Introduction

 As discussed in the write-up on "An Introduction to superelements / sub-structuring (http://ajaytaneja-superelementsfiniteelem.blogspot.in/), it may become necessary at times to divide a large complex model that is to be analyzed, into a series of functional components so that the set of elements in the component have a well defined structural function and each of these set of elements (called as “superelement”) in the component could be analyzed independent of the other.

 A related, but not identical, technique is multiscale analysis. The whole system is first analyzed as a global entity, discarding or passing over details deemed not to affect its overall behavior. Local details are then analyzed using the results of the global analysis as boundary conditions. The process can be continued into the analysis of further details of local models. And so on. When this procedure is restricted to two stages and applied in the context of finite element analysis, it is called global-local analysis in the FEM literature.

 In the global stage the behavior of the entire structure is simulated with a finite element model that necessarily ignores details such as cutouts or joints. These details do not affect the overall behavior of the structure, but may have a bearing on safety. Such details are a posteriori incorporated in a series of local analyses.

Example of global-local analysis

The gist of the global-local analysis can be explained with the help of the following example. Suppose one is faced with the analysis of the rectangular panel shown below in figure 1, which contains three small holes.

Figure 1: Rectangular panel with holes for global-local analysis

 

 

The figure 2 below shows a standard (one-stage) FEM treatment using a largely regular mesh that is refined near the holes.

 

Figure 2: Rectangular panel with holes for a single shot finite element analysis (refined mesh near the holes)

Connecting the coarse and fine meshes usually involves using multifreedom constraints because the nodes at mesh boundaries do not match, as depicted in that figure.

 

Figure 3 illustrates the global-local analysis procedure. The global analysis is done with a coarse but regular FEM mesh that ignores the effect of the holes. This is followed by local analysis of the region near the holes using refined finite element meshes. The key ingredient for the local analyses is the application of boundary conditions (BCs) on the finer mesh boundaries. These BCs may be of displacement (essential) or of force (natural) type. If the former, the applied boundary displacements are interpolated from the global mesh solution. If the latter, the internal forces or stresses obtained from the global calculation are converted to nodal forces on the fine meshes through a lumping process

Figure 3: Global analysis of the whole panel ignoring the effect of the holes

Figure 4: Local analysis of the finer meshes

The global-local technique can be extended to more than two levels, in which case it receives the more encompassing name multiscale analysis.