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Automatic remeshing in finite element simulation of metal forming processes by guide grid method
Journal o[ Materials Processing Technology, 27 (1991) 73-89
73
Elsevier
Automatic remeshing in finite element simulation of metal forming processes by guide grid method J.S. Park and S.M. Hwang Department of Mechanical Engineering, Pohang Institute of Science and Technology, Pohang 790-600, Korea Mechanical Engineering Division, Research Institute of Industrial Science and Technology, Pohang 790-600, Korea
Industrial S u m m a r y An approach to full automation of the remeshing procedure in the finite element simulation of a non-steady metal forming process is presented. The approach is based on a new automatic mesh generation scheme, in which a two-dimensional mesh is generated for an arbitrary domain with the aid of a prespecified mesh referred to as the guide grid. Evaluation of the effect of remeshing on the solution accuracy is made. Backward extrusion with a sharp-cornered punch is simulated, to verify the effectiveness of the proposed approach. The approach is further applied to simulating a rib-web forging process.
Notation Do
EL H
Ho K~ NK R r u
Yd v
Wo
x~ y z t-
initial diameter of specimen force vector current height of specimen initial height of specimen stiffness matrix FEM basis function extrusion ratio coordinate along the radial direction horizontal component of velocity vector die velocity vertical component of velocity vector initial width of specimen ith coordinate of Kth nodal point coordinate along the forging direction coordinate along the axial direction effective strain effective strain rate nodal value of Kth nodal point field variable which is discontinuous at element boundary
0924-0136/91/$03.50 © 1991--Elsevier Science Publishers B.V.
74
// ]
coefficient of Coulomb friction flow stress
1. I n t r o d u c t i o n
Finite element formulations developed for numerical simulation of plastic flow occurring in metal forming are mostly based on the material description of motion in which the finite element mesh deforms with the material. As a result, the element shapes may become severely distorted as the deformation proceeds. In order to continue the simulation at that stage, remeshing is necessary. However, remeshing in general requires a great deal of work. Full automation of the remeshing procedure will eliminate the need for working on remeshing. In addition, it is an essential step toward developing a system for automatic simulation and analysis of a metal forming process. The remeshing procedure consists of two parts. The first part is to rediscretize the deformed domain and generate a new mesh. The second part is to transfer nodal and element variables from the old mesh to the new mesh, in order to define a new reference state from which the computation can be resumed. Because the deforming body often assumes an intricate shape, automation of the remeshing procedure requires a mesh generation scheme that is capable of handling an arbitrary domain. On the other hand, fast mesh generation is strongly desired, since remeshing may often have to be carried out several times during the simulation. Many schemes were proposed for automatic generation of triangular or tetrahedral elements for an arbitrary domain. Cavendish et al. [1] developed a two-stage approach to automatic triangulation of an arbitrary solid model, and it was later refined by Field and Frey [2]. The main feature of the proposed schemes is to inject points into the domain so that a volume triangulation is induced in which the points become nodes of tetrahedral elements. Wordenweber [3] and Woo and Thomasma [4] proposed a different class of schemes for decomposing a solid model into a collection of tetrahedral elements. Their schemes were based on the application of a set of operators that identify and extract tetrahedra from the geometrical and topological information contained in the boundary representation of the domain. However, the tetrahedral element (and the triangular element, in 2-D problems) is not used widely in the finite element analysis of metal forming processes, due to its relatively low computational efficiency. A special attention may be given to a fully automatic scheme which was introduced by Sheppard and Yerry [5,6 ], since the scheme is capable of generating quadrilateral and hexahedral elements for 2-D and 3D finite element analyses, as has been noted by Perucchio and Saxena [ 7 ]. In their approach, a three dimensional domain is approximated with union of variable sized cubes generated by recursively subdividing a spatial region enclosing the domain. The cubes that are not fully inside the domain are cut by
75
the planes which approximately represent the domain boundary. These cubes are then broken into finite element meshes. A drawback of the recursive spatial decomposition is that it is difficult to control the mesh density. In addition, the recursive nature of the process may require substantial amount of computing resources. In this paper, an approach to full automation of the remeshing procedure in the simulation of a non-steady metal forming process is presented. The approach is based on a new automatic mesh generation scheme, which is described in detail in the next section. The effect of remeshing on the solution accuracy is evaluated. Then, using the proposed approach, backward extrusion with a sharp-cornered punch is simulated. The approach is further applied to the simulation of a rib-web forging process. 2. Method of approach
Guide grid method A mesh, comprised of rectangles and used in FEM mesh generation, will be referred to as the guide grid. The guide grid should contain the domain completely within its interior, as shown in Fig. 1. Rectangles in the guide grid are classified into three types. Rectangles which are inside the domain (IN), those which are outside the domain (OUT), and those which are neither IN or OUT (NIO). The process of mesh generation by the guide grid is divided into three stages. They are: (1) Search for IN and NIO rectangles. (2) Extraction of cut segments from NIO rectangles.
I tiliiiiiiiiiliiiiiiiiiiiiiiiiiilliiiiiiiiiiiiiiiiitiiiiiiiiiiiiiiiiiiiiii i iiiiiiiiii!!iiiiiiiiiii!ii!!iiiiiiiiiiiiiiiiii!i!!iiiiiiiiiiiiiiiiiiiiiii
[ iiiiiiiii[iiiiiiiiiiiiiiiii[iiiiiiiiiiiiiiiii!i!iiiiiiiiiii;ii!ii!ij
guide grid
domain
Fig. 1. The guide grid and the domain.
