Adjoint modeling for acoustic inversion

The use of adjoint modeling for acoustic inversion is investigated. An adjoint model is derived from a linearized forward propagation model to propagate data-model misfit at the observation points back through the medium to the medium perturbations not being accounted for in the model. This adjoint model can be used to aid in inverting for these unaccounted medium perturbations. Adjoint methods are being applied to a variety of inversion problems, but have not drawn much attention from the underwater acoustic community. This paper presents an application of adjoint methods to acoustic inversion. Inversions are demonstrated in simulation for both range-independent and range-dependent sound speed profiles using the adjoint of a parabolic equation model. Sensitivity and error analyses are discussed showing how the adjoint model enables calculations to be performed in the space of observations, rather than the often much larger space of model parameters. Using an adjoint model enables directions of steepest descent in the model parameters (what we invert for) to be calculated using far fewer modeling runs than if a forward model only were used.     

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.     

Interlocked optimization and fast gradient algorithm for a seismic inverse problem

We give a nonlinear inverse method for seismic data recorded in a well from sources at several offsets from the borehole in a 2D acoustic framework. Given the velocity field, approximate values of the impedance are recovered. This is a 2D extension of the 1D inversion of vertical seismic profiles [18]. The inverse problem generates a large scale undetermined ill-conditioned problem. Appropriate regularization terms render the problem well-determined. An interlocked optimization algorithm yields an efficient preconditioning. A gradient algorithm based on the adjoint state method and domain decomposition gives a fast parallel numerical method. For a realistic test case, convergence is attained in an acceptable time with 128 processors.     

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.     

声波方程数值反演的FR法

本文提出了声波方程反演的FR方法,FR法是一个应用范围很广的波动方程反演方法,它不受空间维数与边界条件的限制,适应于各类波动反问题。文中推导了算法的基本公式,并给出了一维与二维声波反问题的数值算例。     

Oceanographic inverse problems

Oceanographic inverse problems are generalizations of the concept of linear regression to encompass the fitting of numerical models based on the partial-diffential equations of fluid dynamics to oceanic data by adjusting surface fluxes, initial conditions, and/or transport parameters. The limited resolution of the data requires some sort of regularizing assumptions similar to those needed for spatial interpolation. Computationally, they are nonlinear-least-squares problems involving very many variables. These problems can be ill-conditioned due to the indirect nature of the observations, to the way the dynamics enter, and to the effect of the regularizing assumptions. Solving such large optimization problems requires efficient computation of derivatives of functions defined by computational codes. The adjoint method provides derivatives with respect to all the code's many adjustable inputs for roughly the same cost as computing a single model simulation. Although computationally difficult, model fitting offers the promise of dynamically consistent multivariate analyses of oceanographic data.     

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.     

FDFD: A 3D Finite-Difference Frequency-Domain Code for Electromagnetic Induction Tomography

A new 3D code for electromagnetic induction tomography with intended applications to environmental imaging problems has been developed. The approach consists of calculating the fields within a volume using an implicit finite-difference frequency-domain formulation. The volume is terminated by an anisotropic perfectly matched layer region that simulates an infinite domain by absorbing outgoing waves. Extensive validation of this code has been done using analytical and semianalytical results from other codes, and some of those results are presented in this paper. The new code is written in Fortran 90 and is designed to be easily parallelized. Finally, an adjoint field method of data inversion, developed in parallel for solving the fully nonlinear inverse problem for electrical conductivity imaging (e.g., for mapping underground conducting plumes), uses this code to provide solvers for both forward and adjoint fields. Results obtained from this inversion method for high-contrast media are encouraging and provide a significant improvement over those obtained from linearized inversion methods.     

透平叶栅三维粘性气动反问题的控制理论方法

将基于控制理论的形状优化设计方法应用于粘性可压流动条件下的透平叶栅三维气动反设计,详细推导了三维N-S方程伴随系统的偏微分方程组及其各类边界条件.讨论了伴随系统的解的适定性条件,并由此给出应用N-S方程进行气动优化的目标函数的选取限制.研究了伴随方程的数值求解技术,给出敏感性导数的最终计算式,结合拟牛顿算法发展了三维透平叶栅粘性反问题的气动设计方法.     

