Simulation of plane strain rolling using the Element

Simulation of plane strain rolling using the Element Free Galerkin Method Xiong S., Rodrigues J.M.C. and Martins P.A.F. Instituto Superior Técnico Universidade Técnica de Lisboa Portugal

Summary



This work presents a new approach for analysing plane strain rolling based on the utilisation of the element free Galerkin (EFG) method for slightly compressible rigid-plastic material models

• A detailed description of the method and its numerical implementation is presented with the objective of making clear the fundamental differences to the well-established finite element method for slightly compressible rigid-plastic materials (RPFEM) • The effectiveness of the proposed approach is discussed by comparing theoretical predictions with experimental data found in the literature

Introduction Why meshless methods ?

• Finite elements usually require remeshing procedures for handling nonstationary large plastic deformations owing to the progressive distortion of the mesh and its eventual interference with the dies • Remeshing procedures may contribute to a degradation of the overall accuracy and generally demand extensive computational efforts namely during three-dimensional analysis of complex forming processes

Introduction Why meshless methods ?



Meshless methods perform the discretization of the workpiece entirely in terms of arbitrarily placed nodes without the use of an explicit mesh

• Meshless methods can handle more easily the procedures of refinement and adaptivity frequently utilised in the numerical simulation of bulk metal forming processes • Displacements and/or velocities are described in a less piecewise manner than in finite elements

Classification of meshless methods 

Smoothed particle hydrodynamics (SPH) was the first meshless method. It was proposed by Gingold, Lucy and Monaghan to simulate the astrophysical phenomenon such as stellar collisions, formulation of galaxies and fluctuations of gas and dark matter in an expanding universe Authors

Methods

Gingold, Lucy, Monaghan(1970~80’s)

Smoothed Particle Hydrodynamics (SPH)

Nayroles et al. (1992)

Diffuse Element (DE)

Belytschko et al. (1994)

Element Free Galerkin (EFG)

Laguna (1995)

Smoothed Particle Interpolation (SPI)

Babushka (1995)

Partition of Unity Finite Element Method

Liu et al. (1995)

Reproducing Kernel Particle Method (RKPM)

Liu et al. (1996)

Moving Least Square Reproducing Kernel (MLSRK)

Sulsky and Schreyer (1996)

Particle-in-Cell Method

Oñate et al. (1996)

Finite Point Method (FP)

Duarte and Oden (1996)

h-p Clouds

State-of-the-art in metal forming 

1998/1999 - Chen et al. (1998/1999) used the reproducing kernel particle method (RKPM) to simulate plane strain stretching of a circular blank and simple axisymmetrical upsetting operations



1999/2000 - Kulasegaram et al. employed the corrected smooth particle hydrodynamics method (CSPH) to perform several two-dimensional simulations of basic metal forming processes, (plane-strain extrusion, rolling and upsetting), but only perfectly-plastic materials were taken into consideration



2000/... - Despite the remarkable progresses achieved by meshless methods in other fields of knowledge, the applications in metal forming process simulation are still limited

State-of-the-art in metal forming Reasons for the unawareness/lack of interest on meshless methods



The utilisation of the finite element method is very well established within the metal forming community

• A significant part of the metal forming community works with commercial softwares. Only a few number of research groups are still engaged in developing computer programs • Major software houses made big investments in the development of sophisticated 2D / 3D finite element computer programs

State-of-the-art in metal forming Reasons for the unawareness/lack of interest on meshless methods



Mathematical foundations and computer programming are more difficult than in finite elements



Meshless methods present an important draw back due to the fact that its shape functions are not interpolation functions and therefore the essential boundary conditions cannot be directly imposed

• Available computer programs based on meshless methods are computationally expensive requiring more CPU time than a conventional rigid-plastic finite element computer program

Theoretical background Moving least-squares



Numerical procedures involving differential equations usually perform the approximation for the unknown function in a domain, by mean of trial functions which can be expressed in terms of base functions p(x) with unknown coefficients a(x), m

u h ( x )   pi ( x ) ai ( x )  pT ( x ) a( x ) i 1

• The unknown coefficients a(x) can be obtained by minimising the following norm, n



