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S W A N S E A U N I V E R S I T Y

College of Engineering

SFEM for structural dynamicsStochastic transient dynamics: A reduced spectral functionapproach

Abhishek KunduPhD student,Department of Aerospace Engineering.

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S W A N S E A U N I V E R S I T Y C O L L E G E O F E N G I N E E R I N G

Outline of the talk1 Introduction

Description of UncertaintyA typical stochastic BVPSPDE for dynamical systems

2 Discretization TechniquesStochastic DiscretizationThe Finite Element ModelDiscretization of the temporal domain

3 Spectral decomposition in vector spaceSpectral MethodMathematical nature of the solutionProjection in a finite dimensional vector-space

4 Numerical illustrationThe Euler-Bernoulli beamComputational Cost

5 Conclusion & Future WorkAbhishek Kundu — SFEM for structural dynamics — 09-10 Jan, 2012

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Sources of Uncertainty

Types of uncertainty in computational analysis can be associated with

• Model form uncertainity: mathematical formulation which areunable to capture the physics of the problem (e.g. lack of highfidelity aerodynamic models, etc.)

• types of input to the mathematical model model (parametric)

The parametric uncertainity can be :

• aleatory : can not be reduced by experiment or measuringdevice, inherent in nature (e.g. lift/drag coefficient, convectivecoefficient, etc)

• epistemic uncertainity : lack of complete knowledge due toinsufficient number of experiments, lack of quality control, etc.(e.g. elastic properties of structural systems)

Our focus is on the parametric uncertainty (aleatory/epistemic) whichis multiplicative in nature.Abhishek Kundu — SFEM for structural dynamics — 09-10 Jan, 2012

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A typical boundary value problem

Consider the following boundary value equation :

− ∑i=1,...,d

∂xi

{

∑j=1,...,d

aij∂xju

}= f on D (1)

u = g, on ∂D 6=Φ

Input set S : domain ‘D’diffusion coeff ‘aij ’load function ‘f ’

Output set G : the quantity of interest Q(u) = u(x0) ∈ R, x0 ∈ D

Eqn. (1) defines a mapping A of the stochastic data set S onto theoutput set G, A : S → G.

Abhishek Kundu — SFEM for structural dynamics — 09-10 Jan, 2012

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Description of Uncertainty

The uncertainty in set S is described with a purely probabilisticapproach .

Other possible approaches:

• Worst scenario approach : Input set S is an ε-ball around a givenfunction ac with norm defined as

Sp ={

a ∈ Lp(D) : ‖a−ac‖Lp(D) ≤ ε}

To find the worst scenario associated with the set relationG = A(S) i.e. determine the uncertainty interval I =

∣∣Q(u),Q(u)∣∣

where Q(u) = supa∈S

Q(u(a)) & Q(u) = infa∈S

Q(u(a))

• Fuzzy sets and possibility theory

• Evidence theory


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Description of a stochastic BVP

A simple stochastic bvp can be desribed as follows:

−∇.(a(θ, ·)∇u(θ, ·)) = f (θ, ·) on D (2)

u(θ, ·) = 0 on ∂D

a, f : Θ×D → R are stochastic functions with continuous, boundedcovariance functions.D : is a convex bounded polygonal domain in R

d and (Θ,F ,P) is acomplete probability space.

We make the following assumptions regarding a :

∃amin,amax ∈ (0,+ inf)

P(θ ∈Θ : a(θ,x) ∈ [amin,amax ],∀x ∈ D) = 1


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Function Space

Let W s,q(D) be a Sobolev space of functions with norm

‖v‖W s,q(D) =

{

∑|α|≤s

∫D|∂αv |q dx

} 1q

,1 ≤ q < inf

With q = 2 and equipped with an inner product (·, ·)W s,2(D), we get theHilbert space Hs(D)≡ W s,2(D).

