STATIC ANALYSIS OF UNCERTAIN STRUCTURESUSING INTERVAL EIGENVALUE DECOMPOSITION

Mehdi Modares Tufts University

Robert L. MullenCase Western Reserve University

Static Analysis

An essential procedure to design a structure subjected to a system of loads.

Static Analysis (Cont.)

In conventional static analyses of a structure, the existence of any uncertainty present in the structure’s geometric or material characteristics as well as loads is not considered.

Uncertainty in Transportation Systems

Attributed to:

Structure’s Physical Imperfections

Inaccuracies in Determination of Loads

Modeling Complexities of Load-Structure Interaction

For reliable design, the presence of uncertainty must be included in analysis procedures

Research Objective

To introduce a computationally efficient finite-element-based method for linear static analysis of a structure with uncertain properties expressed as interval quantities.

Presentation Outline

Fundamentals of engineering uncertainty analysis

Introduction of method of Interval Static Analysis (ISA)

Numerical problem

Work conclusions

Engineering Uncertainty Analysis

Formulation

Modifications on the representation of the system characteristics due to presence of uncertainty

Computation

Development of schemes capable of considering the uncertainty

throughout the solution process

Uncertainty Analysis Schemes

Considerations:

Consistent with the system’s physical behavior

Computationally feasible

Paradigms of Uncertainty Analysis

Stochastic Analysis : Random variables

Fuzzy Analysis : Fuzzy variables

Interval Analysis : Interval variables

Interval Variable

A real interval is a set of the form:

Archimedes (287-212 B.C.)

}|{],[~ ulul zzzzzzZ

],[~ bax

)713

71103(

Deterministic Linear Static Analysis

The equilibrium defined as a linear system of equation as:

: Global elastic stiffness matrix : Displacement vector : External force vector

}{}]{[ PUK

][K}{U}{P

Interval Static Analysis

The equilibrium defined as a linear system of equation as:

: Interval elastic stiffness matrix (presence of uncertainty) : Displacement vector : External force vector

}{}]{~[ PUK

]~[K}{U}{P

Prior WorksIn Interval Static Analysis

Muhanna and Mullen (2001)

Neumaier and Pownuk (2007)

Stiffness Eigenvalue Decomposition

Considering an eigenvalue decomposition of the stiffness matrix, the stiffness is:

: Matrix of eigenvalues : Matrix of eigenvectors

Equivalently:

][][

TK ]][][[][

)( i i

N

i

TiiiK

1

}}{{][

Stiffness Matrix Inversion

The stiffness matrix inverse is obtained as:

Equivalently the inversion is:

TK ][]][[][ 11

N

i

Tii

i

K1

1 }}{{1][

Static Response

The static response (the solution to the linear system if equation is:

Substitution of the decomposed stiffness matrix inverse, the response is:

}{][}{ 1 PKU

}){}}{{1(}{1

PUN

i

Tii

i

Interval Eigenvalue Decomposition

Considering an interval eigenvalue decomposition of the stiffness matrix due to the presence of uncertainty, the interval stiffness is:

: Interval eigenvalues : Interval eigenvectors

)~( i

i~

N

i

TiiiK

1

}~}{~{~]~[

Interval Static Response

The interval static response is:

In which, the bounds on the eigenvalues and eigenvectors of interval system must be obtained.

}){}~}{~{~1(}~{

1

PUN

i

Tii

i

1. Interval Stiffness Matrix

The deterministic structure global stiffness is a linear summation of the element contributions to the global stiffness matrix:

[Li] : Element Boolean connectivity matrix [ki] : Element stiffness matrix

n

i

Tiii LKLK

1

]][][[][

Non-Deterministic System with uncertainty in the stiffness characteristics

Non-deterministic element stiffness matrix:

: Interval of uncertainty

Non-deterministic global stiffness matrix:

]])[,([]~[ iiii KulK

],[ ii ul

N

i

Tiiiii LKulLK

1

]]}[])[,]{([[]~[

Interval Stiffness Matrix

Non-deterministic global stiffness matrix:

Deterministic element stiffness contribution:

Structure’s global interval stiffness matrix:

N

i

Tiiiii LKLulK

1

]][][])[,([]~[

Tiiii LKLK ]][][[][

N

iiii KulK

1

]])[,([]~[

2. Interval Eigenvalue Problem

Interval eigenpair problem using the interval global stiffness matrix

Central (pseudo-deterministic) and Radial (perturbation) stiffness matrices:

}~){~(}~){]])[,([(1

N

iiii Kul

N

ii

iiC KulK

1

])[2

(][

N

ii

iiiR KluK

1

])[2

)(~(]~[ ]1,1[~ i

Updated Interval Eigenvalue Problem due to uncertainty in stiffness

Updated interval eigenvalue problem using central and radial stiffness matrices:

Problem Mathematical Interpretation:

An eigenvalue problem on a central stiffness matrix when it is subjected to a perturbation of radial stiffness matrix (linear summation of non-negative definite matrices).

(Constraint imposed by problem’s physical behavior)

}~){~(}~]){~[]([ RC KK

3. Solution for Eigenvalues

Classical linear eigenpair problem for a symmetric matrix :

Real eigenvalues corresponding eigenvectors

Rayleigh quotient (ratio of quadratics):

)...( 21 n ),...,,( 21 nxxx

xAx

nT

T

xxAxxxR )(1

Minimization of Rayleigh Quotient with no constraint

Performing an unconstrained minimization :

The smallest eigenvalue is the solution to the unconstrained minimization on the scalar-valued function of Rayleigh quotient.

1)(min)(min xx

AxxxR T

T

RxRx nn

Minimization of Rayleigh Quotient with imposed constraints

Imposing a single constraint on the minimization:

x : Trial vector z : Arbitrary vector

Performing a constrained minimization :

0. zxT

210

)(min

xRzxT

Constrained Minimizations on R.Q.to find the second eigenvalue

Minimization of R(x) subject to a single constraint :

choosing the {z} that maximizes the minimum yields the second smallest eigenvalue.

For the next eigenvalues more constraints must be imposed.

0. zxT

)](minmax[0

2 xRzxT

Generalization of Constrained Minimizations on R.Q.to find the next eigenvalues

Generalization of results for the Kth eigenvalue:

Subjected to

This principle is called the “ Maximin Characterization” of eigenvalues for symmetric matrices.

)](max[min xRk

2,1,...1),0( kkizx iT

Matrix Non-Negative Definite Perturbation

Symmetric matrix [A] subject to non-negative definite perturbation matrix [E ]:

First eigenvalue : using unconstrained minimization:

Next eigenvalues : using maximin characterization:

)(min)(min)( 11 Axx

Axxxx

xEAxEA T

T

T

T

)(]minmax[])(minmax[)(1,...,1,0.1,...,1,0.

Axx

Axxxx

xEAxEA kT

T

kizxT

T

kizxkii

Monotonic Behavior of Eigenvalues

All eigenvalues of a symmetric matrix subject to a non-negative perturbation monotonically increase from the eigenvalues of the exact matrix.

Similarly, all eigenvalues of a symmetric matrix subjected to a non-positive perturbation monotonically decrease from the eigenvalues of exact matrix.

)()(~ AEA kk

)()(~ AEA kk

Bounds on Eigenvalues Using minimum and Maximin characterization of eigenvalues

The first eigenvalue:

The next eigenvalues:

)]~[][(min])~[]([~1 xKxxKxKK R

TC

T

RxRC n

)]]~[][(minmax[)~(~1,...,1,0.

xKxxKxKK RT

CT

kizxRCki

Bounds on Eigenvalues Using monotonic behavior of eigenvalues

Upper bound:

Lower bound:

)])[(())])[2

)()((])[2

(()]~(~max[11

max1

n

iiik

n

ii

iii

n

ii

iikRCk KuKluKulKK

)])[(())])[2

)()((])[2

(()]~(~min[11

min1

n

iiik

n

ii

iii

n

ii

iikRCk KlKluKulKK

Bounding Interval Eigenvalue Problems

Solution to interval eigenvalue problem correspond to the maximum and minimum eigenvalues:

Two deterministic problems capable of bounding all eigenvalues

(Modares and Mullen 2004)

}){(}){])[(( max1

n

iii Ku

}){(}){])[(( min1

n

iii Kl

4. Solution for Eigenvectors Invariant Subspaces

The subspace is an invariant subspace of if:

i.e., if is an invariant subspace of and, columns of form a basis for , then there is a unique matrix such that:

matrix is the representation of with respect to .