76
(3) Decomposition of the cut segments. (1) For the representation of the boundary, all the nodal points on the boundary of the old mesh are picked up. Information on the old mesh can be obtained from a data base produced by the simulation. For initial mesh generation, however, a set of such points should be provided by a user. The boundary curve is then constructed by connecting these points with lines. Classification of the rectangles in the guide grid may be performed as follows: All NIO rectangles are found first. This is done by examining intersection of an edge of each rectangle with the boundary curve. Then, IN rectangles are searched, using the criterion that a pair of rectangles that are in contact with each other should be both IN or both OUT if none of them are NIO. (2) Each IN rectangle represents a quadrilateral element with the same rectangular shape. Transformation of a NIO rectangle into finite elements starts with extracting a segment of the domain from the rectangle. In the procedure, the boundary curve is traced along the counterclockwise path, and the section on the right hand side is removed from the NIO rectangle. The boundary of the segment thus obtained is represented by the points which are originated from one of the following sources: intersection of the boundary curve with an
5
/
6
4
1
11
10
3"/~ 1
Fig. 2. Extraction of a cut segment from the NIO rectangle.
6
7
t~
Cr~
~
~
O~
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00
CO
~D
Lr~
~
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O~r
~D
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(~
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.~
b~
78
/!, I\L
(a) old mesh system
(b) new mesh system
Fig. 4. Mesh generation by guide grid method: (a) old mesh system; (b) new mesh system.
edge of the rectangle, a corner of the rectangle, and an old nodal point. These points are then consecutively numbered and connected by lines to form a closed loop. In Fig. 2, the procedure to obtain such a loop from a NIO rectangle is illustrated. It is noted that a NIO rectangle may produce several loops. It is possible that three or more consecutive points in the loop lie on a straight or nearly straight line. In this case, the points defining the ends of the line only remain in the loop. The rest are regarded as redundancies and removed from the loop. (3) Next, decomposition of the cut segments is carried out. The decomposition scheme adopted here is based on recursive extraction of quadrilaterals and triangles, which may be summarized as follows: Four consecutive points in the loop are selected. If a quadrilateral can be defined by these points, it is extracted from the segment and a new loop encompassing the rest of the segment is constructed. If not, the next four consecutive points are investigated. The process is repeated until no more quadrilaterals can be extracted from the segment. Triangles are then extracted in a similar manner. Figure 3 shows an example of decomposing a cut segment using the present scheme. Figure 4 shows the new mesh for a star shaped domain constructed by the guide grid method. The old mesh is also shown. As seen in the figure, decomposition occurs only in the region close to the boundary. This is in contrast with decomposition of the entire domain on which other automatic mesh generation schemes are based.
Interpolation of field variables In metal forming, the flow stress is affected by effective strain and temperature. Therefore, these values should be transferred from the old mesh to the
79 new mesh, prior to the resumption of the computation. Our approach is similar to that proposed by Oh, Bawdawy, and Altan [8]. Transfer of element values consists of the following three steps: S t e p 1. Prediction of nodal values in the old mesh from element values in the old mesh. S t e p 2. Prediction of nodal values in the new mesh from nodal values in the old mesh. S t e p 3. Prediction of element values in the new mesh from nodal values in the new mesh. For Step 1, the error resulting from approximating the discontinuous field variable by a continuous one is defined as follows:
E=f
(Nn¢K--~) 240
(1)
Minimization of the error leads to a matrix equation
KLK¢K=FL
(2)
where
KLK= f NL NK d.Q
(3)
FL= ( NL ~dY2
(4)
which may be solved for CK. In step 2, ¢(Knew) are directly evaluated from the continuous field variable of the old mesh, or, 0(Knew) (xK)=N(L°Id) (X~) 0(L°'d)
(5)
For step 3, ~(new) is interpolated from the new continuous field variable, as follows. 6 ('ew) =N(Ln~w)0 (new)
(6)
Generation of boundary conditions and implementation Prior to the resumption of the simulation, boundary conditions must be transferred from the old mesh to the new mesh. Among the nodal points which are on the boundary of the new mesh, some are originated from the old mesh, and the others are freshly generated. Therefore, it suffices to determine the boundary conditions of the freshly generated nodal points. This has been performed automatically, using a simple criterion that a nodal point between the die touching nodal points should also be a die touching nodal point, etc. The
80 mesh generation scheme and the interpolation scheme, as well as a scheme for generating the boundary conditions, were all implemented into a rigid-viscoplastic finite element code [9 ]. Thus, a new program for the simulation of nonsteady metal forming processes which can automatically perform remeshing without any interaction from the user was obtained. The program required information on the guide grid as well as on when remeshing is to be performed, in addition to the ordinary input information for the simulation. 3. R e s u l t s and d i s c u s s i o n
Cylinder compression In order to validate the present approach to automatic remeshing, the problem of compression of a solid cylinder by a flat die was investigated. The process conditions under which the simulation was conducted are given as follows: Diameter of the billet Do = 60 mm Height of the billet H0 = 40 mm Coefficient of friction ~ = 0.2 Material # = 106.86
(
1-~ 0.05-205]
MPa (Annealed Aluminum 1100)
The guide grid used in the simulation initially consisted of 2.5 mm by 2.5 mm squares. The guide grid was allowed to deform during the simulation, in order to keep up with the varying domain. The deformation of the guide grid was controlled by the velocity field u = 0.0 and v = 2 Vo z/H, where u and v are velocity components in r and z direction, respectively, and Vo is the die speed. Thus, as the simulation went on, the rectangles in the guide grid became smaller, and the total number of elements in the domain was increased. With the maximum step size being equal to 0.25 mm, the simulation continued until 77 percent reduction in height was achieved. Remeshing was performed at each step of deformation. This resulted in carrying out remeshing sixty times prior to the completion of the simulation. Two sets of deformed configurations, one obtained with remeshing, and the other without remeshing, are shown in Fig. 5. It is seen that diverse distortion of elements occurred at large reduction, when remeshing was not carried out. On the contrary, when remeshing was carried out, the shapes of the elements in the domain were mostly rectangles. An exception was the peripheral region where non-rectangular quadrilateral elements and triangular elements were produced from the cut segments. The effective strain distributions obtained with remeshing were compared to those obtained without remeshing. In Fig. 6, it is seen that the distributions are almost identical at small reduction. However, a small discrepancy was noted at a large reduction. This is believed to result from the dissimilarity of the mesh system employed in each simulation. Nevertheless, there was little difference in the predicted wall contours, as shown
81
without remeshing
with remeshing
t (a)
h
25% reduction without remeshing
,
i
!
