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An Implicit Gradient Reproducing Kernel Particle Method: Theory and Applications
J. S. Chen, T. H. Huang
Department of Structural Engineering
Center for Extreme Events Research
University of California, San Diego, USA
M. Hillman, G. Zhou
Department of Civil and Environmental Engineering
The Pennsylvania State University, Pennsylvania, USA
Motivation
2
UCSD Blast Simulator
Continuum Fragmented Solids (Particle like)
Meshfree Methods with Nodal Integration
Approximation, Discretization?
Numerical Challenges
3
Oscillatory Solution, Smeared Shearband
Rank Instability: Rank Deficiency Kernel Instability: Insufficient Neighbors
Numerical Fracture, Solution Divergence
Gibbs Instability: Shock Front Discontinuities
Oscillatory Solution, Incorrect Damage
Discretization Instability: Mesh Dependency
Non-convergent in Refinement, Incorrect Softening
4
How to achieve stability, accuracy, and efficiency under the same
framework for Meshfree modeling of extreme events?
5
1. Kernel Instability
Numerical Fracture, Solution Divergence
I
Moving Least-Squares / Reproducing Kernel Approximation
1Let be a bounded domain and S = { , . . . , } be a set of scattered points.
Let function ( ) be approximated by
N N
u
x x
x
1
NK
I I
I
u u
x x
I a I
n
I ( )) (( b ( ))
xx x xxx
The coefficients ( ) are determined from the following reproducingconditions :b x
I I
I
( ) x x x n I I ,0I
( )( ) x x xor
TI I a
1I M xx H 0 H x x x x
• For M to be nonsingular, sufficient neighbors under the kernel support is needed:
higher order of completeness “n” requires larger kernel support size
TI I a I
I
nTI 1 1I 2 2I 3 3I 3 3I1,x x ,x x ,x x , , x x
M x H x x H x x x x ,
H x x
P. Lancaster, K. Salkauskas, 1981; B. Nayroles, G. Touzot, and P. Villon, 1992; T. Belytschko, Y. Y. Lu and L. Gu, 1995; O˜nate, E.,
Idelsohn, S. R., Zienkiewicz, O. C., and Taylor, R. L., 1996; W. K. Liu, S. Jun, Y. F. Zhang, 1995; J. S. Chen, C. Pan, C. T. Wu, and W. K.
Liu, 1996, A. Duarte & J. T. Oden, 1996; J. M. Melenk & I. Babuaka, 1996; S. N. Atluri and T. Zhu, 1998; S. De and K. J. Bathe, 2000.
6
Quasi-Linear Reproduction Kernel Approximation
Consider the following weighted least-squares residual:
2 2
1 11
( )( ) , 0
kP P IN N
h k
I I a I
N
I a I
I I k
h
I ur u u u
xxxx x xx xx
𝐱𝐼
𝐱𝐼𝑘
1
1( )
( ) ( ) ( ) ( , ) ( )PN
h T
I a I I
I
I
u u
x
x H 0 M x H x x x x
1
( , ) ( ( ))SN
k
I
k
I I
H x x H x Hx x x
*(( ) ( )) M x M x M x
*
1 1
( ) ( ) ( ) ( )SP NN
k T k
I I a I
I k
M x H x x H x x x x
,min r x
x
x x
is non-singular if form a non-zero volume 1
kIN
k
I kx( )M x
Yreux, E. and Chen, J. S., “A Quasi-Linear Reproducing Kernel Particle Method,” International Journal for
Numerical Methods in Engineering, Vol. 109. pp. 1045–1064, 2017.
Wave Propagation
-0.00015
-0.0001
-5e-05
0
5e-05
0.0001
0.00015
0 5e-05 0.0001 0.00015 0.0002 0.00025 0.0003
AnalyticalConstant Basis
Quasi-LinearAutomatic Basis
8
Taylor Bar Impact
9
10
Oscillatory Solution, Smeared Shearband
2. Rank Instability
11
Chen, J. S., Hillman, M., Rüter, M., “International Journal for Numerical Methods in Engineering, Vol. 95, pp.
387–418, 2013.
