An introduction to superelements / sub-structuring in the context of finite element analysis

1. Introduction

A finite element model is rarely analyzed only once. Often the model is modified and re-analyzed time and time again. By analyzing only the part of the structure which changes, the user can save significant time.

 

Therefore, it may become necessary, many times, to divide the complete structure that is to be analyzed into functional components so that the set of elements in the component have a well defined structural function and each of these components can be analyzed independent of the other. These elements constituting the set of the functional component are referred to as “superelements”. Superelement analysis can be described as a form of sub-structuring.

That is, a model is divided into superelements [or sub-structures] in such a way that the finite element software could process each superelement [or sub-structure] independent of all other superelements. The processing of each superelement results in a reduced set of matrices [which may be mass, damping, stiffness or load matrix] that represent of the superelement as seen at its connection to adjacent structures. Once all the superelements have been processed, the reduced matrices are assembled in what is known as the residual structure, and the assembly solution if performed.

 Section 2 of this write-up lists the advantages of using superelement analysis and section 3 details the steps to be followed during a superelement analysis. Section 4 provides an insight into static condensation which is a major step in superelement processing that result in obtaining a reduced set of matrices. An example showing a hand calculation through a simple problem detailing the step by step procedure during a superelement analysis will be available soon!

 

 

2. Advantages of uses superelement / substructuring

Efficiency is the primary reason to use superelements. Some of the advantages that result through sub-structuring can be stated as:

Reduced Cost

Instead of solving the entire model each time, superelements offer the advantage of incremental processing. On restarts this advantage is magnified by the need to process only the parts of the structure directly affected by the change. This means that if the user thinks ahead when defining superelements, it is possible to achieve performance improvements on the order of anywhere from 2 to 30 times faster than non-superelement methods (or more).

 

 

Large Problem Capabilities

 

All computers have hardware limits. When the size of a model becomes too large to be processed on a

computer without using superelements, the user can use multiple computer resources to process each

superelement or instead process one superelement at a time on a single computer resource. The reduced matrices for each superelement can be stored on separate drives and brought together for the residual solution.

 

Parallel processing

Since each set of superelements can be analyzed independent of the other, this allows for parallel processing.

 

 

3. Steps in a superelement analysis

Overview of the steps in superelement analysis

The steps during a superelement analysis can be classified as;

I.      Firstly, the input is partitioned into [a separate set for processing each superelement. This is accomplished in most software’s based on user instructions] sets of superelements

Degrees of freedom of superelement:

 

The degrees of freedom of a superelement can be portioned into sets of: Exterior and Interior degrees of freedom

Exterior DOFs: Exterior degrees of freedom are those which are retained for further analysis. These are the boundary degrees of freedom where the superelement connects to the other superelement.

Interior DOFs: Interior degrees of freedom are those that are condensed out during superelement processing i.e. these are condensed out of the matrices during the reduction process.

II.    Once the superelements are partitioned, following computations are carried out in the software;

·         Superelement processing

·         Residual structure processing

Each of the above computations is detailed below.

Superelement processing:

The computations involved in the processing of each superelement are listed below;

1. Once the input is portioned into sets of superelements; structural matrices are formulated for each superelement.
2. The structural matrices, then, go through a reduction process until the only remaining degrees of freedom are the exterior degrees of freedom.
3. The boundary / exterior degrees of freedom are then obtained through the solution process.

Residual structure processing

 

Residual structure processing comprises of the following computations;

• Phase 1 processing.
• Phase 2 processing.
• Phase 3 processing.

Phase 1 processing;

The phase 1 matrices are generated for the residual structure, based on any elements or loads remaining, and then the reduced matrices from the superelements are added at the appropriate dof.

Phase 2 processing;

After the combined (or assembled) matrix for the residual is formed, and constraints applicable to the remaining DOFs are applied and the residual structure problem is solved as part of phase 2 operations.

 

 Phase 3 Processing;

Phase 3 represents the data recovery. The internal degrees of freedom which were condensed out are now recovered. The process may be carried out directly through a sequence of matrix operations, or equation by equation as a back-substitution process.

4. Static condensation

Static condensation is one of the major steps during superelement analysis. This process is carried out during superelement processing wherein the matrices generated for each superelement is reduced so that to eliminate the internal degree(s) of freedoms o that the only dof(s) remaining are the external / boundary degrees of freedom.

Static condensation may be presented in terms of explicit matrix operations, as shown in the next subsection. A more practical technique based on symmetric Gauss elimination is discussed later.

  Static Condensation by Explicit Matrix Operations

 Consider the following system of matrix equation;

 where subvectors ub and ui collect boundary and interior degrees of freedom, respectively. Take second matrix equation:

If Kii is nonsingular we can solve for the interior freedoms:

 

Replacing the matrix equation 1 for the value of ui as obtained above will result in the condensed stiffness equations:

 

are called the condensed stiffness matrix and force vector, respectively, of the substructure.

From this point onward, the condensed superelement may be viewed, from the standpoint of further operations, as an individual element whose element stiffness matrix and nodal force vector are_Kbb and ˜fb, respectively.

Condensation by Symmetric Gauss Elimination

In the computer implementation of the the static condensation process, calculations are not carried out as outlined above. There are two major differences. The equations of the substructure are not actually rearranged, and the explicit calculation of the inverse of Kii is avoided. The procedure may be in fact coded as a variant of symmetric Gauss elimination. To convey the flavor of this technique, consider the following stiffness equations of a superelement:

Suppose that the last two displacement freedoms: u3 and u4, are classified as interior and are to be statically condensed out. To eliminate u4, perform symmetric Gauss elimination of the fourth row and column:

 

Repeat the process for the third row and column to eliminate u3:

 

 

These are the condensed stiffness equations.