~
remeshing
L (b)
50% reduction without remeshing
with remeshing
(c)
75% reduction
Fig. 5. Deformed configurations at intermediate stages of flat die compression of a cylindrical billet.
in Fig. 7. In addition, load-displacement curves were in excellent agreement, seen in Fig. 8. Thus, it may be concluded that the degree of solution accuracy is not very much affected by using the present schemes for remeshing. Backward extrusion of cans A difficulty in simulating a metal forming process is associated with the sharp die corner. Due to the abrupt change in the direction of metal flow in the
82
without
remeshing
---;---
0.1 0.2 O.B 0.35 0.4
5
3
(a)
rerneshing
25Z r e d u c t i o n 2
~ ------
5
~
without remeshing
0.3 0.5 0.6 0.7 0.9
rerneshing
(b) reduction 50%
1 w
-
_
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-
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5
4 3
1.0 1.3 1.6 1.7
5
1
,
5
__~-. 5
remeshing
4
4
~
....
'S5 reme.htng
4 (c)
75% r e d u c t i o n
Fig. 6. E f f e c t i v e s t r a i n d i s t r i b u t i o n s a t i n t e r m e d i a t e s t a g e s of f l a t di e c o m p r e s s i o n of a c y l i n d r i c a l billet.
vicinity of the sharp die corner, elements near the die corner are severely distorted as the deformation proceeds. In addition, these elements often penetrate into or are separated from the die, resulting in accumulation of volume error. In an effort to treat the sharp die corner problem, Mori et al. [10] suggested using a singular element. Another possible approach is to approximate the sharp die corner by a round corner with very small radius of curvature, and use a large number of fine elements to represent the flow near the sharp die corner.
83 65-:
E E
602
5000-
• • • • • without remeshinQ - with remeshin 9
• • • • . without remeshing - with remeshing v
4000 0
50-
rnoximun radius
45:
30OO
2000
40!
° •
• minimum radius
1000
35
36 ° o
.
reduction ratio (~)
Fig. 7. Change in the m i n i m u m and maximum radii during flat die compression of a cylindrical billet.
.
.
.
I
25
.
.
.
.
I
50
.
.
.
.
I
75
reduction ratio (~)
Fig. 8. Load-displacement curves obtained from the simulation of flat die compression of a cylindrical billet.
guide grid
i ~t
it~
, , , ! ,~!,,JJ~ ', ', ', I ', ~ ':', ', ',I~
punchl i i i i~ii!!!~ ', I I I ',~li',l',~
.iilllilllll/liillil/illiliiillillliiiillli
container Fig. 9. The process geometry and the guide grid used in the simulation of axisymmetric backward extrusion.
In both cases, however, frequent remeshing is necessary. T h e present approach to automatic remeshing can provide a very effective mean with regard to the second case. A backward extrusion process was selected for investigating the sharp die corner problem. The process conditions were found from Mori [10]. T h e y are given to be: Diameter of the billet Do-- 30 mm Height of the billet Ho = 20 mm Extrusion ratio R = 1.8
84
..... Ill
iiii
l Ill l
Ill
III III III
III I IIII
I1[ Ill
f
fill
m
ill
III
I
(b) 25 % reduction
(a) 7 % reduction
llllll lliili Bill liilill
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(c) 50 % reduction
,
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i i
I I I
l I
[ I III I i Ill I I flU i i IH I I fll I ! ltl
l I
! n
Ill l I Ill ]] . . . . . .
(d) 62 % reduction
Fig. 10. D e f o r m e d c o n f i g u r a t i o n s a t several stages o f t h e b a c k w a r d e x t r u s i o n process.
Coefficient of friction/~ = 0.13 ( P u n c h and Container) Flow stress ~ = 127 ~o.27 M P a T h e process geometry at the initial stage of forming is described in Fig. 9. T h e guide grid is also shown. Deformation of the guide grid during the simulation was controlled as follows: Divide the guide grid by a horizontal line drawn at the b o t t o m of the punch. T h e upper part moves as a rigid body with the punch, while the lower part is subject to the same deformation mode as used in the simulation of the compression of a cylinder. Figure 10 shows the deformed configurations at several deforming stages. T h e y were obtained with
85 1 2 3 4 5
------
0.05 0.1 0.15 0.3 0.5
1--
0.1
2--
0.3
3--
0.5
4--
03
5--
0.9
P'
f
(a) 7 % reduction
1 -2
--
3
--
4 5
---
(b) 25 % reduction
0.8 0.0 0.8
1-2-8--
0.5 0.8
1.1
4--
1.8
1.6
5--
1.9
(c) 50 % reduction
1.0
I
(d) 62 % reduction
Fig. 11. Effective strain distributions at several stages of the backward extrusion process.
remeshing at every step of deformation. It is seen that the decomposition of the cut segments took place only at the die corner and at the top of the extruded material, indicating that most of the old nodal points on the boundary were redundancies and thus removed. In Fig. 11, the effective strain distributions are shown. They were found to agree well with the measurements made by Mori et al. [ 10 ].
