Application of discrete adjoint method to sensitivity and uncertainty analysis in steady-state two-phase flow simulations

Abstract

Verification, validation and uncertainty quantification (VVUQ) have become a common practice in thermal-hydraulics analysis. An important step in the uncertainty analysis is the sensitivity analysis of various uncertainty input parameters. The efficient method for computing the sensitivities is the adjoint method. The cost of solving an adjoint equation is comparable to the cost of solving the governing equation. Once the adjoint solution is obtained, the sensitivities to any number of parameters can be obtained with little effort. There are two methods to develop the adjoint equations: continuous method and discrete method. In the continuous method, the control theory is applied to the forward governing equation and produces an analytical partial differential equation for solving the adjoint variable; in the discrete method, the control theory is applied to the discrete form of the forward governing equation and produces a linear system of equations for solving the adjoint variable. In this article, an adjoint sensitivity analysis framework is developed using both the continuous and discrete methods. These two methods are assessed with the one transient test case based on the BFBT benchmark. Adjoint sensitivities from both methods are verified by sensitivities given by the perturbation method. Adjoint sensitivities from both methods are physically reasonable and match each. The sensitivities obtained with discrete method is found to be more accurate than the sensitivities from the continuous method. The continuous method is computationally more efficient than the discrete method because of the analytical coefficient matrices and vectors. However, difficulties are observed in solving the continuous adjoint equation for cases where the adjoint equation contains sharp discontinuities in the source terms; in such cases, the continuous method is not as robust as the discrete adjoint method. arXiv:1805.08083v1 [physics.comp-ph] 18 May 2018 popular in computational fluid dynamics field [1,2]. Within the field of aeronautical computational fluid dynamics, the use of adjoint method has been seen in [3,4,5,6]. Adjoint problems arise naturally in the formulation of methods for optimal aerodynamic design and optimal error control [7,2,8,9]. Adjoint solution provides the linear sensitivities of an objective function (e.g. lift force and drag force) to a number of design variables. These sensitivities can then be used to drive an optimization procedure. In a sequence of papers, Jameson developed the adjoint method for the potential flow, the Euler equation, and the Navier-Stokes equation [3,4,5,6]. The application of the adjoint method to optimal aerodynamic design is very successful. However, to the author's best knowledge, successful adjoint sensitivity analysis to two-phase flow problems is rare. Cacuci performed an adjoint sensitivity analysis to two-phase flow problems using the RELAP5/MOD3.2 numerical discretization [10,11,12]. An application of Cacuci's method was illustrated by [13], where the method was applied to th...