deformed shape and stress reconstruction in plate and ...deformed shape and stress reconstruction in...

National Aeronautics and Space

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Deformed Shape and Stress Reconstruction in Plate and Shell Structures Undergoing Large Displacements: Application of Inverse Finite Element Method using Fiber-Bragg-Grating Strains

A. Tessler, J. Spangler, M.Gherlone, M. Mattone, and M. Di Sciuva NASA Langley Research Center, U.S.A. & Politecnico di Torino, Italy

WCCM 2012 Sao Paulo, Brazil (8-13 July 2012)

Inverse problems of wing deflection

FBG (Fiber Bragg Grating) sensor is glued on top of wing to measure surface strain along axis (NASA Dryden)

• Using discrete strain

measurements, ɛ ɛ, determine

full-field solutions for

‒ displacements u

‒ strains ɛ (u)

‒ stresses Ϭ(u)

• Ill-posed problem

• Uniqueness

• Stability

Inverse FEM

Variational Formulation based on First-Order Shear Deformation Theory (Mindlin)

Kinematic Assumptions of First-Order Shear Deformation Theory

• Deformations

– Membrane

– Bending

– Transverse shear

( , ) ( ) ( )

( , ) ( ) ( )

( ) ( )

x y

y x

z

u z u z

u z v z

u w

x x x

x x x

x x

( , ) (strain-measurement directions)

[ , ] (thickness coordinate)

x y

z t t

x

2t

z, w

y, v y

x

x, u

4

Strain rosettes or FBG fiber along x direction

• Displacement components

Top-surface measured strains

Bottom-surface measured strains

xx

yy

xy

xx

yy

xy

2t

z

Reference frame (aligned with strain-measurement directions)

Strains and Section Strains

1 4

2 5

3 6

xx

yy

xy

z

5

• Section strains

1

2

3

0 0 0 0

0 0 0 0

0 0 0

x

m

y

xy x

y

u

v

w

e u L u

4

5

6

0 0 0 0

0 0 0 0

0 0 0

x

b

y

xx y

y

u

v

w

k u L u

3 membrane section strains

3 bending section strains

7

8

0 0 0 1

0 0 1 0

x s

y

x

y

u

v

w

g u L u

• Inplane strains (=6) • Transverse-shear strains (=2)

Strain measurements relate to membrane & bending section strains

4

5

6

1

2

xx xx

i yy yy

xy xy

t

k

1

2

3

1

2

xx xx

i yy yy

xy xy

e

top rosette

bottom rosette

xx

yy

xy

xx

yy

xy

2t

z

1 4

2 5

3 6

xx

yy

xy

z

Express measured strains in terms of FSDT

Evaluating at top and bottom ( )

Surface strains measured at location

z t

6

x

7

8

Cannot be obtained from surface strains

iFEM variational formulation

Minimize an element functional (a weighted least-squares

smoothing functional ) with respect to the unknown displacement degrees-

of-freedom

7

2 2 2

( )h h h h

e e k gw w w u e u e k u k g u g

where the squared norms are

2 2

1

22 2

1

2 2

1

1( ) x y,

(2 )( ) x y,

1( ) x y

e

e

e

nh h

i iA

i

nh h

i iA

i

nh h

i iA

i

d dn

td d

n

d dn

e u e e u e

k u k k u k

g u g g u g

Positive valued weighting constants associated with individual section strains (=8). They place different importance on the adherence of strain components to their measured values.

( , , )e k gw w w

n Number of strain sensors per element

( )h

e u

• Variational statement

iFEM matrix equations

Symmetric, positive definite matrix

dofu

( )iK x

Nodal displacement vector

( )f εRHS vector, function of measured strain values

1dof

( ) 0N

h

e

e

u

u

dof K u f

• Linear Eqs (displ. B.S.’s prescribed)

• iFEM integrates and smoothes strain data

• Higher accuracy than forward FEM

1dof

u K f

• Displacement solution

iFEM’s selective, element-level (local) regularization

1. An element is missing measured transverse-shear section strains

(standard case); Let

9

2

2( ) x y ( ; 1)e

h h

g e kA

d d w w w g u g u

2. An element is missing all measured section strains (in addition to (1))

22

22 2

( ) x y ( )

(2 ) ( ) x y ( )

e

e

h h

eA

h h

kA

d d w

t d d w

e u e u

k u k u

3. An element is missing some measured-strain components

‒ apply forms (2) to the missing components only

Important special cases

410 (small positive constant)

Simple and efficient inverse-shell element: iMIN3

• Anisoparametric interpolations (Tessler-Hughes, CMAME 1985)

, , , : linear shape functions

: quadratic

x yu v

w

10

, ( ) : constant

: linear

h h

h

e u k u

g u

• Section-strain fields

• 3 nodes, 5 or 6 dof/node

( , , )

[ , ]

x y z

z t t

x

2t

z, w

y, v y

x

x, u

h

, , , , : 5 dof/plate

, , , , , : 6 dof/shell

x y

x y z

u v w

u v w

Demonstration problem

11

FEM shell model: Aluminum stiffened flap with two rectangular cut-outs

12

F = (1, 50)

Un-Deformed

Deformed

Elastostatic deformations (ABAQUS/STRI3 (Batoz) 3-node element, no shear deformation; 6 section strains only)

causes geometrically nonlinear response

12 in

• Model has 10 planar element groups

• Each group has its own material reference frame to define strain orientations

3 iFEM modeling and stabilization schemes

13

Model A: Six FEM section strains are mapped onto all iFEM elements • One-to-one (high-fidelity) • All elements have strain data but no

shear strain measurements

Model B: Six FEM section strains are mapped onto perimeter iFEM elements • Simulates tri-axial strain rosettes

along the perimeter • Interior elements have no strain

data including the stiffener (local regularization)

