LS-DYNA Smoothed Particle Galerkin Method for Severe...
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LS-DYNA ® Smoothed Particle Galerkin Method for Severe Deformation and Failure Analyses in Solids C. T. Wu, Y. Guo and W. Hu Livermore Software Technology Corporation
Transcript of LS-DYNA Smoothed Particle Galerkin Method for Severe...
LS-DYNA®Smoothed Particle Galerkin
Method for Severe Deformation and Failure
Analyses in Solids
C. T. Wu, Y. Guo and W. Hu
Livermore Software Technology Corporation
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OutlineOutline
1. Methods in LS-DYNA for solids and structure
analyses.
2. Numerical issues in conventional particle methods.
3. Smoothed particle Galerkin method for solid
applications.
4. Benchmarks and numerical examples.
5. Keyword input format.
6. Conclusions and future plans.
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Methods for Solid and Structural Methods for Solid and Structural
Analyses in LSAnalyses in LS--DYNADYNA®®
Rubber Materials: FEM, EFG; MEFEM
Foam materials: FEM, SPH, EFG, SPG
Metal materials: FEM, SPH, EFG, MEFEM, Adaptive FEM and EFG
Quasi-brittle material fracture: FEM, SPH, EFG, State-based Peridynamic
method
E.O.S. materials and high speed applications: ALE, SPH, SPG
State-based Peridynamic method
Shells: FEM, EFG, SFEM
Soil: ALE, SPH, EFG, SPG
Discrete materials: discrete element method
Composites and Unit cell analysis: FEM, EFG, Immersed Particle Galerkin
method
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Lack of approximation consistency
Impose first-order reproducing condition
Tension instability
Ensure material failure occurs before numerical fracture
Material diffusion
Use higher-order integration scheme
Presence of spurious or zero-energy modes
Need stabilization
Difficulty in enforcing the boundary conditions
Special treatments (Convex approximation…)
Numerical Issues in Conventional Particle Numerical Issues in Conventional Particle
Analysis of Solids and StructuresAnalysis of Solids and Structures
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3D Smoothed Particle 3D Smoothed Particle GalerkinGalerkin MethodMethod
FEM
SPG
• Solid applications
• Read all FEM input formats
• A purely particle computation
• Handle severe deformation + failure
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Smoothed Particle Galerkin (SPG) Method
Has explicit/implicit versions. Currently only explicit method
implemented.
A pure particle integration method without integration cell.
Removes low-energy modes due to rank deficiency in nodal
integration.
Related to residual-based Galerkin meshfree method.
Can be related to non-local or gradient types inelasticity.
Without stabilization control parameters.
Stability analysis via Variational Multi-scale analysis.
First-order convergence in energy norm.
Capable of providing a physical-based failure analysis.
Has the trial version (will be formally released in this year).
Main FeaturesMain Features
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Residual-based Meshfree Galerkin Principle
Wu et. al submitted to J.
Comput. Physics. (2014)
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )
ˆ ˆ
ˆ
ˆ ˆ
ˆ
(2)2(2)
(2)2(2)
Ψ ; dΩ Ψ ; dΩ
1Ψ ; dΩ
2!
Ψ ; dΩ Ψ ; dΩ Ψ ; dΩ
1Ψ ; dΩ
2!
