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Research Papers

Distributed Load Estimation From Noisy Structural Measurements

[+] Author and Article Information
Ben T. Dickinson

Air Force Research Laboratory,
Munitions Directorate,
101 West Eglin Boulevard, Suite 213,
Eglin AFB, FL 32542, FL 32542
e-mail: benjamin.dickinson.1@us.af.mil

John R. Singler

Assistant Professor
Missouri University of Science and Technology,
Department of Mathematics and Statistics,
400 West 12th Street,
Rolla, MO 65409
e-mail: singlerj@mst.edu

We define a structural measurement as a scalar quantity that is related to the deformation of a structure from a reference configuration.

Asymmetry leads to the solution of an adjoint problem.

Strictly speaking, we are under no mathematical obligation to estimate loads that produce a structural response that complies with assumptions of the mathematical model.

1Corresponding author.

Manuscript received May 2, 2012; final manuscript received September 14, 2012; accepted manuscript posted October 10, 2012; published online May 16, 2013. Assoc. Editor: Professor John Lambros.

J. Appl. Mech 80(4), 041011 (May 16, 2013) (16 pages) Paper No: JAM-12-1176; doi: 10.1115/1.4007794 History: Received May 02, 2012; Revised September 14, 2012; Accepted October 10, 2012

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.

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Figures

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Fig. 1

Illustration of the general problem, to find the best estimate of surface forces given a limited set of structural measurements

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Fig. 2

Diagram of circular membrane with embedded structural measurements taken over circular subsets of Ω

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Fig. 3

Benchmark distributed loads I, II, and III belonging to V

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Fig. 4

Measurement arrays A (a), B (b), and C (c) used throughout this study with a superimposed 5067 element triangular mesh

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Fig. 5

Measurement convergence with grid refinement for loads I, II and III

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Fig. 6

Example of nodal alignment of mesh on sensor boundaries to aid finite element convergence of the measurement data

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Fig. 7

H norm estimates for loads I,II, and III for measurement arrays A, B, and C

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Fig. 8

V norm estimates for loads I,II, and III for measurement arrays A, B, and C

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Fig. 9

Benchmark loads IV and V that do not belong to V

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Fig. 10

H norm estimates for loads IV and V for measurement arrays A, B, and C

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Fig. 11

V norm estimates for loads IV and V for measurement arrays A, B, and C

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Fig. 12

Estimate of a uniform load (load IV) where X = H1(Ω) and Θ = 10-10 with nonzero boundary values

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Fig. 13

Load III estimate X norm global relative error versus noise estimate magnitude relative error with 5% random measurement error showing the concurrent minimization of both errors

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Fig. 14

Load III noise estimate error (a) and load estimate error (b) versus regularization parameter for the H, V, and E   norms with 5% measurement noise

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Fig. 15

Load IV estimate X norm global relative error versus noise estimate magnitude relative error with 5% random measurement error showing the concurrent minimization of both errors

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Fig. 16

Load IV noise estimate error (a) and load estimate error (b) versus regularization parameter for the H, V, and E norms with 5% measurement noise

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Fig. 17

Load III estimates with 5% measurement noise; (a) H norm, noise magnitude error = 0.0748; (b) V norm, noise magnitude error = −0.115; (c) E norm, noise magnitude error = 0.0565

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Fig. 18

Comparison of noise estimate and applied 5% noise signal computed at each measurement location; top: x-direction noise estimate; bottom: y-direction noise estimate

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