Model C: Two FEM section strains (axial) are mapped onto perimeter iFEM elements • Simulates linear strain gauges or

FBG strain sensors • Incomplete strain data • Interior elements have no strain

data including the stiffener (local regularization)

Axial strain measurements only in perimeter elements

Tri-axial strain measurements in perimeter (red) elements

Tri-axial strain measurements in every element

Linear problem: % error in reconstructed displacement, uz

14

Model A: Six FEM section strains are mapped onto all iFEM elements

Model B: Six FEM section strains are mapped only onto perimeter iFEM elements

Model C: Two FEM section strains (axial) are mapped only onto perimeter iFEM elements

Ref i Est i

Max

Ref

u u% Error (u ) =

u100z

zz

z

0.45%

0.70%

1.1%

Linear problem: Deviations in uz

15

iFEM 0% noise 5% noise model in strains in strains A 1.00000 0.99998 B 0.99999 0.99998 C 0.99998 0.99985

Pearson correlation, r

iFEM 0% noise 5% noise model in strains in strains A 0.00017 0.00096 B 0.00020 0.00074 C 0.00035 0.00098

RMS

iFEM 0% noise 5% noise model in strains in strains A 0.1951 1.0505 B 0.1934 0.8353 C 0.3575 1.0769

Mean % error

Linear problem: % error in reconstructed von Mises stress (bottom shell surface)

16

Model A: Six FEM section strains are mapped onto all iFEM elements

Model B: Six FEM section strains are mapped only onto perimeter iFEM elements

Model C: Two FEM section strains (axial) are mapped only onto perimeter iFEM elements

Ref i Est i

Max

Ref

% Error ( ) = 100

Linear problem: Deviations in von Mises stress

17

iFEM 0% noise 5% noise model in strains in strains A 0.9994 0.9993 B 0.9842 0.9844 C 0.9903 0.9896

Pearson correlation, r

iFEM 0% noise 5% noise model in strains in strains A 4.0149 5.72724 B 21.9314 23.4736 C 15.7996 16.9735

RMS

N

Ref  i Ref EST i Esti 1

N N2 2

Ref  i Ref Est  i Esti 1 i 1

( )r  

  ( )

2

REF i EST i1σ σ  

N

iRMSN

Ref i Est i

M x

ef1 a

R

σ σ

σ

1100

N

iN

• Mean % error

• Pearson correlation, r

• Root-Mean-Square error

iFEM 0% noise 5% noise model in strains in strains A 0.3786 0.5553 B 1.6796 1.8404 C 1.3501 1.5133

Mean % error

iFEM incremental algorithm for nonlinear deformations

• Use Nonlinear FEM as a virtual experiment (Lagrangean reference

frame)

– At each load increment of NL-FEM, compute the incremental section

strains (6 components) that represent measured strain increments

– Perform iFEM analysis using the strain increments to obtain the

displacements and rotations

– Update the geometry of iFEM mesh due to deformation using iFEM

determined displacements, i.e., x1=x0+u1

– Perform iFEM using the strain increments of the next load increment, and

update the geometry for the next step x2=x1+u2

18

Nonlinear problem, F=50: Displacement magnitude (full load)

19

Model C: Two FEM section strains (axial) are mapped only onto perimeter iFEM elements (simulating FBG strains)

Reference: Nonlinear FEM/ABAQUS (STRI3)

Max Max

Ref(1-u /u ) 100% 1.2%

FEM

iFEM (Model C)

No measured strains in the stiffener

Nonlinear problem: Displacement magnitude (full load) of the stiffener

20

Model C: Two FEM section strains (axial) are mapped only onto perimeter iFEM elements (simulating FBG strains)

Reference: Nonlinear FEM/ABAQUS (STRI3)

Max Max

Ref(1-u /u ) 100% 1.5%

FEM

iFEM (Model C)

Nonlinear problem: Von Mises stress (full load)

21

Model C: Two FEM section strains (axial) are mapped only onto perimeter iFEM elements (simulating FBG strains)

Reference: Nonlinear FEM/ABAQUS (STRI3)

FEM

iFEM (Model C)

Max Max

Ref(1- / ) 100% 8.2%

Summary

• On-board SHM of nextgen aircraft, spacecraft, large space

structures, and habitation structures

– Safe, reliable, and affordable technologies

• Inverse FEM algorithms for FBG strain measurements

– Real-time efficiency, robustness, superior accuracy

– Stable full-field solutions

• Inverse FEM theory

– Strain-displacement relations & integrability conditions

fulfilled

– Independent of material properties

– Solutions stable under small changes in input data

– Linear and nonlinear response

22

Summary (cont’d)

• Inverse FEM’s architecture/modeling

– Architecture as in standard FEM (user routine in ABAQUS)

– Superior accuracy on coarse meshes

– Frames, plates/shell and built-up structures

– Thin and moderately thick regime

– Low and higher-order elements

• Inverse FEM applications

– Computational studies: plate and built-up shell structures

– Experimental studies with plates: FBG strains and strain

rosettes

23

Summary (cont’d)

• Inverse FEM’s architecture/modeling

– Architecture as in standard FEM (user routine in ABAQUS)

– Superior accuracy on coarse meshes

– Frames, plates/shell and built-up structures

– Thin and moderately thick regime

– Low and higher-order elements

• Inverse FEM applications

– Computational studies: plate and built-up shell structures

– Experimental studies with plates: FBG strains and strain

rosettes

24