Ω Ω
Ω
Ω Ω Ω
Ω
≈ + ⋅
+ ⋅
= + −
+ ⋅
∫ ∫
∫
∫ ∫ ∫
∫
ɶ ɶ
ɶ
ɶ ɶ ɶ
ɶ
u X Y X u X Y X u X Y - X
Y X u X Y - X
u X Y X u X Y X Y X Y X
u X Y X Y - X
∇∇∇∇
∇∇∇∇
∇∇∇∇
∇∇∇∇
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ))
ˆ ˆ
ˆ ˆ
(2)2(2)
2(2
1Ψ ; dΩ Ψ ; dΩ
2!Ω Ω
= + ⋅
= + ⋅
∫ ∫ɶ ɶu X Y X u X Y X Y - X
u X u X η X
∇∇∇∇
∇∇∇∇
( ) ( ) ( ) ( ) ( )( ) ( )
ˆ ˆ ˆ ˆ ˆ ˆ, : : : :
ˆ ˆ ˆ ˆ , ,
(2) (2)h s s
h h
stan stab
a d d
a a
δ δ δ
δ δ
Ω Ω= Ω+ Ω
= +
∫ ∫u u u C u u C u
u u u u
∇ ∇ ∇ ∇∇ ∇ ∇ ∇∇ ∇ ∇ ∇∇ ∇ ∇ ∇
( ) ( ) hh la Vuuuu ∈∀= ˆ ˆˆ,ˆ δδδ
( ) ( ) ( )( )
ˆ ˆ ˆ ˆ, : :
1ˆ ˆ ˆ: :
2
(2) (2)h
stab
(2)(2) (2)
a dδ δΩ
= Ω
= +
∫u u u C u
u η u u η
∇ ∇∇ ∇∇ ∇∇ ∇
∇ ∇ ∇ ∇ ∇∇ ∇ ∇ ∇ ∇∇ ∇ ∇ ∇ ∇∇ ∇ ∇ ∇ ∇
( ) ( )( )ˆ ˆ ˆ ˆ :N
2l d d dδ δ δ δ
Ω Γ Ω= ⋅ Ω+ ⋅ Γ − ∇ ⋅ Ω∫ ∫ ∫u u f u t u η f
Displacement approximation
Variational formulation
Gradient type nonlocal strain
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Well-defined Mathematical Property
( )( ) ( )( )
( )
22 22
1 11 0 00
1
min
2 1 2
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ , ,
ˆ ˆ ˆ , , , 0,
(2)s s
h h
stan stab
h h
c c
ca a
c a c c
γ
≤ ≤ +
≤ +
= > ∈
u u u u
u u u uC
u u u V
∇ ∇ ∇∇ ∇ ∇∇ ∇ ∇∇ ∇ ∇
( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( )( )
1/ 2 1/ 22 2
max 0 0
1/ 2 1/ 22 2
3 0 0
max 4 51 1 1 1
ˆ ˆ ˆ ˆ ˆ ˆ, : : : :
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ,
(2) (2)h s s
3 4 5
a d d
d d
c h d h d
c c c ,c ,c
γ
γ
Ω Ω
Ω Ω
Ω Ω
≤ Ω+ Ω
≤ Ω + Ω
+ Ω + Ω
≤ ≤
∫ ∫
∫ ∫
∫ ∫
u v u C v u C v
C ε u ε v
ε u ε v
C u v u v
∇ ∇ ∇ ∇∇ ∇ ∇ ∇∇ ∇ ∇ ∇∇ ∇ ∇ ∇
∇ ∇∇ ∇∇ ∇∇ ∇
ˆ ˆ
h0 , > ∀ ∈u v V
Coercivity
Continuity
Unique solution !
