Submodelling by Substructuring – Static Condensation (Guyan Reduction) using Superelements

structure will be analyzed separately. All the applied loads must eventually find their way to the constraints. For instance, in large structural engineering projects where a multiple story building structure is to be analyzed, the higher stories may be analyzed separately from that below. The higher story may incorporate a shear wall as a structural load-resisting element being rested upon say a beam in the story below. It will be incorrect to restrain rigidly all the supports of the higher story, i.e. the base of the shear wall must not be all supported on rigid supports (with the intention of applying the reaction forces onto the beam of the story below in the opposite direction). If this was done, the shear wall would not arch, but all the load would simply transfer straight down into the rigid supports. Hence, instead, the relative support stiffness under the shear wall must be incorporated, by say modelling the beam as well. The shear wall will then arch and the load distribution will be far more realistic with more of the load transferring directly to the supports of the shear wall. The reactions on these flexible supports (here the forces on the beam and the ends of the shear wall) can then be transferred onto the model containing the story below. If on the other hand, had all the load-resisting elements been columns supported on equal stiffness foundations, the supports of the upper story could well have been rigid and the reaction applied onto the story below. To model the boundaries of the local model with appropriate stiffness is substructuring. Here we have performed manual substructuring as the stiffness of the remaining part of the structure was approximated manually. To sum up, when manual substructuring is undertaken, the supports of a substructure should be modeled to incorporate their relative stiffness such as to obtain the correct distribution of reactions, which in turn will be applied (of course in the opposite direction – Newton’s Third Law) onto the model containing the adjacent substructure. This of course is an approximate procedure as the substructuring is performed manually. Automatic substructuring using superelements to provide the stiffness at the boundaries of the local model is on the other hand exact.

2 MATINEZ, Angel. Submodelling Techniques for Static Analysis. MSC. Software First South European Technology

Modelling the stiffness of the adjacent (to the substructure) elements is fundamental to substructure analysis. The static condensation technique does just that by reducing the equations to only involve the DOFs of the substructure and its interface, but in doing so incorporates the stiffness and internal forces of the remaining part of the whole model onto the interface DOFs. The most important characteristic of static condensation is that the boundary conditions at the interface of the substructure depends only on the stiffness and the loading of the remaining part of the whole model and is totally independent of the stiffness and loading of the submodel. Hence any modifications to the loading and stiffness of the submodel are perfectly acceptable and the answers will be exact. Thus, for static analysis, the static condensation technique produces the exact same results as that obtained from the full analysis.

Consider a finite element model of a structure that has been properly constrained and loaded. Total number of d.o.f.’s is denoted as g, normally equal to 6x(number of nodes). This g set can be subdivided in c (constrained) and f (free).

After applying constraint conditions, static analysis equations become in matrix form as follows:

[ ]

K_ff ⋅

{ } { }

δ_f = F_f (1)

Free (f) d.o.f.’s are going to be subdivided in two sets: m (masters) and s (slaves). Masters d.o.f.’s are the ones to be kept in the reduction process, and slave set will be reduced. This partition gives expression (1) the following

that results in these two equations:

[ ] { } [ ] { } { }

expression (4) in first equation of (3), and reordering terms, the following system of equations is obtained:

[ ] [ ][ ] [ ]

(

K_mm − K_ms K_ss ⁻¹K_sm

)

⋅

^{{ } { }}

δ_m = F_m −

[

K_ms

][ ]

K_ss ⁻¹

{ }

F_s (5) that can be expressed in the compact form:

[ ] ^{{ }}

m

{ }

m^*s

[ ] ^{[ ] [ ][ ] [ ]}

Expression (4) can be used then, to obtain displacements at slave d.o.f.’s (s).

Static condensation applied to linear static analysis gives an exact solution, equal to the solution of the global equation system (2). Consider a finite element model of a structure that is subdivided in two parts or substructures A and B. Finite element model can be subdivided in the following d.o.f.’s sets:

• a: internal d.o.f.’s of substructure A.

• b: internal d.o.f.’s of substructure B.

• i: interface d.o.f.’s between substructures A and B.

Stiffness matrices of each substructure take the form:



Assembling the stiffness matrices of each substructure results in the global system of equations for static analysis of the whole structure:

Static reduction is going to be applied to set (b) as slave d.o.f.’s and (a+i) as masters. Stiffness matrix and load vector are partitioned in the following way:



Recovering the expressions (7) for the calculation of reduced stiffness matrix and reduced load vector:

[ ] [ ] [ ][ ] [ ]

[ ] ^{[ ] [ ]}

Decomposing each matrix as sum of two terms:

[ ]

The reduced equations systems to d.o.f.’s set (a+i) has the following form:

( )

i^*b

Reduced stiffness matrix is composed of the stiffness matrix of substructure A and the reduced stiffness matrix of substructure B to the interface d.o.f. (set i) expanded (completed with zeros) to (a+i) size. In the same way, reduced load vector is the applied load to internal d.o.f.’s of substructure A (set a) plus the reduced load vector of substructure B to set i, and expanded, again, to (a+i) size. In other words, to perform a local static analysis of substructure A by static condensation technique, the following items are needed:

• Finite element model of substructure A.

