the application research of inverse finite element method...

Research ArticleThe Application Research of Inverse Finite Element Method forFrame Deformation Estimation

Yong Zhao,1 Hong Bao,1 Xuechao Duan,1 and Hongmei Fang2

1Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, Xidian University, Xi’an 710071, China2Nanjing Research Institute of Electronics Technology, Nanjing 210039, China

Correspondence should be addressed to Hong Bao; [email protected]

Received 6 February 2017; Revised 16 July 2017; Accepted 23 August 2017; Published 5 November 2017

Academic Editor: Vaios Lappas

Copyright © 2017 Yong Zhao et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A frame deformation estimation algorithm is investigated for the purpose of real-time control and health monitoring of flexiblelightweight aerospace structures. The inverse finite element method (iFEM) for beam deformation estimation was recentlyproposed by Gherlone and his collaborators. The methodology uses a least squares principle involving section strains ofTimoshenko theory for stretching, torsion, bending, and transverse shearing. The proposed methodology is based on stain-displacement relations only, without invoking force equilibrium. Thus, the displacement fields can be reconstructed without theknowledge of structural mode shapes, material properties, and applied loading. In this paper, the number of the locations wherethe section strains are evaluated in the iFEM is discussed firstly, and the algorithm is subsequently investigated through a simplesupplied beam and an experimental aluminum wing-like frame model in the loading case of end-node force. The estimationresults from the iFEM are compared with reference displacements from optical measurement and computational analysis, andthe accuracy of the algorithm estimation is quantified by the root-mean-square error and percentage difference error.

1. Introduction

Aircraft flexible wings with embedded conformal antennasand large frame structures that carry antennas require accu-rate real-time deformation estimation to provide feedbackfor their actuation and control systems [1–3]. Using the mea-sured strain to reconstruct the shape of the structure is a keytechnology in the accurate real-time deformation estimation,which has been studied by many researchers.

The computation of the displacement field of thedeformed structure is commonly performed on the basis ofstrain data measured in real time by a network of straingauges [4–6]. For example, fiber Bragg grating (FBG) sensorshave been extensively researched for deformation estimationdue to their lightness, accuracy, and easy embedment. Thestrategies that reconstruct the deformed shape displacementfield of the structure with in situ strain data can be dividedinto two kinds. One of the two kinds trains the mapping rela-tion between the displacement field and the measured strainby model learning algorithms, for example, neural networkmodel and fuzzy network algorithm [7, 8]. When the training

system has enough measured strains and displacement fielddata of the structure, a certain relation matrix can be deter-mined and the stable relationship between the measuredstrain and the displacement field can be obtained. But thestrategy requires a large number of training data, and themapping relation is easy to fail when the actual loading isbeyond the range of the training cases.

The other establishes the mapping relation between thedisplacement field and in situ strain data without the modellearning. In the literature [9–11], the global or piecewise con-tinuous basis function methods were employed to fit thesurface-measured strain into the structure strain field, andthen, the structure deformation displacement was obtainedfrom the strain-displacement relationship. These methodsare easy to implement, but the reconstruction accuracy ofdeformation estimation depends on the appropriate selectionof basis function and weight coefficients. Mode shapes havebeen used as basis function in [12, 13]. The deformation dis-placements are reconstructed frommeasured strains by usingthe modal transformation method. However, there exist thefollowing disadvantages in this method. (1) The detailed

HindawiInternational Journal of Aerospace EngineeringVolume 2017, Article ID 1326309, 8 pageshttps://doi.org/10.1155/2017/1326309

https://doi.org/10.1155/2017/1326309

material elasticity and inertial parameters are needed toprecisely construct mode shapes. (2) The accuracy ofdeformation reconstruction is severely limited to themodeling precision of the structure, and it is quite difficultto precisely model the complex structure. Maincon [14]developed a finite element-based methodology involving aninverse interpolation formulation that employs the surface-measured strain to determine the loads and structuralresponse of aerospace vehicles, while this algorithm needsan appropriate quality function to adapt the differentloading cases and this function is constructed based on amass of computer simulation and experimental statistics.

Based on the Euler-Bernoulli beam equation, Jute et al.evaluated the deflection of beam by the integration of dis-cretely measured strains directly [15]. The algorithm appliedthe classical beam equation and piecewise continuous poly-nomials to approximate beam curvature through integration.Derkevorkian et al. [16] compared the algorithm with themodal transformation method and demonstrated theadditional benefits of the algorithm to achieve a robustmethod for monitoring ultralightweight flying wings ornext-generation commercial airplanes. Though this one-dimensional scheme has displayed high accuracy in predict-ing deflection, it fails to estimate the element deformationunder multidimensional complex loads.

