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# The Art of Matrix Reduction

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The Art of Matrix ReductionCreating Super-elements in ANSYSME 501 Final ProjectJune 22, 2001Michael Tonks1 Introduction:2 Background2.1 Matrix Reduction2.2 Substructuring Background3 Creating Super-Elements in ANSYS3.1 Create and mesh geometry3.2 Specify Analysis Type3.3 Select Option to Print Stiffness Matrix3.4 Choose Master Nodes and Master DOF’s3.5 Solve Model4 Example Problem5 Conclusions5.1 Future Work5.2 ContributionsThe Art of Matrix ReductionCreating Super-elements in ANSYSME 501 Final ProjectJune 22, 2001Michael TonksAlan Mortensen1 Introduction: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. Background2. How to reduce matrices in Ansys3. Example Problem4. Conclusions2 Background2.1 Matrix ReductionThe 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 constraintsFigure 1 shows a simple plate model with the boundary and internal nodes labeled.boundary nodesinterior nodesFigure 1: Boundary and Interior NodesThe matrix stiffness equation for a flexible part is shown in Eq. 2-1:[ ] }{}{00δKF =Eq.Background-1Once 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 (db) and forces (Fb) are on the top and the interior displacements (di) and forces (Fi) are on the bottom. The matrix is then partitioned as shown in Eq. 2-2. ⎭⎬⎫⎩⎨⎧=⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡ibibiiibbibbFFKKKKδδEq.Background-2Because there are no forces applied to the internal nodes Eq. 2-2 can be simplified:{ } [ ]{ } [ ]{ }ibibbbbKKF δδ +=Eq.Background-3and:{ } [ ]{ } [ ]{ }0=+=iiibibiKKF δδEq.Background-4Eq. 2-4 can be solved for di:[ ][ ]{ }bibiiiKK δδ −=Eq.Background-5The result is substituted into Eq. 2-3 to find a final expression for Fb:{ } [ ]{ } [ ][ ][ ]{ }bibiibibbbbKKKKF δδ1−−= [ ] [ ][ ][ ]( ){ }bibiibibbKKKK δ1−−=Eq.Background-6By comparing Eq. 2-6 to Eq. 2-1 we can find an expression for the reduced stiffness matrix:[ ] [ ] [ ][ ][ ]( )ibiibibbredKKKKK1−−=Eq.Background-72.2 Substructuring BackgroundSuper-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 superelement 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.3 Creating Super-Elements in ANSYSIn 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-inchby ten-inch aluminum plate fixed on one end and with a vertical force on the other. It was meshed using 1-inch shell elements.3.1 Create and mesh geometryThe 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 beretrieved and applied. Figure 3 shows the meshed plate example with boundary conditions and applied forces:Figure 2: Sample model3.2 Specify Analysis TypeThe 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 Type3.3 Select Option to Print Stiffness MatrixThe 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:Select File NameFigure 5: Selecting option to print stiffness matrix3.4 Choose Master Nodes and Master DOF’sIn 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 arechosen, the model can be further reduced by limiting the

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