872

0 | sum119872

119898=1 120572119898radic120582119898 le 119871where 120582

119898infin



E2 [Λ 120572] = 120575

1205720

E2 [Λ 120572Λ

120573] = 120575

120572120573

(27)



119888119898120573120574

= E2 [119910119898Λ 120572Λ

120573] 119898 isin N 120572 120573 isin N

119872

0

119888120572120573120574

= E2 [Λ 120572Λ

120573Λ

120574] 120572 120573 120574 isin N

119872

0 (28)








(119899 + 1) 119875119899+1 (119909) minus (2119899 + 1) 119909119875119899 (119909) + 119899119875119899minus1 (119909) = 0 (29)


recursion

(1minus1199092) 1198751015840119899(119909) = minus 119899119909119875

119899 (119909) + 119899119875119899minus1 (119909) (30)


120595119899 (120585) = radic

2119899 minus 12

int

120585

minus1119875119899minus1 (119905) 119889119905 119899 = 2 3 (31)


120595119899 (120585) =


(119875119899 (120585) minus 119875119899minus2 (120585))

119899 = 2 3 (32)


int

1

minus1

119889120595119894 (120585)

119889120585

119889120595119895 (120585)

119889120585

119889120585 = 120575119894119895 119894 119895 ge 2 (33)



1198731 (120585 120578) =14(1minus 120585) (1minus 120578)

1198732 (120585 120578) =14(1+ 120585) (1minus 120578)

1198733 (120585 120578) =14(1+ 120585) (1+ 120578)

1198734 (120585 120578) =14(1minus 120585) (1+ 120578)

(34)


119873(1)119894(120585 120578) =

12(1minus 120578)120595

119894 (120585)

119873(2)119894(120585 120578) =

12(1+ 120585) 120595119894 (120578)

119873(3)119894(120585 120578) =

12(1+ 120578)120595

119894(120578)

119873(4)119894(120585 120578) =

12(1minus 120585) 120595119894 (120585)

(35)


1198730119894119895(120585 120578) = 120595

119894 (120585) 120595119895

(120578)

119894 = 2 119901 119895 = 2 119901(36)



119899(minus119909) =





119909 = 119909119894 (119905)

119910 = 119910119894 (119905)


(37)



119894 119884



119909 =

12(1minus 120578) 1199091 (120585) +

12(1+ 120585) 1199092 (120578)

+

12(1+ 120578) 1199093 (120585) +

12(1minus 120585) 1199094 (120578)

minus


14(1+ 120585) (1+ 120578)1198832

minus

14(1+ 120585) (1+ 120578)1198833 minus

14(1minus 120585) (1+ 120578)1198834

(38)





119864119872(1205962 x) = 119864 (x) +

119872

sum

119898=1radic120582

119898120601119898 (

x) 119910119898 (1205962) (39)


D (120582 (1205961 1205962 x) 120583 (1205961 1205962 x))

= D0 (1205961 x) +119872

sum

119898=1D119898(1205961 x) 119910119898 (1205962)

(40)

We assume that 119910119898119872




119873


119863times 119881

119863 Here we




int

119863

120598V (u)119879D0120598V (120593119896) +

119872

sum

119898=1119910119898120598V (u)

119879D119898120598V (120593119896) 119889x

= 120581int

119863

u sdot120593119896119889x 119896 = 1 119873

(41)


A(119898)

119894119895= int

119863

120598V (120593119894)119879D

119898120598V (120593119895) 119889x (42)


M119894119895= int

119863

120593119894sdot120593

119895119889x (43)



rarr

R119873 such that

(A(0)+

119872

sum

119898=1A(119898)

119910119898) u (y) = 120581 (y)Mu (y) (44)


Denote by 119911(119899)119896119899


119896119899


119896119899


119899with



(I119899V) (119911) =

119899

sum

119896=0V (119911(119899)

119896) ℓ

(119899)

119896(119911) for 119899 ge 0 (45)

Here ℓ(119899)119896119899




I119865

120574= I

(119877)

1198991otimesI

(120579)

1198992 (46)

whereI(119877)

1198991andI(120579)



(I119865

120574V) (119877 120579)

=

1198991

sum

1198961=0

1198992

sum

1198962=0V (119911(1198991)

1198961 119911

(1198992)1198962

) ℓ(1198991)1198961

(119877) ℓ(1198992)1198962

(120579)

(47)


E1 [V] = int

Ω1

V (119877 (1205961) 120579 (1205961)) 119889P (1205961)

= int

[minus11]2V (119877 120579) 2minus2119889119877119889120579

(48)


E1 [I119865

120574V] =

1198991

sum

1198961=0

1198992

sum

1198962=0V (119911(1198991)

1198961 119911

(1198992)1198962

)119908(1198991)1198961

119908(1198992)1198962

(49)








(120581(119896) u(119896))


120572(y) | 120572 isin A sub N119872



(A (y) minus 120581M) k (y) = Mu (y) (50)


A = A(0)+

119872

sum

119898=1A(119898)

119910119898

(51)


uA (y) = sum

120573isinA

u120573Λ

120573(y)

kA (y) = sum

120573isinA

k120573Λ

120573(y)

(52)


defined as


] 120572 isin A (53)


Ψ120578 = 120577 (54)


120572] = Mu

120572 [120578

120573] = k

120573and

[Ψ120572120573] = (A(0)

minus 120581M) 120575120572120573+

119872

sum

119898=1A(119898)

119888119898120572120573

(55)


w (y) =k (y)1003817100381710038171003817k (y)100381710038171003817

1003817

(56)



E2 [1198992AΛ 120572

] = E2 [1003817100381710038171003817kA1003817100381710038171003817

2Λ

120572] 120572 isin A (57)


and reduces to

sum

120573isinA

sum

120574isinA

119899120573119899120574119888120572120573120574

= sum

120573isinA

sum

120574isinA

k119879120573Mk

120574119888120572120573120574

120572 isin A (58)





given explicitly by

J120575120598= 2119899

120598sum

120574isinA

119888120575120598120574 (59)


E2 [wA119899AΛ 120572] = E2 [kAΛ 120572

] 120572 isin A (60)


Φ120579 = 120578 (61)

where [120578120572] = k

120572 [120579

120573] = w

120573 and

[Φ120572120573]119894119895= 120575

119894119895sum

120574isinA

119899120574119888120572120573120574

(62)


119903 (y) = u (y)119879A (y)u (y) (63)


119903120572

= sum

120573isinA

sum

120574isinA

(u119879120573A(0)u

120574+ sum

120575isinA

119872

sum

119898=1u119879120573A(119898)u

120575119888119898120574120575

)119888120572120573120574

(64)





