satya n. atluri, uci - math.nist.gov · (symmetric galerkin boundary element ... any available...
TRANSCRIPT
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Multi-Scale Analysis of AircraftMulti-Scale Analysis of Aircraft Structural Longevity
(R h C d t d i th l 1990 )(Research Conducted in the early 1990s)
Satya N. Atluri, UCI
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Life Cycle of an Aircraft
Design
MarketRequirements
ProductionDesign
DesignPrototype
Certification
O
MaintenanceAGILE
Operations
& OverhaulsRetirement
& Overhauls
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Structural Integrity of R t ft C t (DTA?)Rotorcraft Components (DTA?)
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Aircraft Fatigue Failure: Loss of Integrity
1988, a Boeing 737-297 serving the flight suffered extensive damage after an explosive decompression in
4-28-1988 After 89,090 flight cycles on a 737-200, metal fatigue lets the top go in flightexplosive decompression in flight, but was able to land safely.
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Micro Crack Level: 10-5 mDTALE: MLPG-SGBNM AlternatingDTALE: MLPG SGBNM Alternating
h
a
c
h
45 degc
h
a
c
h
a
c
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Mega- to Micro-Level Multiple-Scale A lAnalyses
Finite volume
Finite ElementMicro
Cracks
Finite Element
Panel Methods
MeshlessMethodsMethods
BEM
MDO
IPPDInverse Problems
AGILE…
Global Deformation
System Level: 102m
Component Level: 1~ 10-2 m
Micro Crack Level: 10-4 ~ 10-6 m
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Initial Detected Crack Level: 10-4 mAGILE Alternating TechniquesAGILE Alternating Techniques
h
a
c
h
45 degc
h
a
c
h
a
c
Thi k 10 3Thickness: 10-3m
Initial Crack: 10-4m
Initial Crack: 10-4mInitial Crack: 10 m
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Multi-Scale Damage Tolerance for Initially Detectable CracksInitially Detectable Cracks
thickness = 0.09"
D
CB
A
0.4 0.4 0.44 0.44 0.4 0.41.18
0.75
Rivet Diameter =5/32 "
0.75
0.75
0.75
D
skin thickness =0.063"
doubler thickness =0.025"
Rivet Diameter =3/16 "
5/32
Rivet Diameter =5/32 "
0.04
0.044
0.048
0.052
0.056
0.06
0.024
0.028
0.032
0.036
1.712 1.714 1.716 1.718 1.72 1.722 1.724 1.726 1.728 1.73
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Micro-Crack Initiation?Simply using continum-stress mechanicsp y g
0.04
0.044
0.048
0.052
0.056
0.06
0.024
0.028
0.032
0.036
1.712 1.714 1.716 1.718 1.72 1.722 1.724 1.726 1.728 1.73
Micro Structure InclusionMicro-Structure Inclusion
Shot-peening
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AGILE: Model at 10-6 Level with Continuum Detailswith Continuum Details
AGILE: Boundary surface mesh only, without refining FEM mesh. Higher order boundary-elements fit curved surfaces much better!
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AGILEAGILE
• Continum Damage MechanicsContinum Damage Mechanics• Anisotropic Damage Mechanics
G i B d F t M h i• Grain Boundary Fracture Mechanics• Gradient Theories of Material Behavior• _______________? Far in the Future• Ab Initio Dislocation DynamicsAb Initio……Dislocation Dynamics• MD
St ti ti l M h i• Statistical Mechanics• DFT……..
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AGILE (LOCAL): SGBEM-FEM AlternatingAlternating
(Symmetric Galerkin Boundary Element – FEM Alternating Method) (Overall Accuracies of KI, KII,KIII, Jk are the best of any available method)
P
SGBEM
P
FEM SGBEM
+FEM
=
I fi it b dLoaded Finite body with a crack
Infinite body with a crack
Loaded Finite bodywithout a crack
FEM Stiffness matrix inverted only ONCE, Faster!