86
i
i
H Fig. 12. The rib-web forging die and the preform used in the simulation. The guide grid is also shown.
I
[
i
J
i
!
I
|
|
| 11 1[I
]
II
II
iiFFEi I E
il[J ILl/ ]111
i i
I~-I I\ i ~ J I [ I I I~ IL[lll
Fig. 13. Deformed configurations at several stages of forming.
Rib-web forging In order to evaluate the capability of the present approach for handling an arbitrary domain, an F E M simulation for a rib-web forging process was conducted. The simulation was performed utilizing the following process conditions. Width of the stock Wo = 100 mm
87
7•
1
--
0.2
2
--
0.4
3
--
0.6
4
--
0.8
5
--
0.9
6
--
1.0
7
--
1.2
8 :: 1..4
--
10.0
--
15.0
--
20.0
--
30.0
--
50.0
--
80.0
~58 Fig. 14. Effective strain and effective strain rate contours predicted at the final stage of the ribweb forgingprocess. Height of the stock Ho= 120 mm Friction condition ~ = 0.15 Material # = ~o.15 M P a (rate sensitive) Die velocity Vd = 100 m m / s The contoured die, the preform, and the guide grid selected for the simulation are shown in Fig. 12. Deformation of the guide grid during the simulation was controlled in the same manner as employed in the simulation of the backward extrusion process. A horizontal line drawn at the fiat b o t t o m of the die was selected for defining the upper and the lower part of the guide grid. The upper part moved down with the die, while the lower part was deformed by the velocity field u = 0.0, v = 2 V~/H. M a x i m u m step size allowed was 1 mm. Remeshing was performed at every fifth step of deformation. The process of filling the die cavity predicted by the simulation is shown in Fig. 13. It is seen
58
that, regardless of the complexity of the deformed configurations, the new meshes were successfully generated. The effective strain and the effective strain rate contours reveal that maximum deformation occurred in the regions near the die corners, as shown in Fig. 14. 4. Concluding remarks
A mesh generation algorithm referred to as the guide grid method was presented. It was shown that fast mesh generation can be achieved by using the method, since the method does not require the recursive decomposition of the entire domain. The schemes for transferring the nodal and element variables, and the boundary conditions from the old mesh to the new mesh were described. It was shown that on the basis of the guide grid method and these schemes, the remeshing procedure can be fully automated in the simulation of a metal forming process. The approach was implemented into a rigid-viscoplastic finite element code. The effect of remeshing on the solution accuracy was investigated by simulating upsetting of a cylinder by a flat die. The deformed configurations and the load-displacement curve obtained by the simulation with remeshing at every step of deformation were found to be in close agreement with those obtained without remeshing, and it was concluded that the present remeshing procedure does not affect the degree of accuracy of the finite element solution. A backward extrusion process was simulated by using the new approach for automatic remeshing. It was shown that the difficulty arising from simulating a process with a sharp die corner can be overcome by properly constructing a guide grid and enforcing remeshing at every deformation step. The approach was further applied to simulating a rib-web forging process. A contoured die with relatively deep die cavity was used. It was shown that the new meshes can be successfully generated for the complex configurations predicted by the simulation. It is to be emphasized that the present investigation demonstrates the possibility of automatic simulation of various metal forming processes. Acknowledgments The authors wish to thank Pohang Iron and Steel Corporation for its contract 0156A under which the present investigation was possible.
References 1 J.C. Cavendish, D.A. Field and W.H. Frey, An approach to automatic three-dimensional finite element mesh generation, Int. J. Num. Meth. Eng., 21 (2) (1985) 329. 2 D.A. Field and W.H. Frey, Automation of tetrahedral mesh generation, GMR-4967, G. M. R. L., Michigan, 1985.
89 3
B. Wordenweber, Finite-element analysis for the naive user, Solid Modelling by Computers, Plenum Press, New York, 1984, p. 81. 4 T.C. Woo and T. Thomasma, An algorithm for generating solid elements in objects with holes, Comp. Struct., 18 {1984) 333. 5 M.A. Yerry and M.S. Shephard, Automatic three-dimensional mesh generation by the modified-octree technique, Int. J. Num. Meth. Eng., 20 ( 11 ) (1984) 1965. 6 P.L. Baehmann, S.L. Wittchen, M.S. Shephard, K.R. Grice and M.A. Yerry, Robust, geometrically based, automatic two-dimensional mesh generation, Int. J. Num. Meth. Eng., 24 {6 ) {1987) 1043. 7 R. Perucchio and M. Saxena, Automatic mesh generation from solid models based on recursive spatial decompositions, Int. J. Num. Meth. Eng., 28 (11 ) (1989) 2469. 8 S.I. Oh, A. Badawy, T. Altan, A remeshing technique to simulate large plastic flow by FEM, Interim Report on Development of a Computer Aided Method for Warm Forging of Steel, NSF Grant No. MEA-8112622, January 1983. 9 S.M. Hwang, M.S. Joun and J.S. Park, A penalty rigid-plastic finite element method for the determination of stress distributions at the tool-workpiece interfaces in metal forming, Proc. 18th NAMRC, University Park, Pennsylvania, 1990, p. 13. 10 K. Mori, K. Osakada and M. Fukuda, Simulation of severe plastic deformation by finite element method with spatially fixed finite elements, Int. J. Mech. Sci., 25 (1983) 775.