(v ,w)W s,2(D) ≡ ∑|α|≤s

(∂αv ,∂αw)L2(D)

We denote the subspace of all functions in Hs(D) which vanish on theboundary ∂D as Hs

0(D)


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Solution in Tensor product space

The solution of the stochastic bvp Eqn. (2) exists in a tensor productspace. If H1 and H2 are two Hilbert spaces and u ⊂ H1

⊗H2 then

u =n

∑i=1

viwi where vini=1 ⊂ H1 wi

ni=1 ⊂ H2

and the tensor inner product is given by

(u, u)H1⊗

H2 = ∑i,j

(vi , vj)H1(wi , wj)H2

For Eqn. (2), we define the tensor product Hilbert space asH = L2

p(Θ;H10(D)) with the inner product as

(v ,w)H ≡ E[(∇v ,∇w)L2(D)] ∀v ,w ∈ H

The Bilinear form B : H ×H → R is thus

B(v ,w) = E[(a∇v ,∇w)L2(D)]∀v ,w ∈ H (3)

B is continuous and coercive (following assumptions regarding ‘a’).Abhishek Kundu — SFEM for structural dynamics — 09-10 Jan, 2012

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Variational Statement

The variational statement of the problem can now be stated as :

Find u ∈ H such that

B(u,v) = L(v) ∀ v ∈ H (4)

where L(v)≡ E[(f ,v)L2(D)],∀v ∈ H, is a continuous linear form in H

The existence and uniqueness for the solution of the above variationalproblem is ensured by the Lax-Milgram Lemma .


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Stochastic PDEs for structural dynamics

We consider the stochastic partial differential equation (PDE)pertinent to the structural dynamics problem as

ρ(r,θ)∂2U(r, t,θ)

∂t2+Lα

∂U(r, t,θ)∂t

+LβU(r, t,θ) = p(r, t) (5)

The stochastic operator Lβ can be

• Lβ ≡∂

∂x AE(x ,θ) ∂∂x axial deformation of rods

• Lβ ≡∂2

∂x2 EI(x ,θ) ∂2

∂x2 bending deformation of beams

Lα denotes the stochastic damping, which is mostly proportional innature.Here α,β : Rd ×Θ→ R are stationary square integrable randomfields, which can be viewed as a set of random variables indexed byr ∈ R

d . Based on the physical problem the random field a(r,θ) can beused to model different physical quantities (e.g., AE(x ,θ), EI(x ,θ)).Abhishek Kundu — SFEM for structural dynamics — 09-10 Jan, 2012

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Stochastic Discretization

The probabilitic content of the problem is defined with a finite set ofrandom variables, ξ = (ξ1, . . . ,ξm) : Θ→ R

m.For second order random fields, this consists the K.L. Expansion, suchthat if Tg : L2(D)→ L2(D) is a compact self-adjoint operator, then

Tgv(·) =∫

Dcov [g](x , ·)v(x)dx ∀v ∈ L2(D)

where cov [g] : D×D → R is the continuous covariance function.The eigenvalue problem

Tgbi = λibi , (bi ,bj)L2(D) = δij

provides a set of unique eigenfunctions, from which a fewcorresponding to the largest eigenvalues are chosen. Hence in thereduced probability space (Θ(m),F (m),P(m))

gm(θ,x) = E[g](x)+m

∑i=1

√λibi(x)Γi(θ) ∀m ∈ N+


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Finite Element Model

The variational form of the stochastic pde, Eqns. (3) and (4), is suitablefor the FE approximation, a posteriori error estimation, adaptivity, etc.

Using the well-established techniques of variational formulation of thedisplacement-based deterministic finite-element methods we have theglobal matrices

A(θ) =∫

DBT C(x ,θ)Bdx

where,• B is the matrix relating strains to nodal displacements and• C is a constitutive matrix influenced by the mass, stiffness and/or

damping properties of the system and thus is random in nature.Thus the stochastic system matrices can be written as

A i(θ) = A i0 +m

∑j=1

Γi(θ)A ij ∀m ∈ N+


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Discretized FE system

The discretized FE system can be expressed as

2

∑i=0

A i(θ)∂iu(t,θ)

∂t i= f(t); θ ∈Θ; ∀t ∈ [0,T ]; A i ∈ R

n×n; u, f ∈ Rn,

(6)where coefficient matrices A i(θ) inherits the uncertainty of the randomparameters involved in the governing pde.