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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 jcomputational algorithms.Most advanced closed form–Most advanced closed-form mathematics, and only minimal
inumerics
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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.
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AGILE: Faster and more accurate than traditional BIE
• Weakly-singular integrals are numericallyWeakly singular integrals are numerically tractable, with Gaussian quadrature algorithms using q g glower order integrations
• Higher-order elements with curved sides gcan be used, because of its requirement of only C0
ti it hi h i i ll f l fcontinuity, which is especially useful for modeling 3D non-planar cracks with less elementselements.
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AGILE: More applicable than pure BIE
• Built-in FE solver handles moreBuilt in FE solver handles more complicated geometries, including structural elements such as beamsstructural 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 andwalled structures, manifold domains, and bi-material parts.2 D 2 D/3 D t iti & 3 D d li f• 2-D, 2-D/3-D transition, & 3-D modeling of structures w/ mixed-mode crack-growth
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SGBEM: Fundamental SolutionsSolutions
3D Problems xSourcePoint
1
])43[()1(16
1),( ,,*
piipp
i rrr
u
ξxr
Point
field]3))(21[(
)1(81),( ,,,,,,2
*pjiijpjippij
pij rrrrrr
r
ξx u*, *
field
2D Problems2D Problems
]ln)43([)1(8
1),( ,,*
piipp
i rrru
ξx
]2))(21[()1(4
1),( ,,,,,,*
pjiijpjippijp
ij rrrrrrr
ξx
xξr where
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Displacement BIEUsing the fundamental solution u* as the test function ,
Displacement BIEg
we obtain:
DBIE:
dStudSutu p
mmp
jjp ),()(),()()( ** ξxξξxξx
in which, displacements u are determined from the boundary displacements and
Singularity O(1/r2) the boundary tractions
Singularity O(1/r )
when differentiated directly, this leads to a Traction BIE, which is, unfortunately, hyper-singular: O(1/r 3)
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New Non-hyper Singular O(1/r2)T i BIETraction BIE
uUsing the test function, the global weak form of solid mechanics becomes
0)(
,,,,
duEudSuuEn
dSuuEndSuuEn ijnmijmnkkjnmijmni
Replacing the test function with the gradients of fundamental solution we obtain:
0)( ,,,,, duEudSuuEn nijijmnkmijkmijmnn
dSuDdSt bqbb ),()(),()()( ** ξxξξxξx
TBIE:solution, we obtain:
in which, stresses are determined fromthe boundary displacements and
dSuDdSt abpqqpabqab ),()(),()()( ξxξξxξx
Singularity O(1/r2) the boundary displacements and the boundary tractions
Singularity O(1/r )
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De-sigularizationof Symmetric Galerkin Form
Applying Stoke’s Theorem to Symmetric Galerkin formpp y g y
p
pjjxpxpp
dSGuDdSt
dSutdStdSut
)()()()(ˆ
),()()(ˆ)()(ˆ21
*
*
ξxξξx
ξxξxxx
1
CPV p
ijjixp
ijjixp
dSundSt
dSGuDdSt
),()()()(ˆ
),()()()(
* ξxξξx
ξxξξx
dSundSt
dSGtdSuDdSut
CPV
xq
abbaq
qabqxbaxbb
),()(ˆ)()(
),()()(ˆ)(ˆ)(21
*
*
ξxxxξ
ξxξxxx
Singularity O(1/r)
dSHuDdSuD abpqqpxba ),()()(ˆ * ξxξx
H Z D Atl i S N (2003) O Si l F l ti f W kl Si l T ti &Han. Z. D.; Atluri, S. N. (2003): On Simple Formulations of Weakly-Singular Traction & Displacement BIE, and Their Solutions through Petrov-Galerkin Approaches, CMES: Computer Modeling in Engineering & Sciences, vol. 4 no. 1, pp. 5-20.
