Seven Techniques For Finding FEA Errors

Transcription

1 Seven Techniques For Finding FEA Errors by Hanson Chang, Engineering Manager, MSC.Software Corporation Design engineers today routinely perform preliminary first-pass finite element analysis (FEA) on new products or configurations to shorten design cycles. These part-time FEA users are responsible for delivering significant time savings to their companies while improving the product designs. By better understanding the requirements for refining the mesh, design engineers can further reduce the design cycle time by reducing the time spent performing FEA simulations. Mesh refinement is a major and time-consuming step in FEA. Today's FEA packages are capable of predicting a good starting mesh size. After running an initial simulation, the user is faced with a tough decision: Will refining the mesh deliver more accurate results or just use up more time? A mesh that is too coarse (elements too large) can produce inaccurate results, leading to poor product designs. A mesh that is too fine can waste valuable computer resources and lengthen the design cycle. We have identified seven techniques to help the user determine the optimal mesh size. This is a collection of tricks of the trade both new and seasoned FEA users will find useful. For ease of memorization, these seven techniques have been alphabetized from A to G as shown in Fig. 1. Average stress Compare the Average stress fringe plot with the Un-averaged stress fringe plot. The two should be in close agreement for an acceptable mesh. Broken color bands Discretization error causes the stress fringe color bands to appear discontinuous (broken) going from one element to another. Keep this to a minimum. Convergence Plot the maximum model displacement and stress vs. iteration as you refine the mesh. The changes in these two quantities from iteration to iteration will approach zero as you approach the ideal mesh density. Discontinuity Element corner stresses meeting at a common node will not have the same value if the mesh is too coarse. The difference (discontinuity) between these stresses at a node is a measure of discretization error. Keep this to a minimum. Energy norm Formula Gradient Element corner stresses meeting at a common node will not have the same value if the mesh is too coarse. The work done by this stress discontinuity is called the Energy Norm. The total energy norm divided by the total strain energy is a measure of discretization error. This ratio should be kept small. Use mesh density formulas (rules of thumb) to help you make sound meshing decisions and minimize discretization error. The number of color bands across an element is a measure of stress gradient within that element. This number should be kept low. Figure 1. Seven techniques for finding errors in design analysis software.

2 When a mesh is too coarse, it causes discretization error (Fig. 2), which is characterized by element corner stresses that disagree at the shared node (stress jumps). Discretization error represents only one type of FEA error. Modeling, user, and round-off error are other types of FEA errors. The seven techniques presented in the following paragraphs can help the design engineer make mesh refinement decisions by specifically addressing discretization errors. Figure 2. Discretization errors are characterized by "stress jumps" across elements Averaged vs. Un-Averaged Stress An averaged stress fringe plot is created by taking an average of the element corner stresses at each shared node. The averaged stress values are then assigned to the nodes. These average nodal stress values are used to create a color fringe plot for the entire model. On the other hand, an un-averaged fringe plot, plots stress as is, without any averaging. The averaged and unaveraged stress fringe plots are compared. For an acceptable mesh, the two plots should be in close agreement. In the example in Fig. 3, the un-averaged fringe plot on the left shows a maximum stress value of 13,700 psi. The averaged fringe plot on the right shows a maximum stress value of 8,290 psi. The large difference between the two plots indicates that further mesh refinement is required.

3 Figure 3. CAD-embedded SimDesigner generated the two stress fringe plots - the un-averaged stress fringe plot on the left and the averaged stress fringe plot on the right. Broken Color Bands Discretization error causes the stress color bands to appear broken (discontinuous) between elements. This behavior should be kept to a minimum for an acceptable mesh. The typical default visualization settings in some CAD programs may actually mask this behavior. In order to visualize the broken color bands, use the following settings: Element edge display on Color smoothing off Stress averaging off Figure 4. Broken color bands can be clearly seen in the image on the right, where the bands don't line up. Convergence Plot A convergence study is a common method for determining if further mesh refinement is required. In this method, key analysis results are tracked as the mesh is repeatedly refined.

4 Results from each iteration are compared to the previous iteration. Convergence is reached when the percent change in results between two iterations becomes acceptably small. Typically the maximum displacement and maximum stress in the model are tracked in a convergence study. Fig. 5 shows a typical stress convergence plot. Figure 5. Maximum Von Mises stress is plotted against mesh refinement iteration. The curve approaches a horizontal line as the mesh is refined. Discontinuity in Stress A clearly visible stress jump or discontinuity will be visible between elements (Fig. 2) if too few elements are used. In practice, several different methods are used to compute discontinuity in stress: 1. Subtract the lowest element stress value from the highest element stress value at a shared node. This stress difference is taken as the stress discontinuity error. 2. Compute the average stress at a node and calculate the difference between neighboring element stresses and this average nodal stress. The largest stress difference is taken as the stress discontinuity error at this node. 3. This method is similar to the second method above, except the values are massaged by additional root mean square procedures to account for the statistical nature of this error measurement. Users of MD Nastran will recognize this method as Grid Point Stress Discontinuity. 4. MD Nastran also calculates a slightly different element-based stress discontinuity called the Element Stress Discontinuity. Fig. 6 shows a fringe plot of this discontinuity error in SimDesigner.

5 Figure 6. Element Stress Discontinuity Plot generated by SimDesigner. Energy Norm In this method, the stress discontinuities are integrated inside each element as an energy norm, i.e. the work done by the stress jumps. Summing up all the energy norms over the entire finite element model provides the total energy norm due to stress error. To get a sense of how much discretization error is in a model, divide the total energy norm by the total strain energy. This ratio is commonly known as the Global Estimated Error Rate. This method of measuring error is commonly known as the Z2 method named after its authors Messrs. Zienkiewicz and Zhu. This error rate is usually presented as a percentage number and should be kept low for an acceptable mesh. Formula for Mesh Density Mesh density formulas (rule of thumb) is an experience-based technique. Many companies, departments, or even projects have their own set of rules of thumb for determining mesh density. For demonstration purposes only, several examples are listed below: For a flange in bending, the number of elements through the thickness should be at least 3 to 4 elements. A hole should have at least 12 elements around its circumference. The number of elements around a cylindrical shell should be at least 72 elements (every 5 degrees). FEA users should consult company manuals, senior analysts or fellow design analysts for such rules of thumb. Additionally, simple test cases can be run to develop a set of rules of thumb for determining an acceptable mesh. Caution: rules of thumb tend to be subjective and applicable only to specific types of structures and load paths.

6 Gradient within an Element The number of color bands within an element is a measure of how fast stress changes within that element. Too many color bands within an element indicates the element is too big and the mesh should be refined. Figure 7. Five color bands within one element indicate a very high stress gradient within this element. For example, in figure 7 one element contains five color bands. This indicates a rapidly changing stress field (high stress gradient). On the right, the overall stress range includes ten color bands. This means within one element the stress varies from 100% to 50% of maximum model stress, which indicates mesh refinement is necessary. Conclusion Seven tricks of the trade were presented to help the design engineer determine discretization error and when and where to refine the mesh. The skills for deciding when to refine and when not to refine are developed through training, practice and lots of patience. It is well worth the effort, because it will enhance the design engineer's skill set, reduce design cycle time, improve product performance and ultimately maximize the company's return on investment.