3D Crack Detection Using an XFEM Variant and GlobalOptimizaion Algorithms

K. Agathos1 E. Chatzi2 S. P. A. Bordas1,3,4

1Research Unit in Engineering ScienceLuxembourg University

2Institute of Structural EngineeringETH Zurich

3Institute of Theoretical, Applied and Computational MechanicsCardiff University

4Adjunct Professor, Intelligent Systems for Medicine Laboratory, The University of WesternAustralia

June 1, 2016

K. Agathos et al. XFEM based crack detection 1/6/2016 1 / 21

Outline

1 Inverse problem formulation

2 Global enrichment XFEM

3 Parametrization and constraints

4 Numerical examples

5 Conclusions

K. Agathos et al. XFEM based crack detection 1/6/2016 2 / 21

Background

Nondestructive evaluation (NDE)Methods used to examine an object, material or system without impairingits future usefulness

Available Techniques:impedance tomography,radiography, ultrasounds,acoustic emission

SHM - Damage Detection:Monitor changes in the dynamicproperties of a structure

K. Agathos et al. XFEM based crack detection 1/6/2016 3 / 21

Inverse problem

→ Detection of cracks in existing structures

→ Measurements are available

→ A computational model is employed

→ The difference between the two is minimized

→ Information regarding the cracks is obtained

K. Agathos et al. XFEM based crack detection 1/6/2016 4 / 21

Inverse problemMathematical formulation:

Find βi such thatF (r (βi ))→ min

where

βi Parameters describing the crack geometry

r (·) Norm of the difference between measurements and computedvalues

F Some function of the residual

The CMA-ES algorithm is employed to solve the problem.K. Agathos et al. XFEM based crack detection 1/6/2016 5 / 21

Inverse problem

Solution process:

→ Generation of initial population (βi ) with CMA-ES

→ Fitness function (F (r (βi ))) evaluation using XFEM andmeasurements

→ Population is updated with CMA-ES

→ The procedure is repeated until convergence

K. Agathos et al. XFEM based crack detection 1/6/2016 6 / 21

Inverse problem

During the optimization proccess:

A large number of crack geometries is tested

The computational model is solved several times

An efficient and robust method is required

K. Agathos et al. XFEM based crack detection 1/6/2016 7 / 21

XFEM

FEM vs XFEM for fracture:

FEM⇒

XFEM⇒

No crack

Crack 1 Crack 2

K. Agathos et al. XFEM based crack detection 1/6/2016 8 / 21

XFEM

FEM vs XFEM for fracture:

FEM⇒

XFEM⇒

No crack Crack 1

Crack 2

K. Agathos et al. XFEM based crack detection 1/6/2016 8 / 21

XFEM

FEM vs XFEM for fracture:

FEM⇒

XFEM⇒

No crack Crack 1 Crack 2

K. Agathos et al. XFEM based crack detection 1/6/2016 8 / 21

XFEM approximation

XFEM approximation:

u (x) =∑∀I

NI (x) uI︸ ︷︷ ︸FE approximation

+∑∀I

N∗I (x) Ψ (x) bI︸ ︷︷ ︸enriched part

where:

NI (x) are the FE shape functions

uI are the nodal displacements

N∗I (x) are functions forming a PU

Ψ (x) are the enrichment functions

bI are the enriched dofs

K. Agathos et al. XFEM based crack detection 1/6/2016 9 / 21

Enrichment functions

Jump enrichment functions:

H(φ) ={

1 for φ > 0− 1 for φ < 0

Tip enrichment functions:

Fj (r , θ) =[√

r sin θ2 ,√

r cos θ2 ,√

r sin θ2 sin θ,√

r cos θ2 sin θ]

K. Agathos et al. XFEM based crack detection 1/6/2016 10 / 21

XFEM

Some drawbacks of XFEM:

The use of tip enrichment in a fixed area around the crack front(geometrical enrichment) is required for optimal convergence

The use of geometrical enrichment causes conditioning problems

Blending problems the enriched and the standard part of theapproximation

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Global enrichment XFEM

An XFEM variant is employed which:

Enables the application of geometrical enrichment to 3D

Employs weight function blending

Employs enrichment function shifting

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Global enrichment XFEM

Special front elements are introduced:

crack surface

crack front

front elementboundary

front node front element

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Global enrichment XFEM

Front element shape functions:

boundaryfront elementnode

front element

= 2η

= 3η

1−=ξ

= 0ξ

= 1ξ21=ξ

21−=ξ

Ng (ξ) =[1− ξ

21 + ξ

2

]

K. Agathos et al. XFEM based crack detection 1/6/2016 14 / 21

Global enrichment XFEMDisplacement approximation:

u (x) =∑I∈N

NI (x) uI + ϕ (x)∑

J∈N j

NJ (x) (H (x)− HJ)bJ+

+ ϕ (x)

∑K∈N s

NgK (x)

∑j

Fj (x)−∑

T∈N tNT (x)

∑K∈N s

NgK (xT )

∑j

Fj (xT )

cKj

where:

ϕ, ϕ are weight functions

NgK are front element shape functions

HJ ,Fj are nodal values of the enrichment functionsK. Agathos et al. XFEM based crack detection 1/6/2016 15 / 21

Problem parametrization

Elliptical cracks are considered:

x

y

z

0x

2tn

1t

a

b

Parameters:

Coordinates of centerpoint x0 ({x0, y0, z0})

Rotation about the threeaxes θx , θy and θz

Lengths a and b

K. Agathos et al. XFEM based crack detection 1/6/2016 16 / 21

Problem parametrization

Scaling of parameters:

pi = pi1 + pi22 + pi2 − pi1

2 sin(βi10 ·

π

2

)where:

βi are design variables

pi are geometrical parameters of the crack

pi1 , pi2 are lower and upper values for the parameters

K. Agathos et al. XFEM based crack detection 1/6/2016 17 / 21

Penny crack in a cube

Geometry and sensors:

a

xL

yLzL

Sensor locations

4xL

4xL

4xL

4xL 4

yL4

yL4

yL4

yL

4zL

4zL

4zL

4zL

K. Agathos et al. XFEM based crack detection 1/6/2016 18 / 21

Penny crack in a cubeOptimization problem convergence:

evaluations500 1000 1500 2000

fitn

ess

func

tion

10-2

10-1

100

K. Agathos et al. XFEM based crack detection 1/6/2016 19 / 21

Penny crack in a cubeBest solution after different numbers of iterations

Actual crack

Detected crack

Initial guess 500 evaluations 1000 evaluations

1500 evaluations 2000 evaluations

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Conclusions

→ A 3D crack detection scheme was presented

→ Promising results were obtained

→ Extension to practical problems would increase computational cost

→ Computational cost of forward problem solutions should be reduced

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