Adjoint-based sensitivity analysis of flames

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

Viscous attenuation can have a strong impact on seismic wave propagation, but it is rarely taken into account in full waveform inversion (FWI). When viscoelasticity is considered in time domain FWI, the displacement formulation of the wave equation is usually used instead of the popular velocity–stress formulation. However, inversion schemes rely on the adjoint equations, which are quite different for the velocity–stress formulation than for the displacement formulation. In this paper, we apply the adjoint state method to the isotropic viscoelastic wave equation in the velocity–stress formulation based on the generalized standard linear solid rheology. By applying linear transformations to the wave equation before deriving the adjoint state equations, we obtain two symmetric sets of partial differential equations for the forward and adjoint variables. The resulting sets of equations only differ by a sign change and can be solved by the same numerical implementation. We also investigate the crosstalk between parameter classes (velocity and attenuation) of the viscoelastic equation. More specifically, we show that the attenuation levels can be used to recover the quality factors of P and S waves, but that they are very sensitive to velocity errors. Finally, we present a synthetic example of viscoelastic FWI in the context of monitoring CO₂ geological sequestration. We show that FWI based on our formulation can indeed recover P- and S-wave velocities and their attenuation levels when attenuation is high enough. Both changes in velocity and attenuation levels recovered with FWI can be used to track the CO₂ plume during and after injection. Further studies are required to evaluate the performance of viscoelastic FWI on real data.

For low-aspect-ratio turbine blades secondary loss reduction is important for improving performance. This paper presents the application of a viscous adjoint method to reduce secondary loss of a linear cascade. A scalable wall function is implemented in an existing Navier-Stokes flow solver to simulate the secondary flow with reduced requirements on grid density. The simulation result is in good agreement with the experimental data. Entropy production through a blade row is used as the objective function in the optimization of blade redesign and endwall contouring. With the adjoint method, the complete gradient information needed for optimization can be obtained by solving the governing flow equations and their corresponding adjoint equations only once, regardless of the number of design parameters. Three design cases are performed with a low-aspect-ratio steam turbine blade tested by Perdichizzi and Dossena. The results demonstrate that it is feasible to reduce flow loss through the redesign of the blade while maintaining the same mass-averaged turning angle. The effects on the profile loss and secondary loss due to the geometry modification of stagger angle, blade shape and endwall profile are presented and analyzed.

The adjoint method with dynamic checking point scheme is used to enable the sensitivity analysis capability of GEBT (Geometrically Exact Beam Theory), a general-purpose code for nonlinear analysis of composite beams. Such a capability can meet the design challenges associated with future air vehicles featuring highly-flexible slender components made of composites. The adjoint method computes the gradient of a user defined objective function to arbitrarily many design and control parameters by backpropagating the sensitivity information through the physics of a simulation. As a first step, we implemented a simplified version of GEBT using the Chebyshev spectral method in a code named Hydra, which solves the nonlinear integral-differential governing equation governing nonlinear bending, together with its adjoint equation. An example in optimal control is used to demonstrate the efficiency of this method in performing very high dimensional optimization problems in nonlinear beam dynamics.

The purpose of this paper is to present a parameter identification method to determine the force of a blast and the elastic modulus of the ground using the measurements of a dynamic elastic wave, the adjoint equation method of optimal control theory, and the finite element method. Before the excavation of rocky ground, it is important to estimate the ground properties. In this paper, the elastic modulus is determined as the performance function is minimized using a technique based on the first‐order adjoint method. The performance function is a square sum of the discrepancies between the computed and the observed values of the velocities. After the determination of the magnitude of the blasting force, we can determine the elastic modulus of the rock. As the basic equation to calculate the velocities of dynamic elastic body, elastic equilibrium equations with linear viscosity are employed. The adjoint equation method has been utilized in order to calculate the gradient of the performance function with respect to the parameters. The gradient of the performance function is calculated using the first‐order adjoint equation. The weighted gradient method is applied for minimization. In order to solve the state equations in space and time, the finite element method and the Newmark \documentclass{article}\usepackage{amssymb}\footskip=0pc\pagestyle{empty}\begin{document}$\frac{1}{4}$\end{document} method are used. In this paper, we tested the practical application of our proposed method for determination of the elastic modulus of rock at the Ikawa tunnel located in the Tokushima prefecture, Japan. Copyright © 2009 John Wiley & Sons, Ltd.