J. S. Chen, C. T. Wu, S. Yoon, and Y. You 2001; I. Babuška, U. Banerjee, J. E. Osborn, Q. Li 2008; Q. Duan, X.
Li, H. Zhang, T. Belytschko, 2012; J. S. Chen, M. Hillman, M. Rüter, 2013
2 sin( )sin( )
0
u x y in
u on
( 1,1) ( 1,1)
Quadrature
Nodal Integration in Galerkin Approximation
Integration Constraints
( ) ( )^ ^
I Iˆ ˆd d
x x n
(1st order Galerkin Exactness)
31 2
1 2 3 1 2 3ˆ ˆ ˆ, , , , , , 0,1, ,I I Ia L B x x x n
x x x x (higher order G. E.)
First Order Correction: Stabilized Cooforming Nonal Integration (SCNI)
( ) ( ) ( )
L L
^ ^
I L I I
L L
1 1ˆ ˆ ˆd dV V
x x x n
( ) ( )^ ^
I Iˆ ˆd d
x x n
ˆ, ˆ,h h
I I I I
I I
I I I I
n
u u v v
Higher Order Correction: Variationally Consistent Integration (VCI)
ˆ ˆ ˆ, , , , 0,1, ,I I Ia L B n
x x x
1
, , , , , ,n
I I I I I I Ia L B B L a
x x x x x x
Violation of integration constraint
Chen, J. S., Wu, C. T., Yoon, S., and You, Y., International Journal for Numerical Methods in Engineering, Vol. 50, pp.
435-466, 2001; Chen, J. S., Hillman, M., Rüter, M., International Journal for Numerical Methods in Engineering, Vol.
95, pp. 387–418, 2013.
Lu + s = 0 inW
u = g on ¶Wg
Bu = h on ¶Wh
0, ,h h h h ha u v F v v V
-4
-3.5
-3
-2.5
-2
-1.5
-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2
log(
||u
-uh||
0)
log(h)
SCNI: 1.90SNNI: 0.24VC-SNNI: 2.03DNI: 0.28VC-DNI: 1.77
Galerkin Meshfree with Nodal Integration
Naturally stabilized nodal integration (NSNI) First order implicit gradient expansion of strain energy
1 1 2 2 3 3
0
( ; ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )n
i j k T
a ijk a a
i j k
w x s x s x s b w w
x x s x x s H x s b x x s
Strain approximation
Implicit gradient
Gradient reproducing conditions (S. Li, W. K. Liu, 1999, Chen, J. S., Zhang, X., Belytschko, T., 2004)
( ( ))( ) ( ; ) , 0 , 1, 2,3a
i
mm P
w d m ix
P n
x
s x x s s
1
: : d : : : :i i
NP 3i
L L L L L L
L i 1SCNI , VC SNNI
NSNI
V M
C x C x x C x
14
Ti
i
1
I I a I x M x H -H x x x x
1
2
3
[ 0 1 0 0 ]
[ 0 0 1 0 ]
[ 0 0 0 1 ]
, , ,
, , ,
, , ,
H
H
H
Hillman, M., Chen, J. S., International Journal for Numerical Methods in Engineering, Vol. 107, pp. 603–
630, 2016.
Comparison of RK Approximation and Implicit Gradient RK Approximation
15
RK Approximation
I I
I
u x x d
Implicit Gradient RK Approximation
i i
I I
I
x x
Ti
i
1
I I a I x M x H -H x x x x
1
0I I a I
T x M x H x x xH - x
1
2
3
]
[ 0 1 0 0 ]
[ 0 0 1 0 ]
[ 0 0 0 1 ]
T
0
T
T
T
0, 0, 0
, , ,
, , ,
, , ,
H
H
H
H
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.5 -1.0 -0.5
log(
L2 e
rro
r)
log(h)
DNI: 0.47SNNI: 0.21NSNI: 1.81VC-NSNI: 2.05
sin( )sin( ) in
0 on
2u x y
u
Eigenmodes and Eigenvalues for first non-zero eigenvalue
VC-NSNI :
1.325
Fully integrated FEM: 1.30
SNNI: 0.77
Tension test
DNI SNNI VC-NSNI (present)
16
Taylor bar impact
Method Final radius (cm) Final height (cm)
SNNI 0.839 1.649
DNI 0.838 1.660
VC-NSNI 0.760 1.654
Experimental - 1.651
DNI SNNI VC-NSNI (present)
17
18
3. Discretization Instability
Non-convergent in Refinement, Incorrect Softening
Implicit Gradient as a Nonlocal Regularization Nonlocal Strain: Construct ( ; ), such thataw x x s
1 1 2 3
( ) ( ; ) ( ) ( ) ( ),i j kn
a ijk ijk ijk i j ki j k
w d D Dx x x
x x x s s s x x
Gradient Reproducing Conditions:
0
( ) ( ; ) ( ) ( ( )), 0n
m m m
a ijk ijk
i j k
p w d p D p m n
s x x s s x x
Chen, J. S., Zhang, X., Belytschko, T., Computer Methods in Applied Mechanics and Engineering, Vol. 193, pp. 2827-2844, 2004.
