变分伴随正则化方法从雷达回波反演海洋波导(Ⅱ):实际反演试验

在理论推导的基础上,分别利用模拟和实测的雷达回波资料进行反演试验.在模拟雷达回波资料反演时,分别对不引入正则化项和引入正则化项的反演结果进行讨论;在实际雷达回波资料反演时,讨论了反演结果对初猜值精度的依赖性,分别考虑初猜值为经验模型给定时和初猜值为统计反演结果时对最后反演精度的影响.最后比较了变分伴随正则化方法与传统统计反演算法的优缺点,指出下一步反演算法的改进方向.     

垂直天线阵观测信息反演大气折射率廓线

利用垂直天线阵观测得到的电磁场信息建立目标泛函,从电磁波抛物方程传播模式出发,利用偏微分方程最优控制中的伴随方法研究大气折射率廓线反演问题. 针对反演的不适定性,采用正则化思想对目标泛函进行改造,并根据变分同化思想构建反演迭代格式. 数值模拟试验验证了理论算法的可行性. 由于采用分步Fourier算法求解电磁波抛物方程传播模式和伴随模式过程中产生的固有误差随着传播距离的增加累积增大,反演廓线在传播距离较远时不能收敛于观测廓线. 在这种情况下,引入较好的初猜廓线和背景场可以有效地改进反演结果. 关键词： 大气折射率 电磁波抛物方程 变分伴随 正则化     

中子深度分析反演算法研究

针对中子深度分析技术,研究几类反演算法：概率迭代法、奇异值分解求解最小二乘法、线性正则化方法及约束线性正则化法.开展相关数值实验,分别在欠定方程、超定方程情况下,对反演算法的结果进行比较.一般情况下几种算法都能得到较为理想的结果,概率迭代法和约束线性正则化法由于有迭代过程,因此在离子源强度发生阶跃处得到的结果并不理想.在超定方程情况下,当选取的反演范围较为随意时,线性正则化方法可能使得解的稳定性较差,而约束线性正则化法能够很好地抑制求解过程中的不稳定性.对实际测量能谱进行反演时,由于能谱有随机误差,线性正则化方法不能很好地抑制误差的影响,反演结果振荡较强,其余方法的结果与参考值符合很好.     

基于逆散射理论的地震波速度正则化反演

本文基于逆散射理论利用正则化有限差分对比源反演算法对地震波传播速度进行反演, 该方法是基于波动方程的频率域波形反演算法, 利用非线性共轭梯度法, 通过最小化目标优化函数不断迭代更新速度模型. 由于地球物理反演问题的病态性和不稳定性, 通过基于反演参数总变差的正则化处理, 使反演问题变为良性问题且算法具有较强的抗噪声干扰能力. 反演过程中使用了频率-空间域9点差分正演算子以及PML吸收边界条件. 与其他反演算法相比, 由于背景模型在反演迭代过程中保持不变, 可以避免在每次迭代过程中重新构造正演算子及矩阵分解等相关计算过程, 使得该算法非常适合于大规模三维反演计算. 此外, 本文采用基于MPI的并行计算, 进一步提高了反演计算的效率. 二维CSEG模型反演结果表明该方法可以反演得到高分辨率的地震波速度重建结果, 为地震勘探数据处理及解释提供准确的速度信息.     

基于变分方法的有限区域风场分解与重构I: 理论框架和仿真实验

众所周知, 风场分解与重构最有效的方法就是引入速度势和流函数, 其一般通过求解两个Poisson 方程得到. 由于速度势和流函数在边界上的耦合性质,有限区域风场分解是不唯一的, 这对风场分解带来了很大困难. 本文采用变分伴随结合正则化方法来克服风场分解的不唯一性, 其核心是把速度势和流函数的边值作为控制变量来反演. 目标泛函由两部分组成, 一是衡量重构风场误差大小的观测项; 二是保证风场分解问题适定的正则化项, 其目的在于寻求具有气象意义的稳定正则化解. 数值试验结果表明, 在正确选取正则化参数后, 利用变分伴随结合正则化方法进行有限区域风场分解与重构是有效可行的.     