73
Elsevier
Automatic remeshing in finite element simulation of metal forming processes by guide grid method J.S. Park and S.M. Hwang Department of Mechanical Engineering, Pohang Institute of Science and Technology, Pohang 790-600, Korea Mechanical Engineering Division, Research Institute of Industrial Science and Technology, Pohang 790-600, Korea
Industrial S u m m a r y An approach to full automation of the remeshing procedure in the finite element simulation of a non-steady metal forming process is presented. The approach is based on a new automatic mesh generation scheme, in which a two-dimensional mesh is generated for an arbitrary domain with the aid of a prespecified mesh referred to as the guide grid. Evaluation of the effect of remeshing on the solution accuracy is made. Backward extrusion with a sharp-cornered punch is simulated, to verify the effectiveness of the proposed approach. The approach is further applied to simulating a rib-web forging process.
Notation Do
EL H
Ho K~ NK R r u
Yd v
Wo
x~ y z t-
initial diameter of specimen force vector current height of specimen initial height of specimen stiffness matrix FEM basis function extrusion ratio coordinate along the radial direction horizontal component of velocity vector die velocity vertical component of velocity vector initial width of specimen ith coordinate of Kth nodal point coordinate along the forging direction coordinate along the axial direction effective strain effective strain rate nodal value of Kth nodal point field variable which is discontinuous at element boundary
0924-0136/91/$03.50 © 1991--Elsevier Science Publishers B.V.
74
// ]
coefficient of Coulomb friction flow stress
1. I n t r o d u c t i o n
Finite element formulations developed for numerical simulation of plastic flow occurring in metal forming are mostly based on the material description of motion in which the finite element mesh deforms with the material. As a result, the element shapes may become severely distorted as the deformation proceeds. In order to continue the simulation at that stage, remeshing is necessary. However, remeshing in general requires a great deal of work. Full automation of the remeshing procedure will eliminate the need for working on remeshing. In addition, it is an essential step toward developing a system for automatic simulation and analysis of a metal forming process. The remeshing procedure consists of two parts. The first part is to rediscretize the deformed domain and generate a new mesh. The second part is to transfer nodal and element variables from the old mesh to the new mesh, in order to define a new reference state from which the computation can be resumed. Because the deforming body often assumes an intricate shape, automation of the remeshing procedure requires a mesh generation scheme that is capable of handling an arbitrary domain. On the other hand, fast mesh generation is strongly desired, since remeshing may often have to be carried out several times during the simulation. Many schemes were proposed for automatic generation of triangular or tetrahedral elements for an arbitrary domain. Cavendish et al. [1] developed a two-stage approach to automatic triangulation of an arbitrary solid model, and it was later refined by Field and Frey [2]. The main feature of the proposed schemes is to inject points into the domain so that a volume triangulation is induced in which the points become nodes of tetrahedral elements. Wordenweber [3] and Woo and Thomasma [4] proposed a different class of schemes for decomposing a solid model into a collection of tetrahedral elements. Their schemes were based on the application of a set of operators that identify and extract tetrahedra from the geometrical and topological information contained in the boundary representation of the domain. However, the tetrahedral element (and the triangular element, in 2-D problems) is not used widely in the finite element analysis of metal forming processes, due to its relatively low computational efficiency. A special attention may be given to a fully automatic scheme which was introduced by Sheppard and Yerry [5,6 ], since the scheme is capable of generating quadrilateral and hexahedral elements for 2-D and 3D finite element analyses, as has been noted by Perucchio and Saxena [ 7 ]. In their approach, a three dimensional domain is approximated with union of variable sized cubes generated by recursively subdividing a spatial region enclosing the domain. The cubes that are not fully inside the domain are cut by
75
the planes which approximately represent the domain boundary. These cubes are then broken into finite element meshes. A drawback of the recursive spatial decomposition is that it is difficult to control the mesh density. In addition, the recursive nature of the process may require substantial amount of computing resources. In this paper, an approach to full automation of the remeshing procedure in the simulation of a non-steady metal forming process is presented. The approach is based on a new automatic mesh generation scheme, which is described in detail in the next section. The effect of remeshing on the solution accuracy is evaluated. Then, using the proposed approach, backward extrusion with a sharp-cornered punch is simulated. The approach is further applied to the simulation of a rib-web forging process. 2. Method of approach
Guide grid method A mesh, comprised of rectangles and used in FEM mesh generation, will be referred to as the guide grid. The guide grid should contain the domain completely within its interior, as shown in Fig. 1. Rectangles in the guide grid are classified into three types. Rectangles which are inside the domain (IN), those which are outside the domain (OUT), and those which are neither IN or OUT (NIO). The process of mesh generation by the guide grid is divided into three stages. They are: (1) Search for IN and NIO rectangles. (2) Extraction of cut segments from NIO rectangles.
I tiliiiiiiiiiliiiiiiiiiiiiiiiiiilliiiiiiiiiiiiiiiiitiiiiiiiiiiiiiiiiiiiiii i iiiiiiiiii!!iiiiiiiiiii!ii!!iiiiiiiiiiiiiiiiii!i!!iiiiiiiiiiiiiiiiiiiiiii
[ iiiiiiiii[iiiiiiiiiiiiiiiii[iiiiiiiiiiiiiiiii!i!iiiiiiiiiii;ii!ii!ij
guide grid
domain
Fig. 1. The guide grid and the domain.