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Intrinsic Features of the SGBEMIntrinsic 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” matrixth ibilit f i hi h d• the possibility of using higher-order elements with curved sides
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AGILE-2D: Cracks Emanating from F t H l i F l L J i tFastener Holes in a Fuselage Lap-Joint
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FEM Model with Boundary and Load C diti b t NO C kConditions but NO Crack
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2-D Infinite bodywith loaded arbitrarily-shaped line cracks y p
ONLY: Singular Integral equations
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Alternating Procedure: Apply the id l t ti b k t th FEMresidual tractions back on to the FEM
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AGILE-2D Mixed Mode Crack GrowthAGILE 2D Mixed Mode Crack Growth
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AGILE-2D: Multiple HolesAGILE 2D: Multiple Holes
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2D/3D Mixed Analyses withP i C k S dParametric Crack Study
GRIP
P
JOINT
56"
40"
thickness = 0.063"
GRIP
22"
Skin Thickness 0 063”
P
Skin Thickness = 0.063”
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AGILE: Mixed 2D/3D Crack P i A l iParametric Analysis
Existing FE Model with ABAQUS results
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Intermediate FE Model (Joint)Intermediate FE Model (Joint)
Rivet Holes
Local deformed skin
3D FE model with LBCs transferred from the global shell analysis by using AGILE
GUI
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Local FE Model of Rivet HoleLocal FE Model of Rivet Hole
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Multiple Crack Location studyh
h=0.0315"
ac
t-h=0.0315"
R=3/32 " = 0.09375"
t = 0.063"
t-h=0.0315"
h
h=0.0315"
a
=45 degc
t = 0.063"
R=3/32 " = 0.09375"
AGILE FE model
h
h=0.0315"
t = 0.063"
t-h=0.0315"
R=3/32 " = 0.09375"
a
c
h
h=0.0315"
t = 0 063"
t-h=0.0315"
R=3/32 " = 0.09375"
a
c
t = 0.063
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Possible Crack DevelopmentPossible Crack Development
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Experiment Report by Air ForceExperiment Report by Air Force
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CPU TimeCPU Time
• Global AnalysisGlobal Analysis3 Minutes
• Intermediate Analysis (Joint)y ( )21.5 Minutes
• Local Analysis (Rivet Hole)y ( )4.5 Minutes
• Crack Analysis (AGILE)100 Minutes for 31 cases
Total CPU Time 2 Hours in a normal lap-top! (in 2003!)
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Bridge Collapse: Catastrophic Failure
In 2007, a highway bridge over the Mississippi River in Minneapolis collapsed into the river and onto the riverbanks beneath during evening rush hour.
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Application of AGILE-3D in the Fatigue Crack-Growth Analyses of Orthotropic Deck Bridges
Orthotropic Deck Bridges
Fatigue crack at the rib-deck welded joint
dynamic load at the U-rib joint
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The Computational Model (XFEM) used for the Fatigue Crack Analysis of the Rib-Deck Welded Jointy
2-D Plane Strain Model which implies that the crack at the rib-deck is “infinitely” long, across the whole span of two horizontal floor beams / stiffenersstiffeners
An extremely fine mesh has to be used at the crack tip
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Using AGILE-3D for the Prediction of Fatigue Life of Orthotropic Deck Bridges
finite size fatigue crack at the rib-deck joint
M M The advantages of using AGILE-3D for the fatigue crack analysis of orthotropiccrack analysis of orthotropic deck bridges:1) 3-D model can be used to
account for the different sizesaccount for the different sizes and geometries of cracks;
2) Computationally efficient asa coarse mesh is able to givea coarse mesh is able to give accurate results.