The solution vector u(t,θ) : T ×Rm → R

n in can be expressed as :

u(t,ξ(θ)) =p

∑i=0

u i(t)L(ξ(θ)) (7)

where L(ξ(θ)) forms some kind of trial basis in L2P(Θ).

The aim here is to solve for u(t,ξ(θ)) efficiently.


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Time Integration Scheme

We choose to solve the problem every discretized time step in thedomain t ∈ [0,T ]. Hence for a fixed time step ∆T .

2

∑i=0

A i(θ)∂iut+∆t(θ)

∂t i= ft+∆t . (8)

Time integration is performed using the Newmark method based onconstant-average-acceleration scheme, which reduces the equationdown to

A(θ)ut+∆t(θ) = feqvt+∆t(θ) (9)

along with

ut+∆t(θ) = a0[ut+∆t −ut(θ)]−a2ut(θ)−a3ut(θ),and ut+∆t(θ) = ut(θ)+a6ut(θ)+a7ut+∆t(θ)

It can be seen that the values of system response is a linearcombination of the present and previous step values.Abhishek Kundu — SFEM for structural dynamics — 09-10 Jan, 2012

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Time Integration Scheme

The expressions for the modified coefficient matrices and forcingfunctions under Newmark method is given by

A(θ) = a0A1(θ)+a1A2(θ)+ A0(θ)

feqvt+∆t(θ) = ft+∆t + A2[a0ut(θ)+a2ut(θ)+a3ut(θ)]

+ A1[a1ut(θ)+a4ut(θ)+a5ut(θ)] .

The equivalent force at time step t +∆t consists of contributions ofthe system response at previous time steps and thus inherits theparametric stochasticity.

The constants ai , i = 1,2, . . . ,7 are absolutely indepedent of theparametric values of the input set S.


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Spectral Method of Solution

According to the approximate basis building techniques whichexpands the solution vector using some polynomial functions

u(t,θ) = ∑a∈Iq

Hα(θ)uα(t); uα(t) ∈ R, (10)

where Hα are the basis in ϒp ⊂ L2P(Θ).

ϒp is a tensor product space of the discretized stochastic domain,ϒ1 ⊗ϒ2 ⊗ . . .⊗ϒm.uα(t) are the set of unknown coefficients to be evaluated and Iq is asubset of I with cardinal q.


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Polynomial Chaos expansion

• Using the Polynomial Chaos expansion, the solution (a vectorvalued function) can be expressed as

u(t,θ) = u i0h0 +∞

∑i1=1

u i1(t)h1(ξi1(θ))

+∞

∑i1=1

i1

∑i2=1

u i1,i2(t)h2(ξi1(θ),ξi2(θ))

+∞

∑i1=1

i1

∑i2=1

i2

∑i3=1

u i1 i2 i3(t)h3(ξi1(θ),ξi2(θ),ξi3(θ))

+∞

∑i1=1

i1

∑i2=1

i2

∑i3=1

i3

∑i4=1

u i1 i2 i3 i4(t) h4(ξi1(θ),ξi2(θ),ξi3(θ),ξi4(θ))+ . . .

Here u i1(t),...,ip ∈ Rn are deterministic vectors to be evaluated.


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Polynomial Chaos expansion

• After the finite truncation, concisely, the polynomial chaosexpansion can be written as

u(t,θ) =P

∑k=1

Hk(ξ(θ))uk(t) (11)

where Hk(ξ(θ)) are the polynomial chaoses.

• The number of terms P depends on the number of basic randomvariables M and the order of the PC expansion r as

P =r

∑j=0

(M + j −1)!

j!(M −1)!(12)


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Polynomial Chaos expansion: Some Observations

P increases exponentially with M:M 2 3 5 10 20 50 100

2nd order PC 5 9 20 65 230 1325 51503rd order PC 9 19 55 285 1770 23425 176850

• The basis is a function of the pdf of the random variables only.For example, Hermite polynomials for Gaussian pdf, Legender’spolynomials for uniform pdf.