Tessler and Spangler [17] proposed the inverse finitemethodology (iFEM), which can be used to reconstruct thedisplacement field of shear-deformable structures, not onlybeam structure but also plate and shell structures. The mainidea is reconstructing a three-dimensional displacement fieldof beam structure from the surface-measured strains accord-ing to a least squares approach. Due to the fact that only thedisplacement-strain relationship is used, the deformationreconstruction can be accomplished by the methodologywithout the prior knowledge of loads, materials, and inertialand damping properties. To model arbitrary plate and shellstructures [18, 19], Tessler and his partners developed theiFEM algorithm using the first-order shear deformation the-ory and a three-node inverse shell element. FBG sensors wereapplied to measure the surface strains on the slender beams,and then, the deformed displacement was reconstructed byusing an iFEM shell model.

The beam-deformed displacement and cross-sectionaltorsion were reconstructed by Gherlone et al. who employedthe inverse finite element formulation to achieve high recon-struction accuracy of deformed displacement [20–21]. Theauthors first used the Timoshenko beam theory to modelthe beam kinematic accurately, then used the C0 or C1

inverse frame elements and least squares formulation toestablish the relationship between the measured strain andthe arbitrary node displacement field of the beam elementwith no prior knowledge of the finite element model andloads. The reconstruction equation is nonsingular when theboundary conditions are applied; that is, the status of oneend node of the beam element must be known. As the dis-placement field of the beam element in iFEM is constructedbased on an isotropic straight beam structure, whose crosssection is invariable along the whole beam, the deformationof the tapered beam cannot be estimated. Meanwhile, in view

of different load cases, the authors discussed different typesof shape functions that are used to interpolate the kine-matic variables in the beam element; then, the number ofsurface strain measurements and the deformation recon-struction equation are confirmed. Unfortunately, the mini-mum number of locations where the section strains areevaluated, which is critical for obtaining a correct solutionto the algorithm, is not discussed clearly in different loadcases. If the minimum number of the locations where thesection strains are evaluated is inappropriately set, the solu-tion will be nonunique; that is, the reconstructed displace-ment may be incorrect.

The contribution of this paper is twofold. (i) The rela-tionship between the loading case and the number of the sec-tion strain locations is discussed in detail, and the followingconclusions are verified; on the one hand, the minimumnumber of locations should be 2 in the loading case of end-node forces; on the other hand, the minimum number shouldbe 3 under the uniform distributed loading. (ii) In order toexamine the minimum number which is 2, estimation studiesare carried on a beam structure and a wing-like three-dimensional frame structure, in the loading case of end-node force. Six fiber optic strain sensors are placed at twonodes of each beam to capture in situ strain dates. To assessthe estimation error of the iFEM algorithm, the RMS and dif-ference percent parameters are employed.

2. iFEM Algorithms for Beam Estimation

A beam deflection estimation algorithm was developed byGherlone and his collaborators [20]. In the algorithm, theTimoshenko beam theory is first used to analyze theexpression of the displacement field in a straight beamelement (Figure 1).

ux x, y, z = u x + zθy x − yθz x ,vy x, y, z = v x − zθx x ,uz x, y, z =w x + yθx x ,

1

where ux, vy, and wz are the point displacements along the x,y, and z axes, respectively. u x , v x , andw x denote thedisplacements at y = z = 0; θx x , θy x , and θz x are therotations about the three coordinate axes. The six kinematicvariables in the middle axes can be grouped in vector formas follows:

u = u, v, w, θx, θy, θzT 2

L

X

Y

Z

O

v uu (x,y,z)

W

𝜃z𝜃y

𝜃x

Figure 1: Beam geometry and kinematic variables.

2 International Journal of Aerospace Engineering

The arbitrary section strains e u = e1, e2, e3, e4, e5, e6 T

can be obtained by (1).

e1 x = ux x ,e2 x = θy,x x ,e3 x = −θz,x x ,e4 x =wx x + θy x ,e5 x = vx x − θz x ,e6 x = θx,x x

3

The six kinematic variables u can be interpolated by theright shape functions

u =N x ue, 4

where N x and ue denote the shape function and nodaldegrees of freedom, respectively. Substituting (4) into (3)gives arbitrary section strains in terms of the nodal degreesof freedom as follows:

e u = B x ue, 5

where the matrix B x = B1 x , B2 x ,…, B6 x containsthe derivatives of the shape functions N x . Once the sectionstrains e u are obtained, the nodal displacement ue isdetermined; then, the kinematic variables u can be acquiredby (4). However, the section strains e u are derived fromthe kinematic variables u theoretically, rather than the strainmeasurements. So, iFEM uses in situ section strains eεcomputed from the measured strains to replace e u whenthe least squares error function φ u reaches the minimum.