120572120572isinA



(1) With u = u(119896minus1)120572


k120572120572isinA



(3) Set 120581(119896) = 119903120572(u(119896))


120572isinA u(119896)120572



(119896) u(119896))


VA (119877 120579 y) = sum

120572isinA

V120572 (119877 120579) Λ 120572

(y) (65)


E2 [VA (119877 120579 sdot)] = V0 (119877 120579)

E2 [(VA (119877 120579 sdot))2] = sum

120572isinA

(V120572 (119877 120579))

2

(66)


E [V] = E1 [E2 [V]]

E [V2] = E1 [E2 [V2]]

(67)













0

1

2

minus1

minus2

0 1 2minus1minus2

(a) Contour plot 1199061 (119877 = 45)

0 1 2

0

1

2

minus1

minus2

minus1minus2

(b) Contour plot 1199062 (119877 = 45)

0

1

2

minus1

minus2

0 1 2minus1minus2

(c) Contour plot 1199061 (119877 = 45 120579 = 12058712)

0

1

2

minus1

minus2

0 1 2minus1minus2

(d) Contour plot 1199062 (119877 = 45 120579 = 12058712)










12058712

12058724

012 35

120579

R


12058712

0

12

35

120579

R

120582



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17Grid

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Rela

tive e

rror

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

(a) Expected value

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Grid

Rela

tive e

rror

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance




119872119871=

120572 isin N119872

0 | sum119872









1 2 5 10 20Re

lativ

e err

or

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

(a) Expected value

1 2 5 10 20

1 2 5 10 20


Rela

tive e

rror

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance










120572120572isinA119872119871






7 Conclusions




Ener

gy1 2 5 10 20

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8


1 2 5 10 20

1 2 5 10 20


Ener

gy

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7







References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120590 120598 =

2sum

119894119895=1120590119894119895120598119894119895 (8)


(u k)119863 = int

119863

1199061V1 +1199062V2119889x (9)


120598V (u) =

[

[

[

[

[

[

[

[

120597

12059711990910

0 120597

1205971199092120597

1205971199092

120597

1205971199091

]

]

]

]

]

]

]

]

[

1199061

1199062] (10)


D (120582 120583) =[

[

[

120582 + 2120583 120582 0120582 120582 + 2120583 00 0 120583

]

]

]

(11)


119886 (u k) = int

119863

120598V (u)119879D (120582 120583) 120598V (k) 119889x (12)



119894is a 120590-


119894for


(ΩFP) =

2⨂

119894=1(Ω

119894F

119894P

119894) (13)







119863(1205961)times 119881

119863(1205961)such that

119886 (120596 u (1205961 1205962 sdot) k)

= 120581 (1205961 1205962) (u (1205961 1205962 sdot) k)119863(1205961)

forallk isin 119881119863(1205961)

times 119881119863(1205961)

(15)


119886 (120596 u k) = int

119863(1205961)120590 (120596 u) 120598 (k) 119889x = int

119863(1205961)120598V (u)

119879




E [V] = int

Ω

V (1205961 1205962) 119889P (120596) (17)

and its variance is

Var [V] = E [(VminusE [V])2] (18)







119864 (x) = int

Ω2

119864 (1205962 x) 119889P (1205962) (19)


119862119864(x1 x2) = int

Ω2

(119864 (1205962 x1) minus 119864 (x1))

sdot (119864 (1205962 x2) minus 119864 (x2)) 119889P (1205962)

(20)


(C119864V) (x1) = int

119863

C119864(x1 x2) V (x2) 119889x2 (21)


2(119863) rarr 119871



119898infin



119898infin


119898infin



119864 (1205962 x) = 119864 (x) +infin

sum

119898=1radic120582

119898120601119898 (

x) 119910119898 (1205962) (22)





119898infin





E2 [V] = int

Ω2

V (y (1205962)) 119889P (1205962)

= int

[minus11]119872V (y) 2minus119872119889y

(23)


Λ120572(y) =

119872

prod

119898=1119871120572119898(119910

119898) (24)



2120572] = 1 for all 120572 isin N119872

0 The system Λ

120572(y) | 120572 isin N119872


V (y) = sum

120572isinN1198720

V120572Λ

120572(y) (25)





VA (y) = sum

120572isinA

V120572Λ

120572(y) (26)


119872119871= 120572 isin N119872

0 | sum119872

119898=1 120572119898radic120582119898 le 119871where 120582

119898infin



E2 [Λ 120572] = 120575

1205720

E2 [Λ 120572Λ

120573] = 120575

120572120573

(27)



119888119898120573120574

= E2 [119910119898Λ 120572Λ

120573] 119898 isin N 120572 120573 isin N

119872

0

119888120572120573120574

= E2 [Λ 120572Λ

120573Λ

120574] 120572 120573 120574 isin N

119872

0 (28)








(119899 + 1) 119875119899+1 (119909) minus (2119899 + 1) 119909119875119899 (119909) + 119899119875119899minus1 (119909) = 0 (29)


recursion

(1minus1199092) 1198751015840119899(119909) = minus 119899119909119875

119899 (119909) + 119899119875119899minus1 (119909) (30)


120595119899 (120585) = radic

2119899 minus 12

int

120585

minus1119875119899minus1 (119905) 119889119905 119899 = 2 3 (31)


120595119899 (120585) =


(119875119899 (120585) minus 119875119899minus2 (120585))

119899 = 2 3 (32)


int

1

minus1

119889120595119894 (120585)

119889120585

119889120595119895 (120585)

119889120585

119889120585 = 120575119894119895 119894 119895 ge 2 (33)



1198731 (120585 120578) =14(1minus 120585) (1minus 120578)

1198732 (120585 120578) =14(1+ 120585) (1minus 120578)

1198733 (120585 120578) =14(1+ 120585) (1+ 120578)

1198734 (120585 120578) =14(1minus 120585) (1+ 120578)

(34)


119873(1)119894(120585 120578) =

12(1minus 120578)120595

119894 (120585)

119873(2)119894(120585 120578) =

12(1+ 120585) 120595119894 (120578)

119873(3)119894(120585 120578) =

12(1+ 120578)120595

119894(120578)

119873(4)119894(120585 120578) =

12(1minus 120585) 120595119894 (120585)

(35)


1198730119894119895(120585 120578) = 120595

119894 (120585) 120595119895

(120578)

119894 = 2 119901 119895 = 2 119901(36)



119899(minus119909) =





119909 = 119909119894 (119905)

