Development and assessment of adjoint sensitivity analysis method for transient two-phase flow simulations

Gain-margin based discrete-continuous method for the stability analysis of real-time hybrid simulation systems

Adjoint-based techniques can provide the sensitivity of an objective function to any number of parameters of a simulation inexpensively at roughly the cost of a single additional flow calculation. This information can be used to perform sensitivity analyses, aerodynamic shape optimization, and to estimate the error in the objective function due to numerical discretization. Existing approaches to derive the numerically discretized adjoint equations involve the so-called discrete and continuous methods, which differ in the order at which discretization and linearization steps are performed. The effect of these contrasting approaches is that they have both strengths and weaknesses over each other in the form of complexity of the formulation and computational expense of the solution. In this paper, we further develop the hybrid approach of Taylor et al. (2012) that combines elements of the continuous and discrete methods with the intention of capturing both their advantages: reducing the time spent on mathematical derivation of the continuous adjoint equations, lowering the computational requirements of the discrete adjoint equations, and generally improving the quality of the adjoint solution. The specific approach investigated in this paper treats the flow conservation equations in a continuous manner and the turbulence transport equations discretely. The framework is designed such that additional transport equations for any turbulence model can be seamlessly included in a discrete fashion for coupling with the mean flow equations. The methodology is demonstrated in an optimization problem of lift-constrained drag minimization of an airfoil in transonic turbulent flow.

Adjoint methods are widely used in various areas of computational science to efficiently obtain sensitivities of functionals which result from the solution of partial differential equations (PDEs). In addition, adjoint methods have been used in other settings including error estimation, uncertainty quantification and inverse problem formulations. When deriving the adjoint equations, there are two main approaches one can follow: the discrete and the continuous methods, which differ principally in the order of the linearization and discretization steps. The discrete adjoint method starts from the discretized form of the partial differential equation, which is then linearized. On the other hand, the continuous method linearizes the continuous governing equations first and then discretizes the resulting problem. Each of these approaches is found to have advantages and disadvantages over the other. In this paper we consider a hybrid approach between these two methods that aims to combine the better qualities of both: reducing the time spent on the mathematical derivation while also lowering the computational requirements of the discrete method and increasing the overall quality of the adjoint solution.

Effective adjoint approaches for computational fluid dynamics

Background:Appropriate nutritional support is effective on achievement of expected outcomes in intensive care unit (ICU) patients. Although several studies have suggested different conclusions about the effectiveness of tube feeding methods, there is no specific program of nutritional support for patients who have been hospitalized. There is a possibility for complications due to an inadequate nutrition. The aim of our study is to compare the effects of enteral feeding through the bolus and continuous methods on blood sugar and prealbumin level among the ICU inpatients.Materials and Methods:Fifty subjects were selected by convenient sampling from April to Aug 2013 in the ICU wards of Alzahra Hospital, Isfahan, Iran, and randomly assigned to study and control groups in this clinical trial. The subjects in the study group received infusion pump feeding while the control group received bolus feeding for 72 h. Blood sugar was checked for every 4 h for 72 h and the prealbumin level was assessed on the first and the fourth day in two groups.Results:In the study group, the mean blood sugar significantly decreased on the fourth day, compared with the first (P = 0.03, F = 3.85) and third (P = 0.01, F = 3.15) day. In the control group, the mean blood sugar increased from the first day. It was significantly higher in the control group on the second day (P = 0.02, F = 3.55), compared with the study group. In the study group, there was a significant difference in the mean prealbumin before and after intervention (P = 0.048, t = 1.97), but no significant difference was observed in the control group. There was a significant difference between two groups after intervention (P = 0.04, t = 2.05).Conclusion:The obtained results showed that supportive nutrition through a continuous method had an effect on critical patients’ blood sugar control and made a better nutritional status for these patients through an increase of prealbumin level. Positive effects of this feeding method can result in appropriate outcomes for patients’ recovery and reduce the complications.

