A 3D contact smoothing method based on quasi-C1
interpolation and normal voting –Application to 3D Forging
and Rolling
Maha HACHANI and Lionel FOURMENT
Mines ParisTech, CEMEF - Centre for Material Forming, CNRS UMR 7635,
BP 207, 06904 Sophia Antipolis Cedex, France
Abstract. This paper describes the effect of tool discretization accuracy on the simulation of forming processes,
especially for processes where the contact area is quite small with respect to the component size. For smoothing contact
surface discretized by linear triangles, an algorithm is followed to develop a higher order quadratic interpolation of the
curved surface from the positions and normal vectors of the nodes, as proposed by Nagata. Normal vectors are calculated
at each node from the existing discretized surface by considering a patch of surrounding elements. This is accomplished
by the mean of normal voting strategy. The efficiency and reliability of the resulting contact model are checked through
several examples like indenting and ironing a bulk parallelepiped. It is also applied to complex ring rolling, form rolling
and extrusion problems.
Keywords: Contact, surface smoothing, normal surface, forging, rolling, extrusion.
INTRODUCTION
Finite element simulation of forming processes is now widely recognized as an efficient tool for designing actual
forming processes in industry. However, for metal forming processes where the contact area is quite small with
respect to the component size, like wire drawing, extrusion and rolling, there is still an extensive need for increasing
the accuracy and robustness of the results while decreasing the computational time. In fact, despite using parallel
computers during several weeks, results might not be accurate enough to properly reproduce the experiments. The
analysis shows that in general simulation results depend on mesh size in the contact area. The main sources of errors
lie so in the contact treatment which is not accurate enough even if a proper mesh refinement is introduced. The
main aim of this study is to improve the contact algorithms without increasing the computational cost, in other
words, without decreasing the mesh size or the time step. This study is focused on ring rolling process and then
extended to other processes.
Ring Rolling Process
Ring rolling is a hot forming process that produces rings with varied size features. It is schematically shown in
Figure 1 (left). The initial ring is placed over mandrel roll, which is forced toward the drive roll. The drive roll
rotates continuously, reducing the wall thickness, and increasing the diameter.
R o ta tio n
C en terin g
R e d u c e d C o n ta c t
A re a
F la tte n in g o f
flan ks
T ran s la tio n
FIGURE 1. Ring rolling: process scheme (left), mesh refinement for F.E. simulation and resulting reduced contact area (right)
As shown in Figure 1, the key challenge for the computer simulation of this process is that the material deformation
occurs in a much reduced part of the computational domain. Preliminary simulations have provided abnormally high
strain values ( ε ≥ 5 ), which appeared to be closely related to the contact treatment. Due to the coarseness of the
mesh (see figure 1 right), the contact area is limited to very few nodes, contact is not detected at the beginning of the
process, and temporal oscillations are observed. This is regarded to come from the coarse description of the
contacting cylindrical tools that are discretized by linear C0 triangular facets. It is so important to improve the
contact surface discretization. The first idea was to use an analytical description, which is applicable to processes
like ring rolling where shapes are cylinders and cones. Simulation results (see Figure 2) show that it significantly
increases the results quality. Even though the tools are quite finely discretized, the analytical formulation still
improves the contact treatment by establishing the contact earlier and over a larger area (left part of Figure 2), which
results into lower, more homogenous and more physical values of the cumulated strains (right part of Figure 2).
facetized Tools
Analytical Tools
t=0,4s
t=3,7s
0.15
0.144
0.128
Analytical Tools
Discrete F.E Tools
0.112
0.096
0.064
0.046
0.08
0.032
0.015
0
FIGURE 2. Comparison of contact surfaces (left) and equivalent cumulated strains (right)
with facetized and analytical tool descriptions
This preliminary work shows how important it is to improve the tool description, for such processes. For more
complex shapes that might be encountered in bulk forming, the analytical approach is not possible, but there is still a
need to develop a simple, robust and efficient method to smooth the contact surfaces. Various techniques have been
proposed by many researchers, such as NURBS, Bézier patches, Grégory patches and local Hermite diffuse
interpolations, respectively in [2,3,4,5] for instance. In spite of their efficiency to provide good approximations of
the original surface and to ensure a high continuity degree, their application to the finite element method raise some
difficulties. In fact, they often require free parameters or assumptions, which have to be given a priori, which is
hardly possible since the analytic properties of the original surface are generally unknown [1]. Furthermore, the
smoothed surfaces are generally described by complex expressions (cubic or higher polynomials or rational
functions), so the contact search algorithms becomes significantly more complex and more computationally
expensive. Besides, highly continuous interpolation results in coupling a large number of triangle patches, which
might become a difficulty in a parallel computational environment. In this study, the rather simple smoothing
procedure proposed by Nagata [1] is followed. A corresponding contact search algorithm is developed and the
approach is first applied to specific and discriminating problems, like indenting and ironing a bulk parallelepiped
with a cylinder. It is then applied to the rolling of long products.
