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G. Y. Li
e-mail: mgyli@ntu.edu.sg
M. J. Tan
K. M. Liew
Centre for Advanced Numerical
Engineering Simulation,
School of Mechanical and Production
Engineering,
Nanyang Technological University,
Nanyang Avenue,
Singapore 639798
Z. H. Zhong
Department of Mechanical Engineering,
Hunan University,
Changsha 410082, P.R.China
1
Sheet Forming Analysis Based
on Improved Contact Searching
Algorithm and Simple Approach
for Contact Force Evaluation
Explicit finite element procedures for the simulation of sheet-metal-forming processes are
reviewed. Improved techniques for contact searching and contact force evaluation are
then proposed and discussed. In the contact searching algorithm, a new contact territory
definition is introduced and only node-to-segment contact pairs are defined, which makes
the post-contact searching more efficient and easy to implement. For contact force evaluation, a novel approach is employed to make the simulation of sheet forming processes
more accurate. The algorithms are implemented and examined via experiments.
关DOI: 10.1115/1.1288594兴
Keywords: Explicit Finite Element, Contact-Impact, Contact Searching, Sheet Forming
Introduction
The usefulness of computer simulation techniques in the tool
design and manufacturing is more and more widely recognized.
The mechanical phenomenon of sheet forming is contact-impact
with large elastoplastic deformation. With the finite element
method, such contact-impact processes can now be simulated with
good accuracy for practical purposes. Simulation of metalforming process can help not only to trouble shooting in forming
processes under operation, but also to shorten design time and to
reduce costs for manufacturing for new sheet-metal components.
It should therefore not be too difficult to understand the advantages of the simulation techniques. However, one should also be
very cautious in using the simulation techniques and the results
they provided. As the simulation results are obtained from numerical solutions of mathematical models designed for certain
types of problems, they may be subjected to various errors such as
computer truncation errors and finite element discretization errors
etc. The purpose of this paper is to discuss finite element simulation for contact-impact processes and with application to sheetforming simulation, and special attention is paid to the reliability
of the simulation techniques. In this study, emphasis will be put
on the treatment of contact-impact interfaces and the evaluation of
contact force.
Treatment of contact between the blank and the tools is a key
problem in the simulation of sheet-forming processes. There are
many algorithms which can be used for contact searching. Among
them, the so-called master-slave algorithm has been widely used
关1兴. In master-slave algorithm, one must specify which two
boundaries may come into contact prior to the solution. The simplest way is to consider that every segment in the blank may come
into contact with every segment in the tools. However, it will
increase the computing time for contact searching. In reference
关2兴, a general contact searching algorithm based on the concepts
of hierarchy and territory, named HITA, was presented. The contact searching is automatic and no definition of contact boundaries
is needed. However, in HITA, the contact treatment is symmetric
which means that the contact searching must perform twice by the
exchange of the definition of master and slave surface. Furthermore, the single surface contact is also considered. Therefore, the
Contributed by the Materials Division for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials
Division February 18, 1998; revised manuscript received March 14, 2000. Associate
Technical Editor: E. Busso.
contact searching is very time consuming. In this paper, the original HITA algorithm has been modified for the purpose of sheetforming analysis. A new contact territory for contact segment was
presented and only the nodes in the blank are checked with the
segments in the tools to determine whether or not the contact is
possible because the tools can be considered as rigid and their
positions are known during the forming process. Furthermore, the
single surface contact is not considered. Thus, the contact searching in this paper is more efficient without the loses of accuracy.
In the following, a general review is first given to explicit finite
element procedures for contact-impact problems. Then, improved
techniques for contact searching and contact force calculations
will be discussed. After which, numerical examples are given and
reliability and efficiency issues are discussed.
2 Explicit Finite Element Procedures for ContactImpact Processes
With sheet-metal-forming problems in mind, let us assume that
the contact-impact process concerned here involves large displacements, finite rotations, and large plastic deformation. Considering the extremely high grade of nonlinearity involved, explicit finite element approaches will be used. Then, the simulation
of contact-impact processes may be considered the solution of the
following nonlinear equation system
MA⫽R⫺F⫹RC
(1)
with properly given initial conditions, boundary conditions, loading conditions, and material properties including friction properties on material surfaces, see reference 关2兴, where M is the mass
matrix, A the acceleration vector, R the external load vector, F the
internal load vector, RC the nodal force vector due to contact
force. In Eq. 共1兲, the mass matrix M maybe considered as a constant, the external load vector R maybe considered prescribed.