76
(3) Decomposition of the cut segments. (1) For the representation of the boundary, all the nodal points on the boundary of the old mesh are picked up. Information on the old mesh can be obtained from a data base produced by the simulation. For initial mesh generation, however, a set of such points should be provided by a user. The boundary curve is then constructed by connecting these points with lines. Classification of the rectangles in the guide grid may be performed as follows: All NIO rectangles are found first. This is done by examining intersection of an edge of each rectangle with the boundary curve. Then, IN rectangles are searched, using the criterion that a pair of rectangles that are in contact with each other should be both IN or both OUT if none of them are NIO. (2) Each IN rectangle represents a quadrilateral element with the same rectangular shape. Transformation of a NIO rectangle into finite elements starts with extracting a segment of the domain from the rectangle. In the procedure, the boundary curve is traced along the counterclockwise path, and the section on the right hand side is removed from the NIO rectangle. The boundary of the segment thus obtained is represented by the points which are originated from one of the following sources: intersection of the boundary curve with an
5
/
6
4
1
11
10
3"/~ 1
Fig. 2. Extraction of a cut segment from the NIO rectangle.
6
7
t~
Cr~
~
~
O~
Cr~
00
CO
~D
Lr~
~
t~
p~
cD
O~r
~D
cO
"'....~" C~2
C~
(~
c0
C~ ~s
.~
b~
78
/!, I\L
(a) old mesh system
(b) new mesh system
Fig. 4. Mesh generation by guide grid method: (a) old mesh system; (b) new mesh system.
edge of the rectangle, a corner of the rectangle, and an old nodal point. These points are then consecutively numbered and connected by lines to form a closed loop. In Fig. 2, the procedure to obtain such a loop from a NIO rectangle is illustrated. It is noted that a NIO rectangle may produce several loops. It is possible that three or more consecutive points in the loop lie on a straight or nearly straight line. In this case, the points defining the ends of the line only remain in the loop. The rest are regarded as redundancies and removed from the loop. (3) Next, decomposition of the cut segments is carried out. The decomposition scheme adopted here is based on recursive extraction of quadrilaterals and triangles, which may be summarized as follows: Four consecutive points in the loop are selected. If a quadrilateral can be defined by these points, it is extracted from the segment and a new loop encompassing the rest of the segment is constructed. If not, the next four consecutive points are investigated. The process is repeated until no more quadrilaterals can be extracted from the segment. Triangles are then extracted in a similar manner. Figure 3 shows an example of decomposing a cut segment using the present scheme. Figure 4 shows the new mesh for a star shaped domain constructed by the guide grid method. The old mesh is also shown. As seen in the figure, decomposition occurs only in the region close to the boundary. This is in contrast with decomposition of the entire domain on which other automatic mesh generation schemes are based.
Interpolation of field variables In metal forming, the flow stress is affected by effective strain and temperature. Therefore, these values should be transferred from the old mesh to the
79 new mesh, prior to the resumption of the computation. Our approach is similar to that proposed by Oh, Bawdawy, and Altan [8]. Transfer of element values consists of the following three steps: S t e p 1. Prediction of nodal values in the old mesh from element values in the old mesh. S t e p 2. Prediction of nodal values in the new mesh from nodal values in the old mesh. S t e p 3. Prediction of element values in the new mesh from nodal values in the new mesh. For Step 1, the error resulting from approximating the discontinuous field variable by a continuous one is defined as follows:
E=f
(Nn¢K--~) 240
(1)
Minimization of the error leads to a matrix equation
KLK¢K=FL
(2)
where
KLK= f NL NK d.Q
(3)
FL= ( NL ~dY2
(4)
which may be solved for CK. In step 2, ¢(Knew) are directly evaluated from the continuous field variable of the old mesh, or, 0(Knew) (xK)=N(L°Id) (X~) 0(L°'d)
(5)
For step 3, ~(new) is interpolated from the new continuous field variable, as follows. 6 ('ew) =N(Ln~w)0 (new)
(6)
Generation of boundary conditions and implementation Prior to the resumption of the simulation, boundary conditions must be transferred from the old mesh to the new mesh. Among the nodal points which are on the boundary of the new mesh, some are originated from the old mesh, and the others are freshly generated. Therefore, it suffices to determine the boundary conditions of the freshly generated nodal points. This has been performed automatically, using a simple criterion that a nodal point between the die touching nodal points should also be a die touching nodal point, etc. The
80 mesh generation scheme and the interpolation scheme, as well as a scheme for generating the boundary conditions, were all implemented into a rigid-viscoplastic finite element code [9 ]. Thus, a new program for the simulation of nonsteady metal forming processes which can automatically perform remeshing without any interaction from the user was obtained. The program required information on the guide grid as well as on when remeshing is to be performed, in addition to the ordinary input information for the simulation. 3. R e s u l t s and d i s c u s s i o n
Cylinder compression In order to validate the present approach to automatic remeshing, the problem of compression of a solid cylinder by a flat die was investigated. The process conditions under which the simulation was conducted are given as follows: Diameter of the billet Do = 60 mm Height of the billet H0 = 40 mm Coefficient of friction ~ = 0.2 Material # = 106.86
(
1-~ 0.05-205]
MPa (Annealed Aluminum 1100)
The guide grid used in the simulation initially consisted of 2.5 mm by 2.5 mm squares. The guide grid was allowed to deform during the simulation, in order to keep up with the varying domain. The deformation of the guide grid was controlled by the velocity field u = 0.0 and v = 2 Vo z/H, where u and v are velocity components in r and z direction, respectively, and Vo is the die speed. Thus, as the simulation went on, the rectangles in the guide grid became smaller, and the total number of elements in the domain was increased. With the maximum step size being equal to 0.25 mm, the simulation continued until 77 percent reduction in height was achieved. Remeshing was performed at each step of deformation. This resulted in carrying out remeshing sixty times prior to the completion of the simulation. Two sets of deformed configurations, one obtained with remeshing, and the other without remeshing, are shown in Fig. 5. It is seen that diverse distortion of elements occurred at large reduction, when remeshing was not carried out. On the contrary, when remeshing was carried out, the shapes of the elements in the domain were mostly rectangles. An exception was the peripheral region where non-rectangular quadrilateral elements and triangular elements were produced from the cut segments. The effective strain distributions obtained with remeshing were compared to those obtained without remeshing. In Fig. 6, it is seen that the distributions are almost identical at small reduction. However, a small discrepancy was noted at a large reduction. This is believed to result from the dissimilarity of the mesh system employed in each simulation. Nevertheless, there was little difference in the predicted wall contours, as shown
81
without remeshing
with remeshing
t (a)
h
25% reduction without remeshing
,
i
!
