Discrete Adjoint Solution Research Articles

An Adjoint Sensitivity Model for Transient Sequentially Coupled Radionuclide Transport in Porous Media

AbstractWe present an efficient mathematical/numerical adjoint sensitivity analysis model for transient radionuclide transport through heterogeneous porous media. This work extends our previous research on radionuclide transport in a steady‐state regime. Both continuous and discrete adjoint approaches have been developed in this current research. The mathematical equations associated with the adjoint system and required for the continuous approach have been derived. The methodology for deriving the discrete adjoint solution is also presented.For the homogeneous case, the adjoint system for radionuclide decay chain is solved analytically. The four‐member decay chain (238Pu → 234U → 230Th → 226Ra) is considered for validation. The validity of the analytical adjoint sensitivity model has been shown using two illustrative examples associated with two performance measures. The analytical adjoint states are compared with calculated numerical results and show excellent agreements. The sensitivity coefficients are also computed numerically using the perturbation method. The numerical sensitivity coefficients successfully reproduce the analytically derived sensitivities based on adjoint states. The adjoint model has been coupled with a derivative‐based global sensitivity method and applied to a real field case involving the performance assessment of a site currently being considered for underground nuclear storage in northern Switzerland, Zürich Nordost. An illustrative case study of spent fuel and high‐level waste (SF/HLW) repository involving 6 radionuclides and 48 parameters has been considered for identifying the most important parameters with respect to the maximum radionuclide dose at the interface geosphere/biosphere. Based on the results of sensitivity analysis, the uncertainty of the maximum radionuclide dose has been performed using Monte Carlo simulations.

Watch Your Adjoints! Lack of Mesh Convergence in Inviscid Adjoint Solutions

It has been long known that 2D and 3D inviscid adjoint solutions are generically singular at sharp trailing edges. In this paper, a concurrent effect is described by which wall boundary values of 2D and 3D inviscid continuous and discrete adjoint solutions based on lift and drag are strongly mesh dependent and do not converge as the mesh is refined. Various numerical tests are performed to characterize the problem. Lift-based adjoint solutions are found to be affected for any flow condition, whereas drag-based adjoint solutions are affected for transonic lifting flows. A (laminar) viscous case is examined as well, but no comparable behavior is found, which suggests that the issue is exclusive to inviscid flows. It is argued that this behavior is caused by the trailing edge adjoint singularity.

Adjoint-based airfoil optimization with adaptive isogeometric discontinuous Galerkin method

In this work, an adjoint-based airfoil shape optimization algorithm is developed based on the adaptive isogeometric discontinuous Galerkin method for compressible Euler equations to investigate the significance of each design variable of airfoil B-spline parameterization. We first parameterize the airfoil by B-spline curve approximation with some control points viewed as design variables, and build the B-spline representation of the flow field with the curve to apply the goal-oriented h-adaptive isogeometric DG method for flow solution. Then we compute and employ the discrete adjoint solutions for both multi-target error estimation in adaptive mesh refinement. With the isogeometric nature, not only all the geometrical cells but also the numerical basis functions can be analytically expressed by the design variables, indicating that the numerical solutions and objective could be differentiable with respect to those variables. Consequently, the gradient is totally computed in an accurate approach, and the sensitivity analysis is thus improved, by reducing the spatial discretization error and introducing the analytical expression of derivative, to reveal the key parameters for optimization in an intuitive and efficient manner. Although the SQP optimization algorithm is adopted in the paper, the given accurate gradient can be applied to any gradient-based optimization algorithm. The proposed algorithm is demonstrated on RAE2822 airfoil with inviscid transonic flow, where the shape is optimized to minimize the drag coefficient at a constrained lift and airfoil area. The numerical results show that the drag is much more sensitive to the design variables near the tailing edge at the beginning but sensitivity is reduced when optimal.

Adjoint analysis of Buckley-Leverett and two-phase flow equations

This paper analyzes the adjoint equations and boundary conditions for porous media flow models, specifically the Buckley-Leverett equation, and the compressible two-phase flow equations in mass conservation form. An adjoint analysis of a general scalar hyperbolic conservation law whose primal solutions include a shock jump is initially presented, and the results are later specialized to the Buckley-Leverett equation. The non-convexity of the Buckley-Leverett flux function results in adjoint characteristics that are parallel to the shock front upstream of the shock and emerge from the shock front downstream of the shock. Thus, in contrast to the behavior of Burgers’ equation where the adjoint is continuous at a shock, the Buckley-Leverett adjoint, in general, contains a discontinuous jump across the shock. Discrete adjoint solutions from space-time discontinuous Galerkin finite element approximations of the Buckley-Leverett equation are shown to be consistent with the derived closed-form analytical solutions. Furthermore, a general result relating the adjoint equations for different (though equivalent) primal equations is used to relate the two-phase flow adjoints to the Buckley-Leverett adjoint. Adjoint solutions from space-time discontinuous Galerkin finite element approximations of the two-phase flow equations are observed to obey this relationship.