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( ) ( ) ( ) hhhbhhhhh laa Vvvuvuv ∈∀=+ ,,
( ) ( ) ( ) hbbbbhhbh laa Bvvuvuv ∈∀=+ ,,
Error Estimation via Variational Multi-scale Method
( ) Γ=∈=Ω ,:: 1 onbbbh 0vHvvB
( ) ( )
( ) ( )
( ) ( )
( )∑
∑∑
∑ ∑
∑
=
= =
= =
=
=
=
=
=
NP
1K
KK
NP
1K
NP
1J
KJJK
NP
1J
NP
1K
KJKJ
NP
1J
JJ
h
ΨΨ
ΨΨ
Ψ
ux
uxx
uxx
uxxu
~
~~
~~
ˆ~
φ
( ) ( )( ) ( ) ( ) ( ) b
III
BP
1I
b
I
b
I
BP
I
I
b
III
b ZΨΨ ∈∀=
−≈ ∑∑
==
xxuxxuxxu ,~~
1
φ
RKΨRRKKKΨuuu~~~~~ 111 −−− +
+−=+=
TbbTbhh T
( ) ( ) ( )( )
( ) ( ) ( )2
,2
2
,
2
,2
2
,
2
,2
2
,
22
2
~
,,,
ΩΩΩ≤+≤
−∆+−≤
−−+−−=−−
uuu
uuuu
uuuuuuuuuuuu h
hchchc
h
aaa
e
h
e
h
hh
stab
hhhhhh
λµλµλµError estimation in energy-norm
fine-scale approximation
coarse-scale equation
fine-scale equation
global residual-free fine-scale space
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Nonlinear SPG Implementation
,, Γ∆−Ω∆−Ω∆+Ω∆=Π∆ ∫∫∫∫ ΓΩΩΩdtudfuduTudC
Nxxxiiiilkijkljiklijklij δδδεδεδ
( ) v
1n
Tv
n
v
1n
T δδ +
+
++ =∆ RUUKU~~~ 1
1
( ) v
1n
-Tv
n
v
1n
-T
+
+
+−
+ =∆ RAUAKA1
1
1
UAU-1=
~
( ) ( ) ( )IXXIXA ∑=
==NP
1K
KJIKIJIJ ΨΨφ
( )int1ffAUMAA −=− ext-T-T ɺɺ
( )intffAUM −= ext-Tɺɺ
∑ ∑ −−==NP
J
MLKM
T
IK
NP
J
IJ
RS
I
1AMAMM
Explicit dynamic formulation
( ) ( )∑=
⋅−=⋅∇−=NP
J
IJJIIII Ψ
dt
d
1
,
~~ xuu xɺɺ ρρ
ρ
Implicit formulation
Corresponding coordinate
systems in SPG computation
Currently implemented in LS-DYNA©
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( )1+− k
IΨ x
( )1++ k
IΨ x
( )k
IΨ x+
Updated Lagrangain /Eulerian Kernels
( ) ( ) ( ) 1
1
−
+
+
+
++
∂∂
=∂
∂
∂∂
=∂
∂= k
jik
j
I
k
j
k
j
k
j
1k
I
1k
i
1k
I1k
iI, fx
-Ψ
x
x
x
-Ψ
x
-Ψ-Ψxx
x
( ) ( )i
1k
I1k
iI,x
ΨΨ
∂
+∂=+
++ x
x
Consistency Stability Convergence
First-order rate of convergence in energy norm !
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Material fracture v.s. Numerical fracture
physical material fracture before numerical fracture Enlarge numerical support !
Reference configuration
Deformed configuration
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Prandtl’s punch problem
zV
Dimension: 4x2x1
Particles: 21x11x6
Elastic material: E=6.9x104
v=0.3
Vz=2
ρ0=2.7×103
Updated
Lagrangian
Eulerian
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Updated Lagrangian
Eulerian
t=0.2 0.4 0.6 0.8 1.0
Fixed ∆t=3.0×10-5
Prandtl’s punch problem
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Nodal force comparison with different kernel
approximations
Prandtl’s punch problem
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Taylor Bar Impact
R=3.91 mm
H=23.46 mm
ρ0=2.7×10-6 kg/mm3
E=78.2GPa
v=0.3
σy=0.29(1+125ep)0.1
V0=373 mm/ms
Particles: 2263
DX=DY=DZ=1.4 SMSTEP=25
Final H=18.07mm
Exp. H=16.51mm
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t=0.004 0.012 0.020 0.028
∆t=1.76×10-5 1.72×10-5 1.66×10-5 1.66×10-5
Progressive deformation with effective plastic strain contour
Bottom
view
Taylor Bar Impact
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Plate Impact (Ductile Material)
Ball: rigid, R=5.0
Plate: R=20.0, thickness=5.0
Particles: 25721, updated Lagrangian
Elastic perfectly-plastic material:
ρ0=7.85×10-3
E=6.9x104
v=0.3
σy=200.0
Vz=-600.0, -400.0, -300.0
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Plate Impact
t=0.02
∆t=7.47×10-6
t=0.04
∆t=7.44×10-6
t=0.06
∆t=7.44×10-6 t=0.08
∆t=7.39×10-6
t=0.10
∆t=7.36×10-6
Bottom
view
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Front view
Bottom view
Lagrangian Eulerian Eulerian CMD Eulerian PSD
Effective plastic strain (v=600, t=0.06)
