This research investigates the application of sum-of-squares (SOS) optimization method on finite element model updating through minimization of modal dynamic residuals. The modal dynamic residual formulation usually leads to a nonconvex polynomial optimization problem, the global optimality of which cannot be guaranteed by most off-the-shelf optimization solvers. The SOS optimization method can recast a nonconvex polynomial optimization problem into a convex semidefinite programming (SDP) problem. However, the size of the SDP problem can grow very large, sometimes with hundreds of thousands of variables. To improve the computation efficiency, this study exploits the sparsity in SOS optimization to significantly reduce the size of the SDP problem. A numerical example is provided to validate the proposed method.

References

References
1.
Wu
,
M.
, and
Smyth
,
A. W.
,
2007
, “
Application of the Unscented Kalman Filter for Real-Time Nonlinear Structural System Identification
,”
Struct. Control Health Monit.
,
14
(
7
), pp.
971
990
.
2.
Ebrahimian
,
H.
,
Astroza
,
R.
, and
Conte
,
J. P.
,
2015
, “
Extended Kalman Filter for Material Parameter Estimation in Nonlinear Structural Finite Element Models Using Direct Differentiation Method
,”
Earthquake Eng. Struct. Dyn.
,
44
(
10
), pp.
1495
1522
.
3.
Hoshiya
,
M.
, and
Saito
,
E.
,
1984
, “
Structural Identification by Extended Kalman Filter
,”
J. Eng. Mech.
,
110
(
12
), pp.
1757
1770
.
4.
Yang
,
J. N.
,
Lin
,
S.
,
Huang
,
H.
, and
Zhou
,
L.
,
2006
, “
An Adaptive Extended Kalman Filter for Structural Damage Identification
,”
Struct. Control Health Monit.
,
13
(
4
), pp.
849
867
.
5.
Astroza
,
R.
,
Ebrahimian
,
H.
,
Li
,
Y.
, and
Conte
,
J. P.
,
2017
, “
Bayesian Nonlinear Structural FE Model and Seismic Input Identification for Damage Assessment of Civil Structures
,”
Mech. Syst. Signal Process.
,
93
, pp.
661
687
.
6.
Salawu
,
O. S.
,
1997
, “
Detection of Structural Damage Through Changes in Frequency: A Review
,”
Eng. Struct.
,
19
(
9
), pp.
718
723
.
7.
Sanayei
,
M.
,
Arya
,
B.
,
Santini
,
E. M.
, and
Wadia Fascetti
,
S.
,
2001
, “
Significance of Modeling Error in Structural Parameter Estimation
,”
Comput. Aided Civ. Infrastruct. Eng.
,
16
(
1
), pp.
12
27
.
8.
Jaishi
,
B.
, and
Ren
,
W. X.
,
2006
, “
Damage Detection by Finite Element Model Updating Using Modal Flexibility Residual
,”
J. Sound Vib.
,
290
(
1–2
), pp.
369
387
.
9.
Koh
,
C.
, and
Shankar
,
K.
,
2003
, “
Substructural Identification Method Without Interface Measurement
,”
J. Eng. Mech.
,
129
(
7
), pp.
769
776
.
10.
Zhang
,
D.
, and
Johnson
,
E. A.
,
2013
, “
Substructure Identification for Shear Structures I: Substructure Identification Method
,”
Struct. Control Health Monit.
,
20
(
5
), pp.
804
820
.
11.
Moaveni
,
B.
,
Stavridis
,
A.
,
Lombaert
,
G.
,
Conte
,
J. P.
, and
Shing
,
P. B.
,
2013
, “
Finite-Element Model Updating for Assessment of Progressive Damage in a Three-Story Infilled RC Frame
,”
J. Struct. Eng., ASCE
,
139
(
10
), pp.
1665
1674
.
12.
Nozari
,
A.
,
Behmanesh
,
I.
,
Yousefianmoghadam
,
S.
,
Moaveni
,
B.
, and
Stavridis
,
A.
,
2017
, “
Effects of Variability in Ambient Vibration Data on Model Updating and Damage Identification of a 10-Story Building
,”
Eng. Struct.
,
151
, pp.
540
553
.
13.
Zhu
,
D.
,
Dong
,
X.
