Adjoint-based sensitivity analysis of flames

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.     

Conservation laws of the complex short pulse equation and coupled complex short pulse equations

In this paper, the complex short pulse equation and the coupled complex short pulse equations that can describe the ultra-short pulse propagation in optical fibers are investigated. The two complex nonlinear models are turned into multi-component real models by proper transformations. Lie symmetries are obtained via the classical Lie group method, and the results for the coupled complex short pulse equations contain the existing results as particular cases. Based on the linearizing operator and adjoint linearizing operator for the two real systems, adjoint symmetries can be obtained. Explicit conservation laws are constructed using the symmetry/adjoint symmetry pair (SA) method. Relationships between the nonlinear self-adjointness method and the SA method are investigated.     

An adjoint method for shape optimization in unsteady viscous flows

A new method for shape optimization for unsteady viscous flows is presented. It is based on the continuous adjoint approach using a time accurate method and is capable of handling both inverse and direct objective functions. The objective function is minimized or maximized subject to the satisfaction of flow equations. The shape of the body is parametrized via a Non-Uniform Rational B-Splines (NURBS) curve and is updated by using the gradients obtained from solving the flow and adjoint equations. A finite element method based on streamline-upwind Petrov/Galerkin (SUPG) and pressure stabilized Petrov/Galerkin (PSPG) stabilization techniques is used to solve both the flow and adjoint equations. The method has been implemented and tested for the design of airfoils, based on enhancing its time-averaged aerodynamic coefficients. Interesting shapes are obtained, especially when the objective is to produce high performance airfoils. The effect of the extent of the window of time integration of flow and adjoint equations on the design process is studied. It is found that when the window of time integration is insufficient, the gradients are most likely to be erroneous.     

Continuous and Discrete Adjoint Approach Based on Lattice Boltzmann Method in Aerodynamic Optimization Part I: Mathematical Derivation of Adjoint Lattice Boltzmann Equations

The significance of flow optimization utilizing the lattice Boltzmann (LB) method becomes obvious regarding its advantages as a novel flow field solution method compared to the other conventional computational fluid dynamics techniques. These unique characteristics of the LB method form the main idea of its application to optimization problems. In this research, for the first time, both continuous and discrete adjoint equations were extracted based on the LB method using a general procedure with low implementation cost. The proposed approach could be performed similarly for any optimization problem with the corresponding cost function and design variables vector. Moreover, this approach was not limited to flow fields and could be employed for steady as well as unsteady flows. Initially, the continuous and discrete adjoint LB equations and the cost function gradient vector were derived mathematically in detail using the continuous and discrete LB equations in space and time, respectively. Meanwhile, new adjoint concepts in lattice space were introduced. Finally, the analytical evaluation of the adjoint distribution functions and the cost function gradients was carried out.     

升力系数不变时跨音速机翼阻力优化设计

基于共轭方程的优化设计理论,应用三维欧拉方程进行了升力系数不变时跨音速机翼阻力优化设计研究,根据给定的目标函数推导了在物理空间上表述的共轭方程及边界条件,研究了共轭方程的数值求解方法及目标函数对设计变量的敏感性导数求解问题,发展了一种跨音速机翼阻力优化设计方法,应用该设计方法进行了跨音速机翼阻力优化设计研究,优化后机翼表面的激波强度减弱很多,有效减少了波阻.     

Analytical study of the splitting process of a multiply-quantized vortex in a Bose–Einstein condensate and collaboration of the zero and complex modes

We study the dynamics of a trapped Bose–Einstein condensate with a multiply-quantized vortex, and investigate the roles of the fluctuations in the dynamical evolution of the system. Using the perturbation theory of the external potential, and assuming the situation of the small coupling constant of self-interaction, we analytically solve the time-dependent Gross–Pitaevskii equation. We introduce the zero mode and its adjoint mode of the Bogoliubov–de Gennes equations. Those modes are known to be essential for the completeness condition. We confirm how the complex eigenvalues induce the vortex splitting. It is shown that the physical role of the adjoint zero mode is to ensure the conservation of the total condensate number. The contribution of the adjoint mode is exponentially enhanced in synchronism with the exponential growth of the complex mode, and is essential in the vortex splitting.     