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t=0.02
∆t=7.48×10-6
t=0.04
∆t=7.44×10-6
t=0.06
∆t=7.47×10-6t=0.08
∆t=7.48×10-
6
t=0.09
∆t=7.44×10-
6
Bottom
view
Progressive effective plastic strain plots with phenomenological
strain-based damage (v=400)
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v=300
t=0.12
v=400
t=0.09
v=600
t=0.06
Effective plastic strain with phenomenological strain-based damage
Bottom
view
Top
view
Front
view
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v=300 v=400 v=600
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Plate Impact (Brittle Material)
Front View
Back View
Final Crack Pattern
Damage Indicator
(movie)
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Metal cutting analysis (1)
Cutting Speed = 10 m/s
Fixed ∆t=3.0×10-5
Aluminum
ρ0=2.7×10-6 kg/mm3
E=78.2GPa
v=0.3
σy=0.29(1+125ep)0.1
Strain-based failure criteria εfail = 0.5
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Metal cutting analysis (2)
Cutting Speed = 10 m/s
with different cutting
angle
Fixed ∆t=3.0×10-5
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Metal cutting analysis (3)
Cutting Speed = 10 m/s
with different depth
Fixed ∆t=3.0×10-5
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Metal cutting analysis (4)
Cutting Speed = 100 m/s
Fixed ∆t=3.0×10-5
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Metal shearing analysis
Effective plastic strain is monotonically increased w/o diffusion !
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Card1 1 2 3 4 5 6 7 8
Variable SECID ELFORM AET
Type I 47 I
Default
*SECTION_SOLID_SPG
Keyword Input FormatKeyword Input Format
Card2 1 2 3 4 5 6 7 8
Variable DX DY DZ ISPLINE KERNEL LSCALE SMSTEP SWTIME
Type F F F I I F I F
Default 1.5 1.5 1.5 0 3 15
Card3 1 2 3 4 5 6 7 8
Variable IDAM FS STRETCH
Type I F
Default 0
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Failure strain if IDAM=1; maximum principal strain if IDAM=2FS
Damage option.
EQ.0: Continuum damage mechanics (default)
EQ.1: Phenomenological strain damage
EQ.2: Maximum principal strain damage
IDAM
VARIABLE DESCRIPTION
SECID Section ID.
ELFORM Element formulation options. Set to 47 to active SPG method.
DX, DY, DZ Normalized dilation parameters of the kernel functions in X, Y and Z directions.
ISPLINE Option for kernel functions.
EQ.0: Cubic spline function (default).
EQ.1: Quadratic spline function.
EQ.2: Cubic spline function with circular shape.
KERNEL Type of kernel approximation.
EQ.0: updated Lagrangian kernel.
EQ.1: Eulerian kernel.
EQ.2: Semi-Lagrangian kernel.
EQ.3: Pseudo-Lagrangian kernel.
LSCALE Length scale for displacement regularization.
SMSTEP Interval of time steps to conduct displacement regularization.
SWTIME Time to switch from updated Lagrangian kernel to Eulerian kernel.
STRETCH Stretching parameter if IDAM=1
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Conclusions and Future PlansConclusions and Future Plans
1. Smoothed Particle Galerkin (SPG) Method is implemented in LS-DYNA®
and a SMP trial version is available.
2. Mathematical and numerical properties have been provided.
3. The method is able to handle severe deformation involving material
failure for various solid applications.
4. The application to compressible fluids or fluid-type solids is currently
excluded.
5. Official SMP and MPP versions will be released in this year.
6. The extension to adaptive FEM/EFG method will be considered.
7. The switch from FEM to SPG method for severe deformation analysis
will be implemented.