119910 = 119910119894 (119905)


(37)



119894 119884



119909 =

12(1minus 120578) 1199091 (120585) +

12(1+ 120585) 1199092 (120578)

+

12(1+ 120578) 1199093 (120585) +

12(1minus 120585) 1199094 (120578)

minus


14(1+ 120585) (1+ 120578)1198832

minus

14(1+ 120585) (1+ 120578)1198833 minus

14(1minus 120585) (1+ 120578)1198834

(38)





119864119872(1205962 x) = 119864 (x) +

119872

sum

119898=1radic120582

119898120601119898 (

x) 119910119898 (1205962) (39)


D (120582 (1205961 1205962 x) 120583 (1205961 1205962 x))

= D0 (1205961 x) +119872

sum

119898=1D119898(1205961 x) 119910119898 (1205962)

(40)

We assume that 119910119898119872




119873


119863times 119881

119863 Here we




int

119863

120598V (u)119879D0120598V (120593119896) +

119872

sum

119898=1119910119898120598V (u)

119879D119898120598V (120593119896) 119889x

= 120581int

119863

u sdot120593119896119889x 119896 = 1 119873

(41)


A(119898)

119894119895= int

119863

120598V (120593119894)119879D

119898120598V (120593119895) 119889x (42)


M119894119895= int

119863

120593119894sdot120593

119895119889x (43)



rarr

R119873 such that

(A(0)+

119872

sum

119898=1A(119898)

119910119898) u (y) = 120581 (y)Mu (y) (44)


Denote by 119911(119899)119896119899


119896119899


119896119899


119899with



(I119899V) (119911) =

119899

sum

119896=0V (119911(119899)

119896) ℓ

(119899)

119896(119911) for 119899 ge 0 (45)

Here ℓ(119899)119896119899




I119865

120574= I

(119877)

1198991otimesI

(120579)

1198992 (46)

whereI(119877)

1198991andI(120579)



(I119865

120574V) (119877 120579)

=

1198991

sum

1198961=0

1198992

sum

1198962=0V (119911(1198991)

1198961 119911

(1198992)1198962

) ℓ(1198991)1198961

(119877) ℓ(1198992)1198962

(120579)

(47)


E1 [V] = int

Ω1

V (119877 (1205961) 120579 (1205961)) 119889P (1205961)

= int

[minus11]2V (119877 120579) 2minus2119889119877119889120579

(48)


E1 [I119865

120574V] =

1198991

sum

1198961=0

1198992

sum

1198962=0V (119911(1198991)

1198961 119911

(1198992)1198962

)119908(1198991)1198961

119908(1198992)1198962

(49)








(120581(119896) u(119896))


120572(y) | 120572 isin A sub N119872



(A (y) minus 120581M) k (y) = Mu (y) (50)


A = A(0)+

119872

sum

119898=1A(119898)

119910119898

(51)


uA (y) = sum

120573isinA

u120573Λ

120573(y)

kA (y) = sum

120573isinA

k120573Λ

120573(y)

(52)


defined as


] 120572 isin A (53)


Ψ120578 = 120577 (54)


120572] = Mu

120572 [120578

120573] = k

120573and

[Ψ120572120573] = (A(0)

minus 120581M) 120575120572120573+

119872

sum

119898=1A(119898)

119888119898120572120573

(55)


w (y) =k (y)1003817100381710038171003817k (y)100381710038171003817

1003817

(56)



E2 [1198992AΛ 120572

] = E2 [1003817100381710038171003817kA1003817100381710038171003817

2Λ

120572] 120572 isin A (57)


and reduces to

sum

120573isinA

sum

120574isinA

119899120573119899120574119888120572120573120574

= sum

120573isinA

sum

120574isinA

k119879120573Mk

120574119888120572120573120574

120572 isin A (58)





given explicitly by

J120575120598= 2119899

120598sum

120574isinA

119888120575120598120574 (59)


E2 [wA119899AΛ 120572] = E2 [kAΛ 120572

] 120572 isin A (60)


Φ120579 = 120578 (61)

where [120578120572] = k

120572 [120579

120573] = w

120573 and

[Φ120572120573]119894119895= 120575

119894119895sum

120574isinA

119899120574119888120572120573120574

(62)


119903 (y) = u (y)119879A (y)u (y) (63)


119903120572

= sum

120573isinA

sum

120574isinA

(u119879120573A(0)u

120574+ sum

120575isinA

119872

sum

119898=1u119879120573A(119898)u

120575119888119898120574120575

)119888120572120573120574

(64)





120572120572isinA



(1) With u = u(119896minus1)120572


k120572120572isinA



(3) Set 120581(119896) = 119903120572(u(119896))


120572isinA u(119896)120572



(119896) u(119896))


VA (119877 120579 y) = sum

120572isinA

V120572 (119877 120579) Λ 120572

(y) (65)


E2 [VA (119877 120579 sdot)] = V0 (119877 120579)

E2 [(VA (119877 120579 sdot))2] = sum

120572isinA

(V120572 (119877 120579))

2

(66)


E [V] = E1 [E2 [V]]

E [V2] = E1 [E2 [V2]]

(67)













0

1

2

minus1

minus2

0 1 2minus1minus2

(a) Contour plot 1199061 (119877 = 45)

0 1 2

0

1

2

minus1

minus2

minus1minus2

(b) Contour plot 1199062 (119877 = 45)

0

1

2

minus1

minus2

0 1 2minus1minus2

(c) Contour plot 1199061 (119877 = 45 120579 = 12058712)

0

1

2

minus1

minus2

0 1 2minus1minus2

(d) Contour plot 1199062 (119877 = 45 120579 = 12058712)










12058712

12058724

012 35

120579

R


12058712

0

12

35

120579

R

120582



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17Grid

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Rela

tive e

rror

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

(a) Expected value

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Grid

Rela

tive e

rror

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance




119872119871=

120572 isin N119872

0 | sum119872









1 2 5 10 20Re

lativ

e err

or

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

(a) Expected value

1 2 5 10 20

1 2 5 10 20


Rela

tive e

rror

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance










120572120572isinA119872119871






7 Conclusions




Ener

gy1 2 5 10 20

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8


1 2 5 10 20

1 2 5 10 20


Ener

gy

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7







References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119864 (x) = int

Ω2

119864 (1205962 x) 119889P (1205962) (19)


119862119864(x1 x2) = int

Ω2

(119864 (1205962 x1) minus 119864 (x1))

sdot (119864 (1205962 x2) minus 119864 (x2)) 119889P (1205962)

(20)