1 1 2 2 3 3
0
1
1
(
( ; ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( )
ni j k T
a ijk a a
i j
T
I a I I
n
ijk ijk
i j k
w
D R gradient reproductio
w x s x s x w
n
s b w
x x s x x s H x s b x x s
gx M x H x x x
x
x x
x
Tg
Order of basis
functions n
T 1 T( ) ( ) ( ) ( ) ( )bw d
x g M x H x s s x s s
Implicit gradient model
[1] 1 ( ) ( ) x x
[1, 0, 2c] 2 2( ) ( ) ( )c x x x
[1, 0, 2 1c , 0, 24 2c ] 4 2 4
1 2( ) ( ) ( ) ( )c c x x x x
1( ) ( ) ( ) ( )( ) a I I
T
I w x M x H x x x xg x
Elastic Damage Tensile Model
FEM
RKSR
Elastic Damage Tensile Model
Stabilization of Advection-Diffusion Equation
22
1. Brooks and Hughes. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the
incompressible Navier-Stokes equations. Comput. Method. Appl. M. 1982.
2. Hughes et al. A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive
equations. Comput. Method. Appl. M. 1989.
3. Franca et al. Stabilized finite element methods: I. Application to the advective-diffusive model. Comput. Method. Appl. M. 1992.
*
=
=
=
adv
Subgrid scale methods
Galerkin/Least-squares
Streamline upwind/Petrov-Galerkin
Stabilized Petrov-Galerkin method: 1
0[H ], find [H ] such thath h
gw u
1
1
,
.
NPh
I I
I
NPh
I I I
I
u u
w c
x x
x x x
( ) ( ) ( )h h h h hw ,u B w ,u L w
2
adv diff
u u s
u k u
a
Strong form of PDE:
23
Brooks and Hughes. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible
Navier-Stokes equations. Comput. Method. Appl. M. 1982.
Hughes et al. A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive
equations. Comput. Method. Appl. M. 1989.
Franca et al. Stabilized finite element methods: I. Application to the advective-diffusive model. Comput. Method. Appl. M. 1992.
Stabilized Petrov-Galerkin method:
1
1
,
.
NPh
I I
I
NPh
I I I
I
u u
w c
x x
x x x
( ) ( ) ( )h h h h hw ,u B w ,u L w
0
1[ ], find [H ] such tH hath h
gw u
Subgrid scale methods, α=3
Galerkin/Least-squares, α=3
Streamline upwind/Petrov-Galerkin, α=2
*
=
=
=
adv
2
adv diff
u u s
u k u
a
Strong form of PDE:
Stabilization of Advection-Diffusion Equation
1
,NP
h
I I
I
w w
x x T
I I a I x H x x b x x x
Test function construction:
24
1 n
I I I Ix x y y z z H x x
Implicit Gradient RKPM (IG-RKPM)
Gradient reproducing conditions for :
,I I
I
n x x xx : multi-index
h h hww w
T
I I a I
I
M x H x x H x x x x
T 1
I I a I x M xH H x x x x
1 2 31, , , , , , , 0,... , 0a a a K K K
1 2 31, , , ,0, ... , 0a a a
1 2 31, , , , , , ,0, ... , 0a a a K K K
T H
Subgrid scale methods
Galerkin/Least-squares
Streamline upwind/Petrov-Galerkin
T 1
I I a I x M x H x x x0 xH
Virtually no extra cost!
Chen JS, Zhang X, Belytschko T. An implicit gradient model by a reproducing kernel strain regularization in strain localization
problems. Comput. Methods Appl. Mech. Eng., 2004.
