An Application of 3-D Solidification Analysis to Large Complex Castings

Chung-Whee M. Kim
Frank J. Sant


[Presented at the 2nd Pacific Rim International Conference on Modeling of Casting and Solidification Processes, January, 1995]


Abstract

We have provided a tool to casting engineers for solidification analysis of complexly shaped parts for use on the foundry floor. This tool is accurate, timely, flexible, and user friendly.

First, we chose to model the process as a heat conduction problem with latent heat of fusion. Then, the Finite Element Method is chosen for a numerical method since it is geometrically flexible. A unique time integration scheme is used to obtain stable, accurate, and efficient results.

A new meshing technique is developed specifically to utilize the geometric flexibility of the Finite Element Method to obtain accurate results for the casting and die/mold. Also, the meshing technique provides convenient editing and updating of the three-dimensional finite element mesh for subsequent analyses, allowing the engineer to quickly and easily obtain the ideal process.

The new technique, with an efficient solidification analysis program, allows large complex automobile castings to be analyzed with minimum requirements of computer resources. A large (725,000) finite element model of an automobile V6 engine block casting can be analyzed on mid-range engineering workstations.


1.0 Introduction

Casting solidification analysis has become an important part of everyday foundry practice. Yet, a large scale three-dimensional (3-D) solidification analysis, such as an automobile engine block, is still impractical, mostly because of the geometrical complexity of the casting.

Applications of FEM to shaped castings have been limited. This is because most mesh generation methods have been developed for stress analysis, which requires modeling of the part only. Foundry applications require inclusion of the mold/die in process modeling. Therefore, it was necessary to develop an enmeshing technique with the geometric flexibility of FEM, plus the ability to model both the part and the mold/die.


2.0 Solidification Modeling

Although there exist different levels of complex solidification modeling for casting processes, we chose to model the casting solidification process as a heat conduction problem with latent heat of fusion. This model is simpler than heat and mass transfer models, yet more accurate than a pure geometric approach. Thus, one can write: [1]


3.0 Numerical Method

To solve equation (1) for a complexly shaped casting, one needs to use a numerical method. The Finite Difference Method (FDM) and the Finite Element Method (FEM) are both particularly suitable for this task. A simple but not complete comparison of these methods is illustrated here. (Note that the Control Volume/Direct Difference/ Finite Volume methods are another possibility, but they may suffer model construction difficulties similar to those of FEM in practical applications.)

First, FDM offers much easier formulation than FEM, therefore, an FDM program can be simpler and require less computer resources. On the other hand, an FEM mesh has inherent geometric flexibility, unlike an orthogonal FDM mesh. Thus, FEM is a more suitable method for casting solidification analysis, due to the geometric complexity of most castings. However, in practice, FDM programs offer simpler, although crude, meshing schemes, in whihch the pre-set orthogonal mesh and casting geometries are overlapped. This may cause difficulties in sorting out the inside mesh from the remainder. Also, satisfactory enmeshment of curved surfaces and/or thin walled castings is inherently very difficult, if not impossible, with FDM programs.

Most FDM programs depend upon the mesh generation technique developed originally for structural analysis, where only the part needs to be meshed. For casting solidification analysis, both the part (casting) and the surrounding material (mold/die) need to be modeled for accurate results. The existence of the mold/die makes the FEM mesh creation procedure very difficult with conventional meshing techniques, in addition to making the total number of elements very large. Thus, we could summarize that meshing of FDM for casting solidification is simple, but is limited. Also, meshing for FEM is flexible, but is difficult. At this point, there are more commercially available FDM software programs for casting solidification than FEM based ones.


4.0 Meshing Technique

Geometric flexibility of the FEM is essential to casting engineers to obtain accurate solidification analysis results. In addition, stress analysis of parts and/or dies can be performed with FEM with little or no modification, along with fluid flow analysis.

Therefore, we decided to use FEM over FDM. As mentioned earlier, one of the biggest obstacles to use FEM effectively for casting solidification is enmeshment, becasue traditional meshing techniques have been developed for stress analysis. To ease this difficulty, we have developed a new meshing technique.

Our new meshing technique utilizes two-dimensional (2D) meshes to build a three-dimensioanl (3D) mesh. First, a 2D base mesh is created, which describes most of the part's complicated geometry. Next, one or more 2D extension meshes are created. See Figure 1. The extension mesh(es) 'grab' the base mesh elements, adn 'grow' them into a 3D mesh. The 3D mesh has a layered structure. Every 3D element is created by extension of a 2D element in the base mesh. Each element in the 3D model is related to the elements topologically 'above' and 'below' it, which are all created from the same original 2D element. Finally, a batch file is created, whic can be edited and reach time a change is made to the model. The batch file is a great benefit of the meshing technique, since it greatly simplifies editing of the model for subsequent analyses of part an/or die changes.