扩展卡尔曼滤波和不敏卡尔曼滤波在实时雷达回波反演大气波导中的应用

为了改善雷达回波反演大气波导(RFC)方面存在的单时次、单方位角反演的问题,提出利用扩展卡尔曼滤波和不敏卡尔曼滤波的反演算法对大气波导结构的多方位角实时跟踪反演. 在卡尔曼滤波方法中分别给出大气波导结构的参数化方程、观测方程、滤波算法的状态转移方程,最后导出滤波反演算法的迭代求解流程. 在大气波导结构不随时间变化和随时间变化的两种条件下,对扩展卡尔曼滤波和不敏卡尔曼滤波算法进行数值实验. 实验结果表明,不敏卡尔曼滤波更适用于RFC这高度非线性反演问题,它可能今后为大气波导结构多方位角实时跟踪反演的业务化运行提供理论基础与技术保证. 关键词： 大气波导 雷达回波 扩展卡尔曼滤波 不敏卡尔曼滤波     

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     

厄尔尼诺和南方涛动海气耦合模型中参数估计的变分方法

提出一种基于变分原理估计厄尔尼诺和南方涛动海气耦合模型中未知参数的方法. 首先将所研究的非线性海气耦合动力方程引入到目标泛函中; 接着利用变分方法导出伴随方程和待辨识参数泛函梯度的公式; 然后设计了估计未知参数的算法.数值试验结果表明变分方法是一 种能有效估计海气耦合非线性系统未知参数的方法.     

导热几何形状反演的变分原理及边界条件的确立

本文采用变域变分原理,建立了导热几何形状反演问题的变分原理,同时获得了该问题所需满足的边界条件和附加条件.该变分原理能将未知形状的几何变量及控制方程结合在一个变分泛函中,使得数学描述简洁、紧凑,且几何变量及控制方程的求解能耦合地进行.介绍了运用该变分原理并结合有限元方法进行数值计算的方法. 关键词： 几何形状反演 变分原理 有限元 导热     

合成孔径雷达资料反演海面风场的正则化方法研究

针对基于散射计地球物理模型函数的合成孔径雷达资料反演海面风场中背景风向存在很大误差的问题,提出了利用正则化方法反演海面风场,并通过L曲线准则对正则化参数进行最优选择,然后对该方法进行仿真试验.试验结果显示,正则化方法反演海面风场的风速和风向的反演精度均优于变分同化方法,分析不同真实风向情况下的反演精度表明,真实风向在45°和135°附近时,风速反演的精度高,真实风向在0°和90°时,风向反演的精度高,这种现象与地球物理模型函数中后向散射截面相对于风向的导数相关. 关键词： 合成孔径雷达 正则化 L曲线准则 变分同化     

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.     

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.     

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.     

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.     

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.     

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.     

Adjoint wall functions: A new concept for use in aerodynamic shape optimization

The continuous adjoint method for the computation of sensitivity derivatives in aerodynamic optimization problems of steady incompressible flows, modeled through the k–ε turbulence model with wall functions, is presented. The proposed formulation leads to the adjoint equations along with their boundary conditions by introducing the adjoint to the friction velocity. Based on the latter, an adjoint law of the wall that bridges the gap between the solid wall and the first grid node off the wall is proposed and used during the solution of the system of adjoint (to both the mean flow and turbulence) equations. Any high Reynolds turbulence model, other than the k–ε one used in this paper, could also profit from the proposed adjoint wall function technique. In the examined duct flow problems, where the total pressure loss due to viscous effects is used as objective function, emphasis is laid on the accuracy of the computed sensitivity derivatives, rather than the optimization itself. The latter might rely on any descent method, once the objective function gradient has accurately been computed.     

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.     

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。     

An Effective Method for Seeking Conservation Laws of Partial Differential Equations

This paper introduces an effective method for seeking local conservation laws of general partial differential equations （PDEs）. The well-known variational principle does not involve in this method. Alternatively, the conservation laws can be derived from symmetries, which include t/he symmetries of t/he associated linearized equation of t/he PDEs, and the adjoint symmetries of the adjoint eqUation of the PDEs.