However, nodal vectors A, F, and RC are all deformation dependent. Suppose that there are N equations in Eq. 共1兲, then there are
2N⫹M unknowns, i.e., N unknowns in A, N unknowns in F and
M unknowns in RC. To be able to solve Eq. 共1兲, N⫹M extra
equations are required, which are provided by the central difference method, contact constraints and friction law as explained in
reference 关2兴. To calculate F, material constitutive models and
element techniques will be employed. To calculate RC, three basic
problems must be solved, i.e., the searching of contacting nodes,
the enforcement of contact constraints and the employment of
Journal of Engineering Materials and Technology
Copyright © 2001 by ASME
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⫹ ⫹4
X ic
Fig. 1 The contact territory of a segment
friction law. While all the above-mentioned topics are of great
importance in the simulation of contact—impact processes, we
choose only to discuss the searching of contacting nodes and the
enforcement of contact constraints. It should also be noted that
those two topics are critical to efficient and successful simulation
of sheet forming processes.
3
Improved Techniques for Contact Searching
It is of great practical importance to search contacting nodes
efficiently and reliably for large-scale industrial simulation such
as metal forming. There are now various algorithms available for
contact searching. For example, the master-slave algorithm 关1兴,
the single surface algorithm 关3兴, the hierarchy-territory algorithm
关4,2兴, to name a few. The first version of hierarchy-territory algorithm 关4兴 presented several novel ideas in constructing new efficient contact searching algorithm. Most important concepts include the concept of hierarchical objects, the concept of territory,
and the concept of pre- and post-contact searching. These concepts have been used to construct various high performance contact searching algorithms. However, further improvement of the
algorithm can still be made. In the following, an improved version
of the algorithm is described. For contact searching, let us define
a contact system as a collection of contact segments defined with
four or three nodes. The geometry of contact segment may be
defined as
n
x⫽
兺 ␸ 共 ␰,␩ 兲x
i⫽1
i
(2)
i
where x denotes the position vector of a point on the segment, x i
the position vector of contact node i, n the total number of nodes
on the segment, and ␸ i ( ␰ , ␩ ) the interpolation function associated
with node i.
The unit normal vector of the segment can be defined as
N s ⫽x , ␰ ⫻x , ␩ / 兩 x , ␰ ⫻x , ␩ 兩
(3)
where x , ␰ and x , ␩ denote the derivative of x with respect to ␰ and
␩, respectively.
For each contact node P, a unit normal vector can be defined as
follows
m
N d⫽
兺
i⫽1
冒 冏兺 冏
m
N si
i⫽1
N si
(4)
⫹
i⫽1,2,3,4,
120 Õ Vol. 123, JANUARY 2001
i⫽1,2,3,4,
0⭐ ␨ ⭐1
⫺1⭐ ␨ ⭐0
(5)
(6)
where h i denotes the thickness of node i, D p the maximum penetration allowed, D c the prescribed control distance. It should be
noted that h i is zero if node i is from a solid element.
With the contact territory defined, the precontact searching can
be performed in the usual way as described in reference 关4兴 except
that a different algorithm will be required to judge whether a
given node lies within the contact territory or not. However, the
post-contact searching will be different. In this case, there is only
one type of contact pairs, i.e., the node-to-segments contact pair.
The contact territory of the whole system can be defined as the
collection of contact territories of all contact segments in the system. If a node in an existing contact pair is found not within the
contact territory of its current target segment, a neighboring segment is immediately peaked as its new target segment without the
need of testing the node with any edge contact territory or node
contact territory. That is a desirable advantage especially for parallel processing.
4
Contact Force Evaluation
The penalty method has been widely used to calculate contact
force in explicit analysis. However, the method has two major
drawbacks. First, it may affect the numerical stability of the explicit method which is only conditional stable. Second, it introduces artificial penetration which is not desirable from an accuracy point of view. To overcome these shortcomings, the defense
node algorithm as described in reference 关2兴 may be used. In the
defense node algorithm, the Lagrange multiplier method is used to
calculate contact forces and contact constraints are enforced accurately. Unfortunately, the defense algorithm is not compatible
with the rigid modes. The reason is given as follows. If the defense node algorithm is invoked before rigid body calculations,
forces/motion of contact nodes from a rigid body will be modified
due to rigid body constraints, which would violate contact conditions. If the defense node algorithm is invoked after rigid body
calculations, rigid body conditions will be violated due to contact
force calculations.
