4.2. PHILOSOPHY OF MATHEMATICAL MODELING 49
(a) not removing all rigid body translation and rotation or
(b) overly constraining degrees of freedom along a particular direction that preclude de-
formation and Poisson effects in orthogonal directions.
2. One chooses inappropriate finite element formulations, such as
(a) planar or one-dimensional elements that are not appropriate for the observed behavior,
(b) finite elements with inadequate degrees of freedom, or
(c) finite elements with inadequate order of interpolation.
3. Lower-order interpolations appear to predict behavior more accurately than higher-order
interpolations.
4. Meshes with fewer active degrees of freedom appear to predict more accurately than meshes
with more active degrees of freedom.
We wish to illustrate these points with two examples where finite element modeling can go wrong.
Remember, whether or not simplified theory is appropriate, incorrect finite element results are
typically cases of analyst error.
4.2 PHILOSOPHY OF MATHEMATICAL MODELING
e great masters do not take any model quite so
seriously as the rest of us. ey know that it is, after all,
only a model, possibly replaceable.
C.S. Lewis
e game I play is imagination in a tight straightjacket.
at straightjacket is called the laws of physics.
Richard Feynman
S.L. Hayakawa is noted for pointing out that “the symbol is not the thing symbolized; the word is
not the thing; the map is not the territory it stands for” [Dym, 2004], echoing Richard Feynman
who recalled that his father “knew the difference between knowing the name of something and
knowing something” [Public Broadcasting System–NOVA, 1993]. When engineers attempt to
formulate models for systems and processes, it is incumbent upon us to remember that the process,
the system is “the thing,” “the territory.” e model is a symbol, word, or map that in some way
names the thing. ey are not the same. To model some process well requires recasting its real
nature into a simplified shell that allows its basic nature to be captured in mathematical form, a
set of equations whose solutions tell us something about how the model system behaves under a
given set of controlled conditions. An abstraction of the process is shown in Fig. 4.1.
50 4. IT’S ONLY A MODEL
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Figure 4.1: A mathematical model is devised by sufficiently simplifying a problem statement such
that its formulation can be cast in equation form.
Similar conceptualizations have been illustrated elsewhere and these overviews of model-
ing are well worth reading: Carson and Cobelli [2000], Dym [2004], Greenbaum and Chartier
[2012]. In order to numerically model a system, we must observe the system in nature. We must:
1. collect all information relevant to how the system behaves,
2. detail what we need to find out or predict,
3. specify how well we need to know or predict this behavior, and
4. seriously ask a singularly important question: “What do we expect to happen?”
4.2. PHILOSOPHY OF MATHEMATICAL MODELING 51
We should have a knowledgeable, informed expectation of how the system will respond to dis-
turbances, excitation, or loading based on practical experience, prudent observation, and one’s
understanding of the relevant physics.
In any given process, only a few physical mechanisms tend to dominate the behavior. Mak-
ing physically simplifying assumptions means deciding what physical mechanisms to retain (recall
the baby) and which to neglect (recall the bathwater). e modeler needs to retain the dominant
physics and neglect all higher-order effects, making the model as simple as possible, but no sim-
pler. Making appropriate simplifying assumptions is an art whose mastery comes only gradually
with continued experience.
After appropriate simplifying assumptions are made, application of a conservation or bal-
ance principle results in a differential equation for the boundary value problem. Finite element
methods provide a piecewise approximation to the solution of this differential equation. In con-
structing finite element models, the major inputs from the user are
1. the choice of finite element, which dictates the incremental solution interpolation between
nodes and
2. the specific prescription of boundary conditions for the global domain.
We’ve already learned that beam behavior can be approximated using one-dimensional and three-
dimensional models. Here we will use both and compare the results to experimentally measured
values.
Recall that boundary value problems are described fully by a governing differential equa-
tion coupled with an admissible set of appropriate boundary conditions. For static analyses, the
boundary conditions must remove all rigid body translations and rotations.
Upon applying admissible boundary conditions, we solve for displacements throughout
the global region. Most commercial finite element software then post-processes the displacement
solution to compute
1. reaction forces corresponding to applied displacement constraints and
2. internal stresses which may be displayed or contoured.
One goal in model development is to start with the simplest approximation that captures
the physics and provides perhaps crude, but reliable qualitative predictions of system behavior.
We will seek to iterate on the model to provide more quantitative results, and then to validate the
numerical predictions with experimental observations and test results. All models are approxima-
tions whose errors most commonly arise from
1. expectation failures,
2. faulty simplifying assumptions,
3. poor discretization of the domain,
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