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<p>17.6. Substructuring Analysis</p><p>Page 1 of 6</p><p>Theory Reference Chapter 17. Analysis Procedures </p><p>17.6. Substructuring Analysis</p><p>The substructure analysis (ANTYPE,SUBSTR) uses the technique of matrix reduction to reduce the system matrices to a smaller set of DOFs. Matrix reduction is also used by the reduced modal, reduced harmonic and reduced transient analyses. The following substructuring analysis topics are available: Assumptions and Restrictions (within Superelement) Description of Analysis Statics Transients Component Mode Synthesis (CMS)17.6.1. Assumptions and Restrictions (within Superelement)</p><p>1. Any degree of freedom (DOF) may be used. 2. The elements have constant stiffness, damping, and mass effects (e.g., material properties do not change with temperature). 3. Coupled-field elements using load-vector coupling and elements with Lagrange multipliers cannot be used.17.6.2. Description of Analysis</p><p>A superelement (substructure) may be used in any analysis type. It simply represents a collection of elements that are reduced to act as one element. This one (super) element may then be used in the actual analysis (use pass) or be used to generate more superelements (generation or use pass). To reconstruct the detailed solutions (e.g., displacements and stresses) within the superelement, an expansion pass may be done. See the Basic Analysis Guide for loads which are applicable to a substructure analysis.17.6.3. Statics</p><p>Consider the basic form of the static equations ((Equation 171)):(1791)</p><p>{F} includes nodal, pressure, and temperature effects. It does not include {Fnr} (see NewtonRaphson Procedure). The equations may be partitioned into two groups, the master (retained) DOFs, here denoted by the subscript m, and the slave (removed) DOFs, here denoted by the subscript s.</p><p>(1792)</p><p>or expanding:</p><p>mk:@MSITStore:C:Program%20Files%20(x86)ANSYS%20Incv110commonfiles.. 28.10.2010</p><p>17.6. Substructuring Analysis</p><p>Page 2 of 6</p><p>(1793) (1794)</p><p>The master DOFs should include all DOFs of all nodes on surfaces that connect to other parts of the structure. If accelerations are to be used in the use pass or if the use pass will be a transient analysis, master DOFs throughout the rest of the structure should also be used to characterize the distributed mass. The automatic selection of master DOFs is discussed in more detail in Automatic Master DOF Selection, and guidelines for their selection are given in Modal Analysis of the Structural Analysis Guide. Solving (Equation 1794) for {us},(1795)</p><p>Substituting {us} into (Equation 1793)(1796)</p><p>or,(1797)</p><p>where:(1798) (1799) (17100)</p><p>and</p><p>are the superelement coefficient (e.g., stiffness) matrix and load vector, respectively.</p><p>In the preceding development, the load vector for the superelement has been treated as a total load vector. The same derivation may be applied to any number of independent load vectors, which in turn may be individually scaled in the superelement use pass. For example, the analyst may wish to apply thermal, pressure, gravity, and other loading conditions in varying proportions. Expanding the right-hand sides of (Equation 1793) and (Equation 1794) one gets, respectively:</p><p>(17101)</p><p>(17102)</p><p>where:N = number of independent load vectors.</p><p>mk:@MSITStore:C:Program%20Files%20(x86)ANSYS%20Incv110commonfiles.. 28.10.2010</p><p>17.6. Substructuring Analysis</p><p>Page 3 of 6</p><p>Substituting into (Equation 1799):</p><p>(17103)</p><p>To have independently scaled load vectors in the use pass, expand the left-hand side of (Equation 17103)</p><p>(17104)</p><p>Substituting (Equation 17104) into (Equation 17103) :(17105)</p><p>If the load vectors are scaled in the use pass such that:</p><p>(17106)</p><p>where bi is the scaling factor (FACT on the LVSCALE command), then (Equation 1795) becomes:</p><p>(17107)</p><p>(Equation 17107) is used in the expansion pass to obtain the DOF values at the slave DOFs if the backsubstitution method is chosen (SEOPT command). If the resolve method is chosen for expansion pass, then the program will use (Equation 1792) to resolve for {us}. In doing so, the program makes {um} as the internally prescribed displacement boundary conditions since {um} are known in expansion pass. As the program treats DOFs associated with {um} as displacement boundary conditions, the reaction forces by resolve method will be different from that computed at those master DOFs by the backsubstitution method. However, they are all in self-equilibrium satisfying (Equation 1792). The above section Statics is equally applicable at an element level for elements with extra displacement shapes. The master DOFs become the nodal DOFs and the slave DOFs become the nodeless or extra DOFs.17.6.4. Transients</p><p>The general form of the equations for transients is (Equation 175) and (Equation 1729):(17108)</p><p>For substructuring, an equation of the form:</p><p>mk:@MSITStore:C:Program%20Files%20(x86)ANSYS%20Incv110commonfiles.. 28.10.2010</p><p>17.6. Substructuring Analysis</p><p>Page 4 of 6</p><p>(17109)</p><p>is needed. and are computed as they are for the static case ((Equation 1798) and (Equation 1799)). The computation of the reduced mass matrix is done by:</p><p>(17110)</p><p>This simplification was suggested by Guyan(14) because direct partitioning and condensation are not practical (the condensed matrices would be functions of the time derivatives of displacement and very awkward to implement). The damping matrix is handled similarly:</p><p>(17111)</p><p>(Equation 17107) is also used to expand the DOF values to the slave DOFs in the transient case if the backsubstitution method is chosen. If the resolve method is chosen, the program will use (Equation 1792) and make {um} as displacement boundary conditions the same way as the static expansion method does.17.6.5. Component Mode Synthesis (CMS)</p><p>Component mode synthesis is an option used in substructure analysis (accessed with the CMSOPT command). It reduces the system matrices to a smaller set of interface DOFs between substructures and truncated sets of normal mode generalized coordinates (see Craig(344)). For a undamped system, each CMS substructure is defined by a stiffness and a mass matrix. The matrix equation of the motion is:(17112)</p><p>Partitioning the matrix equation into interface and interior DOFs:</p><p>(17113)</p><p>where subscripts m and s refer to:m = master DOFs defined only on interface nodes s = all DOFs that are not master DOFs</p><p>The physical displacement vector, (u), may be represented in terms of component generalized coordinates (see Craig(344)) as in (Equation 17114).(17114)</p><p>mk:@MSITStore:C:Program%20Files%20(x86)ANSYS%20Incv110commonfiles.. 28.10.2010</p><p>17.6. Substructuring Analysis</p><p>Page 5 of 6</p><p>where:y = truncated set of generalized modal coordinates [T] = transformation matrix.</p><p>Fixed-Interface Method For the fixed-interface method (see Craig and Bampton(345)), the transformation matrix has the form:(17115)</p><p>where:[Gsm] = -[Kss]-1[Ksm] = redundant static constraint modes (see Craig and Bampton(345)) s = fixed-interface normal modes (eigenvectors obtained with interface nodes fixed) [I] = identity matrix</p><p>Free-Interface Method For the free-interface method, the transformation matrix has the form:</p><p>(17116)</p><p>where:[sr] = matrix of inertia relief modes</p><p>[m] = matrix of the master dof partition of the free-interface normal modes (eigenvectors obtained with interface dofs free). [s] = matrix of the slave dof partition of the free-interface normal modes.</p><p>Residual Flexibility Free Interface Method For the Residual Flexiblility Free interface (RFFB) method, the transformation matrix has the form:</p><p>(17117)</p><p>where:</p><p>mk:@MSITStore:C:Program%20Files%20(x86)ANSYS%20Incv110commonfiles.. 28.10.2010</p><p>17.6. Substructuring Analysis</p><p>Page 6 of 6</p><p>[Rmm], [Rsm] = submatrices of residual vectors [R]</p><p>(see Residual Vector Method)</p><p>After applying the transformation in (Equation 17114) into the matrix equation of motion (Equation 17112) , the equation of motion in the reduced space is obtained. The reduced stiffness and mass matrices of the CMS substructure will be:(17118) (17119)</p><p>In the reduced system, master DOFs will be used to couple the CMS superelement to other elements and/or CMS superelements.</p><p>Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.</p><p>mk:@MSITStore:C:Program%20Files%20(x86)ANSYS%20Incv110commonfiles.. 28.10.2010</p>
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