J (a( x ))   w(x-x I ) p ( x I ) a( x )  uI I 1

T



2

Theoretical background Moving least-squares This implies that the unknown coefficients a(x) can be determined from,

 J (a( x )) 0 a where,



a( x )  A 1( x ) B( x ) u

n

A( x )   w I ( x ) p( x I ) pT ( x I ) I 1

B( x )   w 1( x ) p( x1 ) w 2 ( x ) p( x 2 )    w n ( x ) p( x n )  uT   u1 u2    un 

Theoretical background Moving least-squares



The moving least-squares approximation for the unknown function u(x) can be written as, n

u ( x )   I ( x ) u I h

I 1



Comparing the aforementioned relationship with the classical finite element interpolants it can be concluded that the meshless shape functions of the moving least squares approximation methods can generally be written as, m

I ( x )   p j ( x ) ( A 1( x ) B( x )) jI j 1

Theoretical background Moving least-squares



The partial derivatives of the shape functions can be obtained applying the usual rules of partial differentiation, m



I ,i   p j ,i ( A 1B) jI  p j ( A,i 1B  A 1B,i ) jI



j 1



By selecting a vector p(x) of linear base functions the explicit form of the meshless shape functions becomes similar to the kernel with first degree of correction that was proposed by Liu et al in the reproducing kernel particle method (RKPM),

I ( x )  w I ( x ) ( x ) ( 1  β( x )T (x - x I ))

Theoretical background Moving least-squares



In general the Kernel functions w(x,h) can be written as,

 W ( x, h )  d ( ), h



x h

Example: The spline kernel function

where

3

  x /h

if   1 if 1    2 if   2

0 .6

W(x)

 1  1.5  0.75 2  3 W ( x, h )  0.25(2  ) 3h   0 2

S pline

0 .4 0 .2 0 -2

-1

0

x/h

1

2

Theoretical background Moving least-squares



In meshless methods, unlike finite elements, the shape functions are not interpolants

I (x J )  IJ



The value of the approximating function generally does not coincide with the value of the unknown parameter at the sampling points. This creates extra difficulties in enforcing the essential boundary conditions

u h (x a )  ua

Theoretical background Slightly compressible rigid-plastic formulation



The slightly compressible rigid-plastic formulation introduced by Osakada et al. is derived from the theory of plasticity for porous materials and offers the possibility of obtaining the stress-strain rate relationship directly by avoiding the problems related with the uncertainty of the hydrostatic pressure

ij , j  qi  0  ij (u ) 

 ij 

1  ui  u j 1 (  )  (ui , j  u j ,i ) 2  x j  xi 2

 2 1 2 (  ij  (  )  k k ij )  3 g 9

Theoretical background Slightly compressible rigid-plastic formulation



The four boundary conditions associated with the slightly compressible rigid-plastic formulation are the following,

on Su

on St

 ij n j  t i

ui  ui on Sf

(u  utool )i ni  0 u~i ti   ij n j  (( kl nl ) nk ) ni  fi   f  uf

Theoretical background Slightly compressible rigid-plastic formulation



The simplest procedure to satisfy the equations and the boundary conditions is to express the weak form of the variational principle entirely in terms of an arbitrary variation in the velocity v,

  (u)  (v) d   t v ij

ij

i



St

i

dS   qi v i d  ku  (ui  ui )v i dS 

Su

  ti v i dS  k f  (u  utool )i ni  v j n j dS  0 Sf

Sf

where, n

u( x )   I ( x ) uI I 1

n

v( x )   I ( x ) vI I 1

Theoretical background Slightly compressible rigid-plastic formulation



By employing a similar procedure as in finite elements it is possible to obtain the following set of non-linear equations,

