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.