# Damage detection and model refinement using elemental stiffness perturbations with constrained connectivity Page: 4 of 13

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matrix update," solves a closed-form equation for

the matrix perturbations which minimize the

modal force error or constrain the solution to sat-

isfy it (see, for example, Baruch and Bar Itzhack

[5], Kabe [6], Berman and Nagy [7], Smith and

Beattie [8], [9], Kaouk and Zimmerman [10], [111)

It is this type of method which is of interest in

this paper. Much of the research done in optimal

matrix update has focused on estimating pertur-

bations to the mass and stiffness properties di-

rectly. In the context of structural damage

detection and health monitoring, the perturba-

tions to the stiffness properties are usually the

most relevant. In this paper, only the perturba-

tion of the structural stiffness properties will be

considered.

Computing the stiffness property perturba-

tions which eliminate the modal force error is of-

ten an underdetermined problem, since the

number of unknowns in the perturbation set can

be much larger than the number of measured

modes and the number of measurement degrees

of freedom. In this case, the property perturba-

tions which satisfy the modal force error equation

are non-unique. Optimal matrix update methods

thus apply a minimization to the property pertur-

bation to select a solution to the modal force error

equation subject to constraints such as symme-

try, positive definiteness, and sparsity. Typically,

this minimization applies to either a norm or the

rank of the perturbation property matrix or vec-

tor.

The main distinction between optimal update

methods which minimize some measure of the

stiffness property perturbations can be drawn

based on two characteristics: First, the stiffness

property which is varied, and second, the objec-

tive function that is used to select the solution.

The stiffness properties can be categorized as the

global stiffness matrix, the elemental stiffness

matrices, or the elemental stiffness parameters

(e.g. E, I, etc.). The objective functions are either

the minimum of a norm of the property perturba-

tion or the minimum of the rank of the property

perturbation. Table (1) shows how several of the

most widely known optimal matrix update proce-

dures can be categorized according to these char-

acteristics. The columns in this table categorize

methods (and cite examples from the literature)

according to which model parameter is used in

the update procedure. The rows categorize the

methods by whether a minimum norm (e.g. least-squares) or a minimum rank function is used as

the objective of the optimization.

As shown in Table (1), the majority of the ear-

ly work in optimal matrix update used the mini-

mum norm perturbation of the global stiffness

matrix [5], [6], [8], [7]. The motivation for using

this objective function is that the desired pertur-

bation is the one which is "smallest" in overall

magnitude. Later work by Kaouk and Zimmer-

man [10], as shown in the second row of Table (1),

used the minimum rank perturbation of the glo-

bal stiffness matrix. This was motivated by the

application of damage detection, where the per-

turbations could be assumed to be limited to a

few isolated locations. The minimum rank stiff-

ness matrix perturbation can be thought of as the

stiffness matrix perturbation with the smallest

number of nonzero values. An extension of this

work computes the perturbations at the element

stiffness matrix level, to limit the computed per-

turbations to certain structural DOF. [11]

A common drawback of the methods listed in

the first two columns of Table (1) is that the com-

puted perturbations are made to stiffness matrix

values at the structural DOF, rather than at the

element stiffness parameter level. There are

three main advantages to computing perturba-

tions to the elemental stiffness parameters rath-

er than to stiffness matrix entries: 1) The

resulting updates have direct physical relevance,

and thus can be more easily interpreted in terms

of structural damage or errors in the FEM; 2) The

connectivity of the FEM is preserved, so that the

resulting updated FEM has the same load path

set as the original; and 3) A single parameter

which affects a large number of structural ele-

ments can be varied independently. This advan-

tage is especially relevant, for example, in civil

engineering applications, where a parameter

such as the Young's modulus of concrete may be

uniform throughout a number of elements but

not precisely known. Previous techniques to com-

pute perturbations at the element parameter lev-

el have been proposed by Chen and Garba [12]

and Li and Smith [13]. These techniques use the

sensitivity of the entries in the stiffness matrix to

the elemental stiffness parameters so that the

minimum norm criterion can be applied directly

to the vector of elemental stiffness parameters.

Thus the resulting update consists of a vector of

elemental stiffness parameters that is a mini-

mum norm solution to the optimal update equa-2

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Doebling, S.W. Damage detection and model refinement using elemental stiffness perturbations with constrained connectivity, article, April 1, 1996; New Mexico. (digital.library.unt.edu/ark:/67531/metadc668175/m1/4/: accessed January 16, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.