static analysis of uncertain structures using interval eigenvalue decomposition mehdi modares tufts...
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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
top related