SMOOTHING TECHNIQUE
Nagata’s Method
The central idea of Nagata’s algorithm [1] is to increase the interpolation order of the triangularized surface, from
linear to quadratic facets, by only using local operators. It has been successfully applied it to sheet metal forming
problems [6]. The main idea is to recover the curvature of triangular or quadrilateral patches (see Figure 6) by only
using the positions and normal vectors at the vertices of polyhedral meshes. This approach is simple and local and so
computationally inexpensive and easy to parallelize.
n3
New curved
surface
x3
Discrete face
n2
n1
n2
n1
(a)
P2
P1
x1
x2
(b)
(c)
x1
x2
FIGURE 3. Nagata Patch: (a-b): interpolation of a face procedure (c): interpolation of a curved segment
Each triangle is composed of three edges; each edge is independently interpolated. The original line is replaced
by a quadratic curve that is orthogonal to the normal given at the end points. The interior of the triangle is then filled
with a parametric polynomial surface reproducing the modified boundary (see Figure 3).
The segment curvature is locally and independently recovered by computing the quadratic curve passing through
the end nodes M1 and M2 which coordinates are respectively X1 and X2 and that is orthogonal to the normal vectors
n1 and n2 respectively at these points (see Figure 3 c). This is achieved by a hierarchical interpolation. A midsegment node is added to the mesh edge, so allowing the second order interpolation. The equation of this curve is:
X ( ξ ) = X 1 N 1 ( ξ ) + X 2 N 2 ( ξ ) + XN 3 ( ξ )
(1)
where ξ is the local coordinate satisfying the condition 0 ≤ ξ ≤ 1 , N1, N2 are the linear interpolation functions
( N 1 ( ξ ) = 1 − ξ ; N 2 ( ξ ) = ξ ) and N3 is the hierarchical quadratic function: N 3 ( ξ ) = 4ξ ( 1 − ξ ) , X is the
position of the mid-edge, the unknown. It is be determined by writing that the curve is orthogonal to the normal
vectors n1 and n2 given at nodes X1 and X2, respectively, and by minimizing the mean least square norm of the
errors with respect to X , using the generalized inverse:
X = ( At A + ε E )−1 At B
(2)
u1 v1 w1 
where A = 
 , where ( u1 , v1 , w1 ) and ( u 2 v 2 w2 ) respectively are the coordinates of n1 and n2 ,
u2 v2 w2 
1
1 B1 = − ( M1M 2 .n1 ) , and B2 = ( M1M 2 .n2 ) ; E is the unit matrix and ε is a small numerical parameter.
4
4
After replacing each edge of the triangular facet by the new quadratic curve by determining the different midedge points of the triangle, X 4 , X 5 , X 6 , the interpolated surface is given by the quadratic equation:
X ( ξ ,η ) = X 1 N1 ( ξ ,η ) + X 2 N 2 ( ξ ,η ) + X 3 N 3 ( ξ ,η ) + X 4 N 4 ( ξ ,η ) + X 5 N 5 ( ξ ,η ) + X 6 N 6 ( ξ ,η )
(3)
 N1 ( ξ ,η ) = 1 − ξ − η
 N4 ( ξ ,η ) = 4(1 − ξ −η )ξ


and hierarchical quadratic functions  N5 ( ξ ,η ) = 4ξη
.
with the original linear function  N 2 ( ξ ,η ) = ξ
 N ( ξ ,η ) = η
 N ( ξ ,η ) = 4(1 − ξ −η )η
 3
 6
n1
x1
n2
n'
x2
1
n'2
(c)
FIGURE 4. (a-b): Surface features: sharp edge (c): interpolation of a curved sharp edge.
Singular points of the domain (edges and corners) are handled by taking into account multiple normal vectors at
these points (see Figure 4) and so by increasing the number of equations that the mid-edge points X has to satisfy.
Validation
Validity and efficiency of the interpolation are first assessed by applying the algorithm to a sphere and cylinder
as typical analytical shapes. Normal vectors are provided by the analytical description of these shapes. Figure 5
shows how this method is able to recover smooth surfaces from coarsely discretized ones.
