The Art of Matrix Reduction

The Art of Matrix Reduction

Creating Super-elements in ANSYS

ME 501 Final Project

June 22, 2001

Michael Tonks

Alan Mortensen

1Introduction:

Finite Element Analysis (FEA) is an important tool in engineering. FEA programs such as ANSYS give an engineer the ability to analyze models with complex geometry where previously it was not possible. One disadvantage with FEA is that as the model size goes up, the analysis time also goes up. Especially in industries that design complicated assemblies, such as the aerospace industry, the models used can get so large that FEA analysis can be very calculation intensive.

A method that has been developed to deal with this problem is matrix reduction. By eliminating all nodes in the FEA stiffness matrix that are not important to the analysis, a super element is created that much smaller and easier to use, but still includes all the important information. In this project, our main goal was find how to use matrix reduction in ANSYS. A secondary goal was to find a way to export the reduced stiffness matrix in ANSYS.

In this paper we will explain the process in ANSYS to create a super element or reduced matrix, and to export that matrix. To accomplish this, the paper is divided into five sections:

1. Background
2. How to reduce matrices in Ansys
3. Example Problem
4. Conclusions

2Background

2.1Matrix Reduction

The first step in matrix reduction is to choose all the nodes that contain important information for the analysis. The important nodes are called boundary nodes while the unimportant nodes are internal nodes. Reasons for choosing the boundary nodes are:

• The nodes are points where other parts will be connected
• The nodes are locations for applied forces
• The nodes are locations for boundary node constraints

Figure 1 shows a simple plate model with the boundary and internal nodes labeled.

Figure 1: Boundary and Interior Nodes

The matrix stiffness equation for a flexible part is shown in Eq. 2-1:

Eq. 21

Once the boundary nodes have been chosen, the stiffness equation is organized such that all the boundary nodes (Kbb) are in the top right corner and all the interior nodes (Kii) are in the bottom left corner. The Force and displacement vectors are also organized and divided such that boundary displacements (b) and forces (Fb) are on the top and the interior displacements (i) and forces (Fi) are on the bottom. The matrix is then partitioned as shown in Eq. 2-2.

Eq. 22

Because there are no forces applied to the internal nodes Eq. 2-2 can be simplified:

Eq. 23

and:

Eq. 24

Eq. 2-4 can be solved for i:

Eq. 25

The result is substituted into Eq. 2-3 to find a final expression for Fb:

Eq. 26

By comparing Eq. 2-6 to Eq. 2-1 we can find an expression for the reduced stiffness matrix:

Eq. 27

2.2Substructuring Background

Super-elements can be used in FEA analysis using a system called substructuring. In substructuring a large model is divided into several sections. Each section is analyzed separately to produce a stiffness matrix. Each stiffness matrix is reduced to create a super element matrix. The separate super-elements are then reassembled and the analysis is performed. The simplified model can find any information needed at all the boundary nodes, but no information at the interior nodes can be found. To find the forces and stresses inside the super-element, subsequent analysis pass can be performed on the detailed substructure model with displacement boundary conditions taken from the complete super-element solution. This additional pass is called an expansion analysis pass. By using substructuring, a very large model with thousands of DOF’s can be reduced to make the analysis simpler.

3Creating Super-Elements in ANSYS

In this section, we will show the step-by-step process to create a super-element matrix in ANSYS. We will also show how to export the super-element matrix. To better illustrate the process we will show a simple plate example for each step. The example is a ten-inch by ten-inch aluminum plate fixed on one end and with a vertical force on the other. It was meshed using 1-inch shell elements.

3.1Create and mesh geometry

The first step is to create the geometry to be used. In can be imported from another program or created in ANSYS. The geometry is then meshed using any element desired. Any boundary conditions and applied forces can be applied. In the plate example, the model was meshed using the shell 63 element in ANSYS.