• The physics of the underlying problem (static, dynamic, heatconduction, transients....) cannot be incorporated in the basis.

• For an n-dimensional output vector, the number of terms in theprojection can be more than n (depends on the number ofrandom variables).

• The functional form of the response is a pure polynomial inrandom variables.


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Mathematical nature of the solution

• The elements of the solution vector are not simple polynomials,but ratio of polynomials in ξ(θ).

Remark

If A(θ) ∈ Rn×n is a stochastic matrix (Gaussian) of rank n, then the

elements of u(θ) are the ratio of polynomials of the form

p(n−1)(ξ1(θ),ξ2(θ), . . . ,ξM(θ))p(n)(ξ1(θ),ξ2(θ), . . . ,ξM(θ))

(13)

where p(n)(ξ1(θ),ξ2(θ), . . . ,ξM(θ)) is an n-th order completemultivariate polynomial of variables ξ1(θ),ξ2(θ), . . . ,ξM(θ).


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Suppose we denote

A(θ) =

[A0 +

M

∑i=1

Γi(ξ(θ))A i

]∈ R

n×n (14)

by its mean and perturbation parts, so that

u(t,θ) = A−1(θ)f(t) (15)

From the definition of the matrix inverse we have

A−1 =Adj(A)det(A)

=CT

a

det(A)(16)

where Ca is the matrix of cofactors. The determinant of A contains amaximum of n number of products of Akj and their linearcombinations. Note from Eq. (14) that

Akj(θ) = A0kj +M

∑i=1

Γi(ξ(θ))Aikj (17)


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• Since all the matrices are of full rank, the determinant contains amaximum of n number of products of linear combination ofrandom variables in Eq. (17). On the other hand, each entries ofthe matrix of cofactors, contains a maximum of (n−1) number ofproducts of linear combination of random variables in Eq. (17).From Eqs. (15) and (16) it follows that

u(θ) =CT

a f

det(A)(18)

Therefore, the numerator of each element of the solution vectorcontains linear combinations of the elements of the cofactormatrix, which are complete polynomials of order (n−1).

• The result derived in this theorem is important because thesolution methods proposed for stochastic finite element analysisessentially aim to approximate the ratio of the polynomials givenin Eq. (13).


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Projection in a finite dimensional vector-space

Lets consider a non-singular linear system of algebraic equationAx = b in a Krylov space whose dimension is the degree of theminimal polynomial.

Definition

A minimal polynomial of A is a unique monic polynomial of minimaldegree such that (A) = 0. This can be constructed with the distincteigenvalues(λj ) of A as

(A) =d

∏j=1

(A −λj I)mj and m ≡

d

∑j=1

mj . (19)

This idea can be used to construct the inverse of a non-singular matrixA in terms of the powers of A.


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Hence, the inverse of A can be written as

A−1 =−1

α0

m−1

∑j=0

αj+1A j (20)

αi are evaluated from the minimal polynomial given in Defn. (1).As a result, the solution vector x of the equation Ax = b lies in theKrylov subspace of order m as

Km(A,b) = span{b,Ab ,A2b, . . . ,Am−1b}. (21)

The Krylov subspace dimension determines the accuracy of thecomputed response. It also dictates the computational efficiency ofthe solution technique, i.e.a low degree of the minimal polynomialwould imply a small Krylov solution space.


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The solution at each time step t +∆t can be projected on to a finitenumber of basis spanning a stochastic Krylov space :

Km

(A0 +

M

∑i=1

ξi(θi)A i

)

︸︷︷︸A(θ)

, feqvt+∆t

. (22)

A choice of a finite number of Krylov basis depends on theeigen-spectrum of the coefficient of the system matrix A(θ). Theeigenvalues of A(θ) are distributed over a long interval on the realaxis. Hence the required number of basis functions (m) is high, whichincreases the computational cost substantially.


