Accurate estimates of flow induced surface forces over a body are typically difficult to achieve in an experimental setting. However, such information would provide considerable insight into fluid-structure interactions. Here, we consider distributed load estimation over structures described by linear elliptic partial differential equations (PDEs) from an array of noisy structural measurements. For this, we propose a new algorithm using Tikhonov regularization. Our approach differs from existing distributed load estimation procedures in that we pose and solve the problem at the PDE level. Although this approach requires up-front mathematical work, it also offers many advantages including the ability to: obtain an exact form of the load estimate, obtain guarantees in accuracy and convergence to the true load estimate, and utilize existing numerical methods and codes intended to solve PDEs (e.g., finite element, finite difference, or finite volume codes). We investigate the proposed algorithm with a two-dimensional membrane test problem with respect to various forms of distributed loads, measurement patterns, and measurement noise. We find that by posing the load estimation problem in a suitable Hilbert space, highly accurate distributed load and measurement noise magnitude estimates may be obtained.

References

References
1.
Chock
,
J.
, and
Kapania
,
R.
,
2003
, “
Load Updating for Finite Element Models
,”
AIAA J.
,
41
, pp.
1667
1673
.10.2514/2.7312
2.
Li
,
J.
, and
Kapania
,
R.
,
2007
, “
Load Updating for Nonlinear Finite Element Models
,”
AIAA J.
,
45
, pp.
1444
1458
.10.2514/1.19073
3.
Maniatty
,
A.
,
Zabaras
,
N.
, and
Stelson
,
K.
,
1989
, “
Finite Element Analysis of Some Inverse Elasticity Problems
,”
J. Eng. Mech.
,
115
, pp.
1303
1317
.10.1061/(ASCE)0733-9399(1989)115:6(1303)
4.
Coates
,
C.
, and
Thamburaj
,
P.
,
2008
, “
Inverse Method Using Finite Strain Measurements to Determine Flight Load Distribution
,”
J. Aircr.
,
45
, pp.
366
370
.10.2514/1.21905
5.
Shkarayev
,
S.
,
Krashantisa
,
R.
, and
Tessler
,
A.
,
2001
, “
An Inverse Interpolation Method Utilizing In-Flight Strain Measurements for Determining Loads and Structural Response of Aerospace Vehicles
,”
Structural Health Monitoring: The Demands and Challenges
, Proceedings of 3rd International Workshop on Structural Health Monitoring Stanford, CA, September 12–14, pp. 336–343.
6.
White
,
J.
,
Douglas
,
D.
, and
Rumsey
,
M.
,
2009
, “
Operational Load Estimation of a Smart Wind Turbine Rotor Blade
,” Health Monitoring of Structural and Biological Systems,
SPIE Proceedings
, Vol. 7295.10.1117/12.815802
7.
Stanford
,
B.
,
Albertani
,
R.
, and
Ifju
,
P.
,
2007
, “
Inverse Methods to Determine the Aerodynamic Forces on a Membrane Wing
,” in 48th Structures, Structural Dynamics, and Materials Conference, Honolulu, HI, April 23–26, AIAA Paper No. 2007-1984.
8.
Stanford
,
B.
, and
Ifju
,
P.
,
2008
, “
The Validity Range of Low Fidelity Structural Membrane Models
,”
Exp. Mech.
,
48
, pp.
697
711
.10.1007/s11340-008-9152-2
9.
Banks
,
H.T.
,
Smith
,
R.C.
, and
Wang
,
Y.
,
1996
,
Smart Material Structures: Modeling, Estimation and Control
,
Wiley
,
New York
.
10.
Becker
,
E. B.
,
Carey
,
G. F.
, and
Oden
,
J. T.
,
1981
,
Finite Elements
, Vol. 1,
Prentice Hall
,
Englewood Cliffs, NJ
.
11.
Gockenbach
,
M. S.
,
2006
,
Understanding and Implementing the Finite Element Method
,
Society for Industrial and Applied Mathematics (SIAM)
,
Philadelphia, PA
.
12.
Hughes
,
T. J. R.
,
1987
,
The Finite Element Method: Linear Static and Dynamic Finite Element Analysis
,
Prentice Hall
,
Englewood Cliffs, NJ
.
13.
Lapidus
,
L.
, and
Pinder
,
G. F.
,
1999
,
Numerical Solution of Partial Differential Equations in Science and Engineering
,
Wiley
,
New York
.
14.
Strang
,
G.
, and
Fix
,
G.
,
2008
,
An Analysis of the Finite Element Method
,
Wellesley-Cambridge Press
, Wellesley, MA.
15.
Engl
,
H. W.
,
Hanke
,
M.
, and
Neubauer
,
A.
,
1996
,
Regularization of Inverse Problems, Mathematics and its Applications
, Vol. 375,
Kluwer
,
Dordrecht
, Germany.
16.
Kaipio
,
J.
, and
Somersalo
,
E.
,
2005
,
Statistical and Computational Inverse Problems
,
Springer-Verlag
,
New York
.
17.
Vogel
,
C. R.
,
2002
,
Computational Methods for Inverse Problems
,
SIAM
,
Philadelphia, PA
.
18.
Kreyszig
,
E.
,
1989
,
Introductory Functional Analysis With Applications
,
Wiley
,
New York
.
19.
Luenberger
,
D. G.
,
1969
,
Optimization by Vector Space Methods
,
Wiley
,
New York
.
20.
Bertero
,
M.
,
De Mol
,
C.
, and
Pike
,
E. R.
,
1988
, “
Linear Inverse Problems with Discrete Data. II. Stability and Regularisation
,”
Inverse Problems
,
4
(
3
), pp.
573
594
.10.1088/0266-5611/4/3/004
You do not currently have access to this content.