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Effects of Finite Element Meshing on Mold Filling Analysis
Geoffrey Engelstein, GR Technical Services, Mountainside, NJ
Introduction
Computerized mold filling analysis (MFA) is becoming a widely accepted tool in the plastics industry. The basis of MFA, regardless of the software algorithms, is the finite element mesh. The quality of the mesh directly influences the quality of the results. Subtleties of mid-plane models, evaluation of critical areas, selection of mesh density, and adaptive meshing techniques are all critical to the accuracy of the analysis. These factors and their relative influence on the analysis are discussed, in the context of linear elements. Rules of thumb are developed to balance accuracy with analysis time, along with explanations of when to break them.
Finite Element Basics
Finite element analysis is a numeric processing technique that has wide applications in structural, vibrational, thermal, flow, and other types of mechanical analysis. Mold filling analysis of plastic injection molds has been around for many years, but only now is the speed of computers allowing this technique to become available to the masses. Whether you perform the analysis yourself or work with a consultant, it is important to understand how the model that is entered into the computer affects the accuracy of the results. This will allow you to determine how much to lean on the numbers that the computer churns out at the end of processing.
The basis of finite element modelling is the 'node'. Nodes are placed on a three-dimensional model of the part at a certain density, called the 'mesh density'. 'Elements' are then formed by connecting the nodes into polygons. For MFA, the elements are usually triangular. In addition they are two dimensional. The nodes are placed at the center of the wall thickness, and the elements are assigned a constant thickness value. This is called a 'shell' or 'mid-plane' model. The implications of this are discussed later.
During the analysis, the key values (temperature, pressure, % filled, etc) are calculated at the nodes. The computer uses the connectivity information that is provided by the elements to determine how these values change from node to node. For all MFA software out today the elements are linear. This means that the values inside an element are assumed to be linear interpolations of the values at the nodes. For instance a point halfway between two nodes with pressures of 6000 kPa and 3000 kPa would be assumed to have a pressure of 4500 kPa. The linearity of the elements limits the accuracy of the model, and consequently, the results. Another type of element which is moving into use in other types of finite element analysis is the 'p-type' element. The 'p' stands for polynomial. The interior of these elements is represented by a higher order polynomial, so the values can range quite a bit within an element. This allows larger elements to be more accurate, but takes more computing power.
Since today's MFA software relies on linear elements, we restrict our discussion solely to them.
Processing Time
One consideration when designing a mesh is total processing time. While today's faster hardware has brought analysis time down, a significant fraction of any project is spent waiting for the computer to crunch numbers.
For any given part, processing time is dependent solely on the number of elements, and consequently node density. Sample times are shown in Figure 1 for a 30 cm square. The processing time is governed by the following relationship:
t = D**4 where t = processing time and D = node density
This equation is easily derived. Each processing cycle more elements are filled. Good programs will only process elements that have plastic in them. So if one additional element is filled each cycle the computer will process elements equal to:
n(n+1)/2
This is the familiar triangular sum formula, where n is the number of elements. In actuality more than one additional element will be filled each cycle, so this factor will vary between the above value and n2. For the large values of n that are typical in flow analysis the growth of this factor can be treated as n2.
Since n is proportional to the node density squared, the overall time is proportional to the fourth power of the node density. For example, a 30 cm square meshed at 0.78 nodes/cm takes approximately 24 minutes to complete on our computers. Increasing this to 1.18 nodes/cm (a factor of 1.5) increases the processing time by a factor of 5, to over 2 hours.
This exponential growth needs to be balanced with the purpose and accuracy of the analysis to determine an appropriate mesh density. In order to approach this intelligently, the effects of mesh density on result accuracy need to be determined.
Accuracy of Flow Pattern and Weld Line Location
One of the primary uses of flow analysis is determining the flow pattern and location of weld lines and air traps. Weld lines are formed where two flow fronts meet at a sharp angle (the worst case being a head on meeting). Air traps are pockets of gas that are trapped inside the cavity with no place to vent.
Two important results arise from the use of linear elements:
- Weld lines will always be seen on the edge of an element
- Air traps must surround a single node
To picture the roots of these statements, think about a single element. Each node has a time associated with it when it is 100% filled. Any points that are in the interior or along the edge will have a fill time that is an average of the times of the three nodes. Obviously this average must be less than or equal to the highest of the three nodes. A weld line or air trap will occur at the point in an element that is filled last, and so has the highest value. (This is not quite as obvious but if you picture different situations you can convince yourself.) Therefore it must occur at one of the nodes.
So the accuracy of weld line location is directly related to the node density. If you have 2 nodes/cm, the best you can determine the location to is 0.50 cm. A similar relation goes with air traps.
Thus it is key to determine what tolerance is acceptable on appearance defects before beginning to mesh.
An alternative, and effective, technique is to initially perform the analysis on a coarse mesh. Once the approximate location of the welds are determined, the density of the mesh in these areas can be increased. If the initial mesh is too coarse, however, this can lead to 'chasing' a weld around the part as the initial estimate was off by so much that a local increase in accuracy moves the weld to another area. As always experience is the best guide when choosing an approach. Many software programs perform this modification automatically. This is known as 'adaptive meshing'.
