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Inverse Pre-Deformation of Tetrahedral Mesh for Large Deformation Finite Element Analysis
Computer-Aided Design & Applications, Vol. 2, No. 6, 2005, pp 805-814

805
Inverse Pre-Deformation of Tetrahedral Mesh for Large Deformation Finite
Element Analysis

Arbtip Dheeravongkit
1
and Kenji Shimada
2


1
Carnegie Mellon University, aom@cmu.edu
2
Carnegie Mellon University, shimada@cmu.edu


ABSTRACT

In the finite element analysis that involves with large deformation, distorted elements are usually
produced in the later stages of the analysis. These distorted elements lead to inaccurate solutions,
slow convergence and premature termination of the analysis. This paper proposes an inverse pre-
deformation method to generate the input tetrahedral mesh for Lagrangian analysis. By this
method, tetrahedral elements in the input mesh are pre-deformed into shapes approximately
opposite of those produced in the analysis. As a result, the number of inverted and ill-shaped
elements can be reduced at the later stage of the analysis. A pre-analysis is required to collect
geometric information and equivalent plastic strain information. The proposed method then
generates a new optimal mesh on the deformed boundary, considering equivalent plastic strain
information to control mesh sizes, and finally, maps the new elements to the undeformed boundary
using barycentric interpolation to create the pre-deformed tetrahedral mesh.

Keywords:
Large Deformation Analysis, Mesh generation, Inverse Pre-deformation, Bubble Mesh.


1. INTRODUCTION
The process of finite element analysis that deals with large deformation usually produces distorted elements at the later
stages of the analysis. These distorted elements lead to several problems: inaccurate results, slow convergence and
premature analysis termination. Metal-forming processes are the most common applications involved with large
deformation analysis; they include forging, extrusion, rolling, deep drawing, etc. An example of such large deformation
analysis is illustrated in Fig. 1. This is a three-dimensional forging example, containing a sinusoidal die that deforms a
deformable blank into geometry with high-curvature corners. As the finite element analysis is performed on this
problem using pure Lagrangian method, several elements are severely distorted especially around high-curvature
corners. Consequently, the resulting mesh contains many highly-distorted elements at the later stages, leading to
several problems listed earlier.



Fig. 1. An example of large deformation finite element analysis (Forging with a sinusoidal die).

As an alternative solution, this paper proposes an inverse pre-deformation method to pre-deform the input
tetrahedral mesh for Lagrangian analysis to reduce the number of inverted and ill-shaped elements at the later stage of
the analysis. The term inverse pre-deformation is used to illustrate the idea of this method, in which we first predict
the way each mesh element will be deformed during analysis, and then create the new input mesh, which contains
elements that have approximately opposite shapes of those predicted. In conclusion, this method pre-deforms input
Computer-Aided Design & Applications, Vol. 2, No. 6, 2005, pp 805-814

806
mesh elements into shapes approximately opposite of which they will ultimately be deformed. With this approach, the
number of inverted and ill-shaped elements in the later analysis stage can be reduced, because overall element shape
quality tends to improve along the analysis process.
The two-dimensional version of this inverse pre-deformation method was proposed earlier to generate the input
quadrilateral mesh for large deformation analysis. It has been proved that the inverse pre-deformation method can
successfully extend the life of the analysis, as well as reduce the number of ill-shaped elements at the later stage [19].
The work presented in this paper shows that a similar concept can be used to generate the tetrahedral mesh for three-
dimensional problems. However, there are some major differences between the two-dimensional and three-
dimensional pre-deformation methods, which will be explained in detail in section 2. Nevertheless, there are two
conventional techniques for addressing this problem, the adaptive remeshing and the Arbitrary Lagrangian-Eulerian
(ALE). However, both techniques have drawbacks.

Adaptive remeshing is a technique, which replaces an over-distorted mesh with a better-conditioned mesh when the
error approximation of analysis exceeds the tolerance, or the maximum error value allowed [2]. The newly- created
mesh may not necessarily have the same topology as the original mesh, and the number of nodes and elements of the
new mesh may differ from the original mesh. Therefore, state variables and history-dependent variables must also be
transferred from the original to the new mesh. State variables include nodal displacements and variables of the contact
algorithm. History-dependent variables are the stress tensor, strain tensor, plastic strain tensor, etc. The adaptive
remeshing technique usually completely re-meshes the part at every certain number of steps in the analysis [1-3]. The
disadvantage of this method is its high computational cost, especially during the procedure for determining the error
estimator and mapping variables from an old to a new mesh [3]. More importantly, computational costs increase
considerably for analysis of complicated geometries.

The Arbitrary Lagrangian-Eulerian (ALE) method is another technique for addressing the problem of large
deformation in finite element analysis. This method combines the features of pure Lagrangian analysis and Eulerian
analysis--two common types of finite element analysis. In pure Lagrangian analysis, a mesh follows the material
deformation during analysis; the mesh is connected to the material throughout the finite element calculation [5]. Since
the mesh and the material are connected, severe distortion of the mesh can cause difficulty in the finite element
calculation. It is here that adaptive remeshing must be applied to improve the shape quality of the mesh in order to
continue the analysis. ALE was developed to reduce the repetition of complete remeshing [4-8]. Essentially, ALE is a
Lagrangian analysis that takes advantage of the advection techniques of Eulerian analysis. In the ALE method, the
mesh is neither connected to the material nor fixed to a spatial coordinate system. Rather, it is prescribed in an
arbitrary manner [4]. During the analysis, the mesh elements deform according to the Lagrangian method. However,
instead of repositioning the nodes at their original position and performing advection, as in the Eulerian method, the
nodes are placed at other positions to obtain optimal mesh quality. The mesh nodes have velocity associate with them
in order to obtain the updated mesh. Mesh velocity plays an important role in the ALE method, as it reduces the
analysis error, and prevents mesh distortion [4]. Another important characteristic of this method is that it changes the
location of the nodes in the existing mesh, instead of creating a completely new mesh, like the adaptive remeshing
method, and it maintains the same (or similar) mesh topology throughout the analysis [5]. However, because of its
complexity, the computation cost is much more expensive than using pure Lagrangian analysis. There are also other
limitations in ALE analysis. In many cases the mesh suffers considerable distortion, and the same mesh topology
cannot be maintained for the entire analysis. In such cases, complete adaptive remeshing is still required. Another
drawback of ALE is that the state-variables remapping step is much more complicated than in the Lagrangian method.

It should be noted that the proposed inverse pre-deformation method is not meant to be a replacement of the previous
two existing methods. However, because the inverse pre-deformation method reduces the number of ill-shaped
elemen