, and
Wang
,
Y.
,
2016
, “
Substructure Stiffness and Mass Updating Through Minimization of Modal Dynamic Residuals
,”
J. Eng. Mech.
,
142
(
5
), p.
04016013
.
14.
Farhat
,
C.
, and
Hemez
,
F. M.
,
1993
, “
Updating Finite Element Dynamic Models Using an Element-by-Element Sensitivity Methodology
,”
AIAA J.
,
31
(
9
), pp.
1702
1711
.
15.
Kosmatka
,
J. B.
, and
Ricles
,
J. M.
,
1999
, “
Damage Detection in Structures by Modal Vibration Characterization
,”
J. Struct. Eng. ASCE
,
125
(
12
), pp.
1384
1392
.
16.
Parrilo
,
P. A.
,
2003
, “
Semidefinite Programming Relaxations for Semialgebraic Problems
,”
Math. Program.
,
96
(
2
), pp.
293
320
.
17.
Nie
,
J.
,
Demmel
,
J.
, and
Sturmfels
,
B.
,
2006
, “
Minimizing Polynomials Via Sum of Squares Over the Gradient Ideal
,”
Math. Program.
,
106
(
3
), pp.
587
606
.
18.
Lasserre
,
J. B.
,
2001
, “
Global Optimization With Polynomials and the Problem of Moments
,”
SIAM J. Optim.
,
11
(
3
), pp.
796
817
.
19.
Laurent
,
M.
,
2009
, “
Sums of Squares, Moment Matrices and Optimization Over Polynomials
,”
Emerging Applications of Algebraic Geometry
,
Springer
, New York.
20.
Henrion
,
D.
, and
Lasserre
,
J. B.
,
2005
, “
Detecting Global Optimality and Extracting Solutions in GloptiPoly
,”
Positive Polynomials in Control
,
Springer, Berlin
.
21.
Li
,
D.
,
Dong
,
X.
, and
Wang
,
Y.
,
2018
, “
Model Updating Using Sum of Squares (SOS) Optimization to Minimize Modal Dynamic Residuals
,”
Struct. Control Health Monit.
,
25
(12), p.
e2263
.
22.
Nie
,
J.
, and
Demmel
,
J.
,
2008
, “
Sparse SOS Relaxations for Minimizing Functions That Are Summations of Small Polynomials
,”
SIAM J. Optim.
,
19
(
4
), pp.
1534
1558
.
23.
Basu
,
S.
,
Pollack
,
R.
, and
Roy
,
M.-F.
,
2003
,
Algorithms in Real Algebraic Geometry
,
Springer
,
Berlin
.
24.
Lall
,
S.
,
2011
, “
Sums of Squares
,” Lecture at Vanier College, Saint-Laurent, QC, Canada,
Technical Report
.http://www.math.mcgill.ca/goren/PAPERSpublic/Sums.Squares.pdf
25.
Nesterov
,
Y.
,
2000
, “
Squared Functional Systems and Optimization Problems
,”
High Performance Optimization
,
Springer
, Boston, MA, pp.
405
440
.
26.
Parrilo
,
P. A.
,
2000
, “Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization,”
Doctoral dissertation, California Institute of Technology
, Pasadena, CA.
27.
Boyd
,
S. P.
, and
Vandenberghe
,
L.
,
2004
,
Convex Optimization
,
Cambridge University Press
,
Cambridge, New York
.
28.
Sturm
,
J. F.
,
1999
, “
Using SeDuMi 1.02, A MATLAB Toolbox for Optimization Over Symmetric Cones
,”
Optim. Methods Software
,
11
(
1–4
), pp.
625
653
.
29.
Nocedal
,
J.
, and
Wright
,
S.
,
2006
,
Numerical Optimization
,
Springer Science & Business Media
, New York.
30.
MathWorks, Inc.,
2016
,
Optimization Toolbox™ User's Guide
,
MathWorks
,
Natick, MA
.
31.
Coleman
,
T. F.
, and
Li
,
Y.
,
1996
, “
An Interior Trust Region Approach for Nonlinear Minimization Subject to Bounds
,”
SIAM J. Optim.
,
6
(
2
), pp.
418
445
.
You do not currently have access to this content.