In sheet-metal-forming simulation, it is desirable to use the
Lagrange multiplier approach to accurately calculate contact
forces between the tool and the workpiece. At the same time, it is
also desirable to treat all the tool parts as rigid bodies in the
simulation to reduce computational work. To be able to do that, a
novel approach as described below may be used.
Consider a general sheet-metal-forming system with four different types of components, i.e., the punch, the blank holder, the
die, and the blank, as shown in Fig. 2. Suppose that the die is fixed
and the punch can only move in the z-direction with a prescribed
velocity ␯ p . The blank holder is not allowed to move in the xy
plane but can move in z-direction. The external pressure applied
to the blank holder is denoted by p(x,y), which acts in the
z-direction and may vary as a function of position 共x,y兲 in the
blank holder.
The Calculation of the Contact Force Between the Blank
and the Die. Suppose that a node from the blank gets in contact
only with the die. Then, the equation of motion of the node in the
normal direction can be written as
ma n ⫽r cn ⫹ f n
where m denotes the total number of segments sharing the nodes,
N si denotes the unit normal vector of ith segment at the node.
The contact territory of the segment can then be defined as an
eight-nodes solid element, as shown in Fig. 1, whose nodes, de⫹
noted by X ic , are defined as follows
X ic ⫽x i ⫹ ␨ N id 共 h i ⫹D p 兲 ,
⫽x i ⫹ ␨ N id 共 h i ⫹D c 兲 ,
(7)
where m is the mass of the node, and a n , r cn , f n are the normal
component of acceleration, contact force and the internal nodal
force, respectively. According to the central difference method, it
holds that
a n 共 n 兲 ⫽ 关 ␯ n 共 n⫹1/2兲 ⫺ ␯ n 共 n⫺1/2兲 兴 /⌬t
(8)
The contact constraint condition can be expressed as
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Fig. 2 The sheet forming system; 1-punch; 2-blank holder;
3-blank; 4-die
Fig. 4 The deformed configuration of numerical result for
square 100 mm blank. „a… Numerical result; „b… experimental
result.
⫺ ␯ n 共 n⫹1/2兲 ⌬t⫹p 共 n 兲 ⫽0
(9)
where p ( n ) is the penetration at time step n and ␯ n ( n ) is the normal
velocity at time step n. Equations 共7兲–共9兲 give
r cn ⫽m 关 p 共 n 兲 /⌬t⫺ ␯ n 共 n⫺1/2兲 兴 /⌬t⫺ f n
(10)
The Calculation of the Contact Force Between the Blank
and the Punch. If a contact node gets in contact with the punch,
Eqs. 共7兲 and 共8兲 still hold, but Eq. 共9兲 should be changed to
␯ p •N s ⌬t⫺ ␯ n 共 n⫹1/2兲 ⌬t⫹p 共 n 兲 ⫽0
(11)
r cn ⫽m 关 p 共 n 兲 ⌬t⫹ ␯ p •N s ⫺ ␯ n 共 n⫺1/2兲 兴 /⌬t⫺ f n
(12)
which gives
The Calculation of the Contact Force Between the Blank
and the Blank Holder. Consider now the case in which a contact node gets in contact with the blank holder. The local coordinate system defined on the blank holder will be used as the reference coordinate system and the blank holder is assumed to have a
flat contact surface with its normal parallel to the punching direction. The equation of motion of a contact node can then be written
as
ma n ⫽r bn ⫹ f n
(13)
where r bn denotes the normal contact force due to the blank
holder. The contact condition for the blank holder is
␯ b 共 n⫹1/2兲 ⌬t⫺ ␯ n 共 n⫹1/2兲 ⌬t⫹p b 共 n 兲 ⫽0
Fig. 3 The deformed configuration of numerical result for
square 80 mm blank. „a… Result by LS-DYNA3D; „b… result by
present method; „c… experimental result.