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To alleviate this problem

• a preconditioned stochastic Krylov space can be used to arrive ata ‘richer stochastic subspace’.This is done using the mean of the coefficient matrix as thepreconditioner.

• This transforms the equations such that the pdf of theeigenvalues of the modified coefficient matrices show a highdegree of overlap.

However,

• as the variability of the random field increases, it is desirable toincorporate ‘some of the randomness’ of the system matricesinto the preconditioner.

• This motivates us to use a different preconditioner for theproblem as is demonstrated in subsequent discussions.


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Theorem

There exists a finite set of functions Lk : (Rm ×Θ)→ (R×Θ) and an

orthonormal basis φk ∈ Rn for k = 1,2, . . . ,n such that the series

u(θ) =n

∑k=1

Lk(ξ(θ))φk (23)

converges to the exact solution of the discretized stochastic finiteelement equation with probability 1.

Outline of proof: In the first step a complete orthonormal basis isgenerated with the eigenvectors of the φk ∈ R

n of the (generalized)eigenvalue problem involving the coefficient matrices.


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We define the matrix of eigenvalues and eigenvectors

λ0 = diag[λ01 ,λ02 , . . . ,λ0n ] ∈ Rn×n;Φ = [φ1,φ2, . . . ,φn] ∈ R

n×n (24)

Eigenvalues are ordered in the ascending order:λ01 < λ02 < .. . < λ0n . Since Φ is an orthogonal matrix we haveΦ−1 =ΦT so that:

ΦT A0Φ= Λ0 and A−10 =ΦΛ−1

0 ΦT

We also introduce the transformations

A i =ΦT A iΦ ∈ Rn×n; i = 0,1,2, . . . ,M (25)

Note that A0 = Λ0, a diagonal matrix and

A i =Φ−T A iΦ−1 ∈ R

n×n; i = 1,2, . . . ,M (26)


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Suppose the solution of Eq. (9) is given by

ut+∆t(θ) =

[A0 +

M

∑i=1

Γi(ξ(θ))A i

]−1

feqvt+∆t(θ) (27)

Using Eqs. (24)–(26) and the orthonormality of Φ one has

ut+∆t(θ) =

[Φ−TΛ0Φ

−1 +M

∑i=1

Γi(ξ(θ))Φ−T A iΦ−1

]−1

feqvt+∆t(θ)

⇒ ut+∆t(θ) = Φ

[Λ0 +

M

∑i=1

Γi(ξ(θ))A i

]−1

︸︷︷︸Ψ(ξ(θ)

)

Φ−T feqvt+∆t(θ)

(28)

where ξ(θ) = [ξ1(θ),ξ2(θ), . . . ,ξM(θ)]T .Abhishek Kundu — SFEM for structural dynamics — 09-10 Jan, 2012

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Now we separate the diagonal and off-diagonal terms of the A i

matrices asA i = Λi +∆i , i = 1,2, . . . ,M (29)

Here the diagonal matrix

Λi = diag[A]= diag[λi1 ,λi2 , . . . ,λin ] ∈ R

n×n (30)

and ∆i = A i −Λi is an off-diagonal only matrix. We can write :

Ψ(ξ(θ)) =

Λ0 +

M

∑i=1

Γi(ξ(θ))Λi

︸︷︷︸Λ(Γi(ξ(θ))

)

+M

∑i=1

Γi(ξ(θ))∆i

︸︷︷︸∆(Γi(ξ(θ))

)

−1

. (31)


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The diagonal matrix Λ(ξ(θ)) is treated as the preconditioner in thestochastic Krylov space, such that the solution can be projected ontoa very few basis functions.Hence the left preconditioned stochastic Krylov space becomes

Km(Λ−1A(θ),Λ−1feqv

t+∆t) = span{ΦTΛ−1Φfeqvt+∆t ,Φ

T (Λ−1∆)Λ−1Φfeqvt+∆t ,

ΦT (Λ−1∆)2Λ−1Φfeqvt+∆t , . . . ,Φ

T (Λ−1∆)m−1Λ−1Φfeqvt+∆t}

(32)

The equivalent infinite Neumann matrix series representation of theabove equation is

