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 represents the sensitivity of the equation system to changes in the independent (parameters) and dependent (calculated) variables, respectively. A vector representing the 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.

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