Abstract

This paper introduces a general methodology for determining the uncertainty of the solution to implicit systems of equations. Equation systems of this type arise from many practical applications, including the analysis of pipe networks, and in the implementation of complex numerical (finite difference or finite element) solution algorithms. The procedure is applicable to either linear or nonlinear equation systems, and does not require any specific algorithm for solution to the equation system itself. A general sensitivity matrix is constructed from an implicit sensitivity analysis of the equation system. This overall sensitivity matrix is expressed in terms of input and output sensitivity matrices, which represent the sensitivity of the equation system to changes in the independent (parameters) and dependent (calculated) variables, respectively. A vector representing the root-mean-square (RMS) uncertainty of the solution variables in the equation system is then given as a function of the given uncertainty in the input parameters. Two specific examples are presented to illustrate the practical application of the technique: (1) An example from fluid mechanics evaluating the uncertainty in solutions to a pipe network problem and (2) an example evaluating the uncertainty of a thermistor calibration and measurement problem.

References

References
1.
Abernethy
,
R.
,
Benedict
,
R.
, and
Dowdell
,
R.
,
1985
, “
ASME Measurement Uncertainty
,”
Trans. ASME J. Fluids Eng.
,
107
(
2
), pp.
161
164
.10.1115/1.3242450
2.
Coleman
,
H. W.
, and
Steele
,
W. G.
,
1999
,
Experimentation and Uncertainty Analysis for Engineers
,
2nd ed.
,
Wiley
,
New York
, pp.
47
82
.
3.
Brown
,
K. K.
,
Coleman
,
H. W.
, and
Steele
,
W. G.
,
1998
, “
A Methodology for Determining Experimental Uncertainties in Regressions
,”
ASME J. Fluids Eng.
,
120
(
3
), pp.
445
456
.10.1115/1.2820683
4.
Pethe
,
1999
, “
Gear Profiling System
,” M.S. thesis, Department of Industrial & Manufacturing Systems Engineering, Kansas State University, Manhattan, KS.
5.
Adrian,
1999
,
Particle Image Velocimetry, A Practical Guide
,
Springer
,
New York
.
6.
Dunn
,
W. L.
, and
Kenneth Shultis
,
J.
,
2011
,
Exploring Monte Carlo Methods
,
1st ed.
,
Elsevier Science
,
Burlington, MD
.
7.
Papadopoulos
,
C. E.
, and
Yeung
,
H.
,
2001
, “
Uncertainty Estimation and Monte Carlo Simulation Method
,”
Flow Meas. Instrum.
,
12
(
4
), pp.
291
298
.10.1016/S0955-5986(01)00015-2
8.
L'Ecuyer
,
P.
,
2012
, “
Random Number Generation
,”
Handbook of Computational Statistics: Concepts and Methods. Handbook of Computational Statistics
,
J. E.
Gentle
,
W.
Haerdle
, and
Y.
Mori
, eds.,
2nd ed.
,
Springer-Verlag
,
Berlin
, pp.
35
71
.
9.
Lopez
,
R. H.
,
Fadel Miguel
,
L. F.
, and
Souza de Cursi
,
J. E.
,
2013
, “
Uncertainty Quantification for Algebraic Systems of Equations
,”
Comput. Struct.
,
128
, pp.
189
202
.10.1016/j.compstruc.2013.06.016
10.
Jeppson
,
R. W.
,
1974
, “
Steady Flow Analysis of Pipe Networks: An Instructional Manual
,” Reports No.
300
. https://digitalcommons.usu.edu/water_rep/300
11.
Fox
,
R. W.
, and
McDonald
,
A. T.
,
1998
,
Introduction to Fluid Mechanics
,
5th ed.
,
Wiley
,
Hoboken, NJ
, p.
361
.
12.
Steinhart
,
J. S.
, and
Hart
,
S. R.
,
1968
, “
Calibration Curves for Thermistors
,”
Deep-Sea Res. Oceanogr. Abstr.
,
15
(
4
), pp.
497
503
.10.1016/0011-7471(68)90057-0
13.
Rudtsch
,
S.
, and
Von Rohden
,
C.
,
2015
, “
Calibration Self-Validation Thermistors High-Precision Temperature Measurements
,”
Measurements
,
76
, pp.
1
6
.10.1016/j.measurement.2015.07.028
You do not currently have access to this content.