A

A

]][[]][[ 111 LXXA

mnX ][ 1 nnA ][

mmL ][ 1

][ 1L ][A

][]][[][ 111 LXAX T

Theorem of Invariant Subspaces

For a real symmetric matrix :

Matrices and forming bases for complementary subspaces

and .

Then, is an invariant subspace of if and only if:

][ 1X ][ 2X

][A

]0[]][[][ 12 XAX T

][A

Simple Invariant Subspaces

An invariant subspace is simple if the eigenvalues of its representation

are distinct from other eigenvalues of :][A

][ 1L

])([])([ 21 LL

Perturbation of Invariant Subspaces Due to perturbation of matrix

Considering the column spaces of and to span two

complementary subspaces, the perturbed orthogonal subspaces are:

in which [P ] is the matrix to be determined.

]~[ 1X ]~[ 2X

]][[][]~[ 211 PXXX

TPXXX ]][[][]~[ 122

Perturbation Problem

A perturbed matrix:

Using theorem of invariant subspaces, is an invariant subspace of iff :

Substitution and linearization:

in the form of a Sylvester’s equation.

]~[A]~[ 1X

]0[]~][~[]~[ 12 XAX T

][][]~[ EAA

]][[][]][[]][[ 1221 XEXPLLP T

Sylvester’s Equation

A Sylvester’s equation is in the form:

Equivalently defining a linear operator [T ]:

The uniqueness of the solution guaranteed when operator [T] is non-singular. This operator is non-singular when eigenvalues of [A] and [B] are distinct:

mnmmmnmnnn CBXXA ][][][][][

mnmn BXXAXT ])][[]][([][][

])([])([ BA

Perturbation of Eigenvectors Specialization of perturbation of invariant subspaces

The perturbed first eigenvector:

The perturbation problem:

If the eigenvalues are simple the solution for [p] exists and is unique as:

]][[}{}~{ 211 pXxx

}]{[][]][[][ 1221 xEXpLp T

}]{[][])[][(][ 121

21 xEXLIp T

Perturbation of Eigenvectors Solution

The solution for the perturbed first eigenvector:

}]{[][])[][]([}{}~{ 121

21211 xEXLIXxx T

Perturbation of Eigenvectors Solution for static analysis

The perturbation matrix:

The solution:

N

ii

iiiR KluKE

1

])[2

)(~(]~[][

}]){)[2

)(~((][])[][]([}{}~{ 11

21

21211 iii

i

N

i

T KluI

Interval Static Response

The interval static response is:

To attain sharper results, the functional dependency of the intervals must be considered.

}){}~}{~{~1(}~{

1

PUN

i

Tii

i

Numerical Example

Present MethodInterval Static Analysis

Combinatorial Solution (NP-hard)Lower and upper values for each element (2n)

Example Problem

2-D Statically indeterminate truss with material uncertainty

EE ])01.1,99.0([~

Results

Vertical displacement of the top node

AEPLU

Lower BoundPresent Method

Lower Bound

Combination Method

Upper Bound

Combination Method

Upper BoundPresent Method

Error%

-1.6265 -1.6244 -1.5859 -1.5838 % 0.12

Conclusions

An alternate finite-element based method for static analysis of structural systems with interval uncertainty is presented.

This proposed method is simple to implement and computationally efficient.

The method can potentially obtain the sharp bounds on the structure’s static response.

While this methodology is shown for structural systems, its extension to various mechanics problems is straightforward.

Future work

Comparison to other existing schemes for efficiency, sharpness and stability

Questions