~
remeshing
L (b)
50% reduction without remeshing
with remeshing
(c)
75% reduction
Fig. 5. Deformed configurations at intermediate stages of flat die compression of a cylindrical billet.
in Fig. 7. In addition, load-displacement curves were in excellent agreement, seen in Fig. 8. Thus, it may be concluded that the degree of solution accuracy is not very much affected by using the present schemes for remeshing. Backward extrusion of cans A difficulty in simulating a metal forming process is associated with the sharp die corner. Due to the abrupt change in the direction of metal flow in the
82
without
remeshing
---;---
0.1 0.2 O.B 0.35 0.4
5
3
(a)
rerneshing
25Z r e d u c t i o n 2
~ ------
5
~
without remeshing
0.3 0.5 0.6 0.7 0.9
rerneshing
(b) reduction 50%
1 w
-
_
m
-
m
m
~
w
5
4 3
1.0 1.3 1.6 1.7
5
1
,
5
__~-. 5
remeshing
4
4
~
....
'S5 reme.htng
4 (c)
75% r e d u c t i o n
Fig. 6. E f f e c t i v e s t r a i n d i s t r i b u t i o n s a t i n t e r m e d i a t e s t a g e s of f l a t di e c o m p r e s s i o n of a c y l i n d r i c a l billet.
vicinity of the sharp die corner, elements near the die corner are severely distorted as the deformation proceeds. In addition, these elements often penetrate into or are separated from the die, resulting in accumulation of volume error. In an effort to treat the sharp die corner problem, Mori et al. [10] suggested using a singular element. Another possible approach is to approximate the sharp die corner by a round corner with very small radius of curvature, and use a large number of fine elements to represent the flow near the sharp die corner.
83 65-:
E E
602
5000-
• • • • • without remeshinQ - with remeshin 9
• • • • . without remeshing - with remeshing v
4000 0
50-
rnoximun radius
45:
30OO
2000
40!
° •
• minimum radius
1000
35
36 ° o
.
reduction ratio (~)
Fig. 7. Change in the m i n i m u m and maximum radii during flat die compression of a cylindrical billet.
.
.
.
I
25
.
.
.
.
I
50
.
.
.
.
I
75
reduction ratio (~)
Fig. 8. Load-displacement curves obtained from the simulation of flat die compression of a cylindrical billet.
guide grid
i ~t
it~
, , , ! ,~!,,JJ~ ', ', ', I ', ~ ':', ', ',I~
punchl i i i i~ii!!!~ ', I I I ',~li',l',~
.iilllilllll/liillil/illiliiillillliiiillli
container Fig. 9. The process geometry and the guide grid used in the simulation of axisymmetric backward extrusion.
In both cases, however, frequent remeshing is necessary. T h e present approach to automatic remeshing can provide a very effective mean with regard to the second case. A backward extrusion process was selected for investigating the sharp die corner problem. The process conditions were found from Mori [10]. T h e y are given to be: Diameter of the billet Do-- 30 mm Height of the billet Ho = 20 mm Extrusion ratio R = 1.8
84
..... Ill
iiii
l Ill l
Ill
III III III
III I IIII
I1[ Ill
f
fill
m
ill
III
I
(b) 25 % reduction
(a) 7 % reduction
llllll lliili Bill liilill
liilnnll Innill ImllBll Iilill liUnli Inllill limnnmll innnill Ininll
linmll ii HimH i nlmlgg 6innll DmiUal
liUmll Inlnll Illill
I i I
IIIIll IIIIll
I li lli ul ml l
(c) 50 % reduction
,
[ i
I I I
f i
I [ I
i i
I I I
l I
[ I III I i Ill I I flU i i IH I I fll I ! ltl
l I
! n
Ill l I Ill ]] . . . . . .
(d) 62 % reduction
Fig. 10. D e f o r m e d c o n f i g u r a t i o n s a t several stages o f t h e b a c k w a r d e x t r u s i o n process.
Coefficient of friction/~ = 0.13 ( P u n c h and Container) Flow stress ~ = 127 ~o.27 M P a T h e process geometry at the initial stage of forming is described in Fig. 9. T h e guide grid is also shown. Deformation of the guide grid during the simulation was controlled as follows: Divide the guide grid by a horizontal line drawn at the b o t t o m of the punch. T h e upper part moves as a rigid body with the punch, while the lower part is subject to the same deformation mode as used in the simulation of the compression of a cylinder. Figure 10 shows the deformed configurations at several deforming stages. T h e y were obtained with
85 1 2 3 4 5
------
0.05 0.1 0.15 0.3 0.5
1--
0.1
2--
0.3
3--
0.5
4--
03
5--
0.9
P'
f
(a) 7 % reduction
1 -2
--
3
--
4 5
---
(b) 25 % reduction
0.8 0.0 0.8
1-2-8--
0.5 0.8
1.1
4--
1.8
1.6
5--
1.9
(c) 50 % reduction
1.0
I
(d) 62 % reduction
Fig. 11. Effective strain distributions at several stages of the backward extrusion process.
remeshing at every step of deformation. It is seen that the decomposition of the cut segments took place only at the die corner and at the top of the extruded material, indicating that most of the old nodal points on the boundary were redundancies and thus removed. In Fig. 11, the effective strain distributions are shown. They were found to agree well with the measurements made by Mori et al. [ 10 ].