(C119864V) (x1) = int

119863

C119864(x1 x2) V (x2) 119889x2 (21)


2(119863) rarr 119871



119898infin



119898infin


119898infin



119864 (1205962 x) = 119864 (x) +infin

sum

119898=1radic120582

119898120601119898 (

x) 119910119898 (1205962) (22)





119898infin





E2 [V] = int

Ω2

V (y (1205962)) 119889P (1205962)

= int

[minus11]119872V (y) 2minus119872119889y

(23)


Λ120572(y) =

119872

prod

119898=1119871120572119898(119910

119898) (24)



2120572] = 1 for all 120572 isin N119872

0 The system Λ

120572(y) | 120572 isin N119872


V (y) = sum

120572isinN1198720

V120572Λ

120572(y) (25)





VA (y) = sum

120572isinA

V120572Λ

120572(y) (26)


119872119871= 120572 isin N119872

0 | sum119872

119898=1 120572119898radic120582119898 le 119871where 120582

119898infin



E2 [Λ 120572] = 120575

1205720

E2 [Λ 120572Λ

120573] = 120575

120572120573

(27)



119888119898120573120574

= E2 [119910119898Λ 120572Λ

120573] 119898 isin N 120572 120573 isin N

119872

0

119888120572120573120574

= E2 [Λ 120572Λ

120573Λ

120574] 120572 120573 120574 isin N

119872

0 (28)








(119899 + 1) 119875119899+1 (119909) minus (2119899 + 1) 119909119875119899 (119909) + 119899119875119899minus1 (119909) = 0 (29)


recursion

(1minus1199092) 1198751015840119899(119909) = minus 119899119909119875

119899 (119909) + 119899119875119899minus1 (119909) (30)


120595119899 (120585) = radic

2119899 minus 12

int

120585

minus1119875119899minus1 (119905) 119889119905 119899 = 2 3 (31)


120595119899 (120585) =


(119875119899 (120585) minus 119875119899minus2 (120585))

119899 = 2 3 (32)


int

1

minus1

119889120595119894 (120585)

119889120585

119889120595119895 (120585)

119889120585

119889120585 = 120575119894119895 119894 119895 ge 2 (33)



1198731 (120585 120578) =14(1minus 120585) (1minus 120578)

1198732 (120585 120578) =14(1+ 120585) (1minus 120578)

1198733 (120585 120578) =14(1+ 120585) (1+ 120578)

1198734 (120585 120578) =14(1minus 120585) (1+ 120578)

(34)


119873(1)119894(120585 120578) =

12(1minus 120578)120595

119894 (120585)

119873(2)119894(120585 120578) =

12(1+ 120585) 120595119894 (120578)

119873(3)119894(120585 120578) =

12(1+ 120578)120595

119894(120578)

119873(4)119894(120585 120578) =

12(1minus 120585) 120595119894 (120585)

(35)


1198730119894119895(120585 120578) = 120595

119894 (120585) 120595119895

(120578)

119894 = 2 119901 119895 = 2 119901(36)



119899(minus119909) =





119909 = 119909119894 (119905)

119910 = 119910119894 (119905)


(37)



119894 119884



119909 =

12(1minus 120578) 1199091 (120585) +

12(1+ 120585) 1199092 (120578)

+

12(1+ 120578) 1199093 (120585) +

12(1minus 120585) 1199094 (120578)

minus


14(1+ 120585) (1+ 120578)1198832

minus

14(1+ 120585) (1+ 120578)1198833 minus

14(1minus 120585) (1+ 120578)1198834

(38)





119864119872(1205962 x) = 119864 (x) +

119872

sum

119898=1radic120582

119898120601119898 (

x) 119910119898 (1205962) (39)


D (120582 (1205961 1205962 x) 120583 (1205961 1205962 x))

= D0 (1205961 x) +119872

sum

119898=1D119898(1205961 x) 119910119898 (1205962)

(40)

We assume that 119910119898119872




119873


119863times 119881

119863 Here we




int

119863

120598V (u)119879D0120598V (120593119896) +

119872

sum

119898=1119910119898120598V (u)

119879D119898120598V (120593119896) 119889x

= 120581int

119863

u sdot120593119896119889x 119896 = 1 119873

(41)


A(119898)

119894119895= int

119863

120598V (120593119894)119879D

119898120598V (120593119895) 119889x (42)


M119894119895= int

119863

120593119894sdot120593

119895119889x (43)



rarr

R119873 such that

(A(0)+

119872

sum

119898=1A(119898)

119910119898) u (y) = 120581 (y)Mu (y) (44)


Denote by 119911(119899)119896119899


119896119899


119896119899


119899with



(I119899V) (119911) =

119899

sum

119896=0V (119911(119899)

119896) ℓ

(119899)

119896(119911) for 119899 ge 0 (45)

Here ℓ(119899)119896119899




I119865

120574= I

(119877)

1198991otimesI

(120579)

1198992 (46)

whereI(119877)

1198991andI(120579)



(I119865

120574V) (119877 120579)

=

1198991

sum

1198961=0

1198992

sum

1198962=0V (119911(1198991)

1198961 119911

(1198992)1198962

) ℓ(1198991)1198961

(119877) ℓ(1198992)1198962

(120579)

(47)


E1 [V] = int

Ω1

V (119877 (1205961) 120579 (1205961)) 119889P (1205961)

= int

[minus11]2V (119877 120579) 2minus2119889119877119889120579

(48)


E1 [I119865

120574V] =

1198991

sum

1198961=0

1198992

sum

1198962=0V (119911(1198991)

1198961 119911

(1198992)1198962

)119908(1198991)1198961

119908(1198992)1198962

(49)








(120581(119896) u(119896))


120572(y) | 120572 isin A sub N119872



(A (y) minus 120581M) k (y) = Mu (y) (50)


A = A(0)+

119872

sum

119898=1A(119898)

119910119898

(51)


uA (y) = sum

120573isinA

u120573Λ

120573(y)

kA (y) = sum

120573isinA

k120573Λ

120573(y)

(52)


defined as


] 120572 isin A (53)


Ψ120578 = 120577 (54)


120572] = Mu

120572 [120578

120573] = k

120573and

[Ψ120572120573] = (A(0)

minus 120581M) 120575120572120573+

119872

sum

119898=1A(119898)

119888119898120572120573

(55)


w (y) =k (y)1003817100381710038171003817k (y)100381710038171003817

1003817

(56)



