3D Crack Detection Using an XFEM Variant and Global Optimizaion Algorithms K. Agathos 1 E. Chatzi 2 S. P. A. Bordas 1 , 3 , 4 1 Research Unit in Engineering Science Luxembourg University 2 Institute of Structural Engineering ETH Z¨ urich 3 Institute of Theoretical, Applied and Computational Mechanics Cardiff University 4 Adjunct Professor, Intelligent Systems for Medicine Laboratory, The University of Western Australia June 1, 2016 K. Agathos et al. XFEM based crack detection 1/6/2016 1 / 21
Outline Inverse problem formulation 1 Global enrichment XFEM 2 Parametrization and constraints 3 Numerical examples 4 Conclusions 5 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 impairing its future usefulness SHM - Damage Detection: Monitor changes in the dynamic properties of a structure Available Techniques: impedance tomography, radiography, ultrasounds, acoustic emission 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 problem Mathematical formulation: Find β i such that F ( r ( β i )) → min where β i 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. K. Agathos et al. XFEM based crack detection 1/6/2016 5 / 21
Inverse problem Solution process: → Generation of initial population ( β i ) with CMA-ES → Fitness function ( F ( r ( β i ))) evaluation using XFEM and measurements → 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: No crack FEM ⇒ XFEM ⇒ K. Agathos et al. XFEM based crack detection 1/6/2016 8 / 21
XFEM FEM vs XFEM for fracture: No crack Crack 1 FEM ⇒ XFEM ⇒ K. Agathos et al. XFEM based crack detection 1/6/2016 8 / 21
XFEM FEM vs XFEM for fracture: Crack 2 No crack Crack 1 FEM ⇒ XFEM ⇒ K. Agathos et al. XFEM based crack detection 1/6/2016 8 / 21
XFEM approximation XFEM approximation: � � N ∗ u ( x ) = N I ( x ) u I + I ( x ) Ψ ( x ) b I ∀ I ∀ I � �� � � �� � FE approximation enriched part where: N I ( x ) are the FE shape functions u I are the nodal displacements N ∗ I ( x ) are functions forming a PU Ψ ( x ) are the enrichment functions b I are the enriched dofs K. Agathos et al. XFEM based crack detection 1/6/2016 9 / 21
Enrichment functions Jump enrichment functions: � 1 for φ > 0 H ( φ ) = − 1 for φ < 0 Tip enrichment functions: � √ r sin θ � 2 , √ r cos θ 2 , √ r sin θ 2 sin θ, √ r cos θ F j ( r , θ ) = 2 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 the approximation K. Agathos et al. XFEM based crack detection 1/6/2016 11 / 21
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 K. Agathos et al. XFEM based crack detection 1/6/2016 12 / 21
Global enrichment XFEM Special front elements are introduced: front element front node front element boundary crack front crack surface K. Agathos et al. XFEM based crack detection 1/6/2016 13 / 21
Global enrichment XFEM Front element shape functions: ξ = 1 − 1 = ξ − 2 ξ = 0 1 ξ = 2 = 1 ξ = 2 η � 1 − ξ � 1 + ξ N g ( ξ ) = = 3 2 2 η front element node front element boundary K. Agathos et al. XFEM based crack detection 1/6/2016 14 / 21
Global enrichment XFEM Displacement approximation: � � u ( x ) = N I ( x ) u I + ¯ ϕ ( x ) N J ( x ) ( H ( x ) − H J ) b J + I ∈N J ∈N j � � � � � N g N g c Kj + ϕ ( x ) K ( x ) F j ( x ) − N T ( x ) K ( x T ) F j ( x T ) K ∈N s j T ∈N t K ∈N s j where: ϕ, ϕ are weight functions ¯ N g K are front element shape functions H J , F j are nodal values of the enrichment functions K. Agathos et al. XFEM based crack detection 1/6/2016 15 / 21
Problem parametrization Elliptical cracks are considered: Parameters: n Coordinates of center a t z 2 point x 0 ( { x 0 , y 0 , z 0 } ) t b 1 y Rotation about the three x 0 axes θ x , θ y and θ z x Lengths a and b K. Agathos et al. XFEM based crack detection 1/6/2016 16 / 21
Problem parametrization Scaling of parameters: � β i � p i = p i 1 + p i 2 + p i 2 − p i 1 10 · π sin 2 2 2 where: β i are design variables p i are geometrical parameters of the crack p i 1 , p i 2 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: L z 4 L z 4 L L z z 4 a L z 4 L y 4 L y 4 L y 4 y L L y L L L L 4 x x x x 4 4 4 4 L x Sensor locations K. Agathos et al. XFEM based crack detection 1/6/2016 18 / 21
Penny crack in a cube Optimization problem convergence: 10 0 fitness function 10 -1 10 -2 500 1000 1500 2000 evaluations K. Agathos et al. XFEM based crack detection 1/6/2016 19 / 21
Penny crack in a cube Best solution after different numbers of iterations Initial guess 500 evaluations 1000 evaluations Actual crack 1500 evaluations 2000 evaluations Detected crack K. Agathos et al. XFEM based crack detection 1/6/2016 20 / 21
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 K. Agathos et al. XFEM based crack detection 1/6/2016 21 / 21
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