φ u = e u − eε 2, eε = eε1, eε2, eε3, eε4, eε5, eε6 T 6

In view of the effect of the axial stretching, bending,twisting, and transverse shearing, the improved leastsquares error functionalφe u is obtained by the dot productof the weighting coefficient vector W and the original vectorφ u = φe

k .

W = wk = w01, w0

2IeyAe , w0

3IezAe , w0

4,w05, w0

6IepAe ,

φe u =W ⋅φ u ,

φek =

Ln〠n

i=1ek xi − eεk xi

2, k = 1, 2,…, 6 ,

7

where w0k k = 1, 2,…, 6 denote dimensionless weighting

coefficients whose initial values are identically set as 1; Ae,Iey and Iez , and Iep are, respectively, the cross-sectional area,second moments of the area according to the y- and z-axes,and polar moment of the area of the beam element. L is thelength of the beam element; xi 0 ≤ xi ≤ L and n are, respec-tively, the axial coordinate of the locations where the sectionstrains are evaluated and the number of locations, that is, theaxial coordinate of sections where section strains are distrib-uted in, and the number of sections.

For a straight beam member of constant circle crosssection, the in situ section strains at x = xi, eε xi =eε1 xi , eε2 xi ,…, eε6 xi

T , could be derived from the mea-sured strains of the beam surface with appropriate strain-tensor transformations from the θ, x, r to xX , xY , xZcoordinates [22].

ε∗2 xi, θ, β = eε1 xi c2β − vs2β + eε2 xi c2β − vs2β sθR

+ eε3 xi c2β − vs2β cθR + eε4 xi cβsβcθ

− eε5 xi cβsβsθ + eε6 xi cβsβR,

8

with cβ ≡ cos β, sβ ≡ sin β, cθ ≡ cos θ, and sθ ≡ sin θ

In (8), v is the Poisson ratio, ε∗2 xi, θ, β is the in situstrains that are obtained from strain sensors. R denotes theradius of the beam cross section (Figure 2).

Substituting (5) into (7) results in the following quadraticform:

φe u = 12 ue Tkeue − ue Tke + ce 9

Herein, ce is a constant, and ke and fe are indicated asfollows:

ke = 〠6

k=1wkk

ek, kek =

Ln〠n

i=1BTk xi Bk xi ,

fe = 〠6

k=1wkf

ek, f ek =

Ln〠n

i=1BTk xi e

εk xi

10

Finally, the relationship between the deformation and insitu section strains, shown in (11), can be confirmed whenthe minimization of functional φe u is performed.

keue = fe 11Once the appropriate shape functions and the problem-

dependent displacement boundary conditions (e.g., settingthe displacement of one end point to zero, which means thatone of end nodes of the beam is fixed) are given, ue can bederived from a nonsingular system, and the vector fedepends on the measured strain values that change during

R ≡ x3

R

X

Y

Z

⁎

x1

𝛽

𝜃𝜃

𝛽x

⁎

x2⁎

Figure 2: Location and coordinates of a strain gauge placed on thebeam external surface.

3International Journal of Aerospace Engineering

deformation. Once the nodal degrees of freedom ue are con-firmed, the displacements and rotations of every node alongthe centroid axis of the beam element are obtained by (4)and the deformed shape of the whole beam can be recon-structed by (1).

3. Minimum Number of the Section StrainLocation Selection

Although the displacement shape function and the numberof surface strain measurements in different loading caseshave been discussed in [20, 21], n, the number of the sectionswhere the section strains are distributed in, is not shownclearly. In (11), once the problem-dependent displacementboundary conditions are determined, the nonsingularityof the equation will depend on the selection of n. In ourinvestigation, it is found that n can be determined by aspecific loading case. More specifically, the orders of thesection forces and the moments can be obtained by theloading case, which determines the order of the sectionstrains. With the value of the section strain order, n isdetermined immediately.

With the equilibrium equations (12), the section forces(N , Qy, and Qz) and moments (Mx, My , and Mz) can beobtained by the load concentrations qx x , qy x , and qz xalong the x, y, and z directions (Figure 3).