Ψ(ξ(θ)) =∞

∑s=0

(−1)s[Λ−1 (ξ(θ))∆(ξ(θ))

]sΛ−1 (ξ(θ)) (33)


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The following can be noted from the preceeding development:

1 The diagonal matrix Λ(ξ(θ)) is treated as the preconditioner inthe stochastic Krylov space.

2 This enables us to project the solution onto a very few basisfunctions.

3 The diagonal dominance of matrices A i is conducive to thisapproach.

4 If the mean of the coefficient matrix is used as the preconditioner,the terms of the Neumann series also span the leftpreconditioned stochastic Krylov space.


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Taking an arbitrary r -th element of u(t,θ), Eqn. (28) can berearranged to have

urt+∆t(θ) =

n

∑k=1

Φrk

(n

∑j=1

Ψkj (ξ(θ))(φT

j feqvt+∆t

))

(34)

Defining

Lk (t,ξ(θ)) =n

∑j=1

Ψkj (ξ(θ))(φT

j feqvt+∆t

)(35)

and collecting all the elements in Eqn. (34) for r = 1,2, . . . ,n one has

ut+∆t(θ) =n

∑k=1

Lk (t,ξ(θ))φk (36)


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Truncating the series in Eqn. (33) and taking m terms of the minimalpolynomial of the left-preconditioned stochastic Krylov space :

Ψ(m) (ξ(θ)) =m

∑s=0

(−1)s[Λ−1 (ξ(θ))∆(ξ(θ))

]sΛ−1 (ξ(θ)) (37)

From this one can obtain a sequence for different m

u(m)t+∆t(θ) =

n

∑k=1

L(m)k (t,ξ(θ))φk ; m = 1,2,3, . . . (38)

Since θ ∈Θ is arbitrary, u(m)t+∆t(θ) approximates the solution of

Eqn. (8) for every θ when m → ∞. This implies that

Prob{

θ ∈Θ : limm→∞

u(m)t+∆t(θ) = ut+∆t(θ)

}= 1 (39)

at every time step t +∆t , given that the solution of the previous timestep t has converged.Abhishek Kundu — SFEM for structural dynamics — 09-10 Jan, 2012

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A few observations :• No particular pdf of the random variables has been assumed in

this derivation, hence they can be general as long as the solutionexists.

• The matrix power series is different from the classical Neumannseries in that the elements of the former are not simplepolynomials in ξi(θ) but are in terms of the ratio of polynomials.

• The convergence of the series depends on the spectral radius of

R(ξ(θ)) = Λ−1 (ξ(θ))∆(ξ(θ)) (40)

• A generic term of the matrix R is

Rrs =∆rs

Λrr=

∑Mi=1Γi(ξ)∆irs

Λ0r +∑Mi=1Γi(ξ)Λir

=∑M

i=1Γi(ξ)Airs

Λ0r +∑Mi=1Γi(ξ)Airr

; r 6= s

(41)which shows that the spectral radius of R is controlled by thediagonal dominance of the A i matrices.Abhishek Kundu — SFEM for structural dynamics — 09-10 Jan, 2012

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Spectral functions

Definition

The functions Lk (t,ξ(θ)) ,k = 1,2, . . .n are called the spectralfunctions as they are expressed in terms of the spectral properties ofthe coefficient matrices of the governing discretized equation.

• Each of the spectral functions Lk (t,ξ(θ)) contain infinite numberof terms and they are highly nonlinear functions of the randomvariables ξi(θ).

• The truncation of the spectral functions has to be done based ontheir convergence characteristics.

• Different order of spectral functions can be obtained by usingtruncation in the expression of Lk (t,ξ(θ))

• The spectral functions are in general non-Gaussian even if theinput stochasticity has Gaussian distribution.


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Summary of the basis functions (spectral functions)

The basis functions are:

1 not polynomials in ξi(θ) but ratio of polynomials.

2 independent of the nature of the random variables (i.e. applicableto Gaussian, non-Gaussian or even mixed random variables).