The effect of dissipation on coherent quantum tunnelling in a double-well system: Two-level model using an adjoint equation approach

We examine the effect of dissipation on coherent quantum tunnelling between the two lowest levels of a double-well system using an adjoint equation approach developed previously in the treatment of quantum-optical problems. Dissipation is modelled by a linear coupling to a bath of harmonic oscillators. The high frequency portion of the bath is adiabatically eliminated using a cumulant expansion technique, which generates frequency renormalisation terms. Making a low temperature assumption, an approximate two-level model is developed in the form of three coupled stochastic differential equations. With a decorrelation approximation, the two-level equations can be solved to yield familiar results. However, the adjoint equation also permits the use of direct stochastic simulation as a means of solution, and simulations are carried out for a range of parameters. A comparison with the decorrelation approximation is made.     

透平叶栅三维形状反问题研究

随着CFD技术的发展,基于伴随方法的求解Euler和NS方程的气动优化设计已成为流体力学形状反问题研究中的热门领域.本文应用该方法对透平叶栅进行三维气动优化设计,详细推导了Euler方程伴随系统的偏微分方程组及其各类边界条件,首次给出了透平内流伴随方程边界条件的具体形式,并给出伴随变量的物理意义.结合拟牛顿算法发展了三维透平叶栅形状反问题气动优化算法,并给出了算法的流程.     

Efficient evaluation of the direct and adjoint linearized dynamics from compressible flow solvers

The direct and adjoint operators play an undeniably important role in a vast number of theoretical and practical studies that range from linear stability to flow control and nonlinear optimization. Based on an existing nonlinear flow solver, the design of efficient and straightforward procedures to access these operators is thus highly desirable. In the case of compressible solvers, the use of high-order numerical schemes combined with complicated governing equations makes the derivation of efficient procedures a challenging and often tedious undertaking. In this work, a novel technique for the evaluation of the direct and adjoint operators directly from compressible flow solvers is presented and extended to include nonlinear differentiation schemes and turbulence models. The application to the incompressible counterpart is also discussed. The presented method requires minimal additional programming effort and automatically takes into account subsequent modifications in the governing equations and boundary conditions. The introduced methodology is demonstrated on existing numerical codes, and direct and adjoint global modes are calculated for three typical flow configurations. Implementation issues and the performance measures are also discussed. The proposed algorithm presents an easy-to-implement and efficient technique to extract valuable information for the quantitative analysis of complex flows.     

Adjoint sensitivity computations for an embedded-boundary Cartesian mesh method

We present a new approach for the computation of shape sensitivities using the discrete adjoint and flow-sensitivity methods on Cartesian meshes with general polyhedral cells (cut-cells) at the wall boundaries. By directly linearizing geometric constructors of the cut-cells, an efficient and robust computation of shape sensitivities is achieved for problems governed by the Euler equations. The accuracy of the linearization is verified by the use of a model problem with an exact solution. Verification studies show that the convergence rate of gradients is second-order for design variables that do not alter the boundary shape, and is reduced to first-order for shape design problems. The approach is applied to several three-dimensional problems, including inverse design and shape optimization of a re-entry capsule in hypersonic flow. The results show that reliable approximations of the gradient are obtained in all cases. The approach is well-suited for geometry control via computer-aided design, and is especially effective for conceptual design studies with complex geometry where fast turn-around time is required.     

Aerodynamic design optimization by using a continuous adjoint method

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.     

Adjoint algorithms for the Navier–Stokes equations in the low Mach number limit

This paper describes a derivation of the adjoint low Mach number equations and their implementation and validation within a global mode solver. The advantage of using the low Mach number equations and their adjoints is that they are appropriate for flows with variable density, such as flames, but do not require resolution of acoustic waves. Two versions of the adjoint are implemented and assessed: a discrete-adjoint and a continuous-adjoint. The most unstable global mode calculated with the discrete-adjoint has exactly the same eigenvalue as the corresponding direct global mode but contains numerical artifacts near the inlet. The most unstable global mode calculated with the continuous-adjoint has no numerical artifacts but a slightly different eigenvalue. The eigenvalues converge, however, as the timestep reduces. Apart from the numerical artifacts, the mode shapes are very similar, which supports the expectation that they are otherwise equivalent. The continuous-adjoint requires less resolution and usually converges more quickly than the discrete-adjoint but is more challenging to implement. Finally, the direct and adjoint global modes are combined in order to calculate the wavemaker region of a low density jet.     

