3D Crack Detection Using an XFEM Variant and Global Optimizaion - - PowerPoint PPT Presentation

3D Crack Detection Using an XFEM Variant and Global Optimizaion Algorithms

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

1Research Unit in Engineering Science

Luxembourg University

2Institute of Structural Engineering

ETH Z¨ urich

3Institute of Theoretical, Applied and Computational Mechanics

Cardiff University

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

Australia

June 1, 2016

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Outline

1

Inverse problem formulation

2

Global enrichment XFEM

3

Parametrization and constraints

4

Numerical examples

5

Conclusions

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Background

Nondestructive evaluation (NDE) Methods used to examine an object, material or system without impairing its future usefulness Available Techniques: impedance tomography, radiography, ultrasounds, acoustic emission SHM - Damage Detection: Monitor changes in the dynamic properties of a structure

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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

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Inverse problem

Mathematical formulation: Find βi such that F (r (βi)) → min where βi Parameters describing the crack geometry r (·) Norm of the difference between measurements and computed values F Some function of the residual The CMA-ES algorithm is employed to solve the problem.

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Inverse problem

Solution process: → Generation of initial population (βi) with CMA-ES → Fitness function (F (r (βi))) evaluation using XFEM and measurements → Population is updated with CMA-ES → The procedure is repeated until convergence

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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

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XFEM

FEM vs XFEM for fracture: FEM ⇒ XFEM ⇒ No crack

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XFEM

FEM vs XFEM for fracture: FEM ⇒ XFEM ⇒ No crack Crack 1

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XFEM

FEM vs XFEM for fracture: FEM ⇒ XFEM ⇒ No crack Crack 1 Crack 2

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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

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Enrichment functions

Jump enrichment functions: H(φ) =

• 1

for φ > 0 − 1 for φ < 0 Tip enrichment functions: Fj (r, θ) =

√r sin θ

2, √r cos θ 2, √r sin θ 2 sin θ, √r cos θ 2 sin θ

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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 the approximation

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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 element boundary front node front element

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Global enrichment XFEM

Front element shape functions:

boundary front element node front element = 2 η = 3 η

1 − = ξ = 0 ξ = 1 ξ

2 1

= ξ

2 1

− = ξ

Ng (ξ) =

1 − ξ

2 1 + ξ 2

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Global enrichment XFEM

Displacement approximation:

u (x) =

• I∈N

NI (x) uI + ¯ ϕ (x)

• J∈N j

NJ (x) (H (x) − HJ)bJ+ + ϕ (x)

 

K∈N s

Ng

K (x)

• j

Fj (x) −

• T∈N t

NT (x)

• K∈N s

Ng

K (xT)

• j

Fj (xT)

  cKj

where: ¯ ϕ, ϕ are weight functions Ng

K are front element shape functions

HJ, Fj are nodal values of the enrichment functions

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Problem parametrization

Elliptical cracks are considered:

x y z x

2

t n

1

t a b

Parameters: Coordinates of center point x0 ({x0, y0, z0}) Rotation about the three axes θx, θy and θz Lengths a and b

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Problem parametrization

Scaling of parameters: pi = pi1 + pi2 2 + pi2 − pi1 2 sin

βi

10 · π 2

• where:

βi are design variables pi are geometrical parameters of the crack pi1, pi2 are lower and upper values for the parameters

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Penny crack in a cube

Geometry and sensors:

a

x

L

y

L

z

L Sensor locations

4

x

L 4

x

L 4

x

L 4

x

L 4

y

L 4

y

L 4

y

L 4

y

L 4

z

L 4

z

L 4

z

L 4

z

L

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Penny crack in a cube

Optimization problem convergence: evaluations

500 1000 1500 2000

fitness function

10-2 10-1 100

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Penny crack in a cube

Best 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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