3 not general but specific to a problem as it utilizes the eigenvaluesand eigenvectors of the systems matrices.

4 such that truncation error depends on the off-diagonal terms ofthe matrix ∆(ω,ξ(θ)).


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The Euler-Bernoulli beam example

• An Euler-Bernoulli cantilever beam with stochastic bendingmodulus for a specified value of the correlation length and fordifferent degrees of variability of the random field.

0 5 10 15 200

1000

2000

3000

4000

5000

6000

Nat

ural

Fre

quen

cy (

Hz)

Mode number

(a) Natural frequency distribu-tion.

0 5 10 15 20 25 30 35 4010

−4

10−3

10−2

10−1

100

Rat

io o

f Eig

enva

lues

, λ 1 / λ j

Eigenvalue number: j

(b) Eigenvalue ratio of KL decom-position

• Length : 1.0 m, Cross-section : 39 × 5.93 mm2, Young’sModulus: 2 × 1011 Pa.

• Load: Unit impulse at t = 0 on the free end of the beam.Abhishek Kundu — SFEM for structural dynamics — 09-10 Jan, 2012

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Problem details

• The bending modulus of the cantilever beam is taken to be ahomogeneous stationary Gaussian random field of the form

EI(x ,θ) = EI0(1+a(x ,θ)) (42)

where x is the coordinate along the length of the beam, EI0 is theestimate of the mean bending modulus, a(x ,θ) is a zero meanstationary random field.

• The autocorrelation function of this random field is assumed to be

Ca(x1,x2) = σ2ae−(|x1−x2|)/µa (43)

where µa is the correlation length and σa is the standarddeviation.

• A correlation length of µa = L/10 is considered in the presentnumerical study.


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Problem details

The random field is assumed to be Gaussian. The results arecompared with the polynomial chaos expansion.

• The number of degrees of freedom of the system is n = 200.• The K.L. expansion is truncated at a finite number of terms such

that 90% variability is retained.• direct MCS have been performed with 10,000 random samples

and for three different values of standard deviation of the randomfield, σa = 0.05,0.1,0.2.

• Constant modal damping is taken with 1% damping factor for allmodes.

• Time domain response of the free end of the beam is soughtunder the action of a unit impulse at t = 0

• Upto 4th order spectral functions have been considered in thepresent problem. Comparison have been made with 4th orderPolynomial chaos results.


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Mean deflection of the beam

(c) Mean, σa = 0.05. (d) Mean, σa = 0.1. (e) Mean, σa = 0.2.

• Time domain response of the deflection of the tip of the cantileverfor three values of standard deviation σa of the underlyingrandom field.

• The proposed spectral method produces a very close match withthe direct MCS results. The error in the PC solution increasessignificantly with time. For higher values of the standard deviationof the underlying random field, the discrepancy is higher.


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Standard deviation of the beam response

(f) Standard deviation ofdeflection, σa = 0.05.

(g) Standard deviation ofdeflection, σa = 0.1.

(h) Standard deviation ofdeflection, σa = 0.2.

• The standard deviation of the tip deflection of the beam.

• Since the standard deviation comprises of higher order productsof the Hermite polynomials associated with the PC expansion,the higher order moments are less accurately replicated and tendto deviate more significantly.


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Response at earlier time steps

A closer look at the system response in t ∈ [0,0.2].

(i) Mean deflection, σa = 0.20. (j) Std. dev., σa = 0.20.

• The 4th order PC performs satisfactorily in terms of estimatingthe mean deflection within this time interval.

• However, the obtained standard deviation of the response is quiteunsuitable for practical applications.


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Probability density function of the tip deflection

(k) t = 0.125s, σa = 0.05. (l) t = 0.125s, σa = 0.1. (m) t = 0.125s, σa = 0.2.

• The spectral function approach reproduces the pdf of theresponse quite satisfactorily (compare with direct MCS method).

• An increase in order of the PC method to improve the higherorder moments of the response would be of little use, since thedimension of the resulting linear algebraic system increases withit exponentially.