A model of single-phase flow of slightly compressible liquid is studied and the discontinuous and the continuous Galerkin methods for this model are compared. After studying the results of comparison, one may make conclusion that both method produce solutions that mostly are in accord with laws of physics. However, computational costs for the continuous method are less than ones for the discontinuous method, and the discontinuous method may produce non-physical jumps in solutions on initial time layers.

This paper proposes continuous linear multistep methods for the numerical solution of first-order ordinary differential equations (ODEs) with step number k = 1 and k = 2. These methods are used to integrate some first-order initial value problems and the block method developed from the continuous method using interpolation and collocation approach adopting power series approximation as the basis function for the derivation of these methods. These methods are found to be consistent, zero stable, convergent, and accurate. It is noteworthy that the results generated from these methods are significantly accurate and efficient when compared with existing methods, which will be effective in solving first-order Ordinary Differential Equations.

This paper presents an adjoint method for the optimal control of unsteady flows. The goal is to develop the continuous and discrete unsteady adjoint equations and their corresponding boundary conditions for the time-accurate method. First, this paper presents the complete formulation of the time-dependent optimal design problem. Second, we present the time-accurate unsteady continuous and discrete adjoint equations. Third, we present results that demonstrate the application of the theory to a two-dimensional oscillating airfoil. The results are compared with a multipoint approach to illustrate the added benefit of performing full unsteady optimization.

The exhaust volute, an important component of a centrifugal compressor, plays a crucial role in improving overall compressor efficiency. Traditional design methods rely heavily on designer’s experience and often result in low design efficiency. In this study, a topology optimization design method for the exhaust volute flow channel is proposed to address this issue. The optimization objective focuses on minimizing flow energy dissipation within the design domain while considering flow compressibility. The sensitivity of the objective function to design variables is calculated using the discrete adjoint method. In the open-source platform OpenFOAM, the finite volume method is utilized to solve governing equations with darcy source term and adjoint equations, and the Method of Moving Asymptotes (MMA) algorithm is applied to solve the optimization problem. Numerical examples of different volute types in two and three dimensionals are used to demonstrate the effectiveness of the proposed method. The results show that the topology optimization method proposed in the present work can quickly obtain the optimal volute configuration.

A discrete adjoint method for pressure-based algorithms

Local exhaust system is an effective but energy-consuming method for capturing oil mist particles in machining workshops. To reduce the flow resistance of an exhaust system for minimal fan energy consumption, the method of applying individually shape-optimized exhaust hoods, namely mass-production design, is feasible. However, the combined effect of multiple exhaust hoods in an exhaust system may not be optimal in reducing the flow resistance. This investigation thus firstly validated the shape optimization of an individual exhaust hood by a discrete adjoint method. The discrete adjoint method could adjust the shape of an exhaust hood automatically in the direction of reducing the flow resistance. The design variables were the coordinates of wall boundaries of the exhaust hood. The validation used measured data from a small-scale experiment. This study then applied the validated discrete adjoint method to conduct customized design through the shape optimization of multiple exhaust hoods simultaneously in the exhaust system. The flow resistance under customized design was compared with the method of mass-production design. The results revealed that the customized design led to different shapes of individual exhaust hoods and they were different from the shape of the individually optimized exhaust hood. The flow resistance of the exhaust system under customized design was reduced by 57%. However, only 36.5% reduction in flow resistance was achieved when the mass-production design method was employed. The customized design method was more effective in reducing flow resistance of the exhaust system.

This paper presents the fundamentals of a continuous adjoint method and the applications of this method to the aerodynamic design optimization of both external and internal flows. General formulation of the continuous adjoint equations and the corresponding boundary conditions are derived. With the adjoint method, the complete gradient information needed in the design optimization can be obtained by solving the governing flow equations and the corresponding adjoint equations only once for each cost function, regardless of the number of design parameters. An inverse design of airfoil is firstly performed to study the accuracy of the adjoint gradient and the effectiveness of the adjoint method as an inverse design method. Then the method is used to perform a series of single and multiple point design optimization problems involving the drag reduction of airfoil, wing, and wing-body configuration, and the aerodynamic performance improvement of turbine and compressor blade rows. The results demonstrate that the continuous adjoint method can efficiently and significantly improve the aerodynamic performance of the design in a shape optimization problem.