Journal of Engineering Materials and Technology
(14)
where p b ( n ) denotes the penetration into the blank holder. Furthermore, the motion of the blank holder can be described using the
following equation
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Fig. 5 The dimensions of the blank and the tools for s – rail forming
k
M a b⫽
兺r
i⫽1
i
bn ⫹
冕冕
which gives
p 共 x,y 兲 dS
(15)
␯ b 共 n⫹1/2兲 ⫽ 关 F⌬t⫹M ␯ b 共 n⫺1/2兲 兴
where k denotes the total number of contacting nodes.
Equations 共13兲 and 共14兲 give
r bn ⫽m 关 p b 共 n 兲 /⌬t⫹ ␯ b 共 n⫹1/2兲 ⫺ ␯ n 共 n⫺1/2 兲 兴 /⌬t⫺ f n
冕冕
F⫽
兺 兵m 关 p
i
i⫽1
兺 m␯
i
i⫽1
i
i
i
b 共 n 兲 /⌬t⫺ ␯ n 共 n⫺1/2兲 兴 /⌬t⫺ f n 其
b 共 n⫹1/2兲
i
册
(18)
冕冕
k
p 共 x,y 兲 ds⫹
兺 兵m 关 p
i
i⫽1
i
i
i
b 共 n⫺1/2兲 /⌬t⫺ ␯ n 共 n⫺1/2兲 /⌬t⫺ f n 兴 其
The contact force between the blank and the blank holder can be
obtained from Eqs. 共16兲 and 共18兲.
5
k
⫹
i⫽1
(19)
p 共 x,y 兲 ds
k
⫹
兺m
where
(16)
Inserting Eq. 共16兲 into Eq. 共15兲, it can be obtained that
M a b⫽
冒冋
k
M⫹
/⌬t
122 Õ Vol. 123, JANUARY 2001
(17)
Implementation and Application Aspects
The implementation of the contact searching algorithm is similar to that of the unified contact algorithm described in reference
关5兴. The main difference lies in the post-contact searching. In the
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unified contact searching algorithm based on the territory concept,
the post -contact searching needs to deal with three types of contact pairs, i.e., node-to-segment, node-to-edge and node-to node
contact pairs. When a node-to-segment contact pair changes to a
node-to-edge or node-to-node contact pair, the post-contact
searching may need to go through several test pairs before a
proper target edge or target node could be found. In the present
algorithm, that is no longer necessary. If a contacting node slides
away from its current target segment, the node may violate more
than one edge condition. Suppose that the ith edge condition is
mostly violated, then the segment connected to the ith edge of the
current target segment is selected as the new target segment and
no further test needs to be performed. That not only reduces computational work but also facilitates storage management and saves
storage space. As a drawback, it may introduce some error in the
calculation of target point position. However, any such error will
be corrected in the subsequent solution cycle without losing global accuracy.
The implementation of the algorithm for contact force calculation is quite straightforward. Nevertheless, there is one point
which deserves special attention. If a contact node lies within the
contact territory of both die and blank holder, the node can be
regarded in contact with either the die or the blank holder, but not
both. The judgment can be based on nodal force calculations associated with internal stresses. If the nodal force tends to move the
node against the die, the node should then be considered to contact with the die. It is the same for the blank holder.
Let us now look at the performance of the algorithm via two
examples. The first example concerns deep drawing of a square
cup.
In order to check the behavior and the accuracy of the algorithm described above, a square cup drawing test is used. The side
length of the punch is 47.55 mm, the side length of the die is 50.5
mm. The radius of the punch shoulder and the die shoulder are 5.0
mm. Blanks are square with 80 mm or 100 mm side length and
0.7 mm thickness. The material is JIS-SPCE deep drawing steel.
The test is performed until 20 mm of punch stroke without blank
holder.
Figure 3共a兲 shows the deformed configuration of numerical result for square 80 mm blank by LS-DYNA3D in which masterslave contact searching algorithm and penalty method for contact
force evaluation are used. Figure 3共b兲 shows the result by the
present method and Fig. 3共c兲 shows the experimental result. Good
agreement between them is drawn.
Figure 4共a兲 shows the deformed configuration of numerical result by the present method for square 100 mm blank and Fig. 4共b兲
shows the experimental result. The computation is able to predict
the wrinkling mode and is in good agreement with experimental
result quite well.
