A superelement 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 Substructuring Analysis Guide for loads which are applicable to a substructure analysis.

Although a superelement may be used in any type of structural analysis, only static and transient analyses are considered in the following sections to derive the reduced forms of the structural matrices (stiffness , mass , damping , load vector ).

For first-order non-structural analyses (thermal, electric, magnetic, diffusion) and second-order non-structural analyses (fluid), matrices associated with zero order terms (), first order terms (), and second order terms () are reduced with the same logic. For coupled-field analyses, only the reduction of is possible.

Consider the basic form of the static equations (Equation 15–1):

(15–108)

includes nodal, pressure, and temperature effects. It does not include (see Newton-Raphson Procedure). The equations may be partitioned into two groups, the master (retained) DOFs, here denoted by the subscript , and the slave (removed) DOFs, here denoted by the subscript .

(15–109)

or expanding:

(15–110)

(15–111)

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. Solving Equation 15–111 for ,

(15–112)

Substituting into Equation 15–110

(15–113)

or,

(15–114)

where:

(15–115)

(15–116)

(15–117)

and are the superelement coefficient (e.g., stiffness) matrix and load vector, respectively.

This development is equivalent to reduce the nodal displacement vector as:

(15–118)

is a transformation matrix with the form:

(15–119)

where:

= identity matrix

The vectors constituting are also called static constraint modes (see Craig and Bampton [346]).

It follows that:

(15–120)

(15–121)

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 15–110 and Equation 15–111 one gets, respectively:

(15–122)

(15–123)

where:

= number of independent load vectors.

Substituting into Equation 15–116:

(15–124)

To have independently scaled load vectors in the use pass, expand the left-hand side of Equation 15–124

(15–125)

Substituting Equation 15–125 into Equation 15–124 :

(15–126)

If the load vectors are scaled in the use pass such that:

(15–127)

where is the scaling factor ( VAL1 on the SFE,,,SELV command), then Equation 15–112 becomes:

(15–128)

Equation 15–128 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 15–109 to resolve for . In doing so, the program makes as the internally prescribed displacement boundary conditions since are known in expansion pass. As the program treats DOFs associated with 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 15–109.

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.

The general form of the equations for transients is Equation 15–5 and Equation 15–38:

(15–129)

For substructuring, an equation of the form:

(15–130)

is needed. and are computed as they are for the static case (Equation 15–115 and Equation 15–116 or Equation 15–120 and Equation 15–121). As suggested by Guyan ([14]), the mass matrix is also reduced through the transformation matrix given in Equation 15–119. The reduced mass matrix is calculated by:

(15–131)

where:

The damping matrix is handled similarly:

(15–132)

Equation 15–128 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 15–109 and make {u_m} as displacement boundary conditions the same way as the static expansion method does.

Component mode synthesis is an option used in substructure analysis (accessed with the CMSOPT command) when DOFs are structural. It reduces the system matrices to a smaller set of interface DOFs between substructures (components) and truncated sets of normal mode generalized coordinates (see Craig [345]).

For a damped system, each CMS substructure is defined by a stiffness matrix, a mass matrix, and a damping matrix. The matrix equation of motion is:

(15–133)

Partitioning the matrix equation into interface and interior DOFs:

(15–134)

where subscripts m and s refer to:

= master DOFs defined only on interface nodes

= all DOFs that are not master DOFs

Similarly to Equation 15–118, the nodal displacement vector, , may be represented in terms of master DOFs completed by component generalized coordinates (see Craig [345]) as in Equation 15–135.

