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This is the accepted version of this article. Published as: Gu, YuanTong (2010) An adaptive local meshfree updated Lagrangian approach for large deformation analysis of metal forming. Advanced Materials Research, 97-101. pp. 2664-2667.

© Copyright 2010 Please consult the author.

An adaptive local meshfree updated Lagrangian approach for large deformation analysis of metal forming

Y.T. Gu School of Engineering System, Queensland University of Technology,

Brisbane, QLD 4001, Australia

yuantong.gu@qut.edu.au

Keywords: Metal forming, Large deformation, Meshless method, Meshfree method, FEM

Abstract. The large deformation analysis is one of major challenges in numerical modelling and simulation of metal forming. Because no mesh is used, the meshfree methods show good potential for the large deformation analysis. In this paper, a local meshfree formulation, based on the local weak-forms and the updated Lagrangian (UL) approach, is developed for the large deformation analysis. To fully employ the advantages of meshfree methods, a simple and effective adaptive technique is proposed, and this procedure is much easier than the re-meshing in FEM. Numerical examples of large deformation analysis are presented to demonstrate the effectiveness of the newly developed nonlinear meshfree approach. It has been found that the developed meshfree technique provides a superior performance to the conventional FEM in dealing with large deformation problems for metal forming.

Introduction

A geometrically nonlinear analysis of solids is a challenge for researchers in engineering and sciences. Although the finite element method (FEM) is a well-established mesh-based method, it often encounters difficulties for nonlinear analyses due to issues of the mesh distortion. Especially for some geometrically nonlinear problems with very large deformation, for example the metal forming, FEM will fail to give reasonable solutions due to the severe mesh distortions.

Recently, meshfree (or meshless) methods have attracted more and more attention from researchers. The meshfree methods do not require a mesh to discretize the problem domain, and the shape functions are constructed entirely based on a set of scattered nodes. Therefore, they can avoid the disadvantages of FEM. A group of meshfree methods have been developed so far including the smooth particle hydrodynamics (SPH)[1], the element-free Galerkin (EFG) method [2], the reproducing kernel particle method (RKPM) [3], and the point interpolation method (PIM)[4]. Because no mesh is used, meshfree methods show very good potential for the geometrically nonlinear analysis. Chen et al. [5] and Jun [6] concluded that EFG and RKPM based on the global weak-form were very effective for the large deformation analyses. Based on the total Lagrangian (TL) approach, Gu et al. [7][8] developed local meshfree techniques for large deformation analyses. However, the study for the nonlinear analyses by the meshfree methods based on the local weak-forms [9] is few, especially using the updated Lagrangian (UL) approach [10], which refers all variables to the current configuration. One of distinguished advantages of meshfree methods is to perform the adaptive analysis easily. To achieve the adaptive analysis in large deformation simulation, the UL approach is necessary for many cases, so it is important to develop a local meshfree UL approach for large deformation analyses.

In this paper, a local meshfree formulation based on the radial basis function (RBF) interpolation[11], the local weak-form and the updated Lagrangian approach is developed for the large deformation problems. In the UL approach for the large deformation analyses, after several deformation steps, the nodal distribution may become improper and lead to a large computational error. In these cases, the deformed problem domain should be automatically re-discretized (called re-noded, a type of adaptive analysis) to improve the computational accuracy. An effective adaptive

technique is proposed and the procedure to re-node the deformed problem domain is much easier than re-meshing in FEM. Considering the properties of adaptive analysis, the re-noding is much more efficient using the UL approach than using the total Lagrangian (TL) approach [10]. Several numerical examples of the 2-D large deformation analysis are studied to illustrate the effectiveness of the present local meshfree formulation.

Meshfree approach for large deformation analysis

Consider a body, as shown in Fig. 1, which occupies a region Ω0 at the initial stage and occupies a region Ωn at step n. Using the quantities related to the current configuration, the standard equilibrium equation for a solid is given by [10] 0bσ =+⋅∇ ρ in Ωn (1)where σ is the Cauchy stress tensor, ∇ is the gradient operator defined in the current configuration, b are the body forces per unit mass and ρ is the mass density in the current configuration, nΩ.

For a field node L, the governing equation, Eq. (1), is satisfied using the Petrov-Galerkin formulation, leading to a local weak form equation for this node[9][11]: : d d d d d 0

n q n q n qi n qu n qt

n n n n nρΩ Ω Γ Γ Γ

∇ Ω − ⋅ Ω − ⋅ ⋅ Γ − ⋅ ⋅ Γ − ⋅ Γ =∫ ∫ ∫ ∫ ∫v σ v b v n σ v n σ v t (2)

where v is the test (weight) function centred usually at the node L, nΓqi is the internal boundary of the quadrature domain, which does not intersect with the global boundary nΓ; nΓqt is the part of the natural boundary that intersects with the quadrature domain, and nΓqu is the part of the essential boundary that intersects with the quadrature domain.