Hillman M, Chen JS. Nodally integrated implicit gradient reproducing kernel particle method for convection dominate problems.
Comput. Methods Appl. Mech. Eng., 2016.
25
SU/PG vs IG-RKPM
0 in [0 10]
(0) 0 (10) 1
,x ,xxau ku ,
u , u
Strong advection:
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6 7 8 9 10
x
Exact
RKPM
SU/PG RKPM
IG-RKPM
Hillman M, Chen JS. Nodally integrated implicit gradient reproducing kernel particle method
for convection dominate problems. Comput. Methods Appl. Mech. Eng., 2016.
RKPM
RKPM with artificial diffusion IG-RKPM
Strong advection with a boundary layer
Hillman M, Chen JS. Nodally integrated implicit gradient reproducing kernel particle method for convection dominate
problems. Comput. Methods Appl. Mech. Eng., 2016.
RKPM IG-RKPM
After a full rotation:
Rotating cone problem
flow direction
A A x
y
u 0
u 0
A A
uCosine hill
k 0
u 0
u 0
Hillman M, Chen JS. Nodally integrated implicit gradient reproducing kernel particle method for convection dominate
problems. Comput. Methods Appl. Mech. Eng., 2016.
28
4. Gibbs Instability: Shock Front Discontinuities
Oscillatory Solution, Incorrect Damage
SCNI Based Smoothed Flux Divergence
Conservation equation
Galerkin equation
, , , 0h h h
tw ,t u t u t d
x x F x
, , , 0tu t u t x F x
1 1 1
, , ,
I I
h h h h k
I k k
kI I I
t u t d u t d u t lV V V
F F x F x n F x n
Smoothed Nodal Integration
, 0h h h
I I t I
h
II
I
w t u t w t t V F
, , : Flux conservedh k h k
k ku t u t
F x n F x n
Riemann solution at each xk according to characteristic speeds
,
,
i
i
F u t
u t
x
x | 0
,| 0
n
nk
h
IRP h n tk h
I t
uu u t
u
xshock speed
Riemann solutionRPu
1
,RPh k
I k k
kI
t u t lV
F F x n
J. S. Chen, C. T. Wu, S. Yoon, and Y. You, 2001
Roth, M. J., Chen, J. S., Slawson, T. R., Danielson, K. D., Computational
Mechanics, Vol. 57, pp. 773–792, 2016.
Treatment of Shocks in Nonlinear Solids
Divergence operation for volumetric stress
,
1
I
v
I i I ij j
I
S dV
dnV
I
j
v
ijI
I
1
dnPV
I
iI
I
1
I
IX *X JX
n
I
0
,, , , , , , , , ,
, , , , 0
hi
h
d v v
i i i j ij i ij j i ij j
i i i i
S
w t u t d w t t d w t t d w t t n d
w t h t d w t b t d
X
X
X X X X X X X X
X X X X
Variational equation
Roth, M. J., Chen, J. S., Danielson, K. D., Slawson, T. R., International Journal for Numerical Methods in
Engineering, Vol. 108, pp. 1525–1549, 2016.
Rankine-Hugoniot Solution
Cell interface pressure
]][[]][[ 0 uUP S
]][[]])[[(sgn uAuCU Bs
)()}()sgn({)( ***0*
IIIBII uuuuAuuCPP
I
IX *X JX
n
I
Rankine-Hugoniot jump equations
Consistency condition at interface
* 0 * * *( ) { sgn( ) ( )}( )J J B J J JP P C u u A u u u u
x̂IX
II uP ,
JJ uP ,
*X JX
pressure interface basedHugoniot * PSCNI integration cell
velocityinterface basedHugoniot * u
*1
I
I i i
I
S P n dV
propertiesshock material &
cityshock velo
AC
U
b
s
Roth, M. J., Chen, J. S., Danielson, K. D., Slawson, T. R., International Journal for Numerical Methods in
Engineering, Vol. 108, pp. 1525–1549, 2016.
