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Procedía Engineering 207 (2017) 1087-1092

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International Conference on the Technology ofPlasticity, ICTP 2017, 17-22 September 2017,

Cambridge, United Kingdom

Research on Natural Element Metngd and the application to simulate metal forming processes

Ping Lua,b *, Yang Shua,b, Dahai Lua,b, Kaiyong Jianga,b, Bin Liua,b, Changbiao Huanga,b

Cue to the special characteristics of metal forming processes (i.e., large deformadions, complex nonlinear, etc.), finite element method (FEM) often encounters the; mesh distortion problem in simulating l arge ond severe deformation metal forming proc esses. Nnce meshes become severely distorted, the whole simulation proce ss cannot be continued unless the remeshing is used. Nevertheles s, remeshing usually leads to nos only tide deterioration of computational drecision but also the increase of time-consuming. To solve the probl ecc a meshless method known as natural element method (NEM) is introduced to analyze metd aormtng p rocesses in the papor. The shape function of NEM is constructed by emp loying th e natural neighbor interpolation method, and it has the high accuracy of thc approximation. In addition, because the fhape function possesses the Kronecker d property likc fJne FEM, the essentiai boundary conditions can be imposed (directly. Thersfore, the psper combines NEM with rigiddvisco-plastic flow Theory and applies it to simulate mctal forming processes. Furthermore, a numerical analysts procegure is developed . A numeticsl sxccpIs of metal Worming processes is analyzed. The validity of the numesical s imulation program is evaluated by comparing simulation results with those obtained by rigid/visco-plastic finite element method.

© 2017 The Adon. Publiahed by Elsevier Ltd.

Peer-review under responsibility of the scientific committee of the International Conforence on thr Technolo gh o f Plasticity. Kfywords: Metal forming processes; Simulation; Natural Element Method; Rigid/visco-plastic;

* Corresponding author. Tel.: +86-592-6162598; fax: +86-592-6162595. E-mail address: pingluTeiitciu.8du.cn

2Fujian Key Laboratory of ¡Special Energy Manufacturing, Huaqiao University, Xiamen 361021, China bXiamen Key Laboratory of Digital Vision Measurement, Huaqiao University, Xiamen 361021, China

Abstract

1877-7058 © 2017 The Authors. Published by Elsevier Ltd.

Peer-review under responsibility of the scientific committee of the International Conference on the Technology of Plasticity. 10.1016/j.proeng.2017.10.1135

1. Introduction

The finite element method (FEM) as one of the most important numerical methods has been employed widely in the field of computational mechanics and analysis of engineering problems. But FEM often encounters the mesh distortion problem in simulating large and severe deformation metal forming processes. Because both the shape function and integration of FEM depend heavily on the pre-processed meshes, once meshes become severely distorted, the numerical integration precision and the computational accuracy of shape function decline, and even the whole simulation process cannot be continued unless the remeshing is used. Nevertheless, in the remeshing process, it is necessary to transmit the data from the old mesh to the new one by either interpolation or approximation method. Hence, remeshing usually leads to not only the deterioration of computational precision but also the increase of time-consuming.

In order to overcome these above problems from FEM, many scholars have proposed more than twenty types of meshless numerical methods in recent years, such as the smooth particle hydrodynamic method (SPH), the elementfree Galerkin method (EFG), the reproducing kernel particle method (RKPM), the h-p clouds method (HPCM) and so on. Based on the discrete nodes in local domain, the shape functions of these meshless methods have been constructed by interpolation or approach method according to a certain way. Researchers achieved remarkable progresses on the application of the meshless simulation to the metal forming processes. Cueto et al. [1] made a review of the most relevant meshless (or meshfree) approaches to the field of material forming involving both solid and fluids.

Natural element method (NEM) is a type of the meshless numerical method that has developed over the last nearly twenty years [2]. Comparing with other meshless method, the shape function of NEM is constructed by employing the natural neighbor interpolation method, the shape function of NEM possesses the Kronecker 5 property, so the essential boundary conditions can be imposed directly. NEM really has overcome the difficulty of imposing essential boundary conditions for other meshless methods. Meanwhile, in the process of construction the shape function, the number of the natural neighbor nodes has the local adaptive characteristic. That is the number of the influential nodes for calculating point has reached the strict neighbor optimum in the Voronoi diagram. Besides, the number of the natural neighbor nodes is more than the number of the interpolation node of the FEM. The accuracy of the NEM is higher than that of the FEM. Thus, NEM possesses both advantages of the meshless method and FEM; it is an abroad promising meshless method for solving partial differential equation.