Unsteady blade row interactions play an important role in the performance of the compressor stages. However, due to the large cost of the unsteady flow simulation, most aerodynamic optimizations of the compressor are based on the steady flow simulation. This paper adopts the time spectral method to reduce the cost of the unsteady flow simulation and a discrete adjoint solver based on the unsteady flow solver has been developed. The unsteady flow equations and the adjoint equations are solved using an in-house code. The in-house code is based on the finite volume method and solves the URANS (Unsteady Reynolds Averaged Navier-Stokes) equations on the multi-block structured mesh. For spatial discretization the 3rd order WENO (Weighted Essentially Nonoscillatory) upwind scheme is used for reconstruction and the convective flux is computed with Roe’s approximate Riemann solver. The widely used one-equation Spalart-Allmaras turbulence model is adopted for the flow simulation. For the adjoint solution, the constant-eddy viscosity assumption is adopted and only the main flow adjoint equations are solved. The adjoint equations are formed in a discrete manner, which leads to more accurate discrete objective function sensitivity than the continuous adjoint method. The present work serves as an essential part of the system for efficient unsteady aerodynamic optimization of turbomachinery.

Simulation of chemically reacting flows using detailed chemistry introduces a large number of chemistry model parameters. While not all significantly affect the target outcomes of a simulation, the parameters that do are not always known a priori. In order to improve simulations for specified target outcomes, termed quantities of interest (QoIs), the sensitivity of these QoIs to the model parameters are needed. However, evaluating the sensitivities is computationally expensive, especially for complex fuels that may involve many parameters. For these simulations, the forward sensitivity method requires the solution of an additional number of governing equations proportional to the number of parameters. Here, an adjoint sensitivity approach is formulated where the computational cost scales as the number of QoIs and not the number of parameters. Specifically, adjoint equations are derived for laminar, incompressible, variable density reacting flow and applied to hydrogen flame simulations. From the solution of the corresponding adjoint equations, sensitivity of the QoIs to chemistry model parameters is calculated. The one-dimensional simulation results show that the adjoint sensitivity results closely match those of forward sensitivity methods, thus providing validation of the adjoint method. The two-dimensional simulation results indicate the most sensitive parameters for two QoIs, flame tip temperature and NOx emission. For these tests, the adjoint method reduces computational expense compared to forward sensitivity methods by a factor proportional to the number of QoIs over the number of parameters, here 2/172. Such savings can be more drastic for cases that involve complex fuels, such as combustion of jet fuel, requiring thousands of chemistry model parameters. Further, this sensitivity information can be used in development of experiments by pointing out which are the critical chemistry model parameters.

This paper discusses anisotropic mesh adaptation, addressing either a local interpolation error, or the error on a functional, or the norm of the approximation error, the two last options using an adjoint state. This is explained with a Poisson model problem. We focus on metric-based mesh adaptation using a priori errors. Continuous metric-based methods were developed for this purpose. They propose a continuous statement of the mesh optimisation problem, which need to be then discretised and solved numerically. Tensorial metric-based methods produce directly a discrete optimal metric for interpolation error equirepartition. The novelty of the present paper is to extend the tensorial discrete method to addressing (1) L 1 errors and (2) adjoint-based analyses, two functionalities already available with continuous metric. A first interest is to be able to compare tensorial and continuous methods when they are applied to the reduction of approximation errors. Second, an interesting feature of the new formulation is a potentially sharper analysis of the approximation error. Indeed, the resulting optimal metric has a different anisotropic component. The novel formulation is then compared with the continuous formulation for a few test cases involving high-gradient layers and gradient discontinuities.