The above discussed local weak-form is obtained based on Cauchy stresses and the current configuration. The second Piola-Kirchhoff stress (and its’ pair is the Green strain E) is also often used and Eq. (2) can be rewritten as the following matrix format: d d d d d

n q n qi n qu n qt n qΩ Γ Γ Γ Ω

Ω − Γ − Γ − Γ − Ω =∫ ∫ ∫ ∫ ∫wFS vNFS vNFS vT ρvb 0 , where 1 TJ −=σ FSF (3)

Due to the current configuration is unknown in the large deformation analysis, two approaches of the piecewise linear solution are often used [10]. The updated Lagrangian (UL) approach, which refers all the stresses and deformations to the current (deformed) configuration at step n, is used in this study. In addition, the incremental method is often employed in the large deformation analysis. We have uxx Δ+=+ n

nn

n1 , and 1n n

n n+ = + ΔS σ S (4)

Figure 1 Initial and current configurations

Initial configuration

0Ω X2, x2

x

X1, x1

X

Ωn

n+1

Current (deformed) configuration

where uΔ is the displacement increment of this material particle, and ΔS is the increment of the second Piola-Kirchhoff stress. A fundamental measurement of deformation is described by the deformation gradient, F

FIxuIux

xxxF ΔΔΔ +=

∂∂

+=+∂∂

=∂∂

=+

+nn

nnn

nnn

nnn

n )(1

1 (5)

Substituting all the above increment relations into Eq. (3), we can get the final local weak-form for the large deformation analysis based on UL. Using the RBF meshfree shape functions, we can obtain the discretized equations. It should be mentioned that the final system of equations will be nonlinear. Hence, the Newton-Raphson iteration is used to obtain results in the analyses of large deformation problems.

Numerical results

A cantilever beam: A cantilever beam is firstly considered to verify the present meshfree approach. The size of the beam is (10×2) and the state of plane strain is considered. The large deformation analysis is performed and the beam is subjected to a distributed vertical loading along the right end with f=1/Unit. The analysis is carried out using load incremental steps N and the load-scaling factor is β=10.0. It means that at the kth loading step, the distributed loading is fk=10k/Unit.

To study accuracy and convergence, we also solve this problem by FEM using a fine mesh, and the FEM results are taken as the reference solution. The following norm is defined as an error indicator,

( ) ( ) ( )Num Ref Ref

u t t te u u u= − (6)

For easy comparisons, the vertical displacement at point A, which is the middle point of the free end of this beam, is analysed.

In the large deformation analyses, after several deformation steps, the nodal distribution may become improper and lead to a large computational error. In these cases, the deformed problem domain should be re-noded to improve the computational accuracy. It is similar to the re-meshing in the FEM-nonlinear analysis. A simple technique to re-node the deformed problem domain for this beam, as shown in Fig. 2, is developed. The detailed procedure is : a) to form a large rectangle region including the deformed domain; b) to draw several parallel lines regularly distributed on x direction (or y direction); c) to compute intersection points with the global boundary for the deformed domain; d) to get the new field nodes by regularly dividing the line segments between intersection points. It has been found that this method is very easy to be used in the practical simulation. In this case of the cantilever beam, if 33 nodes are used in the 1st loading step to the 8th loading step (without re-noding), the computational error will be 3% at the 8th step. However, if from the 5th loading step, the re-noding technique is used, i.e., to discretize the domain by 48 field nodes. The computational error can be significantly reduced to 0.25% . It has proven that the adaptive analysis is effective.

To study the stability of the present meshfree approach, many loading steps are computed. Fig. 3 plots the results of 30 loading steps. It can be found that very stable results have been obtained by this meshfree simulation. It should be mentioned here that in Fig. 3, the vertical deflection at the free end of the beam is already more than 5 times of the initial depth of the beam. It means that the developed meshfree approach is still stable even in the case of very large deformation.

0 2 4 6 8 10 12

-8

-6

-4

-2

0

2

x

y

Fig. 2 Adaptive analysis Fig. 3 Large deformation for a cantilever beam

A typical metal forming problem-Compression of billet: The compression of a metal solid, which is a typical problem in metal forming, is analyzed. A 4 ×2 solid billet, as shown in Fig. 4, is studied by the newly developed meshfree method. This billet is subjected to compression along the axial direction subjected to a distributed loading along the right end with 40/Unit. The analysis is carried out using load incremental steps N and the load-scaling factor is α=10.0. Fig. 4 plots the progression of deformations obtained by the meshfree formulation for different loading steps. It is seen that the billet can be compressed as much as 75% compared to its original length. The same problem is also solved by FEM and by the local Kriging method [8]. It is found that FEM will converge very slowly once the compression is more than 50%. It demonstrates that the meshfree formulation developed in this paper is more effective than FEM for the large deformation problems.

Conclusions

A local meshfree formulation is developed for the large deformation analysis which is the key for the metal forming modelling. The radial basis function (RBF) is employed to construct the meshfree shape functions. The discrete equations for large deformation analysis are obtained based on the local weak-form and the updated Lagrangian approach. Because this meshfree formulation does not require explicit mesh, it can fully avoid mesh distortion difficulties in the large deformation analysis of metal forming. In addition, a simple and effective adaptive technique to re-node the deformed problem domain is also developed, and this procedure is much easier than the re-meshing in FEM. Several numerical examples are presented to demonstrate the effectiveness of the developed method

Fig. 4 The compression progression for a billet

N=40

p

: Initial nodes; *: Deformed nodes after 40 steps

N=10N=20

N=30N=40

Initial configuration

x

y

N=10

N=40

Ο: Initial nodes; *: Deformed nodes after 40 steps

Initial conf.

in the large deformation analysis. It has been found that the present approach has good performance, and it is also very stable even for irregularly distributed nodes. Hence, the present local meshfree method has good potential for the simulation of metal forming problems.

References

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