The paper studies the fundamental theory of NEM including shape function computation and integration scheme. Moreover, the paper combines NEM with rigid/visco-plastic flow theory and establishes the rigid/visco-plastic NEM. Furthermore, a numerical analysis procedure is developed. The validity of the numerical simulation program is demonstrated by simulating numerical example of the forging process.

2. Fundamental theory of NEM

2.1. Voronoi diagram and dual Delaunay triangle structure

Like other meshless methods, NEM uses nodes to discretize the problem domain. But the difference is that the computation of shape function is based on Voronoi diagrams and Delaunay triangulations structure. And natural neighbor interpolation is calculated based on the Voronoi diagram with the computational point being the kernels. The Voronoi diagram and Delaunay triangulations structure have the strict mathematical definition in computational geometry, which are basic geometric structures obtained by irregular distributed discrete nodes.

Take the two-dimensional space as an example [3], there is a group of discrete node set N = {p1, p2, p3... pn} , and the coordinate vector of arbitrary node is x ,, the plane is divided into several Voronoi cells with each node as the core V(p,), and within the regions V(pr) the distance from any point to the core node pt is closer than that to the other node pj e N(, ^ J) distance. The Voronoi diagram of the discrete node set is composed by these Voronoi cells V(pt). A mathematical expression is defined as the following formula,

V(p,) = {x eRd : d(x, x,) < d(x, xJ )VJ / ,} (1)

where, d(x, xJ ) represents the distance from a point x to xJ , Rd represent a d dimensional space. The Voronoi diagram V(pI) of any node is a convex polygon. Besides, if the Voronoi diagram V(pI ) and V(pj ) share a

common edge, then connect these two nodes Pj and pJ , then Delaunay triangulations structure is obtained by doing

the same work for all nodes. Delaunay triangulations structure meets the empty circumcircle property, which means that no node of the cloud lies within the circle covering a Delaunay triangle.

Figure 1 shows the Voronoi diagram and Delaunay triangulations structure as the number of nodes n = 7 in two-dimensional plane. For any point p within the region, the second-order Voronoi diagram can be generated based on the above definitions. And the node pI which shares the common edge with the point of p is called the natural neighbor node.

2.2. Natural neighbor interpolation

The main natural neighbor interpolation methods are the Sibson interpolant and Laplace interpolation. The Sibson interpolation method is linear in the convex domain boundary, but it cannot realize the exact interpolation of the boundary in the non-convex domain, while the Laplace interpolation can be used to realize the linear interpolation in the non-convex boundary [4-7]. The calculation of Laplace interpolation is a reduced dimension method. In two-dimensional space, the shape function is constructed by calculating the distance. In three-dimensional space, it is calculated by calculating planar area. The computational cost of Laplace interpolation is lower than that of Sibson interpolation. So in the paper Laplace interpolation is adopted.

According to Laplace interpolation method, shape function of node I is defined as

(x) = _«£tt_

E«,(x) (2)

" (x) = hT) (3)

hj (x)

The derivatives of shape function is evaluated as

a, ,i (x)- (x) 'Yjaj i (x)

i (x) =-n-j=--(4)

Yjaj(x)

= (5) hj (x)

where,

where sJ (x) represent the length of the Voronoi segment associated to node J , hJ (x) is the distance between x and its natural neighbor node (see Figure 2).

2.3. Numerical integration schemes

The numerical integration schemes of NEM include stabilized conforming nodal integration, partition of unity integration scheme and the background mesh integration scheme [8, 9]. Because the computation of shape function is based on Voronoi diagrams and Delaunay triangulations structure in NEM. Hence, the paper utilizes the Delaunay triangulations as background integration meshes to carry on numerical integration.

Delaunay triangulation mesh is a subdivision geometrical profile of the problem domain. Vertexes of the mesh are the discrete nodes. Thus all integration points are within triangulation meshes or on the edges of the triangulation meshes. It is no need to judge the position of the integration points within or without the problem domain. Especially in the large metal plastic deformation, the Delaunay triangulation meshes have better shape adaptability to the deformation. Therefore, that using Delaunay triangles as background integration meshes is more suitable for simulating metal forming process by NEM.

3. Numerical example

Combining rigid/visco-plastic theory and NEM, a program for simulating metal forming processes with arbitrarily shaped dies is developed. Researches on rigid/visco-plastic theory and key simulation techniques for metal forming processes can be referred to our preliminary work [10, 11].