E2 [1198992AΛ 120572

] = E2 [1003817100381710038171003817kA1003817100381710038171003817

2Λ

120572] 120572 isin A (57)


and reduces to

sum

120573isinA

sum

120574isinA

119899120573119899120574119888120572120573120574

= sum

120573isinA

sum

120574isinA

k119879120573Mk

120574119888120572120573120574

120572 isin A (58)





given explicitly by

J120575120598= 2119899

120598sum

120574isinA

119888120575120598120574 (59)


E2 [wA119899AΛ 120572] = E2 [kAΛ 120572

] 120572 isin A (60)


Φ120579 = 120578 (61)

where [120578120572] = k

120572 [120579

120573] = w

120573 and

[Φ120572120573]119894119895= 120575

119894119895sum

120574isinA

119899120574119888120572120573120574

(62)


119903 (y) = u (y)119879A (y)u (y) (63)


119903120572

= sum

120573isinA

sum

120574isinA

(u119879120573A(0)u

120574+ sum

120575isinA

119872

sum

119898=1u119879120573A(119898)u

120575119888119898120574120575

)119888120572120573120574

(64)





120572120572isinA



(1) With u = u(119896minus1)120572


k120572120572isinA



(3) Set 120581(119896) = 119903120572(u(119896))


120572isinA u(119896)120572



(119896) u(119896))


VA (119877 120579 y) = sum

120572isinA

V120572 (119877 120579) Λ 120572

(y) (65)


E2 [VA (119877 120579 sdot)] = V0 (119877 120579)

E2 [(VA (119877 120579 sdot))2] = sum

120572isinA

(V120572 (119877 120579))

2

(66)


E [V] = E1 [E2 [V]]

E [V2] = E1 [E2 [V2]]

(67)













0

1

2

minus1

minus2

0 1 2minus1minus2

(a) Contour plot 1199061 (119877 = 45)

0 1 2

0

1

2

minus1

minus2

minus1minus2

(b) Contour plot 1199062 (119877 = 45)

0

1

2

minus1

minus2

0 1 2minus1minus2

(c) Contour plot 1199061 (119877 = 45 120579 = 12058712)

0

1

2

minus1

minus2

0 1 2minus1minus2

(d) Contour plot 1199062 (119877 = 45 120579 = 12058712)










12058712

12058724

012 35

120579

R


12058712

0

12

35

120579

R

120582



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17Grid

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Rela

tive e

rror

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

(a) Expected value

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Grid

Rela

tive e

rror

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance




119872119871=

120572 isin N119872

0 | sum119872









1 2 5 10 20Re

lativ

e err

or

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

(a) Expected value

1 2 5 10 20

1 2 5 10 20


Rela

tive e

rror

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance










120572120572isinA119872119871






7 Conclusions




Ener

gy1 2 5 10 20

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8


1 2 5 10 20

1 2 5 10 20


Ener

gy

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7







References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















(119899 + 1) 119875119899+1 (119909) minus (2119899 + 1) 119909119875119899 (119909) + 119899119875119899minus1 (119909) = 0 (29)


recursion

(1minus1199092) 1198751015840119899(119909) = minus 119899119909119875

119899 (119909) + 119899119875119899minus1 (119909) (30)


120595119899 (120585) = radic

2119899 minus 12

int

120585

minus1119875119899minus1 (119905) 119889119905 119899 = 2 3 (31)


120595119899 (120585) =


(119875119899 (120585) minus 119875119899minus2 (120585))

119899 = 2 3 (32)


int

1

minus1

119889120595119894 (120585)

119889120585

119889120595119895 (120585)

119889120585

119889120585 = 120575119894119895 119894 119895 ge 2 (33)



1198731 (120585 120578) =14(1minus 120585) (1minus 120578)

1198732 (120585 120578) =14(1+ 120585) (1minus 120578)

1198733 (120585 120578) =14(1+ 120585) (1+ 120578)

1198734 (120585 120578) =14(1minus 120585) (1+ 120578)

(34)


119873(1)119894(120585 120578) =

12(1minus 120578)120595

119894 (120585)

119873(2)119894(120585 120578) =

12(1+ 120585) 120595119894 (120578)

119873(3)119894(120585 120578) =

12(1+ 120578)120595

119894(120578)

119873(4)119894(120585 120578) =

12(1minus 120585) 120595119894 (120585)

(35)


1198730119894119895(120585 120578) = 120595

119894 (120585) 120595119895

(120578)

119894 = 2 119901 119895 = 2 119901(36)



119899(minus119909) =





119909 = 119909119894 (119905)

119910 = 119910119894 (119905)


(37)



119894 119884



119909 =

12(1minus 120578) 1199091 (120585) +

12(1+ 120585) 1199092 (120578)

+

12(1+ 120578) 1199093 (120585) +

12(1minus 120585) 1199094 (120578)

minus


14(1+ 120585) (1+ 120578)1198832

minus

14(1+ 120585) (1+ 120578)1198833 minus

14(1minus 120585) (1+ 120578)1198834

(38)





119864119872(1205962 x) = 119864 (x) +

119872

sum

119898=1radic120582

119898120601119898 (

x) 119910119898 (1205962) (39)


D (120582 (1205961 1205962 x) 120583 (1205961 1205962 x))

= D0 (1205961 x) +119872

sum

119898=1D119898(1205961 x) 119910119898 (1205962)

(40)

We assume that 119910119898119872




119873


119863times 119881

119863 Here we




int

119863

120598V (u)119879D0120598V (120593119896) +

119872

sum

119898=1119910119898120598V (u)

119879D119898120598V (120593119896) 119889x

= 120581int

119863

u sdot120593119896119889x 119896 = 1 119873

(41)


A(119898)

119894119895= int

119863

120598V (120593119894)119879D

119898120598V (120593119895) 119889x (42)


M119894119895= int

119863

120593119894sdot120593

119895119889x (43)



rarr

R119873 such that

(A(0)+

119872

sum

119898=1A(119898)

119910119898) u (y) = 120581 (y)Mu (y) (44)


Denote by 119911(119899)119896119899


119896119899


119896119899


119899with



(I119899V) (119911) =

119899

sum

119896=0V (119911(119899)

119896) ℓ

(119899)

119896(119911) for 119899 ge 0 (45)

Here ℓ(119899)119896119899




I119865

120574= I

(119877)

1198991otimesI

(120579)

1198992 (46)

whereI(119877)

1198991andI(120579)



(I119865

120574V) (119877 120579)

=

1198991

sum

1198961=0

1198992

sum

1198962=0V (119911(1198991)

1198961 119911

(1198992)1198962

) ℓ(1198991)1198961

(119877) ℓ(1198992)1198962

(120579)

(47)