Numerical developments for short-pulsed Near Infra-Red laser spectroscopy. Part II: inverse treatment

Indirect optical spectroscopy or tomography, that is, mapping of optical properties in scattering and absorption inside a medium given a set of measurements at the boundaries, is highly dependent on the radiative transfer model used to track radiative energy propagation in semi-transparent materials. In the first part of this study, a numerical tool adapted for treating radiative transfer in the frame of short-pulsed laser beam interaction with non-homogeneous matter has been presented. In this paper, it is intended to show how such numerical tools can undergo inversion through adjoint treatment or reverse differentiation.Adjoint models, as well as reverse differentiation, are used in order to allow an efficient computation of the gradient, in the unknown optical parameters space, of an objective or cost function estimating the residual between data obtained at the boundary and predictions by numerical simulations. This gradient is a crucial indication as to update, through line minimization, the set of internal optical properties of the medium.First, the theoretical background of the inverse treatments, both reverse differentiation and adjoint model, for the transient radiative transfer equation model introduced in Part I is developed. Second, different reconstruction configurations are presented. Time-dependent sampling and time filtering effects of the measurements are addressed. Image reconstructions from simulated data are achieved for material phantoms of simple geometry.     

Burgers方程初边值条件的控制

为将优化控制技巧用于复杂的地球物理环流模型及其它领域,以Burgers方程为模型,描述了初边值条件的优化控制.在一般化的意义上,给出了连续问题及其相应的离散形式.引入伴随变量,并由此导出伴随方程.比较初始控制中不同频率误差对优化控制的影响,分析了用有限观测数据作空间插值后的数据、方法的优化控制能力及效果,并进行了数值实验.     

Elastic-wave identification of penetrable obstacles using shape-material sensitivity framework

This study deals with elastic-wave identification of discrete heterogeneities (inclusions) in an otherwise homogeneous “reference” solid from limited-aperture waveform measurements taken on its surface. On adopting the boundary integral equation (BIE) framework for elastodynamic scattering, the inverse query is cast as a minimization problem involving experimental observations and their simulations for a trial inclusion that is defined through its boundary, elastic moduli, and mass density. For an optimal performance of the gradient-based search methods suited to solve the problem, explicit expressions for the shape (i.e. boundary) and material sensitivities of the misfit functional are obtained via the adjoint field approach and direct differentiation of the governing BIEs. Making use of the message-passing interface, the proposed sensitivity formulas are implemented in a data-parallel code and integrated into a nonlinear optimization framework based on the direct BIE method and an augmented Lagrangian whose inequality constraints are employed to avoid solving forward scattering problems for physically inadmissible (or overly distorted) trial inclusion configurations. Numerical results for the reconstruction of an ellipsoidal defect in a semi-infinite solid show the effectiveness of the proposed shape-material sensitivity formulation, which constitutes an essential computational component of the defect identification algorithm.     

The Adjoint Method for an Inverse Design Problem in the Directional Solidification of Binary Alloys

This paper presents a systematic procedure based on the adjoint method for solving a class of inverse directional alloy solidification design problems in which a desired growth velocityv_fis achieved under stable growth conditions. To the best of our knowledge, this is the first time that a continuum adjoint formulation is proposed for the solution of an inverse problem with simultaneous heat and mass transfer, thermo-solutal convection, and phase change. In this paper, the interfacial stability is considered to imply a sharp solid–liquid freezing interface. This condition is enforced using the constitutional undercooling criterion in the form of an inequality constraint between the thermal and solute concentration gradients,GandG_c, respectively, at the freezing front. The main unknowns of the design problem are the heating and/or cooling boundary conditions on the mold walls. The inverse design problem is formulated as a functional optimization problem. The cost functional is defined by the square of theL₂norm of the deviation of the freezing interface temperature from the temperature corresponding to thermodynamic equilibrium. A continuum adjoint system is derived to calculate the adjoint temperature, concentration, and velocity fields such that the gradient of the cost functional can be expressed analytically. The cost functional minimization process is realized by the conjugate gradient method via the finite element method solutions of the continuum direct, sensitivity, and adjoint problems. The developed formulation is demonstrated with an example of designing the directional solidification of a binary aqueous solution in a rectangular mold such that a stable vertical interface advances from left to right with a desired growth velocity.     

变分伴随正则化方法从雷达回波反演海洋波导 Ⅰ.理论推导部分

针对传统统计反演算法在雷达回波反演海洋波导(RFC)方面计算量过大的问题,提出一种变分伴随正则化物理反演算法.在变分伴随方法中分别导出切线性模式、伴随方程及伴随边界条件、伴随方程求解表达式、以及泛函梯度的数学表达形式;讨论了伴随方法求泛函梯度对复方程如何协调的问题.考虑到雷达电磁波传播的特征,选择了适当的正则化项来解决反演中的不适定问题.最后给出反演算法实施中的迭代格式.     