Nx + qx x = 0,Qy,x + qy x = 0,Qz,x + qz x = 0,

Mx,x = 0,My,x −Qz = 0,Mz,x −Qy = 0

12

The relationship between section forces and momentsand the section strains ei i = 1, 2,…, 6 can be interpretedas the following constitutive equations:

N = Axe1,Mx = Jxe6,Qy = Gye5,My =Dye2,Qz = Gze4,Mz =Dze3

13

where Ax ≡ EA is the axial rigidity; Gy ≡ k2yGA and Gz ≡ k2zGA

are the shear rigidities, with k2y and k2z denoting the shear cor-

rection factors; and Jx ≡ GIp and Dy ≡ EIy and Dz ≡ EIz are,respectively, the torsional rigidity and the bending rigidi-ties. For the uniform section beam element, parametersmentioned above are constant. Then, the order of the sec-tion strains ei will be identical to that of the section forcesand moments.

Substituting (13) into (12) gives the relationship betweenthe section strains and the loads as follows:

e1 =NAx

= −qx x dxAx

,

e2 =My

Dy= Qzdx

Dy= −qz x dxdx

Dy,

e3 =Mz

Dz=

Qydx

Dz=

−qy x dxdx

Dz,

e4 =Qz

Gz= −qz x dx

Gz,

e5 =Qy

Gy=

−qy x dx

Gy,

e6 =Mx

Jx= 0dx

Jx= c1

14

In most cases, the load in the x direction is zero, that is,qx x = 0; then,

e1 =−qx x dx

Ax= 0dx

Ax= c2 15

The section strains e1 and e6 are constant in (14) and(15), and the order of the residual section strains, e2, e3, e4,and e5, will be discussed as the following two loading cases.

Consider a beam element loaded by the end-node forces,qy x and qz x are zero along the x-axis. Then, e2, e3, e4,and e5 are determined by (14).

e2 =−qz x dxdx

Dy= 0dxdx

Dy= a1x + b1,

e3 =−qy x dxdx

Dz= 0dxdx

Dz= a2x + b2,

e4 =−qz x dx

Gz= 0dx

Gz= a3,

e5 =−qy x dx

Gy= 0dx

Gy= a4

16

where a1, a2, a3, a4, b1, and b2 are unknown constantparameters.

As in (15), the highest order of the section strains e u islinear e2, e3 , which means that the distribution of the bend-ing moments is a skew line, that the corresponding errors in(7) can be obtained from the shaded area between two skewlines, and that each line can be confirmed by two differentnodes, that is, two section strains in different sections

Mz My

Mx

Qy

N

QzZ

Y

X

Figure 3: Beam section forces and moments.


(Figure 4(a)). Thus, n = 2 for e2 and e3 in (7). Meanwhile, theother section strains, e1, e4, e5, and e6, are constant, whichmeans that their distributions are lines parallel to the x-axis,that the associative errors in (7) can be obtained from theshaded area between two parallel lines, and that every linecan be confirmed by one node, that is, one section strain(Figure 4(b)). Thus, the number of the sections where sectionstrains are distributed is n = 1. Finally, the number of the sec-tions, n, is 2 in (7) and (10) in the loading case of end-nodeforce and moments.

Another case is the beam element loaded by uniformlydistributed forces, qy x and qz x are constant along thex-axis, and the order of the section strains e2, e3, e4, ande5 can be deduced as follows.

Assumption: qy x = d1, qz x = d2.Then,

e2 =−qz x dxdx

Dy= −d2 dxdx

Dy= −d22Dy

x2 + d3x + d4,

e3 =−qz x dxdx

Dz= −d1 dxdx

Dz= −d12Dz

x2 + d5x + d6,

e4 =−qz x dx

Gz= −d2 dx

Gz= −d2x

Gz+ d7,

e5 =−qy x dx

Gy= −d1 dx

Gy= −d1x

Gy+ d8

17

Herein, di i = 1,…, 8 are unknown constant parame-ters. The highest order of the section strains e u is quadratice2, e3 , and then, the distribution of the bending moments isa parabola. The corresponding errors in (7) can be obtainedfrom the shaded area between two parabolas which are con-firmed by three different nodes (Figure 4(c)), that is, threesection strains in three different sections. Thus, n = 3 for e2and e3 in (7). Meanwhile, e4 and e5 are linear, where the dis-tribution is a skew line, and e1 and e6 are constant. Similarto the loading case of end-node force, the correspondingnumbers of sections are 1 and 2, respectively. Finally, thenumber of the sections, n, is 3 in (7) and (10) in the loadingcase of uniformly distributed forces.