E1 [V] = int

Ω1

V (119877 (1205961) 120579 (1205961)) 119889P (1205961)

= int

[minus11]2V (119877 120579) 2minus2119889119877119889120579

(48)


E1 [I119865

120574V] =

1198991

sum

1198961=0

1198992

sum

1198962=0V (119911(1198991)

1198961 119911

(1198992)1198962

)119908(1198991)1198961

119908(1198992)1198962

(49)








(120581(119896) u(119896))


120572(y) | 120572 isin A sub N119872



(A (y) minus 120581M) k (y) = Mu (y) (50)


A = A(0)+

119872

sum

119898=1A(119898)

119910119898

(51)


uA (y) = sum

120573isinA

u120573Λ

120573(y)

kA (y) = sum

120573isinA

k120573Λ

120573(y)

(52)


defined as


] 120572 isin A (53)


Ψ120578 = 120577 (54)


120572] = Mu

120572 [120578

120573] = k

120573and

[Ψ120572120573] = (A(0)

minus 120581M) 120575120572120573+

119872

sum

119898=1A(119898)

119888119898120572120573

(55)


w (y) =k (y)1003817100381710038171003817k (y)100381710038171003817

1003817

(56)



E2 [1198992AΛ 120572

] = E2 [1003817100381710038171003817kA1003817100381710038171003817

2Λ

120572] 120572 isin A (57)


and reduces to

sum

120573isinA

sum

120574isinA

119899120573119899120574119888120572120573120574

= sum

120573isinA

sum

120574isinA

k119879120573Mk

120574119888120572120573120574

120572 isin A (58)





given explicitly by

J120575120598= 2119899

120598sum

120574isinA

119888120575120598120574 (59)


E2 [wA119899AΛ 120572] = E2 [kAΛ 120572

] 120572 isin A (60)


Φ120579 = 120578 (61)

where [120578120572] = k

120572 [120579

120573] = w

120573 and

[Φ120572120573]119894119895= 120575

119894119895sum

120574isinA

119899120574119888120572120573120574

(62)


119903 (y) = u (y)119879A (y)u (y) (63)


119903120572

= sum

120573isinA

sum

120574isinA

(u119879120573A(0)u

120574+ sum

120575isinA

119872

sum

119898=1u119879120573A(119898)u

120575119888119898120574120575

)119888120572120573120574

(64)





120572120572isinA



(1) With u = u(119896minus1)120572


k120572120572isinA



(3) Set 120581(119896) = 119903120572(u(119896))


120572isinA u(119896)120572



(119896) u(119896))


VA (119877 120579 y) = sum

120572isinA

V120572 (119877 120579) Λ 120572

(y) (65)


E2 [VA (119877 120579 sdot)] = V0 (119877 120579)

E2 [(VA (119877 120579 sdot))2] = sum

120572isinA

(V120572 (119877 120579))

2

(66)


E [V] = E1 [E2 [V]]

E [V2] = E1 [E2 [V2]]

(67)













0

1

2

minus1

minus2

0 1 2minus1minus2

(a) Contour plot 1199061 (119877 = 45)

0 1 2

0

1

2

minus1

minus2

minus1minus2

(b) Contour plot 1199062 (119877 = 45)

0

1

2

minus1

minus2

0 1 2minus1minus2

(c) Contour plot 1199061 (119877 = 45 120579 = 12058712)

0

1

2

minus1

minus2

0 1 2minus1minus2

(d) Contour plot 1199062 (119877 = 45 120579 = 12058712)










12058712

12058724

012 35

120579

R


12058712

0

12

35

120579

R

120582



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17Grid

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Rela

tive e

rror

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

(a) Expected value

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Grid

Rela

tive e

rror

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance




119872119871=

120572 isin N119872

0 | sum119872









1 2 5 10 20Re

lativ

e err

or

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

(a) Expected value

1 2 5 10 20

1 2 5 10 20


Rela

tive e

rror

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance










120572120572isinA119872119871






7 Conclusions




Ener

gy1 2 5 10 20

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8


1 2 5 10 20

1 2 5 10 20


Ener

gy

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7







References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119894 119884



119909 =

12(1minus 120578) 1199091 (120585) +

12(1+ 120585) 1199092 (120578)

+

12(1+ 120578) 1199093 (120585) +

12(1minus 120585) 1199094 (120578)

minus


14(1+ 120585) (1+ 120578)1198832

minus

14(1+ 120585) (1+ 120578)1198833 minus

14(1minus 120585) (1+ 120578)1198834

(38)





119864119872(1205962 x) = 119864 (x) +

119872

sum

119898=1radic120582

119898120601119898 (

x) 119910119898 (1205962) (39)


D (120582 (1205961 1205962 x) 120583 (1205961 1205962 x))

= D0 (1205961 x) +119872

sum

119898=1D119898(1205961 x) 119910119898 (1205962)

(40)

We assume that 119910119898119872




119873


119863times 119881

119863 Here we




int

119863

120598V (u)119879D0120598V (120593119896) +

119872

sum

119898=1119910119898120598V (u)

119879D119898120598V (120593119896) 119889x

= 120581int

119863

u sdot120593119896119889x 119896 = 1 119873

(41)


A(119898)

119894119895= int

119863

120598V (120593119894)119879D

119898120598V (120593119895) 119889x (42)


M119894119895= int

119863

120593119894sdot120593

119895119889x (43)



rarr

R119873 such that

(A(0)+

119872

sum

119898=1A(119898)

119910119898) u (y) = 120581 (y)Mu (y) (44)


Denote by 119911(119899)119896119899


119896119899


119896119899


119899with



(I119899V) (119911) =

119899

sum

119896=0V (119911(119899)

119896) ℓ

(119899)

119896(119911) for 119899 ge 0 (45)

Here ℓ(119899)119896119899




I119865

120574= I

(119877)

1198991otimesI

(120579)

1198992 (46)

whereI(119877)

1198991andI(120579)



(I119865

120574V) (119877 120579)

=

1198991

sum

1198961=0

1198992

sum

1198962=0V (119911(1198991)

1198961 119911

(1198992)1198962

) ℓ(1198991)1198961

(119877) ℓ(1198992)1198962

(120579)

(47)


E1 [V] = int

Ω1

V (119877 (1205961) 120579 (1205961)) 119889P (1205961)

= int

[minus11]2V (119877 120579) 2minus2119889119877119889120579

(48)


E1 [I119865

120574V] =

1198991

sum

1198961=0

1198992

sum

1198962=0V (119911(1198991)

1198961 119911

(1198992)1198962

)119908(1198991)1198961

119908(1198992)1198962

(49)