Initial Geometry
Smoothed Geometry
Comparison
Sphere
Cylinder
FIGURE 5. Effect of the quadratic interpolation on a triangulated surface of a sphere using analytical normals
Normal vector computing
Now, for the case of discretized surfaces where the analytical normals are not provided, additional information is
required to determine surface normal vectors at the mesh vertices. It is observed that the smoothing quality
significantly depends on the normal vectors values, so the normal vectors have to be very accurately calculated at
vertices. Several methods have been investigated in the literature, such as consistent normal [7], the SPR method [8]
and normal voting [9]. Normals are calculated from the existing piecewise values at the center of the surface
triangles, by considering a neighborhood (or patch) of elements surrounding a node, as shown in Figure 6. The
normal voting strategy proposed by Page et al. [9] provides the best results. It was successfully used in imagery [10]
as well as in mesh smoothing [11]. The basic idea of normal vector voting is to select a surface region around a
vertex. Each triangle belonging to this patch of elements votes in order for the estimation of the normal orientation at
the central vertex. In the example of Figure 6 (a), the triangle fi in the neighborhood of vertex m has a normal N that
generates a normal vote Ni at m.
m
m
(a)
(b)
FIGURE 6. (a) The vote of fi in order to estimate m orientation (b) First order patch
In this study, only first order patches are used, as shown in Figure 6 (b), so the normal Ni generated by triangle fi
in the example is equal to N. After collecting all votes into a covariance matrix A, its eigen-analysis is carried out.
The main eigenvector provide the normal orientation of the central vertex m, while the eigen-values are used to
make a distinction between surface nodes and edges and corners.
A=
∑ wV = ∑ w N N
i i
i
i
t
i
 mci 
Si

exp 
where : wi =
 σ 
S max


(3)
The previous test of Figure 5 is run again using normal voting calculations. Results are shown in Figure 7. The
obtained shape is almost similar to the analytical one. The maximum error between the analytical and calculated
(
normals are calculated by Max AnalyticalNormalim − NormalVoting im
m∈∂Ω
) , which provides the value of 0.09. This
shows how the original surface can be recovered without much information.
Initial Shape
Smoothed sphere (Analytical normal)
Smoothed sphere (Normal Voting)
Sphere
FIGURE 7. Impact of using Normal Voting in sphere smoothing.
CONTACT ALGORITHM USING SMOOTHING TECHNIQUE
For the contact algorithm, it is necessary to compute the orthogonal projection P of any spatial point M onto the
smoothed contact surface, which is now quadratic (3). M is first projected on the linear facets of the contact surface,
and then a Newton-Raphson algorithm is used to find the ( ξ ,η ) coordinates of P that satisfy (3).
Cylinder tool
velocity
New curved
surface
vout = −0, 01.u z
C 0 triangle
New projection
P
On curved surface
M
FIGURE 7. Projection on a curved surface (left) – Meshes for the indentation of a bulk parallelepiped (right).
SIMULATION RESULTS
Hertz’ contact: indenting a bulk parallelepiped
A cylinder is indented onto an elastic bulk parallelepiped with a vertical velocity of 0.01 mm.s-1 (see figure 7 right). The thickness of the parallelepiped is 5 mm. The mesh size is approximately h=1 mm. The cylinder radius is
5mm and tool mesh size is approximately h =1.7 mm. The time step is ∆t=1 s and the indentation is 2 mm.
Simulation results are presented in Figure 8. Contact areas resulting from calculations with discretized tool (Figure
7), smoothed tool, analytical tool and refined tool (three time smaller mesh size of the cylinder is h =0.5 mm), show
how smoothing allows a better calculation of the contact surface. Figure 9 presents the evolution of the number of
contact nodes, which shows that smoothing allows a better contact detection, larger and more regular contact
surface, and removing temporal oscillations. Smoothed surface is more accurate. In deed, the contact normal is
almost perfectly calculated, as shown in Figure 10 that represents the maximum error of the normal calculations
between analytical and calculated values; smoothing reduces this error from 25% to 0.05%.
discretised tool
discretised tool:
Refined Cylinder
Analytical tools
Smoothed Tool
FIGURE 8. Comparison of contact surface resulting from facetized, smoothed, analytical and refined tool
nodest
Number of contact
Number of contact nodes=f(t)
160
140
120
100
80
60
40
20
0
Dis cretis ed Tools
Sm oothing Tools
Oscillation
0
t(s)
100
200
300
400
Max Error
FIGURE 9. Comparison of number of contact nodes resulted form using facetized tool and smoothed tool
Max Error Of Contact Norm als =f(t)
0,30
0,25
0,20
0,15
0,10
0,05
0,00
Error Smoothing Normal /Analytical normal
Error Smoothing Normal /Analytical normal
t(s)
0
50
100
150
200
250
FIGURE 10. Comparison of number of Contact Normal using facetized tool and smoothed tool
Ironing a bulk parallelepiped
Cylinder
velocity
Vout = 1.u x
FIGURE 11. Ironing a bulk parallelepiped (left) and shape rolling of a long product (middle) with a zoom on the roll (right).