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Typical structural componentsTypical structural components
High Surface/Volume ratio
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Multiple Level AnalysesMultiple Level Analyses
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AGILE: N l 3D f ti thNon-planar 3D fatigue growth
1.5"
2"
00
1" 1.9"0.5"
0.1"
0.01"
Non-planar 3D fatigue growth of an inclined
i i l f ksemi-circular surface crack
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Nonplanar fatigue growth of an inclined semi circular surface crackan inclined semi-circular surface crack
• ASTM E740 specimen• Mixed-mode fatigue growth
1.5"
1" 1.9"0.5"
2"
00
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AGILE ModelsAGILE Models
Finite BodyFinite Body w/o Crack
2304 El t2304 Elements(Hexa 20)
Crack S fSurface
24 Elementsalong crack front
(Quad 8)
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Stress Intensity FactorsI iti l C k:Initial Crack
0.8
NKI NKII NKIII
0.4
0.6
K0,
KII/
K0,
KIII
/K
K2S K3S Forth, Keat & Favrow (2002)
KI
FEM-SGBEM Alternating
0.2
ss In
tens
ity F
acto
rs K
I/K
KII
Han. Z. D.; Atluri, S. N. (2002): SGBEM (for Cracked Local
-0.2
0
Nor
mal
ized
Str
es
KIII
SGBEM (for Cracked Local Subdomain) – FEM(for uncracked global Structure)AlternatingMethod for Analyzing 3D Surface Cracks and Their
-0.40 15 30 45 60 75 90
Angle, degree
Fatigue-Growth, CMES: Computer Modeling in Engineering & Sciences, vol. 3 no. 6, pp. 699-716.
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Crack in the specimen
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Final CrackInitial Crack
Final Crack Predicted by
Crack
using AGILE
Initial CrackCrack
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Fatigue Loading Cycles0.4
Fatigue Loading Cycles
0.3
0.35 AGILE FEAM
specimen 1
specimen 2
specimen 3
The critical depth of the crack
AGILE 0 29”
0.2
0.25
k de
pth
(in)
p
specimen 4AGILE 0.29Exp. Ave. 0.284”
(0.34”, 0.23”, 0.32”, and 0.25”)
0.1
0.15Cra
ck
0
0.05
1.E+03 1.E+04 1.E+05 1.E+06
Cycles
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The Non-planarly Growing Crack...p y g
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Analysis of Cracks in Solid Propellant R k t G iRocket Grains
P
Mu
Solid Propellant Rocket Grain under tension and inner pressure
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Unsymmetric BE Crack ModelUnsymmetric BE Crack Model
Unsymmetric Crack
Crack Front
Semi-Circular Crack
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Crack Front AdvancementsCrack Front Advancements
Crack Front after 3 Steps Crack Front
after 6 StepsCrack Front after 9 Steps
Crack Front after 11 Steps
Initial Crack
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Center Line of Growing CrackCenter Line of Growing Crack
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Final Crack SurfaceFinal Crack Surface
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Simulation: Growth of the CrackSimulation: Growth of the Crack
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Some Other Fracture CodesSome Other Fracture Codes
• Codes based on analytical/handbookCodes based on analytical/handbook solutions– NASGRO, FASTRAN,
• Full BEM codes– BEASY, FRANC3DBEASY, FRANC3D
• Full FEM codes with specific elements– ABAQUS, MARC, ZenCrack, XFEMABAQUS, MARC, ZenCrack, XFEM
• FEM-SGBEM Alternating Code– AGILE (Most Efficient & Most Accurate)AGILE (Most Efficient & Most Accurate)
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From FEM ZenCrack to XFEMFrom FEM, ZenCrack to XFEM• FEM: Enriched Singular
El t (d l d iElements (developed in 1970’s, pioneered by Atluriand his colleagues,and his colleagues, implemented in ABAQUS, MARC, etc.)
C fi i & d i M h– Confirming & adaptive Meshes.– Accuracy dependent on the
mesh quality.q y– Costly labor of Meshing & Re-
MeshingNo automated crack growth– No automated crack growth.Enrichment Elements are the KEY!
