Satya N. Atluri, UCI Life Cycle of an Aircraft Market - PowerPoint PPT Presentation

Multi-Scale Analysis of Aircraft Multi-Scale Analysis of Aircraft Structural Longevity (R (Research Conducted in the early 1990s) h C d t d i th l 1990 ) Satya N. Atluri, UCI

Life Cycle of an Aircraft Market Requirements Design Design Production Design Prototype Certification AGILE Maintenance O Operations Retirement & Overhauls & Overhauls

Structural Integrity of R t Rotorcraft Components (DTA?) ft C t (DTA?)

Aircraft Fatigue Failure: Loss of Integrity 1988, a Boeing 737-297 serving the flight suffered 4-28-1988 After 89,090 flight cycles on a 737-200, metal fatigue lets the top go in flight extensive damage after an explosive decompression in explosive decompression in flight, but was able to land safely.

Micro Crack Level: 10 -5 m DTALE: MLPG-SGBNM Alternating DTALE: MLPG SGBNM Alternating h c a h c 45 deg h a c h a c

Mega- to Micro-Level Multiple-Scale A Analyses l Finite volume Finite Element Finite Element Micro Panel Methods Cracks Meshless Methods Methods BEM MDO IPPD Inverse Problems AGILE … Global Deformation System Level: Component Level: Micro Crack Level: 1~ 10 -2 m 10 -4 ~ 10 -6 m 10 2 m

Initial Detected Crack Level: 10 -4 m AGILE Alternating Techniques AGILE Alternating Techniques h c a h c 45 deg h a c h a c Thickness: 10 -3 m 10 3 Thi k Initial Crack: 10 -4 m Initial Crack: 10 -4 m Initial Crack: 10 m

Multi-Scale Damage Tolerance for Initially Detectable Cracks Initially Detectable Cracks 0.75 0.75 thickness = 0.09" 0.75 0.75 0.4 0.4 0.4 0.44 1.18 0.44 0 . 4 A B C Rivet Diameter = 5/32 " 5/32 D D Rivet Diameter = doubler thickness = 3/16 " 0.025" Rivet Diameter = 5/32 " skin thickness = 0.063" 0.06 0.056 0.052 0.048 0.044 0.04 0.036 0.032 0.028 0.024 1.712 1.714 1.716 1.718 1.72 1.722 1.724 1.726 1.728 1.73

Micro-Crack Initiation? Simply using continum-stress mechanics p y g 0.06 0.056 0.052 0.048 0.044 0.04 0.036 0.032 0.028 0.024 1.712 1.714 1.716 1.718 1.72 1.722 1.724 1.726 1.728 1.73 Micro Structure Micro-Structure Inclusion Inclusion Shot-peening

AGILE: Model at 10 -6 Level with Continuum Details with Continuum Details AGILE: Boundary surface mesh only, without refining FEM mesh. Higher order boundary- elements fit curved surfaces much better!

AGILE AGILE • Continum Damage Mechanics Continum Damage Mechanics • Anisotropic Damage Mechanics • Grain Boundary Fracture Mechanics G i B d F t M h i • Gradient Theories of Material Behavior • _______________? Far in the Future • Ab Initio Ab Initio……Dislocation Dynamics Dislocation Dynamics • MD • Statistical Mechanics St ti ti l M h i • DFT……..

AGILE (LOCAL): SGBEM-FEM Alternating Alternating (Symmetric Galerkin Boundary Element – FEM Alternating Method) (Overall Accuracies of KI, KII,KIII, Jk are the best of any available method) P P FEM FEM SGBEM SGBEM = + I fi it Infinite body b d Loaded Finite body Loaded Finite body with a crack with a crack without a crack FEM Stiffness matrix inverted only ONCE, Faster!

Why AGILE? Why AGILE? • Accuracy is the best: Accuracy is the best: –State-of-the-art advanced theories & analytical developments are used, in conjunction with the most efficient j computational algorithms. –Most advanced closed-form Most advanced closed form mathematics, and only minimal numerics i

Advanced Theories • Solvers are developed, based on both FEM(for uncracked structure) and SGBEM(for a subdomain w/2- ) ( D or 3-D crack). • SGBEM is developed, using the newly developed weakly-singular BIEs: weakly singular BIEs: – Support higher-order elements for curved surfaces – higher performance and accuracy – Preserve the symmetry of the matrices • FEM & SGBEM are coupled through the Schwartz alternating method: alternating method: – FE mesh, and the SG-BEM crack-model are totally uncoupled – Ease of mesh creation – Very Fast algorithm for automated crack growth FE model is – Very Fast algorithm for automated crack growth, FE model is factorized and solved only once.