(120581(119896) u(119896))


120572(y) | 120572 isin A sub N119872



(A (y) minus 120581M) k (y) = Mu (y) (50)


A = A(0)+

119872

sum

119898=1A(119898)

119910119898

(51)


uA (y) = sum

120573isinA

u120573Λ

120573(y)

kA (y) = sum

120573isinA

k120573Λ

120573(y)

(52)


defined as


] 120572 isin A (53)


Ψ120578 = 120577 (54)


120572] = Mu

120572 [120578

120573] = k

120573and

[Ψ120572120573] = (A(0)

minus 120581M) 120575120572120573+

119872

sum

119898=1A(119898)

119888119898120572120573

(55)


w (y) =k (y)1003817100381710038171003817k (y)100381710038171003817

1003817

(56)



E2 [1198992AΛ 120572

] = E2 [1003817100381710038171003817kA1003817100381710038171003817

2Λ

120572] 120572 isin A (57)


and reduces to

sum

120573isinA

sum

120574isinA

119899120573119899120574119888120572120573120574

= sum

120573isinA

sum

120574isinA

k119879120573Mk

120574119888120572120573120574

120572 isin A (58)





given explicitly by

J120575120598= 2119899

120598sum

120574isinA

119888120575120598120574 (59)


E2 [wA119899AΛ 120572] = E2 [kAΛ 120572

] 120572 isin A (60)


Φ120579 = 120578 (61)

where [120578120572] = k

120572 [120579

120573] = w

120573 and

[Φ120572120573]119894119895= 120575

119894119895sum

120574isinA

119899120574119888120572120573120574

(62)


119903 (y) = u (y)119879A (y)u (y) (63)


119903120572

= sum

120573isinA

sum

120574isinA

(u119879120573A(0)u

120574+ sum

120575isinA

119872

sum

119898=1u119879120573A(119898)u

120575119888119898120574120575

)119888120572120573120574

(64)





120572120572isinA



(1) With u = u(119896minus1)120572


k120572120572isinA



(3) Set 120581(119896) = 119903120572(u(119896))


120572isinA u(119896)120572



(119896) u(119896))


VA (119877 120579 y) = sum

120572isinA

V120572 (119877 120579) Λ 120572

(y) (65)


E2 [VA (119877 120579 sdot)] = V0 (119877 120579)

E2 [(VA (119877 120579 sdot))2] = sum

120572isinA

(V120572 (119877 120579))

2

(66)


E [V] = E1 [E2 [V]]

E [V2] = E1 [E2 [V2]]

(67)













0

1

2

minus1

minus2

0 1 2minus1minus2

(a) Contour plot 1199061 (119877 = 45)

0 1 2

0

1

2

minus1

minus2

minus1minus2

(b) Contour plot 1199062 (119877 = 45)

0

1

2

minus1

minus2

0 1 2minus1minus2

(c) Contour plot 1199061 (119877 = 45 120579 = 12058712)

0

1

2

minus1

minus2

0 1 2minus1minus2

(d) Contour plot 1199062 (119877 = 45 120579 = 12058712)










12058712

12058724

012 35

120579

R


12058712

0

12

35

120579

R

120582



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17Grid

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Rela

tive e

rror

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

(a) Expected value

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Grid

Rela

tive e

rror

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance




119872119871=

120572 isin N119872

0 | sum119872









1 2 5 10 20Re

lativ

e err

or

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

(a) Expected value

1 2 5 10 20

1 2 5 10 20


Rela

tive e

rror

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance










120572120572isinA119872119871






7 Conclusions




Ener

gy1 2 5 10 20

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8


1 2 5 10 20

1 2 5 10 20


Ener

gy

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7







References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











I119865

120574= I

(119877)

1198991otimesI

(120579)

1198992 (46)

whereI(119877)

1198991andI(120579)



(I119865

120574V) (119877 120579)

=

1198991

sum

1198961=0

1198992

sum

1198962=0V (119911(1198991)

1198961 119911

(1198992)1198962

) ℓ(1198991)1198961

(119877) ℓ(1198992)1198962

(120579)

(47)


E1 [V] = int

Ω1

V (119877 (1205961) 120579 (1205961)) 119889P (1205961)

= int

[minus11]2V (119877 120579) 2minus2119889119877119889120579

(48)


E1 [I119865

120574V] =

1198991

sum

1198961=0

1198992

sum

1198962=0V (119911(1198991)

1198961 119911

(1198992)1198962

)119908(1198991)1198961

119908(1198992)1198962

(49)








(120581(119896) u(119896))


120572(y) | 120572 isin A sub N119872



(A (y) minus 120581M) k (y) = Mu (y) (50)


A = A(0)+

119872

sum

119898=1A(119898)

119910119898

(51)


uA (y) = sum

120573isinA

u120573Λ

120573(y)

kA (y) = sum

120573isinA

k120573Λ

120573(y)

(52)


defined as


] 120572 isin A (53)


Ψ120578 = 120577 (54)


120572] = Mu

120572 [120578

120573] = k

120573and

[Ψ120572120573] = (A(0)

minus 120581M) 120575120572120573+

119872

sum

119898=1A(119898)

119888119898120572120573

(55)


w (y) =k (y)1003817100381710038171003817k (y)100381710038171003817

1003817

(56)



E2 [1198992AΛ 120572

] = E2 [1003817100381710038171003817kA1003817100381710038171003817

2Λ

120572] 120572 isin A (57)


and reduces to

sum

120573isinA

sum

120574isinA

119899120573119899120574119888120572120573120574

= sum

120573isinA

sum

120574isinA

k119879120573Mk

120574119888120572120573120574

120572 isin A (58)





given explicitly by

J120575120598= 2119899

120598sum

120574isinA

119888120575120598120574 (59)


E2 [wA119899AΛ 120572] = E2 [kAΛ 120572

] 120572 isin A (60)


Φ120579 = 120578 (61)

where [120578120572] = k

120572 [120579

120573] = w

120573 and

[Φ120572120573]119894119895= 120575

119894119895sum

120574isinA

119899120574119888120572120573120574

(62)


119903 (y) = u (y)119879A (y)u (y) (63)


119903120572

= sum

120573isinA

sum

120574isinA

(u119879120573A(0)u

120574+ sum

120575isinA

119872

sum

119898=1u119879120573A(119898)u

120575119888119898120574120575

)119888120572120573120574

(64)





120572120572isinA



(1) With u = u(119896minus1)120572


k120572120572isinA



(3) Set 120581(119896) = 119903120572(u(119896))