The overall accuracy of the flow path is comparable to weld line accuracy. In general, the straighter the flow path the larger the elements can be. If a flow path curves around bosses, holes, or other obstacles it is important to increase the node density in these areas so the bends can be accurately modelled. Remember, the analysis is based on the fact that the flow through an element is smooth and linear. If it isn't, it is important to break it up into more elements. A numeric treatment of curved areas, flat elements, and the mid-plane model is given later.
Accuracy of Contours
'Contour parameters' include pressure, temperature, shear, and density. These values are used to determine if the part will fill, appearance defects such as blush and sink, and part warpage. Surprisingly the accuracy of these results is not strongly dependent on the mesh density. Figures 2-4 show graphs of some of these values for a range of node densities. These measurements were taken at the center of a 5 cm square. The values do not smoothly converge on a set curve. Instead they bounce around, seemingly at random. However most of the discrepancies occur at the beginning or the end of the fill. The variation through most of the cycle is minimal, and even rather low mesh densities are fairly accurate.
The 'volatility' of a contour parameter is a measure of its sensitivity to mesh density variation. The most common values can be ranked as follows:
| Most volatile: |
Shear
Shear Stress
Pressure
Density
|
| Least volatile: |
Temperature |
An explanation of the causes of this volatility lie outside the scope of this paper. However some basic rules of thumb can be determined. If you are concerned about the precise values of shear stress in a part (determining blush and optical clarity, for example), a higher mesh density is required than if you are interested in temperature and density variations (for determining warpage).
In addition the volatility of a parameter depends on the relative, rather than absolute, mesh density. For example, a 5 cm square with 5 nodes/cm will show the same accuracy as a 25 cm square with 1 node/cm. In general, for accurate results, we try to have at least 30-50 nodes along the length of a part, regardless of its overall size. This value, obviously, varies with the feature density of the part.
Finally, although its location is rarely needed that precisely, the maximum and minimum values of any of these parameters will be located on a node, for the same reasons given above for weld line formation. So if this information is important the mesh density should be increased in the appropriate area.
Mid-Plane Effects
As discussed earlier, MFA is based on a mid-plane model. That means that the elements themselves are two-dimensional, with a numeric (rather than geometric) thickness, although the elements are, of course, linked in three-dimensions. This has a number of important effects on how the part is meshed.
The first thing to consider is the effect of curves, such as fillets and bosses. The most important thing to maintain in a mid-plane model is the part volume. If one area of the part is under-modeled the whole balance of the fill can be inaccurate, and the analysis rendered meaningless.
Figure 5 shows an example geometry which is quite common. Two 5 mm long walls are joined by an outside radius of 6.25 mm radius. The wall thickness is 2.5 mm and the overall width of the part is 10 mm. The volume of this part is 446.3 mm3, derived from standard volume calculations. The mid-plane of this part is shown in the figure. The overall length is 17.85 mm. When multiplied by the wall thickness and width we arrive at the correct volume value of 446.3.
When this part is meshed, the curve will be broken up into a series of lines. The closer their length is to the correct value of 17.85 mm, the more accurate the analysis will be.
Figure 6 shows a graph of number of nodes on this segment versus the volume error percentage. Note the dramatic difference between modelling the segment with two versus three nodes. Higher values introduce only a small difference, considering the increase in processing time (remember D4!). Thus the rule of thumb is to have at least three nodes on a 90E curve -- one at each end and one at the midpoint. This introduces a volume error of about 3%, which is acceptable for most fillets. On a full round boss we typically use 6 nodes. If the curve is large compared to the part, more nodes will be needed.
Another mid-plane consideration is the placement of ribs. This is a mistake that is typically made by beginners to finite-element modelling. Figure 7 shows a standard rib. The natural tendency is to define the midplane as shown, adding half the wall thickness to the rib height. Note however that this will increase the volume, since the area at the base of the rib will be counted twice. The right way to model the rib would be merely to include the rib height that is above the floor of the part. This gives the correct volume.
The last topic is taper. Tapered walls are endemic to plastic parts, and require special care to be modelled properly. Each element has a constant thickness, so modelling a tapered wall requires stepping the thickness from element to element. Two things are important to consider. First, the thickness of an element should be the average of the start and end thickness, not just the starting thickness. This maintains the total volume of the wall. Second, it is important to add extra elements to a tapered wall if the contour values such as shear and pressure are important. The narrowing or widening of a passage has profound effects on the way plastic flows due to the effects of freezing in the cavity. Extra elements in these areas, especially if they are on the main flow path, go a long way to increasing accuracy.
Conclusion
Attention paid to the design of a finite-element mesh for Mold Filling Analysis will go a long way to increasing accuracy and decreasing processing time. Several rules of thumb have been developed:
- Processing time increases with the fourth power of node density
- Weld line accuracy is directly proportional to node density
- Contour accuracy is loosely coupled to node density
- Small curves (like fillets) should have at least 3 nodes per 90E bend
- Rib height should be height above the floor, not true midplane height
- Extra elements should be added to tapered areas on main flow path
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