86
i
i
H Fig. 12. The rib-web forging die and the preform used in the simulation. The guide grid is also shown.
I
[
i
J
i
!
I
|
|
| 11 1[I
]
II
II
iiFFEi I E
il[J ILl/ ]111
i i
I~-I I\ i ~ J I [ I I I~ IL[lll
Fig. 13. Deformed configurations at several stages of forming.
Rib-web forging In order to evaluate the capability of the present approach for handling an arbitrary domain, an F E M simulation for a rib-web forging process was conducted. The simulation was performed utilizing the following process conditions. Width of the stock Wo = 100 mm
87
7•
1
--
0.2
2
--
0.4
3
--
0.6
4
--
0.8
5
--
0.9
6
--
1.0
7
--
1.2
8 :: 1..4
--
10.0
--
15.0
--
20.0
--
30.0
--
50.0
--
80.0
~58 Fig. 14. Effective strain and effective strain rate contours predicted at the final stage of the ribweb forgingprocess. Height of the stock Ho= 120 mm Friction condition ~ = 0.15 Material # = ~o.15 M P a (rate sensitive) Die velocity Vd = 100 m m / s The contoured die, the preform, and the guide grid selected for the simulation are shown in Fig. 12. Deformation of the guide grid during the simulation was controlled in the same manner as employed in the simulation of the backward extrusion process. A horizontal line drawn at the fiat b o t t o m of the die was selected for defining the upper and the lower part of the guide grid. The upper part moved down with the die, while the lower part was deformed by the velocity field u = 0.0, v = 2 V~/H. M a x i m u m step size allowed was 1 mm. Remeshing was performed at every fifth step of deformation. The process of filling the die cavity predicted by the simulation is shown in Fig. 13. It is seen
58
that, regardless of the complexity of the deformed configurations, the new meshes were successfully generated. The effective strain and the effective strain rate contours reveal that maximum deformation occurred in the regions near the die corners, as shown in Fig. 14. 4. Concluding remarks
A mesh generation algorithm referred to as the guide grid method was presented. It was shown that fast mesh generation can be achieved by using the method, since the method does not require the recursive decomposition of the entire domain. The schemes for transferring the nodal and element variables, and the boundary conditions from the old mesh to the new mesh were described. It was shown that on the basis of the guide grid method and these schemes, the remeshing procedure can be fully automated in the simulation of a metal forming process. The approach was implemented into a rigid-viscoplastic finite element code. The effect of remeshing on the solution accuracy was investigated by simulating upsetting of a cylinder by a flat die. The deformed configurations and the load-displacement curve obtained by the simulation with remeshing at every step of deformation were found to be in close agreement with those obtained without remeshing, and it was concluded that the present remeshing procedure does not affect the degree of accuracy of the finite element solution. A backward extrusion process was simulated by using the new approach for automatic remeshing. It was shown that the difficulty arising from simulating a process with a sharp die corner can be overcome by properly constructing a guide grid and enforcing remeshing at every deformation step. The approach was further applied to simulating a rib-web forging process. A contoured die with relatively deep die cavity was used. It was shown that the new meshes can be successfully generated for the complex configurations predicted by the simulation. It is to be emphasized that the present investigation demonstrates the possibility of automatic simulation of various metal forming processes. Acknowledgments The authors wish to thank Pohang Iron and Steel Corporation for its contract 0156A under which the present investigation was possible.
References 1 J.C. Cavendish, D.A. Field and W.H. Frey, An approach to automatic three-dimensional finite element mesh generation, Int. J. Num. Meth. Eng., 21 (2) (1985) 329. 2 D.A. Field and W.H. Frey, Automation of tetrahedral mesh generation, GMR-4967, G. M. R. L., Michigan, 1985.
89 3
B. Wordenweber, Finite-element analysis for the naive user, Solid Modelling by Computers, Plenum Press, New York, 1984, p. 81. 4 T.C. Woo and T. Thomasma, An algorithm for generating solid elements in objects with holes, Comp. Struct., 18 {1984) 333. 5 M.A. Yerry and M.S. Shephard, Automatic three-dimensional mesh generation by the modified-octree technique, Int. J. Num. Meth. Eng., 20 ( 11 ) (1984) 1965. 6 P.L. Baehmann, S.L. Wittchen, M.S. Shephard, K.R. Grice and M.A. Yerry, Robust, geometrically based, automatic two-dimensional mesh generation, Int. J. Num. Meth. Eng., 24 {6 ) {1987) 1043. 7 R. Perucchio and M. Saxena, Automatic mesh generation from solid models based on recursive spatial decompositions, Int. J. Num. Meth. Eng., 28 (11 ) (1989) 2469. 8 S.I. Oh, A. Badawy, T. Altan, A remeshing technique to simulate large plastic flow by FEM, Interim Report on Development of a Computer Aided Method for Warm Forging of Steel, NSF Grant No. MEA-8112622, January 1983. 9 S.M. Hwang, M.S. Joun and J.S. Park, A penalty rigid-plastic finite element method for the determination of stress distributions at the tool-workpiece interfaces in metal forming, Proc. 18th NAMRC, University Park, Pennsylvania, 1990, p. 13. 10 K. Mori, K. Osakada and M. Fukuda, Simulation of severe plastic deformation by finite element method with spatially fixed finite elements, Int. J. Mech. Sci., 25 (1983) 775.