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The Art of Analysis with Hierarchical Modeling

In any finite element analysis, the first step for an analyst is to choose an appropriate mathematical model, and the second step is to solve that model using finite element procedures. In almost all analyses, the first step is most important and also most difficult. To be able to choose an appropriate mathematical model, the analyst must be familiar with the basic mathematical models that are available, and in particular know the hierarchy of such models.

Considering different mathematical models with different levels of complexity, the goal of an analyst is to choose the simplest possible model that provides accurate results given the purpose of the analysis. This mathematical model then needs to be solved using a reliable finite element representation. This finite element representation is called "the finite element model".

Of course, choosing "the best finite element model" is an art. In this Brief we illustrate the importance of selecting a proper model with simple examples — and somewhat surprising results.

The topic of hierarchical modeling is addressed in

K.J. Bathe, Finite Element Procedures, Prentice Hall, 1996, Chapter 1

Lecture 1 of Finite Element Analysis of Solids and Fluids I, M.I.T. OpenCourseWare

and in greater depth and breadth in the recently published book

M.L. Bucalem and K.J. Bathe, The Mechanics of Solids and Structures — Hierarchical Modeling and the Finite Element Solution, Springer, 2011.

We draw the examples below from this reference.

Figure 1 shows the schematic of the first example. A cantilever structure with rectangular cross section is subjected to a lateral tip load. The objective is to assess the shear stress distribution through the depth of the structure at its mid-span and also at its support.



Figure 1  Schematic of the cantilever structure (l = 1.0, h = 0.1)


A small fillet is used at the support (see Figure 2).




Figure 2  Geometry of the first model; the steel support is considered infinitely rigid


The physical situation in Figure 2 is modeled using the linear elastic plane stress mathematical model. The model is solved using 9-node displacement-based elements. All the nodes along the left edge of the model are fixed. A very fine mesh is used in the area near the support to accommodate the small fillet feature.

The resulting shear stress distribution through the depth of the beam is depicted in Figures 3 and 4 and is compared with the shear stress distribution predicted by classical beam theory. According to beam theory, the shear stress distribution through the depth of the beam varies parabolically with zero stress at the top and bottom and its maximum at the center. As seen, the shear stress distribution in the mid-span is in good agreement with the beam theory. However, the shear stress distribution at the support is totally different and exhibits high stress gradients close to the top and bottom faces of the beam. A close up of the high stress gradient region is shown in Figure 5.




Figure 3  Shear stress distribution at the mid-span (finite element vs. beam theory)




Figure 4  Shear stress distribution at section through A, at the support (finite element vs. beam theory)




Figure 5  Close up of the shear stress distribution close to the high stress gradient region


Next we consider an improved mathematical model of this problem, namely, instead of using a rigid support, we include the steel support in the model. A schematic of the model is depicted in Figure 6.




Figure 6  Improved model including both the fillet and the steel support


Figure 7 shows the shear stress distribution at a vertical section passing through point A of this model, while Figure 8 shows a close up of the high stress gradient region. Intuitively, we might expect the shear stresses to decrease since instead of a rigid support now we have introduced a flexible support for the aluminum beam. However, by comparing the results in Figures 4 and 7 we notice that — the peak shear stress has increased!

This is due to the fact that the aluminum beam now punches into the steel support. Still, the qualitative shape of the stress distribution has not changed, and the shear stress distribution near the support predicted using beam theory is totally wrong.




Figure 7  Shear stress distribution at the support (finite element vs. beam theory)




Figure 8  Close up of the shear stress distribution close to the high stress gradient region


Figure 9 shows another interesting analysis example representing a machine tool jig. Here, using mathematical models of different complexity we obtain most interesting results. The details regarding these analyses and the results can be found in the M.L. Bucalem and K.J. Bathe book, referred to above, where also comprehensive theory and additional examples are presented.




Figure 9  Schematic of a machine tool jig




Figure 10  Hierarchical models of the machine tool jig

The important point is that it is crucial for an analyst to

- understand the various hierarchical mathematical models available,
- choose the appropriate model, and
- solve this model with reliable finite element procedures.

In practice, a sequence of models may need to be solved, not only involving linear structural response, but also nonlinearities and indeed temperature, fluid and electromagnetic effects.

ADINA has been developed for use in hierarchical modeling in which increasingly more complex models of physical events can be solved. ADINA can be used for comprehensive analysis of solids and structures, fluids, heat transfer, electromagnetics and their interactionsmultiphysics.

Analysis in engineering and the sciences will move increasingly towards this hierarchical modeling approach — and hence this approach is best taught at universities (see 1 and 2).


Keywords:
Hierarchical modeling, mathematical models, finite element models, reliable finite elements, structures, solids, fluids, temperatures, electromagnetics, fluid-structure interaction, multiphysics, education

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