Extended Riemann-SNNI
, ,
xx x
h hx x
dev
i i i j ij
vol
i
i i
j i
i i
jw u d w d
w h d w b
d
d
w
,
* *
1 1
,
1
x x
I vol
i I j ij
NP NP NPIJ IJ vol IJ IJ vol IJ
j ij j ij i I
vol
i j i
J
J J J
jw d F d
P
, ,
, ,1
( ) ( ) ( ) ( )
x
IJ
j J I j I J j
NP
J L L I L L LI JL
JI
j
j j
d
V
x x x x
Riemann-SNNI
J
IJ
I
I J
IJ
A
IJIJ
IJ n
Iv
Jv
Inv
Jnv
The local Riemann problem of nodal pair I-J
Conservation of linear momentum and energy
,
1( ) ( ) ( )d
LL I jI
L
jn
V x x x
,
1 1
1, 0NP NP
J J j
J J
Riemann-SCNI
,
0
, , ,
, , , , ,
, , , , 0
,
h
h
v v
i i
d
i i
j j i ij j
i j ij
i i i i
w t u t d w t t d
w t h t d w
w t t d w t
t t
n
d
t
b
d
X
X
X
X X X X
X X
X X
X
X
X
Riemann-SNNI
L
LL
1D Elastoplastic Wave Propagation
Material Model: J2 perfect plasticity
Impact vel, 273 m/s
RKPM without shock algorithm RKPM with shock algorithm,
Riemann-SNNI
Oscillatory Smooth
Noh's 2D Implosion Problem
Lagrangian Riemann-SNNI
Density distribution
Pressure distribution
Initial node distribution
Initial condition:
1. All particles move toward the center with a
unit velocity.
2. Initial pressure is zero.
3. Initial density is one.
2D Sedov Blast Problem
Density distribution Pressure contour
High energy
release at the
center
Air
Two-dimensional Plate Impact with Rarefaction Waves
Experimental peak pressure: 8 Gpa
RKPM without shock algorithm RKPM with shock algorithm,
Extended Riemann-SNNI
1000 m/s
Marsh, S. A., LASL Shock Hugoniot Data,
University of California Press, Berkley, 1980
Micro-crack informed Damage Model
0 0(1 ) (1 )d d
dY Y
Tension-compression
Decoupled Damage (M. Ortiz)
Fully Tensorial Damage
Model
Ren, X., Chen, J. S., Li, J., Slawson, T. R., Roth, M. J., International Journal of Solids and Structures, Vol. 48, 1560–1571, 2011.
Rebar Pullout
38
Meshfree RKPM Modeling Shear cone formation
SNNI
(unstable) NSNI
SNNI (unstable) NSNI
near center cross section: dense cones and cracks
Center cross section: a few cones and cracks
Fragmentation, radial and circumferential cracking
23
D. Cargile, Army Engineer Research And Development Center, 1999.
Experimental
RKPM Modeling of Debris cloud shape in thick target
Explosive Welding Modeling
Parent (base) tube
Flyer tube
Explosive
α
2D configuration Capsule used in Mars Sample Return Mission
Grignon, F., Benson, D., Vecchio, K.S. and Meyers, M.A., "Explosive
welding of aluminum to aluminum: analysis, computations and
experiments," International Journal of Impact Engineering, vol. 30, no.
10, pp. 1333-1351, 2004.
Simulation of Explosive Welding
Simulation of RC column subjected to blast loads
Time:
0.08
msec
Time:
0.16
msec
Shock wave propagation in RC column
Reflected tensile wave
Spalling
Tension damage distribution
Explosive test
(K.C. Wu et al. Journal of impact Engineering. 2011)
Numerical result obtained by using RKPM
RKPM Modelling of Levee Failure
H. Mori, Univ. of Cambridge, 2008;
S. Bandara, K. Soga, Computers & Geotechnics, 2015;
Summary and Conclusion
Stabilities in modeling of extreme loading problems based on nodal integration are addressed:
– Kernel stability: Quasi-linear RK approximation
– Rank stability: Naturally stabilized nodal integration
– Shock physics and Gibbs stability: Riemann SCNI and SNNI
– Discretization instability: Implicit gradient or scaling law
Accuracy enhancement and stability in nodal integration are achieved under the Variational Consistency framework.
USACM Thematic Conference
Meshfree and Particle Methods: Applications and Theory
September 17-19, 2018, Santa Fe or Albuquerque NM, USA
Sandia National Laboratories
http://www.usacm.org/conferences.
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