• The effect of substructure B over A at interface d.o.f.’s:

• Reduced stiffness matrix of substructure B to interface d.o.f.’s (set i), and

• Reduced load vector of substructure B to interface d.o.f.’s (set i)

i

The most important characteristic of submodelling analysis by substructuring using static condensation is that the boundary conditions at the interface (reduced model of substructure B) only depends on the stiffness of substructure B, and is totally independent of the stiffness of substructure A. Then, any modification of substructure A will be treated correctly, and the results will be the same than the ones obtained with the complete model.

Static Condensation Submodelling Method for Substructure A

Let us discuss in terms of a-sets and o-sets. Displacement vector sets in NASTRAN define the equations of motion and are partitioned in the following ways.

(i) g-set is the unconstrained set of structural equations

(ii) the g-set is partitioned into the m-set pertaining to the dependent set of DOFs and the n-set pertaining to the independent set of DOFs

g-set − m-set = n-set

(iii) the n-set is then partitioned into s-set pertaining to the constrained DOFs and the f-set pertaining to the unconstrained DOFs

n-set − s-set = f-set

(iv) the f-set is partitioned into the o-set pertaining to the DOFs eliminated by static condensation and the a-set pertaining to the a-set (analysis set) pertaining to the DOFs not eliminated by static condensation

f-set − o-set = a-set

In the static condensation process you select a set of dynamic DOFs called the a-set; these are the retained DOFs that form the analysis set. The complementary set is called the o-set and is the set of DOFs that are omitted from the dynamic analysis through a reduction process. The process distributes the o-set mass, stiffness, and damping to the a-set DOFs by using a transformation that is based on a partition of the stiffness matrix (hence the term static condensation). This reduction process is exact only for static analysis, but leads to approximations in the dynamic response. The a-set DOFs are defined by the ASET or ASET1 Bulk Data entries, and the o-set DOFs are defined by the OMIT or OMIT1 Bulk Data entries.

If you start with the stiffness equation in terms of the set of the unconstrained (free) structural coordinates, you have

Partitioning the free degrees of freedom into two subsets of the f-set, you obtain

Rewriting

On expansion of top equation

On expansion of bottom equation

First, the f-set is reduced to the a-set using Guyan reduction. Second, the analysis set {u_a} is solved for. Third, the omitted set {uo} is solved for. In static analysis, the results using static condensation are numerically exact. The partitioned solution merely changes the order of the operations of the unpartitioned solution.Guyan reduction has special applications in dynamic analysis. In dynamic analysis, the reduction is approximate; the term {uOO} is ignored in this case. The reduction is based solely on static transformation BUT is EXACT provided that no loads are applied to the o-set degrees of freedom.

To utilize static condensation, you can choose either the ASET/ASET1 or the OMIT/OMIT1 entries. With these entries you should specify only the a-set (with ASET/ASET1 entries) or o-set (with OMIT/OMIT1 entries) degrees of freedom. The unspecified remaining f-set DOFs are automatically placed in the complementary set. However, if you specify both the a-set and o-set DOFs, then the unspecified remaining f-set DOFs are automatically placed in the o-set. The same DOF cannot be specified on both the a-set and o-set; otherwise, the job fails with UFM 2101A.

Since the reduction process is performed on an individual degree of freedom, it is possible to have some of the degrees of freedom at a grid point in the a-set and other degrees of freedom at a grid point in one of the other mutually exclusive sets. No additional user input is required.

$ BULK DATA

ASET G1 C1 G2 C2 G3 C3 G4 C4

ASET1 C G1 G2 G3 G4 G5 G6 G7

G8 G9 G10 G11 …etc…

ASET1 C G1 THRU Gi

$ BULK DATA

OMIT G1 C1 G2 C2 G3 C3 G4 C4

OMIT1 C G1 G2 G3 G4 G5 G6 G7

G8 G9 G10 G11 …etc…

OMIT1 C G1 THRU Gi

The choice of whether to use the ASET/ASET1 or OMIT/OMIT1 is really a matter of convenience.

b

In document Finite Element Modelling Techniques in MSC NASTRAN and LS DYNA (Page 27-33)