AGILE: Faster and more accurate than traditional BIE • Weakly-singular integrals are numerically Weakly singular integrals are numerically tractable, with Gaussian quadrature algorithms using q g g lower order integrations • Higher-order elements with curved sides g can be used, because of its requirement of only C 0 continuity, which is especially useful for ti it hi h i i ll f l f modeling 3D non-planar cracks with less elements elements.

AGILE: More applicable than pure BIE • Built-in FE solver handles more Built in FE solver handles more complicated geometries, including structural elements such as beams structural elements, such as beams, plates, shells, and MPCs. • More efficient for problems with high • More efficient for problems with high volume/surface ratios, for example, thin- walled structures manifold domains and walled structures, manifold domains, and bi-material parts. • 2-D, 2-D/3-D transition, & 3-D modeling of 2 D 2 D/3 D t iti & 3 D d li f structures w/ mixed-mode crack-growth

SGBEM: Fundamental Solutions Solutions 3D Problems Source x Point Point 1      x ξ * p u ( , ) [( 3 4 ) r r ]    i ip , i , p 16 ( 1 ) r r  field field 1 1           x ξ * p u*,  * ( , ) [( 1 2 )( r r r ) 3 r r r ]    ij ij , p ip , j jp , i , i , j , p 2 8 ( 1 ) r 2D Problems 2D Problems 1       x ξ * p u ( , ) [ ( 3 4 ) ln r r r ]    i ip , i , p 8 ( 1 ) 1           x ξ * p ( , ) [( 1 2 )( r r r ) 2 r r r ]    ij ij , p ip , j jp , i , i , j , p 4 ( 1 ) r   r ξ x where

Displacement BIE Displacement BIE Using the fundamental solution u * as the test function , g we obtain: DBIE :     x ξ x ξ ξ x ξ * p * p u ( ) t ( ) u ( , ) dS u ( ) t ( , ) dS p j j m m     in which, displacements u are determined from  the boundary displacements and Singularity O(1/r 2 ) Singularity O(1/r )  the boundary tractions when differentiated directly, this leads to a Traction BIE, which is, unfortunately, hyper-singular: O(1/r 3 )

New Non-hyper Singular O(1/r 2 ) Traction BIE T i BIE  u Using the test function, the global weak form of solid mechanics becomes     n E u u dS n E u u dS i ijmn m , n j , k k ijmn m , n j , i              n n E E u u u u dS dS u u ( ( E E u u ) ) d d 0 0 n ijmn m , k j , i m , k ijmn j , i , n    Replacing the test function with the gradients of fundamental solution, we obtain: solution we obtain: TBIE :               x x ξ ξ x x ξ ξ ξ ξ x x ξ ξ * q * ( ( ) ) t t ( ( ) ) ( ( , , ) ) dS dS D D u u ( ( ) ) ( ( , , ) ) dS dS ab b q ab b p q abpq b     in which, stresses are determined from Singularity O(1/r 2 ) Singularity O(1/r )  the boundary displacements and the boundary displacements and  the boundary tractions

De-sigularization of Symmetric Galerkin Form Applying Stoke’s Theorem to Symmetric Galerkin form pp y g y 1     x x x ξ x ξ ˆ ˆ * p t ( ) u ( ) dS t ( ) dS t ( ) u ( , ) dS  p p x p x j j       2       x x ξ ξ ξ ξ x x ξ ξ ˆ * p t t ( ( ) ) dS dS D D ( ( ) ) u u ( ( ) ) G G ( ( , ) ) dS dS  p x i j ij       CPV   x ξ ξ x ξ ˆ * p t ( ) dS n ( ) u ( ) ( , ) dS  p x i j ij     1 1      x x x ξ x ξ * q ˆ ˆ t ( ) u ( ) dS D u ( ) dS t ( ) G ( , ) dS  b b x a b x q ab       2   CPV   ξ x x * q x ξ t ( ) dS n ( ) u ˆ ( ) ( , ) dS Singularity O(1/r)  q a b ab x            x ξ x ξ * ˆ D u ( ) dS D u ( ) H ( , ) dS  a b x p q abpq     H Han. Z. D.; Atluri, S. N. (2003): On Simple Formulations of Weakly-Singular Traction & Z D Atl i S N (2003) O Si l F l ti f W kl Si l T ti & Displacement BIE, and Their Solutions through Petrov-Galerkin Approaches, CMES: Computer Modeling in Engineering & Sciences , vol. 4 no. 1, pp. 5-20.

Intrinsic Features of the SGBEM Intrinsic Features of the SGBEM • weak singularity of the kernel: weak singularity of the kernel: O(1/r) • symmetric structure of the global • symmetric structure of the global “stiffness” matrix • the possibility of using higher-order th ibilit f i hi h d elements with curved sides

AGILE-2D: Cracks Emanating from F Fastener Holes in a Fuselage Lap-Joint t H l i F l L J i t