4. Verification

A simple cantilevered solid beam and a three-dimensionalframe structure were subjected to the end-node static loadsto assess the iFEM potential for the flexible wing deformationestimation. The beam and frame structures were made of6061-T6 aluminum alloy. The Young’s modulus isE=73000Mpa, the Poisson ratio is v = 0 3, and its densityis ρ = 2712 63kg/m3. The frame was composed of three solidbeams and middle plates; the span of each beam L=640mmand the radius R=10mm (see Figure 5). For the solid circularcross section, the shear correction factors are k2y = k2z = 0 887[21]. In our verification, every whole solid beam is regardedas one beam element. Accordingly, the principle axis of framestructure is divided into three elements.

The experimentally measured surface strains areobtained by fiber optic strain sensors. Displacement mea-surements are captured by 3D optical measurement instruc-tion (NDI, Optotrak Certus), which determines the positionof the identification point by using three CCD cameras tocapture the infrared lights emitted by position sensors(Figure 6). The instruction is also used to assess the iFEM-recovered deflections, where the accuracy of the NDI is0.1mm. A position sensor was placed as close as possible to

(a) Clamped-free beam

and loading

A

(b) Identification

points of NDI

(c) Frame structure and loading

Figure 5: Loading case and the distributions of NDI identificationpoints.

Error

XL

0

Section strain

e𝜀

e

(a) Constant section strain

Error

XL

0

Section strain

e𝜀

e

(b) Linear section strain

Error

XL

0

Section strain e𝜀

e

(c) Quadratic section strain

Figure 4: Distribution and error of section strains in different orders. eε is a section strain in eε computed from surface strain measurementsby (8). e is a section strain in e u deduced by (3).


the beam root to verify the effectiveness of the clampingarrangement. The force is achieved by placing several weightson a pothook.

For the frame structure, each beam is regarded as anelement and the optical fiber sensor location scheme usedfor each element is identical. For the end-node staticloads, the displacement field of the whole beam elementis interpolated by C0 continuous shape function and thenumber of required strain sensors is 6 (see [21]). As theradius of the beam element is small (R=10mm) and the gridlength of every fiber grating sensor is 10mm, it is difficultto stick the six strain gauges on one section; six fiber opticstrain gauges are placed on two different sections alongthe beam (Table 1).

The accuracy of the reconstitution is assessed by root-mean-square (RMS) and percentage difference (%Diff).

RMS = 〠m

i=1τiFEM xi − τNDI xi

22 ,

%Dif f τ = 100∗ RMSmax τNDI xi

,18

where τ = v, w is the deformation displacement along they- or z-axis and m is the number of deflection shapedisplacement measurement identification points of NDI(Figure 5(b)A) in the structure. For the beam, m = 6(Figure 5(a)), and for the frame, m = 14 (Figure 5(c)).The superscript “iFEM” refers to the reconstitution byiFEM while “NDI” refers to the experimental measure from3D optical measurement instruction; xi is the ith locationalong the axis where the displacement u is measured.

At first, five different vertical force cases (Table 2)were applied at the free end of the beam (Figure 5(a)).Figure 7 shows the comparison between iFEM recon-struction and NDI measurement along the beam axis inforce direction, for the loading case 68N. Symbol x/Lindicates the location of identification point along thebeam surface. Zero means identification point at theclamped end node, and 1 means identification point at thefree end node. v(y) indicates the corresponding displacementin force direction.

0 0.2 0.4 0.6 0.8 1‒12

‒10

‒8

‒6

‒4

‒2

0

x/L

v(y

) (m

m)

NDI measuredIFEM computed

Figure 7: Comparison of iFEM reconstruction for a beam to NDImeasurement in Y.

Table 2: Loading case for the clamped-free beam. Max deflections are captured from NDI.

Case 1 Case 2 Case 3 Case 4 Case 5

Loads 5N 15N 37N 41N 68N

Max deflection in Y −0.79mm −2.5mm −6.11mm −6.82mm −10.96mm

RMS in Y 0.05mm 0.16mm 0.32mm 0.33mm 0.42mm

%Diff(v) in Y 6% 6.4% 5.2% 4.8% 3.8%

Table 1: Optical fiber sensor location in one element.

eε1 x, θ, β eε2 x, θ, β eε3 x, θ, β eε4 x, θ, β eε5 x, θ, β eε6 x, θ, βL/2, −2π/3, 0 L/2, 0, 0 L/2, 2π/3, 0 4L/5, −2π/3, 0 4L/5, 0, π/4 4L/5, 2π/3, 0

B

A

C

Figure 6: Static test setup. The single beam (A), frame structure (B), and NDI Optotrak Certus (C).