Generation of perturbations by means of decoupled equations and their adjoints

It is shown that the procedure introduced by Wald for constructing solutions of a coupled system of linear partial differential equations from the solution of a single equation, based on the concept of the adjoint of a linear partial differential operator, can be extended to equations involving spinor fields, matrix fields and two or more fields. Some results concerning massless spinor fields are presented and the application of the method to linear perturbations of Yang-Mills fields and of Einstein-Maxwell fields is indicated.     

Adjoint-based sensitivity and ignition threshold mapping in a turbulent mixing layer

Successful ignition in non-premixed turbulent flows remains a fundamental challenge in combustion systems. Current design strategies typically rely on iterative testing to map the spatial distribution of ignition probability. We propose to accelerate this by formulating the adjoint of the perturbed and linearised governing equations in such a way that sensitivity of an ignition indicator can be obtained with a cost comparable to the flow solution. A space–time discrete adjoint method for multi-component chemically reacting flows is developed, and the gradient formed via the corresponding adjoint solution is used to identify regions favourable to ignition in a direct numerical simulation of non-premixed turbulent free shear flow. This approach requires a specific definition of an ignition metric, although this can be problematic because ignition either succeeds or fails after some period and thus gradients for some metrics become ill-defined near the ignition threshold. To this end, a quantity of interest is designed to provide short-time sensitivity in conjunction with an indicator function over a long-time period that informs whether successful ignition occurred. The gradients are used in a line-search algorithm to map the ignition boundary under specific constraints. Finally, parametric sensitivity is evaluated at different flow realisations to analyse factors governing local sensitivity in unsteady chemically reacting flows.     

Data assimilation for the Navier–Stokes- equations

The well-posedness of the data assimilation problem for the Navier–Stokes-α equations on a bounded three-dimensional domain is investigated. The data assimilation procedures under consideration are the adjoint method of variational data assimilation (4D-Var) and the method of continuous data assimilation. Concerning the adjoint method the existence of optimal initial conditions with respect to an observation-dependent cost functional is proven, the optimizers are characterized by a first-order necessary condition involving the adjoint linearized Navier–Stokes-α equations and conditions for the uniqueness of the initial conditions are given. Well-posedness of the continuous data assimilation problem is proven and convergence rates in terms of observational resolution are provided.     

Relationship between different approaches to derive weighting functions related to atmospheric remote sensing problems

The paper is devoted to the investigation of the relationship between different methods used to derive weighting functions required to solve numerous inverse problems related to the remote sensing of the Earth's atmosphere by means of scattered solar light observations. The first method commonly referred to as the forward-adjoint approach is based on a joint solution of the forward and adjoint radiative transfer equations and the second one requires the linearized forward radiative transfer equation to be solved. In the framework of the forward-adjoint method we consider two approaches commonly used to derive the weighting functions. These approaches are referenced as the “response function” and the “formal solution” techniques, respectively. We demonstrate here that the weighting functions derived employing the formal solution technique can also be obtained substituting the analytical representations for the direct forward and direct adjoint intensities into corresponding expressions obtained in the framework of the response function technique. The advantages and disadvantages of different techniques are discussed.     

基于二阶离散变分方法的非线性映射参数估计

提出一种估计非线性映射未知参数的二阶离散变分方法.首先针对非线性离散混沌系统, 利用变分方法导出了伴随方程和目标泛函梯度, 以此为基础利用二阶离散变分方法给出了二阶伴随方程和精确计算Hessian矩阵-向量乘积的显式表达式; 其次设计了估计非线性映射未知参数的新算法, 并以此对Hyperhenón映射和二维抛物映射中的未知参数进行了精确的估计. 数值仿真结果表明了该方法的有效性和优点. 关键词： 非线性映射 参数估计 二阶离散变分方法 伴随方程     