In the second example, the s – rail benchmark problem from
NUMISHEET’96 is considered. The detail description of the tools
and the blank is given in Fig. 5. The material is Draw Quality
Mild Steel. Punch stroke is 37 mm. Figure 6 shows the deformed
configuration after springback with blank holder force of 10 KN.
Some wrinkling is observed. Figure 7 shows the shape of a section. The solid line is for the result by present method and the
dash line is for the result by LS-DYNA3D, and the difference
between them is very little. Figure 8 shows the deformed configuration after springback with the blank holder force of 200 KN. In
this case, no wrinkling occurs and the amount of springback is
very small.
With the present approach, the deformed configuration obtained
may be considered the same as those obtained with the penalty
method with carefully chosen penalty parameter. However, it is
not desirable to use the penalty method in industrial application
since it is very difficult to choose a ‘‘proper’’ penalty parameter
even for an expert, not to mention for normal engineers.
Journal of Engineering Materials and Technology
Fig. 6 The deformed configuration after springback with blank
holder force of 10 KN
Fig. 7 The shape of a section after springback by present
method „solid… and by LS-DYNA3D „dash…
Fig. 8 The deformed configuration after springback with the
blank holder force of 200 KN
6
Concluding Remarks
The improved contact searching algorithm offers an efficient
post-contact searching algorithm even on serial computers. That is
due to the assumption that the sliding velocities of contacting
nodes are sufficiently small as compared to the stress wave speed.
Errors may become significant if the relative sliding velocity becomes large, say over 10 percent of the stress wave speed. The
approach for contact force evaluation is also subjected to the limitation that the contact surface on the blank holder is flat and is
normal to the punch direction. If the contact surface becomes
curved and frictional forces are significant, the situation becomes
much more complicated. Further investigation on both algorithms
JANUARY 2001, Vol. 123 Õ 123
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is still underway, which is believed to bring up a general, efficient
and accurate algorithm set for metal forming simulation.
References
关1兴 Hallquist, J. O., 1992, ‘‘LS-DYNA3D, Theoretical Manual,’’ LSTC, Livermore.
关2兴 Zhong, Z. H., 1993, Finite Element Procedures for Contact—Impact Problems, Oxford Press.
关3兴 Benson, D. J., and Hallquist, J. O., 1987, ‘‘A Single Surface Contact Algorithm for the Post-buckling Analysis of Shell Structures,’’ San Diego, La Jolla,
University of California.
关4兴 Zhong, Z. H., and Nilsson, L., 1989, ‘‘A Contact Searching Algorithm for
General Contact Problems,’’ Comput. Struct., 1, pp. 197–209.
关5兴 Zhong, Z. H., and Nillson, L., 1996, ‘‘A Unified Contact Searching Algorithm
Based on the Territory Concept,’’ Comput. Methods Appl. Mech. Eng., 130,
pp. 1–16.
124 Õ Vol. 123, JANUARY 2001
关6兴 Hallquist, J. O., 1979, ‘‘NIKE2D—A Vectorized, Implicit, Finite Deformation, Finite Element Code for Analysing the Static and Dynamic Response of
2D Solids,’’ RPT. UCRL-52678, LLNL, Livermore.
关7兴 EUROPORT, 1996, ‘‘Insights, Industrial Application Codes on Parallel
Computers—Fluid Dynamics and Structural Mechanics,’’ GMD.
关8兴 Li, G . Y., Yang, D. Y., and Zhong, Z. H., 1996, ‘‘3D Dynamic Explicit Finite
Element Analysis for Sheet Forming by Using a General Contact Searching
Algorithm and Different Shell Elements,’’ NUMISHEET’96, pp. 421–426.
关9兴 Zhong, Z. H., and Li., G. Y., 1998, Computer Simulation of Sheet Metal
Forming Processes with Applications, Beijing, Beijing University of Science
and Technology Press.
关10兴 Li, G. Y., Tan, M. J., and Liew, K. M., 1998, ‘‘An Accurate and Efficient Shell
Element,’’ Proc. HPC’98, Singapore, pp. 226–238.
关11兴 Li, G. Y., Tan, M. J., and Liew, K. M., 1999, ‘‘Springback Analysis for Sheet
Forming Processes By Explicit Finite Element Method in Conjunction with the
Orthogonal Regression Analysis,’’ Int. J. Solids Struct., 36, pp. 4653–4668.
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