EXTENDED FORWARD SENSITIVITY ANALYSIS OF ONE-DIMEN Page: 3 of 13
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Extended forward sensitivity analysis of one-dimensional isothermal flow
The two most popular sensitivity analysis methods are the forward method [2, 3] and the adjoint
method [4, 5]. In the forward method, the governing equations are differentiated with respect to a
parameter of interest, generating an auxiliary system of equations the same size as the original
system. The solution sensitivity is obtained by solving this system of equations. The forward
method requires that a new system of equations is solved for each input parameter of interest. In
a systems level code with thousands of input parameters, it is desirable to construct the
sensitivity equations automatically and solve them in parallel. Several authors have successfully
applied the forward sensitivity analysis method to heat conduction and radiation diffusion
problems. [6, 7, and 8] Forward sensitivity analysis can be extended to include time step
sensitivity and potentially grid size sensitivity as an alternative way to perform convergence
studies. [9, 10] The extended forward sensitivity analysis provides a systematic method to
evaluate the parameter sensitivity effects on solution, along with time and space convergence
information. This "extended forward sensitivity analysis" method had only been demonstrated in
single equation diffusion problems in previous papers. In this paper, we apply the method to 1-D
isothermal single-phase flow problems, which are described by a two-equation hyperbolic
system.
In the adjoint method, an additional system of equations, adjoint to the original equations, is
formed and solved for each system response of interest. Once the adjoint solution is found, it can
be used to compute sensitivities for all input parameters. Since the number of additional systems
of equations is not dependent on the number of input parameters, the adjoint method is desirable
when the number of input parameters is larger than the number of system responses. While there
are examples of the adjoint method applied to simplified problems [11], the authors are unaware
of successful applications to large-scale time-dependent nonlinear equations such as those that
must be solved when analyzing a LOCA (Loss of Coolant Accident).
The most common method of uncertainty quantification is the Monte Carlo method. In the
Monte Carlo method, the computer code is treated like signal generator. It is run many times
while perturbing the input parameters. The results are analyzed and statistical methods are used
to infer confidence bounds on the solution. It is attractive because it is a "black box" method
requiring no changes to existing code. Its major disadvantage is the high computational cost. For
further review of uncertainty quantification methods in nuclear engineering and other disciplines,
readers are directed to the above-mentioned references.
2. OVERVIEW OF FORWARD SENSITIVITY ANALYSIS
For completeness, the general equations describing forward sensitivity analysis will be
summarized below. Consider a general time-dependent partial differential equation given by
G(t,Y, p) =a+ F(t,Y, p) = 0, (1)
where Y is the state vector, t is time, and p is an input parameter. We solve this equation with a
Jacobian-free Newton-Krylov method (JFNK). In the JFNK method, Newton iteration is used for
solving the nonlinear equations and a Krylov subspace method, usually the generalized minimal
residual method (GMRES), is used as the linear solver.
International Conference on Mathematics and Computational Methods Applied to Nuclear Science & 2/12
Engineering (M&C 2013), Sun Valley, Idaho, USA, May 5-9, 2013
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Johnson, Matthew & Zhao, Haihua. EXTENDED FORWARD SENSITIVITY ANALYSIS OF ONE-DIMEN, article, May 1, 2013; Idaho Falls, Idaho. (https://digital.library.unt.edu/ark:/67531/metadc838383/m1/3/: accessed March 29, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.