For the frame structure, five different vertical force cases(Table 3) were applied at the free end of the frame(Figure 5(c)). The two deformation contour plots of theframe are plotted by high-fidelity direct FE analysis (ANSYS12.0), which shows the comparison between iFEM recon-struction and NDI measurement along the beam axis in forcedirection, for the loading case 45.5N (Figure 8). The defor-mation unit in the two figures is meter.

It is seen from the preceding results that the iFEMmethodology shows a good potential as a reliable estimationtechnique. The estimation results gained from the iFEMalgorithm have a better approximation to the NDI measure-ment results. For the beam test, the accuracy of the reconsti-tution assessed by percentage difference does not exceed6.4% and the error brings down with the loading increase.For the frame test in the loading case of 18N, the percentagedifference is 1.3%, but the difference grows up with the load-ing increase; especially from the loading case 23N to 35N,the difference increases 4%. The cause for this phenomenonis that the gripper of the frame structure is not very stable,which leads to the fact that the gap spacing between theframe and its carrier grows up with the loading increase thathas a great influence on the boundary conditions.

5. Conclusion

This study investigates the application potential of the iFEMalgorithm for the flexible wing and other frame structures.This method employs the in situ strain measurements andproper displacement shape function to estimate the deforma-tion shape of the beam and frame structure, without the needto know the applied loads and material properties and to usemodal shapes. When the boundary conditions of the recon-struction equation and the shape function are determined,the deformation shape will be accurately estimated in a non-singular system. Although the boundary conditions areknown in hypothesis, the number of location where the sec-tion strains are evaluated is not shown clearly; thereby, thesingularity of the equation will be influenced.

This paper discusses the minimum number of locationwhere the section strains are evaluated with the loading case,using equilibrium equations in detail, and verifies the resultthat for the loading case of end-node force, the number is2. The results of the static loads in the beam and the framemodel show that the iFEM algorithm has a good potentialfor the flexible frame structure estimation; especially for asimply supported beam, the accuracy of the estimation by

Table 3: Loading case for the frame structure. Max deflections are captured from NDI.

Case 1 Case 2 Case 3 Case 4 Case 5

Loads 18N 23N 35N 45.5N 68N

Max deflection in Y 16.31mm 20.5mm 31.2mm 41.3mm 62.8mm

RMS in Y 0.22mm 0.49mm 2.0mm 3.3mm 6.8mm

%Diff(v) in Y 1.3% 2.4% 6.4% 8.1% 10.9%

Max deflection in Z 2.53mm 3.23mm 5.1mm 6.5mm 10.2mm

RMS in Z 0.32mm 0.41mm 0.73mm 0.75mm 1.1mm

%Diff(w) in Z 12.6% 12.8% 14.6% 11.5% 10.8%

Displacement Max displacement = 37 mm

4.1

Nodal solution

USUM (AVG)

1

Step = 1Sub = 1Time = 1

RSYS = 0DMX = 0.037019SMX = 0.037019

8.2 12.3 16.5 20.6 24.7 28.8 32.9 37

Unit: mm

ANSYS

X

Y

Z

Load

(a) Contour plot of iFEM reconstruction

Nodal solution1

USUM (AVG)

Step = 1Sub = 1Time = 1

RSYS = 0DMX = 0.045279SMX = 0.045279

5 10.1 15.1 20.1 25.2 30.2 35.2 40.2 45.3

ANSYS

Displacement Max displacement = 37 mm Unit: mm

Y

X

Z

Load

(b) Contour plot of NDI measurement

Figure 8: Comparison of iFEM reconstruction for a frame to NDI measurement.


iFEM is around 6%, and for the frame structure in small load(18N), the accuracy reaches up to 1.3%, while as the gripperof the frame is not firm enough, the estimation accuracybrings down with the loading increase. So, the next work isto explore the technology that reduces the influence of thegap spacing of the frame gripper on accuracy of the framedeformation estimation.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

Acknowledgments

This work was financially supported by the National NaturalScience Foundation of China (Grants 51675398 and51775401), the National Key Basic Research Program ofChina (Grant 2015CB857100), and the CAS “Light of WestChina” Program (no. 2016-QNXZ-A-7).

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