A plane strain forging process was analysed by the developed procedure. The height of the billet is 100mm, and the width is 120mm. Due to the symmetry of geometry, only a half of the billet is considered. The upper die moves downward with a velocity of 10 mm/s. The arctangent friction model is employed for modelling the friction behaviour on the workpiece-die interface. The constant frictional factor m is selected as 0.3. The flow stress of the material is

a-=753s0U4Mpa (6)

The billet is discretized with 273 nodes. In order to verify the validity of the procedure, the plane strain forging process is simulated by FEM software-Deform (2D) too. The FEM model is discretized by quadrilateral element with 273 nodes. Also, the other simulation parameters are the same. Figure 3 shows the initial shape of the workpiece and the geometric shape of dies.

Figure 4 gives the geometrical shapes of the deforming workpiece obtained by NEM and FEM-Deform (2D) respectively at 20% reduction in height. It can be seen that metal flowing trends simulated by NEM are in good agreement with that obtained by FEM-Deform (2D). Clearly, the bulge at the stage is obvious.

Figure 5 describe the geometrical shapes of the deforming workpiece at 50% reduction in height. It can be seen the shapes of workpieces (in the right) obtained by the method proposed in this paper are consistent with the ones (in the left) gained by the FEM software. From the right diagrams of the two figures, the streamline that exhibits metal deformation can be observed through the distribution of the nodes, and the flowing of the nodes can be traced clearly. However, from the results obtained by FEM, only the exterior outline of the deforming metal can be seen, the

streamline of metal deformation and the flow rules unable to be observed. That is because remeshing happens once the meshes of FEM occurs serious distortion with dramatically deformation.

f . 1 I 1 I 1 I 1 I 1 I 1 I

Fig.3 The initial shape of the workpiece and the geometric shape of dies.

:::::: : : •. \

............ ......

Fig. 4 The geometrical shapes of the deforming workpiece obtained by NEM and FEM respectively at 20% reduction in height.

Fig. 5 The geometrical shapes of the deforming workpiece obtained by NEM and FEM respectively at 50% reduction in height.

In order to compare the differences of the results obtained by NEM and FEM, the geometrical shapes of the deforming workpiece predicted by two methods at 50% reduction in height were drawn in one picture. Form the following Figure 6, it can be seen that the contact angle predicted with the NEM is some different from the one obtained by FEM. Probably because the FEM and NEM are two kinds different numerical calculation methods, and they have different approximate schemes. In addition, FEM has occurred remeshing in the simulation of the forging process, and free boundary nodes have become less after remeshing. So the contact angles predicted with the NEM and FEM are inconsistency.

The positions, velocity of the characteristic nodes numbered as 1-5 and 1'-5' in Fig. 6 are listed in Tables 1. The quantitative comparisons show that the results of EFG and FEM are in good agreement. From the table, numerical comparisons show that the results of NEM and FEM are in good agreement except the characteristic nodes sequential number are 5and 5'. The x coordinates of 5and 5' are 144.256 and 148.406 respectively, the corresponding absolute

errors is 4.15, but the corresponding relative error of the results calculated by two methods is small about 2.8%.

00 20 40 60 80 100 120 140

Fig. 6 The comparison of the geometrical shapes of the deforming workpiece obtained by NEM and FEM at 50% reduction in height

Tables 1 The coordinates, velocity of the characteristic nodes at 50% reduction in height evaluated by NEM and FEM

_ .. , NEM FEM

Sequential „ . ,,, , , , , Sequential „ .

n Coordinate Velocity value (mm/s) n , Coordinate

-i-1-— number -

number

Velocity value (mm/s)

x у Vx Vv x у Vx Vv

1 59.051 50.286 2.821 -10.005 1' 59.776 50.584 3.144 -10.001

2 64.631 49.312 2.537 -11.331 2' 64.876 49.316 2.613 -11.408

3 142.511 30.485 47.227 -9.994 3' 142.631 30.584 41.085 -9.999

4 148.125 14.079 45.881 -3.457 4' 149.174 13.949 47.082 -4.717

5 144.256 0 43.465 0 5' 148.406 0 47.462 0

4. Conclusions

Natural element method is applied to the simulation of metal forming processes. Laplace interpolation is used to calculate the shape function. The Delaunay triangulations as background integration meshes are adopted to calculate numerical integration. Numerical example analyses for a plane strain forging demonstrate that the rigid/visco-plastic NEM established and the procedure developed by the paper are correct. Method proposed in the paper is capable of simulating metal forming processes with severe deformation and arbitrarily shaped dies. The flowing of the deformation metal can be traced clearly.

Acknowledgements

The authors would like to acknowledge financial support from the National Natural Science Foundation, China (No.51305144, No. 51475174), Fujian Provincial Natural Science Foundation (No. 2015J01203).

References

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