(15–135)

where:

= truncated set of generalized modal coordinates

Fixed-Interface Method

For the fixed-interface method, also commonly referred to as the Craig-Bampton method(see Craig and Bampton [346]), the transformation matrix has the form:

(15–136)

where:

= fixed-interface normal modes (eigenvectors obtained with interface nodes fixed)

= null matrix

Free-Interface Method

For the free-interface method, also commonly referred to as the Herting method (see Hintz [410], Herting [413]), the transformation matrix has the form:

(15–137)

where:

= matrix of the master DOF partition of the free-interface normal modes (eigenvectors obtained with interface DOFs free)

= matrix of the slave DOF partition of the free-interface normal modes

= matrix of inertia relief modes

is included only if rigid body modes are present (see CMSOPT„„,FBDDEF). Any rigid body modes present are not included in

where:

= matrix of the master DOF partition of the rigid body modes

Residual-Flexible Free-Interface Method

For the Residual-Flexible Free-Interface (RFFB) method, also commonly referred to as the Martinez method (see Martinez et al. [411]), the transformation matrix has the form:

(15–138)

where:

= submatrices of the following residual vectors (see Residual Vector Method):

Any rigid body modes present are included in .

The residual vectors are also called residual attachment modes. Partitioning the residual flexibility matrix given in Equation 14–127, the residual attachment modes are defined as:

(15–139)

Transformation

After applying the transformation in Equation 15–135 into the matrix equation of motion Equation 15–133 , the equation of motion in the reduced space is obtained. The reduced stiffness, mass, and damping matrices and the reduced load vector of the CMS substructure will be:

(15–140)

(15–141)

(15–142)

(15–143)

In the reduced system, master DOFs will be used to couple the CMS superelement to other elements and/or CMS superelements.

For the fixed-interface method, if the fixed-interface normal modes are mass normalized, the reduced stiffness, mass, and damping matrices, and the reduced load vector have the final form:

(15–144)

(15–145)

(15–146)

(15–147)

Where:

= a diagonal matrix containing the eigenvalues of retained fixed-interface normal modes.

Recovery of the Slave Displacements

The displacements at slave or interior DOFs are recovered from the lower part of Equation 15–135. For the fixed-interface method, they are:

(15–148)

When reduced load vectors are scaled in the use pass, pseudo-static correction terms are added. Equation 15–148 becomes:

(15–149)

with:

where:

= residual flexibility matrix of the slave DOFs partition of the stiffness matrix

= slave DOFs partition of the reduced stiffness matrix

Equation 15–148 and Equation 15–149 are also valid for the free-interface method except that should be replaced with .

For the RFFB method, is recovered from Equation 15–135 and Equation 15–138 as:

(15–150)

Default Expansion Pass

The default expansion pass is based on the calculation of element results from the complete displacement solution.

Expansion Pass Preceded by Element Results Calculation in the Generation Pass

For the fixed-interface method, if the element results of the component nodes were calculated during the generation pass (ELCALC = YES on CMSOPT), the displacements are recovered according to Equation 15–148, even if reduced load vectors were scaled in the use pass.

However, the loss of the first static correction term can be compensated by adding a residual vector assocated with the first load vector (RESVEC,ON in the first solve of the generation pass).

The element results of the static constraint modes and the fixed-interface normal modes are combined in the expansion pass in the same way as the displacements.

For example, for the stresses, if the stress-displacement matrix of element is noted as , the stress vector of element is:

(15–151)

Partitioning the system into interface and interior DOFs:

(15–152)

For the fixed interface method, introducing Equation 15–135 and Equation 15–136 into Equation 15–152 gives:

(15–153)

The matrix of static constraint mode stresses of element :

(15–154)

and the matrix of fixed-interface normal mode stresses of element :

(15–155)

are calculated and stored during the generation pass. Then in the expansion pass, the stress vector of element is calculated as:

(15–156)

Modal Damping Ratios

Modal damping ratios can be directly included in the reduced damping matrix when the fixed-interface method is used (ANTYPE,SUBSTR with CMSOPT,FIX and SEOPT,,3).The following terms are added to the diagonal of in Equation 15–146:

where:

= modal damping ratio of the ith fixed-interface normal mode (input with MDAMP)

= circular natural frequency of the ith fixed-interface normal mode

Unsymmetric Matrices

The reduction of a skew-symmetric damping matrix due to the effect of gyroscopic damping (CORIOLIS,ON,,,ON) is supported. The damping matrix must contain the gyroscopic effect only. For all three methods, the modes used in the transformation basis are those of the conservative system. The resulting reduced damping matrix is also skew-symmetric.