Ku  f 1  T BI ( D  CCT ) BJ d  ku  I I J dS  k f  I (n nT ) J dS  g  Su Sf

K IJ  

fI   I t dS   I q d   I t dS  ku  I u dS  k f  I (utool n) n dS T

St



Sf

Su

Sf

Theoretical background Element free Galerkin method



The discretization procedure (nodes instead of elements)

NP

~ ~ K u   IJ J fI J 1

(I  1,2,...,NP )

NE

~ ~ K u   m m fm m 1

(m  1,2,...,NE )

Theoretical background Element free Galerkin method where,

~ K IJ 

   (x

cMIJ

c

I

c

)  J ( x c )   c I ( x c )  J ( x c )I

~   c J ( x c )  I ( x c ) J c 

 k  (x u

I

B

~ )IJ ( x B ) J B

BMIJB

  I ~ T     f ( x F )  k f nn I ( x F ) J ( x F ) J F   uf ( x F ) F MIJF  

~ fI 

  (x I

~ J ) t ( x ) T T T 

T M I

  (x I

~ J ) q ( x ) c c c 

cMI

k

~ J  ( x ) u u I B B 

BMIB

 ~ utool ( x F ) T    ( ) ( ) k ( ( ) ( )) ( )  x  x u x n x n x F I F  f F u (x ) f tool F F F  JF F M I f F   

 3 



 1 2 (  )  g 9

Theoretical background Element free Galerkin method



The assembly procedure,

~ Km 

 ...

gpm

~ K IJ 

 ...

cMIJ

Results and discussion Plane strain rolling





The element free Galerkin (EFG) method was compared with available experimental results and those provided by the rigid-plastic finite element method for slightly compressible material models (RPFEM) Experimental data (Shida S. and Awazuhara H.) : Case 1 2 3 4

ho(mm) 0.50 0.50 0.50 0.50

h1(mm) 0.46 0.42 0.38 0.34

Material : Steel Radius of the rolls : 65 mm Linear velocity of the rolls : 0.25 m/s

Reduction (%) 8 16 24 32

Results and discussion Plane strain rolling The numerical evaluation of the integrals is performed in accordance to the cell structure depicted the figure. The nodes are located on the material flow lines and four neighbouring nodal points form a cell This image cannot currently be display ed.





Both complete (2x2) and reduced (1x1) Gauss point integration schemes were utilised for the calculation of the domain integral

Results and discussion Computed distribution of the roll-workpiece normal contact pressure obtained from EFG and RPFEM

Pressure (MPa)

1600

1200

800

400

EFG FEM

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Contact Length (mm)



Case 4, discretization of the domain by 21x3 nodes

3.5

Results and discussion Computed distribution of the equivalent strain obtained from EFG and RPFEM



Case 4, discretization of the domain by 21x3 nodes

Results and discussion Rolling torque (per unit of width) calculated by EFG and RPFEM

6000

6000 EFG

Case 4

Torque (N m/m)

Torque (N m/m)

4500

RPFEM

Case 3

3000

Case 2

4500

Case 4

3000

Case 3

Case 2

1500

1500 Case 1

Case 1

0

0 0

5

10

15

20

Nodes in x direction



25

30

0

5

10

15

20

Nodes in x direction

Solid: 3 nodes, Blank: 2 nodes in the thickness direction

25

30

Results and discussion Comparison of measured and computed roll torque (per unit of width) as a function of the percentage of reduction

Results and discussion Comparison of measured and computed roll separating force (per unit of width) as a function of the percentage of reduction

Conclusions 







The fundamentals of the element free Galerkin (EFG) method for solving metal forming problems were presented The main innovation of this work is the development of a new EFG approach based on the rigid-plastic formulation for slightly compressible material models Numerical simulations of the flat rolling of steel under plane strain assumptions allow us to conclude that EFG ensures a complete description of all the variables of interest to the rolling process and that the computed predictions of the rolling torque and roll separating force are in fair agreement with experimental results However it should be noted that the proposed method is computationally expensive requiring more than twice the CPU time needed for a conventional rigid-plastic finite element approach