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Computational cost

The calculation times are shown for a single time step (done with an80×80 system and 2 Gaussian random variables).

Calculation Avg Time(s) Min Time(s) Max Time(s)Direct MCS 13.589 13.506 13.798

2nd order spectral 1.375 1.345 1.3963rd order spectral 1.445 1.414 1.4654th order spectral 1.500 1.481 1.523

4th order PC 5.117 4.975 5.3274th order PC (8 cores) 1.329 1.201 1.477

• All calculations were performed using a single processor corewhile the optimized ATLAS, LAPACK and BLAS libraries wereused on 8 processor cores for the last case.

• The 4th order spectral function is 9 times more efficient thandirect MCS and 3.5 times more efficient than 4th order PC.

• Computational time increases with spectral function order.Abhishek Kundu — SFEM for structural dynamics — 09-10 Jan, 2012

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Conclusions

• The stochastic partial differential equations for structuraldynamics is considered.

• The solution is projected into a finite dimensional completeorthonormal vector basis and the associated stochasticcoefficient functions are obtained at each time step.

• The coefficient functions, called as the spectral functions, areexpressed in terms of the spectral properties of the systemmatrices.

• The Newmark method is utilized for the time integration scheme.• If p(< n) number of orthonormal vectors are used and M is the

number of random variables, then the computational complexitygrows in O(Mp2)+O(p3) for large M and p in the worse case.

• We consider a problem with 6 random variables and n = 200degrees of freedom. The 4th order PC required solving a linearsystem of 42,000×42,000 for this simple benchmark problem.


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Conclusions

• The proposed spectral function approach utilized a 4th orderspectral function with 10 orthonormal vector basis, resulting in amuch less computationally intensive solution process.

• The deterioration of the PC time domain response beyond acertain interval of time is known as stochastic drift.

• From Eqn. (8), we see the linear ODE has quadratic non-linearityin stochastic domain.

• This implies that only for early times, the solution can beapproximated as a linear continuation of the random input.

• With increasing time, the non-linear development is compoundedmore and more, requiring an increasing amount of terms in thepolynomial chaos expansion in terms of the input expansion.

• However, this has extreme adverse effects w.r.t. thecomputational cost of the linear system.


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Conclusions

• The spectral functions used in this study has a ‘rational form’.• This implies that each of these functions is a infinite polynomial

series in terms of the random variables.• The convergence of the spectral functions is verified from its

comparison with direct MCS simulation results.• The dimension of the left preconditioned stochastic Krylov

subspace reduces due to the particular choice of thepreconditioner which incorporates ‘some of the randomness’ ofthe system matrices into the it.

• The only information used in constructing the PC basis is theprobability space of the random variables involved. In contrast,the spectral functions constructs a ‘customized basis’, utilizingthe spectral information and perturbation characteristics of thesystem matrices.

• This results in the spectral functions to have better adaptivenature to the particular problem at hand.


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Discussions

• The true nature of the solution is a ratio of two polynomials ofrandom variables where the denominator has higher degree thanthe numerator. The proposed spectral basis functions have thiscorrect mathematical form.

• The proposed method utilizes the eigen-spectrum of thedeterministic coefficient matrices A0 to achieve a POD-like modelreduction which helps to work with a significantly smallersubspace dimension.

• The polynomial basis used in the PC method has no adaptivecharacteristics and remains the same for all time steps, however,in reality, the non-linearity in the stochastic domain iscompounded with each incremental time step.

• The spectral functions used in the present approach changeswith each time step which allows a better estimation of theresponse variables.


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Future Work

The future work that can be pursued along this direction may include :

• Trying to reducing the computational burden of integration overthe probability space using the efficient variance reduction and/orsampling techniques.

• A-priori error analysis and a rigorous study of the convergencebehavior can give important intuitive guidance in moving towardsa choice of a more efficient set of basis functions suitable for thisclass of stochastic problems.

• Extension of the presented idea to the class of non-linear(geometric) dynamics problems.

• Extension of the present approach to study the behavior of timedependent diffusion problems using the dynamic loading.


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Thank you for your attention!

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