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From FEM ZenCrack to XFEMFrom FEM, ZenCrack to XFEM• Zen Crack: a crack mesh
generator– Insert a crack into a non-
k d FEM M hcracked FEM Mesh– Create the meshes outside
involving FEM Solversinvolving FEM Solvers.– Reduce labor work in
creating the conforming g gand adaptive meshes
– Algorithm is unstable.
Enriched Elements still play the KEY role!
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From FEM ZenCrack to XFEMFrom FEM, ZenCrack to XFEM• XFEM: Split elements to
match the cracks– Integrate the element
i l ti i t th FEMmanipulation into the FEM Solvers, and HIDE it from the users.
Splitting elements!
– No adaptive meshes– Splitted elements without p
quality.– No accuracy control.
Only 2D Enriched Elements can be used.
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What about XFEM 3D?(up to 2010)
• Only Tet Mesh but No Hexa Mesh.
• No 3D enrichment element for non-planar cracks.
• The accuracy is heavily dependent on the initial FEM MeshFEM Mesh.
FEM without Enrichment Elements!
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What about XFEM 3D?(Rabczuk Bordas Zi (2010): Computers and Structures 88 pp 1391–1411)(Rabczuk, Bordas, Zi (2010): Computers and Structures 88, pp. 1391–1411)
• 30x30x30=27,000 FE initial mesh.
Penny-shaped embedded crack in a tension bar
elements: Error = 3.3% • 60x60x60=216,000
elements: Error = 2.07% • 120x120x120=1,728,000 XFEM3D Results
elements: Error = 1.21%
• AGILE: 20 elementsError = 0.3%
XFEM-3D is NOT suitable for fatigue & fracture analyses AGILE mesh.
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What about XFEM 3D in C i l C d ?Commercial Codes?
Not even close, even in 2D XFEM!
i h i l i i hXFEM3D, without singularity enrichment, is NOT suitable for fracture analysis!
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How to Reach 10-6 Level even using continuum mechanics?continuum mechanics?
• FEM: Zoom-in refined localized mesh, => 10-5
• XFEM: Splitting Elements withoutElements without mesh quality control, => 10-5
• AGILE: Completely de coupled FEM 0 044
0.048
0.052
0.056
0.06
de-coupled FEM-SGBEM LOCAL model, Cracks can be
0.024
0.028
0.032
0.036
0.04
0.044
1.712 1.714 1.716 1.718 1.72 1.722 1.724 1.726 1.728 1.73
two orders lower, => 10-6
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Comparison between CodesCodes Modeling CPU Accuracy Fully 3D Complicate LinkCodes Modeling
Time CPUTime
Accuracy Fully Automated
Growth
3D NonPlanar
Crack
Complicate Model and
LBCs
Link Commercial FE Codes
AGILE Crack only Minutes per step <1% YES YES YES YES step
BEASY Full BEM
Model with Crack
6~10 times slower ~3% Restriction YES Quad
Mesh Limited
F ll BEMFRANC3D
Full BEM Model with
Crack Slower ~3% Unstable YES NO NO
NASGRO Predefined C k l Fast -- YES NO NO NONASGRO Crack only Fast YES NO NO NO
ABAQUS MARC
Full FEM Model with
Crack Fast ~10% NO YES YES Self
ZenCrack Full FEM
Model with Crack
Fast ~10% Unstable YES Unstable NA
XFEM Worse
than YES NO Not for YESXFEM ----- - than ABAQUS
YES NO Cracks YES
AGILE has the BEST Accuracy & can be run on demand in a real-time fashion!
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AGILE Probabilstic Prognostics Toolg
Integrated Structural Health Management System
diagnostics
Mega Level FE Model Damage Accumulat ion
ilist
ic
alys
is
atin
g lo
gy
Prob
abi
FE
ana
alte
rna
tech
nol
crack growthmodel Lib.
