(From IP-Net Oct. 1, 1995)
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From: (Dr. James Beck)
Subject:
Date: Tue, 26 Sep 95
PERSPECTIVE ON THE RELATION OF CURRENT ENGINEERING PRACTICE
TO INVERSE PROBLEMS
This contribution has two purposes. One is to stimulate
conversation among engineers and others who perform experiments and
estimate parameters. It gives a general framework which I see many
engineers (and others) working. The second purpose is to give
mathematicians and others an understanding of the common
experimental-analytical paradigms for unknown processes and their
relationship to the study of inverse problems. I would be happy to hear
from anyone who would like to discuss these ideas
further.
Below are some thoughts on two current research paradigms in
engineering. These paradigms are contrasted with what I consider to be
a more powerful paradigm - which is actually part of the subject of
inverse problems. This third type is familar to the inverse problems
community but it is not widely known or practiced in engineering.
Common Research Paradigms in Engineering
Two types of paradigms in engineering research are commonly used.
Type A involves investigating a "simple" phenomena and a single
parameter is found using a simple algebraic equation. Type B has its
objective to verify that the model is satisfactory to describe a certain
phenomena.
Common Paradigm of Type A
In the type A paradigm, a process has an unknown such as thermal
conductivity, heat transfer coefficient, diffusion coefficient, Young's
modulus, or friction coefficient. Although the mathematical model for
the phenomena may be complex, the final equation for finding the
parameter of interest is usually quite simple, frequently as an
algebraic equation.
The other part of the type A procedure involves an experiment. The
experiments is selected to produce measurements that are compatible with
the model. From these measurements and the model, the parameter is
determined.
Common Paradigm of Type B
In the type B of the common paradigm, an incompletely understood
engineering process is investigated in two distinct and complementary
ways: one uses experiments and the other uses analytical or computer
modeling. The first part involves an analytical model. This can
involve the solution of ordinary or partial differential equations. Any
needed constants are found from the literature or completely separate
experiments of Type A which are found by breaking the problem into
several independent parts. After all the parts are found, they are
assembled into one large model and a prediction is made for some
experimental conditions.
An experimental effort produces measurements for the same process.
No interaction between the analysis and the experiment for the complete
process is allowed. The experimental group in effect "throws over the
wall" the data and description of the experiment to the analytical
group.
Then a figure of overall results is produced, comparing those from
the model and the experiment. Characteristically, the comparison of the
graphical results is visual and not quantitative. Instead the agreement
is usually simply said to be "satisfactory" or even "excellent," showing
that the model is also satisfactory. An important point is that the
results of the experiment and analysis are purposely kept apart until
the last possible moment, and then compared only on same plot. The
intent is to avoid any "knobs" to turn to get agreement between the
model and the measurements. Results of the model may be not used to
modify and improve the experiment; similarly the model may not be
modified based on the experiment.
New Research Paradigm in Engineering - Involving Inverse Problems: Type C
In the "new research paradigm," Type C paradigm, the emphasis is
upon combined and interactive experiments and analysis. The concepts of
experiment design and "stretching and straining" the model enters.
Computers are used both in the experiments, modeling and estimating of
parameters or determining better models. The paradigm is now described
in more detail.
The paradigm is directed toward understanding some physical
engineering process that has some unknown aspects. A first objective is
to identify what is unknown. This in turn leads to the design of an
experiment that will provide measurements that can be used to determine
what is unknown. Two aspects should be considered at this point.
First, the errors (or uncertainty) of the measuring devices(s) should be
understood and quantified. The second aspect is that the experiment
should be optimally designed, as much as possible without precisely
knowing all the parameters or possibly the correct model. A simulation
should be performed to see if the experiment will reveal what is thought
to be unknown. This then requires some interaction with the
analysis/modeling group in the beginning of the investigation. The
purpose is to reveal if the experiment has the potential to determine
the unknowns.
Then the experiment is performed. After that, the analysis is
performed (possibly involving finite differences or elements). Instead
of simply performing a direct calculation and comparing the results in a
graphical fashion, the analysis now includes an inverse algorithm for
estimating some parameters or functions. This estimation algorithm may
be nonlinear and involve iteration. The residual principle may be used
in which the estimated standard deviation between the measurements and
the estimated values are made to be about equal to the expected
measurement errors. The residuals are examined to determine any
systematic trends or signatures. Confidence regions are constructed.
After the experiment has been analyzed, it may be possible to
improve the experiment using optimality concepts. Furthermore the
residuals might give some insight for improving the model.
An important point is that this Type C paradigm does not require
breaking the problem into a number of parts (Type A experiments). In
some cases it may still be very wise to do that. However, there are
cases in which the individual parts are not independent. For example,
some materials change (dry, burn, ablate, cure, etc.) during the
process; in such cases the Type B paradigm is not adequate. In other
cases, the desired result is a function of time, such as a
time-dependent heating condition, which cannot be found by the Type B
paradigm.
I would appreciate any comments.
James V. Beck, Professor (beck@egr.msu.edu)
Department of Mechanical Engineering
A231 Engineering Building
Michigan State University
East Lansing, MI 48824 Tel no. 517-355-8487, Fax: 517-353-1750
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