Necessary and sufficient conditions for the existence of a Lagrangian in field theory. I. Variational approach to self-adjointness for tensorial field equations

In this paper we identify some of the most significant references on the inverse problem of the calculus of variations for single integrals and initiate the study of the generalization of the underlying methodology to classical field theories. We first classify Lorentz-covariant tensorial field equations into nonlinear, quasi-linear, and semilinear forms, and then introduce their systems of equations of variation and adjoint systems. The necessary and sufficient conditions for the self-adjointness of class $\text{C}$², regular, tensorial, nonlinear, quasi-linear and semilinear forms are worked out. We study the Lagrange equations, their system of equations of variations (Jacobi equations) and their adjoint system by proving that, for class $\text{C}$⁴ and regular Lagrangian densities, they are always self-adjoint. We then introduce a concept of analytic representation which occurs when the Lagrange equations coincide with the field equations up to equivalence transformations and refine the definition by particularizing it as direct or indirect and ordered or nonordered. Some of the conventional cases of tensorial fields are considered and we prove, in particular, that the conventional representation of the complex scalar field in interaction with the electromagnetic field is of the ordered indirect type. For the objective of identifying our program we recall the two classes of equivalence transformations of the Lagrangian densities which are primarily used nowadays, namely, the Lorentz (coordinate) transformations and the gauge transformations (transformations of fields within a fixed coordinate system), and postulate the existence of a third class, which we term isotopic transformations of the Lagrangian density and which consist of equivalence transformations within a fixed coordinate system and gauge. We finally outline the objectives of our program, which essentially consist of the identification of the necessary and sufficient conditions for the existence of a Lagrangian in field theories and their first application to the transformation theory within the framework of our variational approach to self-adjointness.     

Adjoint sensitivity analysis of kinetic,thermochemical, and transport data of nitrogen and ammonia chemistry

Many studies apply sensitivity analysis to explore the impact of reaction kinetic parameters on model predictions. The importance of thermochemical and transport data is often assumed to be relatively low. While this is true for specific combustion properties of hydrocarbons, the role of thermochemical and transport data in combustion processes of nitrogen-containing molecules remains to be investigated. Thus, this work applies adjoint sensitivity analysis to the complete set of parameters in combustion models, i.e., kinetics, thermodynamics, and transport data. This integral approach increases the number of parameters considered in the sensitivity analysis drastically. Compared to forward sensitivity analysis, the adjoint approach is very efficient for a large number of parameters, and analysis with several thousand parameters can be performed in seconds. Nitrogen oxide formation in methane/air flames and laminar burning velocities of ammonia/air flames are considered as prediction targets. Sensitivity analysis results for kinetic, thermochemical, and transport data are compared by jointly considering all appearing parameter uncertainties. The comparison reveals that, due to their importance for the equilibrium constants of elementary reactions, the optimization potential of thermodynamic properties is often similarly high as that of the kinetics parameters. Transport parameters are found to be of the lowest priority for the model development due to their low uncertainties, even though high sensitivities are determined for several of them. More specifically, the analysis for the laminar burning velocities of ammonia/air flames reveals a high optimization potential for parameters in the N₂-amine chemistry, including the molar heat capacities of N₂H₂, N₂H₃, and NH. Interestingly, analyses with different mechanisms reveal strongly diverging results, especially regarding the importance of reactions with OH, which is uncommon when considering the combustion of hydrocarbons.     

任意边界板结构的声辐射模态误差传感器的设计

获取声辐射模态伴随系数是基于声辐射模态理论进行主动结构声控制(ASAC)的重要环节。以往PVDF分布式传感器的设计难点是振速展开受边界条件的限制,其设计过程往往是针对特定边界条件展开的。本文在声辐射模态理论和两维分布式传感器的压电方程的基础上,将板表面振速分布用Legendre多项式展开,给出了两维板结构的PVDF传感器形状与边界条件无关的设计方法。这样设计得到的传感器能应用于任意边界条件和任意振速分布的两维板结构,且实时性好,拓宽了其应用范围。本文还分别以固定边界条件板及在该板中任取一小区域两种情况为例,证明了该设计方法的可行性。