High-Order Output-Based Adaptive Simulations of Turbulent Flow in Two Dimensions

Output-based high-order adaptive results are presented for several benchmark two-dimensional turbulent-flow simulations. The discretization is a high-order discontinuous Galerkin finite element method, and the equations solved are compressible Navier–Stokes, Reynolds-averaged with a modified version of the Spalart–Allmaras one-equation model. Mesh refinement requirements are studied through automated output-based adaptation in which a discrete adjoint solution associated with an output (e.g., the drag coefficient) weights a fine-space residual and automatically selects the elements that need more resolution. The roles of high-order and mesh anisotropy are also investigated. Finally, differences are investigated between two mesh refinement strategies: hanging-node refinement of structured meshes versus metric-based remeshing of unstructured triangles.

Adjoint‐based airfoil optimization with discretization error control

SUMMARYIn this article, we develop a new airfoil shape optimization algorithm based on higher‐order adaptive DG methods with control of the discretization error. Each flow solution in the optimization loop is computed on a sequence of goal‐oriented h‐refined or hp‐refined meshes until the error estimation of the discretization error in a flow‐related target quantity (including the drag and lift coefficients) is below a prescribed tolerance. Discrete adjoint solutions are computed and employed for the multi‐target error estimation and adaptive mesh refinement. Furthermore, discrete adjoint solutions are employed for evaluating the gradients of the objective function used in the CGs optimization algorithm. Furthermore, an extension of the adjoint‐based gradient evaluation to the case of target lift flow computations is employed. The proposed algorithm is demonstrated on an inviscid transonic flow around the RAE2822, where the shape is optimized to minimize the drag at a given constant lift and airfoil thickness. The effect of the accuracy of the underlying flow solutions on the quality of the optimized airfoil shapes is investigated. Copyright © 2014 John Wiley & Sons, Ltd.

Adjoint-accelerated statistical and deterministic inversion of atmospheric contaminant transport

In this work we present and compare deterministic and statistical algorithms for efficiently solving large-scale contaminant source inversion problems. The underlying equations of contaminant transport are assumed linear but unsteady and defined over complex geometries. The algorithms presented are accelerated through discrete adjoint solutions that are pre-computed efficiently in an offline stage, yielding savings in the time-critical online stage of several orders of magnitude in computational time. In the deterministic case, adjoints accelerate the application of the Hessian matrix, while in the statistical case, adjoints are used to directly evaluate samples. To address deterioration of statistical sampling efficiency for anisotropic posteriors, we present an application of a recently developed ensemble Markov chain Monte Carlo method. Results for two- and three-dimensional problems demonstrate the feasibility of statistical inversion for large-scale problems and show the advantage of statistical results over single-point deterministic results.

Error estimation and adjoint based refinement for an adjoint consistent DG discretisation of the compressible Euler equations

Adjoint consistency in addition to consistency is the key requirement for discontinuous Galerkin discretisations to be of optimal order in L2 as well as measured in terms of target functionals. We provide a general framework for analysing adjoint consistency and introduce consistent modifications of target functionals. This framework is then used to derive an adjoint consistent discontinuous Galerkin discretisation of the compressible Euler equations. We demonstrate the effect of adjoint consistency on the accuracy of the flow solution, the smoothness of the discrete adjoint solution and the a posteriori error estimation with respect to aerodynamical force coefficients on locally refined meshes.

Adjoint Consistency Analysis of Discontinuous Galerkin Discretizations

This paper is concerned with the adjoint consistency of discontinuous Galerkin (DG) discretizations. Adjoint consistency—in addition to consistency—is the key requirement for DG discretizations to be of optimal order in $L^2$ as well as measured in terms of target functionals. We provide a general framework for analyzing the adjoint consistency of DG discretizations which is also useful for the derivation of adjoint consistent methods. This analysis will be performed for the DG discretizations of the linear advection equation, the interior penalty DG method for elliptic problems, and the DG discretization of the compressible Euler equations. This framework is then used to derive an adjoint consistent DG discretization of the compressible Navier-Stokes equations. Numerical experiments demonstrate the link of adjoint consistency to the accuracy of numerical flow solutions and the smoothness of discrete adjoint solutions.