The same case with same properties is considered here, except that the cylinder has now a horizontal velocity of
1 mm.s-1 (Figure 11 - left). Simulation results show that smoothing significantly reduces the numerical oscillations
of forces. The normals continuity allows smoothing the contact force changes along the facet boundaries, in the
same way that was noticed by Puso and Laursen [4]. Figure 12 illustrates this fact by comparing the time variations
of the vertical force Fz on the tool surface in the previously described different cases. It is noticeable that smoothing
results are almost similar to analytical ones (dark bleu curve and yellow one, respectively).
Smoothing
Discretised Tool
Analytical
Refined Cylinder
Fz=f(t)
250
200
Fz
150
100
50
0
0
2
4
6
8
10
12
14
16
18
20 t(s)
FIGURE 12. Force Fz calculated on surface tool
Shape rolling of a long product
The geometries of tools employed in the shape rolling simulation are shown in Figure 11 – right. In this more
complex and realistic case, the smoothing procedure allows much better calculating the contact conditions (contact
area is smoother and more regular) as can be seen in Figure 13. The equivalent strains (Figure 14) and strain rates
(Figure 15) are lower and more homogenous, which provides results that corresponds better to observations.
Facetised tool
Smoothed tool
t=0.045s
Facetised tool
Smoothed tool
t=0.35s
FIGURE 13. Comparison of contact surface between facetized tool description and smoothed one
Facetised tool
Smoothed tool
FIGURE 14. Comparison of equivalent strain between facetized tool and smoothed one.
Even though the smoothing contact algorithm requires more computational effort for normal calculations, smooth
surface estimation and especially for the projection of contact nodes, the computational time does not significantly
increase, as the simulation of the 0.35 s of this process only needs 12h:38mins instead of 12h with the standard
approach, which only represent an increase of 5%.
Facetised tool
Smoothed tool
t=0.1s
t=0.35s
FIGURE 15. Comparison of equivalent strain rate between facetized tool and smoothed one.
CONCLUSION
The developed contact algorithm allows a much more precise simulation of a wide range of metal forming
processes with reduced contact area. It is based on a quadratic interpolation of the contact surface that combines the
smoothing technique proposed by Nagata and the normal voting strategy utilized in digital imaging. The obtained
contact surfaces are more accurate, more regular without time oscillations, so providing more accurate results in
terms of equivalent strain rates and total strains, which better fit with expectations. Last and not least, this local
approach is well adapted to parallel calculations and the additional computational cost only represents few percents
of the total one.
REFERENCES
1. Nagata T. Simple local interpolation of surfaces using normal vectors. Computer Aided Geometric Design 2005.
2. Stadler M, Holzapfel GA, Korelc J. Cn continuous modeling of smooth contact surfaces using NURBS and application to 2D
problems. International Journal for Numerical Methods in Engineering 2003;57:2177–203.
3. Wriggers P, Krstulovic-Opara L, Korelc J. Smooth C1 interpolations for two-dimensional frictional contact problems, Int. J.
Numer. Methods Engrg. 2001;51:1469–95.
4. Puso MA, Laursen TA. A 3D contact smoothing method using Gregory patches. Int. J. Numer. Methods Engrg. 2002;54:1161–
94.
5. Rassineux, A., Villon, P., Savignat, J.-M., Stab, O., 2000. Surface remeshing by local Hermite diffuse interpolation. Int. J.
Numer. Methods Engrg. 49, 31–49.
6. Hama T., Nagata T., Teodosiu C., Makinouchi A., Takuda H. Finite-element simulation of springback in sheet metal forming
using local interpolation for tool surfaces Int. J. Mech. Sci., vol. 50, pp. 175– 192, 2008
7. Guerdoux S. Numerical simulation of the Friction Stir Welding process , Ph.D. thesis, Mines ParisTech, 2007
8. O. C. Zienkiewicz and J. Z. Zhu, The Superconvergent Patch Recovery (SPR) and adaptive finite element. Computer Methods
in Applied Mechanics and Engineering. Vol. 101, pp.207-224, 1992.
9. D. L. Page, Y. Sun, A. F. Koschan, J. Paik, and M. A. Abidi. Normal Vector Voting: Crease Detection and Curvature
Estimation on Large, Noisy Meshes. Graphical Models 2002
10. Hyun Soo Kim, Han Kyun Choi, Kwan H. Lee, Feature detection of triangular meshes based on tensor voting theory,
Computer-Aided Design 41 (2009) 47_58
11. Takafumi Shimizu, Hiroaki Date, Satoshi Kanai, Takeshi Kishinami, A New Bilateral Mesh Smoothing Method by
Recognizing Features, Ninth International Conference on Computer Aided Design and Computer Graphics (CAD/CG 2005)