120572isinA u(119896)120572



(119896) u(119896))


VA (119877 120579 y) = sum

120572isinA

V120572 (119877 120579) Λ 120572

(y) (65)


E2 [VA (119877 120579 sdot)] = V0 (119877 120579)

E2 [(VA (119877 120579 sdot))2] = sum

120572isinA

(V120572 (119877 120579))

2

(66)


E [V] = E1 [E2 [V]]

E [V2] = E1 [E2 [V2]]

(67)













0

1

2

minus1

minus2

0 1 2minus1minus2

(a) Contour plot 1199061 (119877 = 45)

0 1 2

0

1

2

minus1

minus2

minus1minus2

(b) Contour plot 1199062 (119877 = 45)

0

1

2

minus1

minus2

0 1 2minus1minus2

(c) Contour plot 1199061 (119877 = 45 120579 = 12058712)

0

1

2

minus1

minus2

0 1 2minus1minus2

(d) Contour plot 1199062 (119877 = 45 120579 = 12058712)










12058712

12058724

012 35

120579

R


12058712

0

12

35

120579

R

120582



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17Grid

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Rela

tive e

rror

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

(a) Expected value

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Grid

Rela

tive e

rror

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance




119872119871=

120572 isin N119872

0 | sum119872









1 2 5 10 20Re

lativ

e err

or

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

(a) Expected value

1 2 5 10 20

1 2 5 10 20


Rela

tive e

rror

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance










120572120572isinA119872119871






7 Conclusions




Ener

gy1 2 5 10 20

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8


1 2 5 10 20

1 2 5 10 20


Ener

gy

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7







References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










and reduces to

sum

120573isinA

sum

120574isinA

119899120573119899120574119888120572120573120574

= sum

120573isinA

sum

120574isinA

k119879120573Mk

120574119888120572120573120574

120572 isin A (58)





given explicitly by

J120575120598= 2119899

120598sum

120574isinA

119888120575120598120574 (59)


E2 [wA119899AΛ 120572] = E2 [kAΛ 120572

] 120572 isin A (60)


Φ120579 = 120578 (61)

where [120578120572] = k

120572 [120579

120573] = w

120573 and

[Φ120572120573]119894119895= 120575

119894119895sum

120574isinA

119899120574119888120572120573120574

(62)


119903 (y) = u (y)119879A (y)u (y) (63)


119903120572

= sum

120573isinA

sum

120574isinA

(u119879120573A(0)u

120574+ sum

120575isinA

119872

sum

119898=1u119879120573A(119898)u

120575119888119898120574120575

)119888120572120573120574

(64)





120572120572isinA



(1) With u = u(119896minus1)120572


k120572120572isinA



(3) Set 120581(119896) = 119903120572(u(119896))


120572isinA u(119896)120572



(119896) u(119896))


VA (119877 120579 y) = sum

120572isinA

V120572 (119877 120579) Λ 120572

(y) (65)


E2 [VA (119877 120579 sdot)] = V0 (119877 120579)

E2 [(VA (119877 120579 sdot))2] = sum

120572isinA

(V120572 (119877 120579))

2

(66)


E [V] = E1 [E2 [V]]

E [V2] = E1 [E2 [V2]]

(67)













0

1

2

minus1

minus2

0 1 2minus1minus2

(a) Contour plot 1199061 (119877 = 45)

0 1 2

0

1

2

minus1

minus2

minus1minus2

(b) Contour plot 1199062 (119877 = 45)

0

1

2

minus1

minus2

0 1 2minus1minus2

(c) Contour plot 1199061 (119877 = 45 120579 = 12058712)

0

1

2

minus1

minus2

0 1 2minus1minus2

(d) Contour plot 1199062 (119877 = 45 120579 = 12058712)










12058712

12058724

012 35

120579

R


12058712

0

12

35

120579

R

120582



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17Grid

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Rela

tive e

rror

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

(a) Expected value

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Grid

Rela

tive e

rror

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance




119872119871=

120572 isin N119872

0 | sum119872









1 2 5 10 20Re

lativ

e err

or

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

(a) Expected value

1 2 5 10 20

1 2 5 10 20


Rela

tive e

rror

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance










120572120572isinA119872119871






7 Conclusions




Ener

gy1 2 5 10 20

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8


1 2 5 10 20

1 2 5 10 20


Ener

gy

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7







References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

1

2

minus1

minus2

0 1 2minus1minus2

(a) Contour plot 1199061 (119877 = 45)

0 1 2

0

1

2

minus1

minus2

minus1minus2

(b) Contour plot 1199062 (119877 = 45)

0

1

2

minus1

minus2

0 1 2minus1minus2

(c) Contour plot 1199061 (119877 = 45 120579 = 12058712)

0

1

2

minus1

minus2

0 1 2minus1minus2

(d) Contour plot 1199062 (119877 = 45 120579 = 12058712)










12058712

12058724

012 35

120579

R


12058712

0

12

35

120579

R

120582



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17Grid

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Rela

tive e

rror

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

(a) Expected value

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Grid

Rela

tive e

rror

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance




119872119871=

120572 isin N119872

0 | sum119872









1 2 5 10 20Re

lativ

e err

or

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

(a) Expected value

1 2 5 10 20

1 2 5 10 20


Rela

tive e

rror

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance










120572120572isinA119872119871






7 Conclusions




Ener

gy1 2 5 10 20

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8


1 2 5 10 20

1 2 5 10 20


Ener

gy

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7







References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










12058712

12058724

012 35

120579

R


12058712

0

12

35

120579

R

120582



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17Grid

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Rela

tive e

rror

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

(a) Expected value

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Grid

Rela

tive e

rror

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance




119872119871=

120572 isin N119872

0 | sum119872









1 2 5 10 20Re

lativ

e err

or

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

(a) Expected value

1 2 5 10 20

1 2 5 10 20


Rela

tive e

rror

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance










120572120572isinA119872119871






7 Conclusions




Ener

gy1 2 5 10 20

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8


1 2 5 10 20

1 2 5 10 20


Ener

gy

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7







References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1 2 5 10 20Re

lativ

e err

or

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

(a) Expected value

1 2 5 10 20

1 2 5 10 20


Rela

tive e

rror

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(b) Variance










120572120572isinA119872119871






7 Conclusions




Ener

gy1 2 5 10 20

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8


1 2 5 10 20

1 2 5 10 20


Ener

gy

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7







References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Ener

gy1 2 5 10 20

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8


1 2 5 10 20

1 2 5 10 20


Ener

gy

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7







References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article hybrid stochastic finite element method for...

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