Component Level FE Model Micro Level Crack
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Automated Global, Intermediate, & Local Evaluations for Damage Tolerance Analyses & Life Estimation:
AGILE for DTA & LEAGILE for DTA & LE(Status as of Dec. 2004)Satya N. Atluri, UCI
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Why AGILE?Why AGILE?
• Simple to use:Simple to use:–Easiness of Model Creation–User-Friendly Graphical Interfaces–Least computationally intensive–Least computationally intensive–Automatic re-solution of Intermediate
model, if load-redistribution due to crack-growth occursg
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What is embedded in AGILE?What is embedded in AGILE?
• Open Architecture:Open Architecture:– Various mixed mode loadings.
2 D & 3 D Mi ed Mode Non planar fatig e– 2-D & 3-D Mixed-Mode, Non-planar fatigue-crack-growth modelingSophisticated mathematics + minimal numerics– Sophisticated mathematics + minimal numerics
– Fatigue-crack-growth models.– Probabilistic analyses.
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Support multiple load casesSupport multiple load cases
• Structural components are undergoingStructural components are undergoing several loading cases within one flight , including take-off & landing liftingincluding take off & landing, lifting, carrying. The load spectrums are different.
• The life of the loading components will be• The life of the loading components will be estimated under the combined load cases.
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Easiness of Model CreationEasiness of Model Creation
• Simple FE mesh creation without theSimple FE mesh creation, without the crack surface in the FE model.
• Simple creation of crack model as only a• Simple creation of crack model, as only a surface mesh in SGBEMI d d f th SGBEM d FE• Independence of the SGBEM and FE meshes:– leverage the existing FE models and results– Parametric crack analysis is very simple
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Graphical User InterfaceF ll i d i PATRANFully integrated into PATRAN
• The proficiency of the GUI makes AGILE user-p yfriendly and minimizes human-errors typically associated with data preparation.S i ALL AGILE d l i• Supporting ALL AGILE model creation.
• Seamless integration with MSC.PATRAN, minimizes user trainingminimizes user training.
• Supporting PATRAN session file, i.e. recording and playing back.a d p ay g bac
• Supporting all PATRAN FE model files for NASTRAN, MARC, ABAQUS and so on.
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AGILE ArchitectureFE codes
NASTRANGraphical User Interfaces NASTRANANSYSMARCLoad/BC
f
Graphical User Interfaces
M d l D b …TransferorModel Database
AGILE 2D/3DAnalyses-Codes
Fatigue Models
Results:Result: Life EstimationK Solutions
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Support most crack growth models
• Paris Model• Walker Model• NASGRO ModelNASGRO Model
• Load SpectrumLoad Spectrum• Analytical models for
plasticity-inducedplasticity induced Crack-closure
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AGILE as an Integrated Probabilistic P ti T l i SHM S t
Environmental inputsSensors
) C
Prognostic Tool in an SHM System
1) Controlled Diagnostic Inputs
2) Signal Processingand Filtering
5) Integrated Probabilistic
3) Multi-scale Interrogation
CrackLength
4) Probabilistic Diagnostic Imaging
) gPrognostics
Load
• Damage Formation• Growth• TypeCrackLength
Load
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Probabilistic AnalysisProbabilistic Analysis
• The probabilistic information on pre-crackThe probabilistic information on pre crack damage and macro-crack growth will be analyzed in terms of location, size and type of damage.
• Automatic life prediction in a probabilistic sense for structures will be implemented with probabilistic information of the real environmental conditionsenvironmental conditions.
• Experimental database will be used as one possible probabilistic input as well as otherpossible probabilistic input, as well as other theoretical and numerical models.
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AGILE-2D: DemonstrationAGILE 2D: Demonstration
Support most 2D triangularSupport most 2D triangular and quadrilateral elements
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Mixed Mode Crack Growth:No Changes in FE Mesh
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Dialog-based InterfaceDialog based InterfaceAGILE GUI Dialogs
Agile Menu
Selection from Listsfrom Lists
Intelligent Engine for Automatic Parameter Calculation