Loads and boundary conditions are also applied in this step. It is important to note that the loads are not applied to the super-element matrix; the load vector is simply saved. When the super-element is used in a substructure analysis, the various load vectors can be retrieved and applied. Figure 3 shows the meshed plate example with boundary conditions and applied forces:

Figure 2: Sample model

3.2Specify Analysis Type

The next step is to tell ANSYS that you want to do a substructuring analysis. The menu path is shown in Figure 4 below:

Figure 4: Choosing Analysis Type

3.3Select Option to Print Stiffness Matrix

The default for ANSYS is to not print the Stiffness Matrix. If the stiffness matrix is desired, ANSYS must be told to print it. Figure 5 shows the menu path required:

Figure 5: Selecting option to print stiffness matrix

3.4Choose Master Nodes and Master DOF’s

In order to create the super-element, the boundary nodes must be chosen. In ANSYS, boundary nodes are called master nodes. Following the menu path below, define masters is chosen and each master node can be chosen graphically. Once the masters are chosen, the model can be further reduced by limiting the DOF’s that will be included.

Figure 6: Choosing Master Nodes and Master DOF’s

3.5Solve Model

After the correct settings have been selected, the model is solved. In the solution menu, solve current LS is selected. ANSYS then solves the model and automatically saves the super-element matrix in a file entitled [File Name].sub. This super-element matrix is ready for use in a substructure analysis.

To export the stiffness matrix, the following menu path is followed on the top pull down menu: List – Other – Superelem – Data.

The data box is shown below:

Figure 7: Listing super-element matrix

This brings up a list file that contains a lot of unneeded information but at the end contains the stiffness matrix. Extracting the condensed stiffness matrix from the output file for the simple plate example above, we get the following matrix in Excel and Matlab:

Figure 3:

Here we can see that the eleven masters defined in the model result in a symmetric 11x11 matrix that is fully populated while the original matrix was a 726x726 sparsely populated matrix. The super-element matrix accurately describes the stiffness matrix for the entire plate.

The format of the output file from ANSYS is difficult to use. In this example the matrix was small enough that the data could be entered into Excel by hand, but with a larger super-element matrix this would be very difficult. A possible solution to this difficulty would be to write a Matlab or C program to rewrite the stiffness matrix output into a more useable form.

4Example Problem

Picture of part / Pro/E model of part / ANSYS import of part through IGES

With the basics of creating a simple super-element mastered, we are ready to try a larger problem. The front leading edge of a flap on a Boeing 737 is our target part and our goal is to create a reduced stiffness matrix for it. As seen in figure 9 below, to test our methods we created a rough CAD model of the leading edge part, and imported it into ANSYS using an IGES copy.

Figure 4: Sample Problem

Our next step was to mesh the area using shell elements, and apply masters at the edge in question. We assumed displacements in a single plane, allowing us to further reduce the stiffness matrix complexity by using only three degrees of freedom. The meshed part is shown in figure 10. Solving for the load system in ANSYS, we ended up with a symmetric matrix with 72 rows and 72 columns in the format shown in figure 11. Though a bit unwieldy in format, ANSYS has proven to easily output the reduce stiffness matrix for our leading edge part.

Figure 10: Meshed Part Figure 11: Stiffness Matrix in ANSYS

To further increase the feasibility of using ANSYS to create and output a super-element matrix for use in other data programs, we created an Ansi-C program that converts the ANSYS output file into an ASCII file that contains only the matrix values. This matrix can easily be read into Excel, Matlab, and other programs. With use of this file, a super-element matrix can be created in ANSYS, converted, and inputted into a data analysis program quickly and easily.

5Conclusions

As we stated in the introduction, super-elements are an important tool for analyzing complicated models. We found that super-elements can be generated in ANSYS. This process is simple and can easily be taught.

For some other applications, such as flexible assembly tolerance analysis, it is necessary to use the stiffness matrix. Many commercial FEA programs cannot export the matrix, but we have found that ANSYS can. As we have shown above, the super-element of the stiffness matrix can be exported. If the complete stiffness matrix is needed, every node of the model can be chosen as a master and the entire stiffness matrix can be exported.

5.1Future Work

While working on this project, we found several areas that we feel need further research. These aspects are listed below:

• Investigate how to add specified displacements to the super-element model
• Investigate how to combine the super-elements together for use in a complete substructure analysis
• Find how to use the expansion pass to expand the super-element to find the internal stresses of the part

5.2Contributions

By doing this project we have found several things that are important contributions to the 501 class. We have found how to create a super-element in ANSYS and shown a step-by-step list of how it is done. We have also found how to export the stiffness matrix in ANSYS and shown how that is done. Both these points can be an important tool for FEA analysis, and exporting the stiffness matrix will help further research in flexible assembly tolerance analysis.