Identification of an optimal derivatives approximation by variational data assimilation

Variational data assimilation technique applied to identification of optimal approximations of derivatives near boundary is discussed in frames of one-dimensional wave equation. Simplicity of the equation and of its numerical scheme allows us to discuss in detail as the development of the adjoint model and assimilation results. It is shown what kind of errors can be corrected by this control and how these errors are corrected. This study is carried out in view of using this control to identify optimal numerical schemes in coastal regions of ocean models.     

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.     

Adjoint-based optimization of PDEs in moving domains

In this investigation we address the problem of adjoint-based optimization of PDE systems in moving domains. As an example we consider the one-dimensional heat equation with prescribed boundary temperatures and heat fluxes. We discuss two methods of deriving an adjoint system necessary to obtain a gradient of a cost functional. In the first approach we derive the adjoint system after mapping the problem to a fixed domain, whereas in the second approach we derive the adjoint directly in the moving domain by employing methods of the noncylindrical calculus. We show that the operations of transforming the system from a variable to a fixed domain and deriving the adjoint do not commute and that, while the gradient information contained in both systems is the same, the second approach results in an adjoint problem with a simpler structure which is therefore easier to implement numerically. This approach is then used to solve a moving boundary optimization problem for our model system.     

利用声带动力学模型参数反演方法进行病变嗓音分类

提出一种声带动力学模型参数反演方法,从发声机理角度对声带病变嗓音进行有效区分。依据声带生理组织和伯努利定律构建声带动力学模型,确定模型优化参数向量,耦合声门气流获取模型声门波;利用迭代自适应逆滤波算法获得实际嗓音声门波作为目标声门波;采用遗传优化算法提出通过匹配目标和模型声门波特征参数实现模型参数反演。实验结果表明,表征声门波的各时频域参数匹配相对误差不超过2%;依据反演所获模型参数提出去除声门下压影响的平均归一化缩放系数,克服声带非对称性特征在区分病变嗓音方面的不足,实现病理嗓音的全面有效区分。      

Extraction of the acoustic component of a turbulent flow exciting a plate by inverting the vibration problem

An improvement of the Force Analysis Technique (FAT), an inverse method of vibration, is proposed to identify the low wavenumbers including the acoustic component of a turbulent flow that excites a plate. This method is a significant progress since the usual techniques of measurements with flush-mounted sensors are not able to separate the acoustic and the aerodynamic energies of the excitation because the aerodynamic component is too high. Moreover, the main cause of vibration or acoustic radiation of the structure might be due to the acoustic part by a phenomenon of spatial coincidence between the acoustic wavelengths and those of the plate. This underlines the need to extract the acoustic part. In this work, numerical experiments are performed to solve both the direct and inverse problems of vibration. The excitation is a turbulent boundary layer and combines the pressure field of the Corcos model and a diffuse acoustic field. These pressures are obtained by a synthesis method based on the Cholesky decomposition of the cross-spectra matrices and are used to excite a plate. Thus, the application of the inverse problem FAT that requires only the vibration data shows that the method is able to identify and to isolate the acoustic part of the excitation. Indeed, the discretization of the inverse operator (motion equation of the plate) acts as a low-pass wavenumber filter. In addition, this method is simple to implement because it can be applied locally (no need to know the boundary conditions), and measurements can be carried out on the opposite side of the plate without affecting the flow. Finally, an improvement of FAT is proposed. It regularizes optimally and automatically the inverse problem by analyzing the mean quadratic pressure of the reconstructed force distribution. This optimized FAT, in the case of the turbulent flow, has the advantage of measuring the acoustic component up to higher frequencies even in the presence of noise.     

Design sensitivity analysis for sequential structural-acoustic problems

A design sensitivity analysis of a sequential structural-acoustic problem is presented in which structural and acoustic behaviors are de-coupled. A frequency-response analysis is used to obtain the dynamic behavior of an automotive structure, while the boundary element method is used to solve the pressure response of an interior, acoustic domain. For the purposes of design sensitivity analysis, a direct differentiation method and an adjoint variable method are presented. In the adjoint variable method, an adjoint load is obtained from the acoustic boundary element re-analysis, while the adjoint solution is calculated from the structural dynamic re-analysis. The evaluation of pressure sensitivity only involves a numerical integration process for the structural part. The proposed sensitivity results are compared to finite difference sensitivity results with excellent agreement.