This method is particularly suitable to casting solidification, but it is also useful for fluid flow and stress analyses.


5.0 Analysis

5.1 FEM
After semi-discretization in space using finite elements, equation (1) becomes

Note that latent heat of fusion effects can be included in [M] or {f} depending on the formulation. [2] [3]

5.2 Time Integration Scheme
Equation (2) further reduces to

at each time step, with a time integration scheme.

To reduce equation (2) further, a time integration scheme has to be applied to equation (2). In the case of a casting solidification simulation, the solution domain consists of casting and mold, which have considerably different initial temperatures and material properties. These severe initial conditions, which are assigned at the interface(s) of the sub-domains, could cause some oscillations at early stages even with an unconditionally stable integration method.[4] However, the fully implicit method is truly stable. Therefore the fully implicit time integration method seemed to be preferable for casting solidification analysis. Although, it is well known that this method can be less accurate than the Crank- Nicholson method. Therefore, we combined the advantages of the fully implicit method and the Crank-Nicholson method. That is, we used the fully implicit method at the early calculation stage, then change to the modified Crank-Nicholson method, in order to improve efficiency without losing accuracy.[5]

To solve equation (3), the most general purpose FEM program, originally developed for stress analysis, uses a kind of direct solver. This solver is very powerful to solve relatively small systems of linear equations, in terms of available computer resources, but its memory requirement grows exponentially with the number of unknowns. In casting solidification analysis the system of equation (3) becomes large mainly due to the mold components. The elimination of the mold from casting solidification analysis has been tried with limited success.[6]

Recently, the iterative solver regained its popularity for large systems of equations, arising from the FEM, because they generally require less computer resources compared to direct solvers. In particular, the conjugate gradient (CG) method became popular for FEM application.[7]

This technique minimizes the residual, {r}, of equation (3).

In theory, convergence can be achieved at a finite number of iterations. In real problems, however, convergence is slow or may not be achieved at all. Therefor the CG technique is used with 'preconditioning' to accelerate convergence. With the preconditioning matrix [P], equation (3) becomes

The choice of [P] is very important to obtain results efficiently. For best results, [P] should be close to the inverse of [A], and it should be easy to calculate. We adapted the Symmetric Successive Over-Relaxation (SSOR) method for preconditioning because we did not need to calculate any new matrix coefficients.[8]


6.0 Application

Solidification and fluid flow analyses were performed on a North American V6 engine block. The engine block (to be produced in Europe) is a gravity semi-permanent mold casting with cast-in cast iron liners. Because of an integrated water crossover and high pressure oil lines, casting integrity is a major concern. This engine block will be used in high volumes, so cost and ease of manufacturing were also major considerations.

6.1 Mesh Generation
Due to the size and complexity of the casting, mesh generation of the entire process was a challeng. Using the previously discussed mesh generation techniques, a finite element mesh of the casting, gating, liners, permanent mold, water lines, and cores was produced. The finite element mesh took six weeks to build. The total mesh size was approximately 650,000 elements and nodes. A majority of these elements are in the mold and cores. Interface elements, which are automatically created in the analysis software, made the total mesh size 726,000 elements and 724,000 nodes. Figures 2 and 3 show the casting with the cast in liners.

6.2 Solidification Analysis
The importance of cyclic solidification analysis for permanent molds has been reported.[9] Therefore, cyclic solidification was performed on the V6 block. The casting cycle was divided into three segments: 1)Dwell (solidification), 2)Mold Open (convection to air), and 3) Placement of the cast in liners and cores. Five cycles were simulated to reach a quasi-steady state of the permanent sections of the die.

This analysis was performed on an HP 735/99MHz computer and took 13 hours to run.

Solidification simulation indicated several areas for potential porosity. Potential shrinkage porosity can be identified by iso-contour plots of solidification time. Figure 4 shows the isolated region of liquid metal. Cooling lines and risering are then modified to ensure directional solidification. Other areas of potential porosity can be identified by cut sections seen in figures 5 and 6. These areas will not be identified without cyclic analysis. These areas are caused by an "over heated" section of the die.

Modifications of the mold and process were modeled to ensure a directionally solidified casting. Due to the proprietary nature of this component, we cannot discuss mold and gating configurations.


7.0 Conclusion

A new 3D enmeshing technique for FEM has been developed for shaped casting applications. This technique, with an efficient solidification analysis program, allows large complex castings to be analyzed with minimum requirements of computer resources.

The new meshing technique has been successfully applied to an automobile V6 engine block casting. Several changes have been made to produce quality castings based on these analyses.


8.0 Acknowledgments

Thanks to J. DeLeeuw of EKK, Inc., for review and discussions.