A shape reconstruction method for geophys- ical objects by temperature measurements is presented which uses adjoint equations and a level set function approach. Temperature is measured on subdomains, e.g., representing boreholes. This information is used to reconstruct the shape of the geophysical layers. For this purpose, shape optimization techniques are applied. The method uses a representation of the layers by a so-called level set function. The evolution of this level set function is then used to determine the optimal shape. The "speed" of the evolution is computed using adjoint equations. Synthetic examples demonstrate the use of the inverse method and its behavior in different configurations.

Effective adjoint approaches for computational fluid dynamics

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.

The optimum planning of industrial pollutant sources, which optimizes the economic object without violating environmental constraints, is an important and hard task to be conquered. In this paper, an adjoint method is developed to solve the problem. The penalty function is introduced to deal with the environmental inequality constraints, and Lagrange function is constructed to derive the adjoint equation and the gradient of the object function. In this means, the gradient of the object function can be calculated by solving the adjoint equation, and the information from the gradient is used to make the object function descend and approach to an optimal solution after some iterations. A two-dimensional, simplified model is used for numerical experiments. The theoretical derivations are verified by the results of the experiments. Furthermore, the adjoint method is shown to be of excellent convergence and efficiency, which is adaptive to the fast de-velopment of air quality numerical models and super computers.

Summary Viscous attenuation can have a strong impact on seismic wave propagation, but is rarely taken into account in full waveform inversion (FWI). In the time domain, when viscoelasticity is considered, the displacement formulation of the wave equation is usually used. However, the adjoint state equations are quite different for the velocity-stress formulation than for the displacement formulation. In this paper, we derive the adjoint state equations for the viscoelastic wave equation based on the velocity-stress formulation. Using a modified definition of the memory variables commonly found in the literature, we define a Lagrangian from which the adjoint state equations and the misfit gradient are derived. The resulting expressions are similar to the displacement formulation, but differ by the source term and by the wavefield cross-correlations giving the misfit gradient. To validate our results, the misfit gradient obtained by the adjoint state method is compared to the misfit gradient calculated by finite-difference for a simplified problem, giving an excellent agreement. In short, this work gives the right adjoint state equations for the velocity-stress formulation, which is commonly used for time-domain viscoelastic modeling. Further studies are required to evaluate the performance of this approach in real FWI viscoelastic experiments.

An add-on module for SINTEF's Matlab Reservoir Simulation Toolbox. Uses Proper Orthogonal Decomposition and Trajectory Piecewise Linearization to perform production optimization, history matching and closed-loop reservoir management for reservoir simulation. One simple OW example and Egg model examples are included in order to test the module. Needs to be used in conjunction with MRST (tested with version 2014b)

In this article, a new methodology for sensitivity analysis evaluation of electric potential distribution near transmission lines (TLs) is proposed. The potential is modeled by the finite element method, and the sensitivity related to cable positions is obtained using the adjoint method, and thus the objective function’s sensitivity is adopted during the optimization process using a multiple cuts’ ellipsoid method to get new geometries with high surge impedance loading (SIL). The sensitivities of a large number of conductors by phase are obtained with high precision and very low computational cost, using adjoint sensitivities, which are not dependent on the number of design variables. The approximation using central finite differences (CFDs) to calculate the sensitivity is used for comparisons and validation. Therefore, with this methodology, the finite element method (FEM) model of TLs can be used during the optimization process without compromising the required computational processing time.

Inverse problem solution is an integral part of data interpretation for well testing in petroleum reservoirs. In case of two-phase well tests with water injection, forward problem is based on the multiphase flow model in porous media and solved numerically. The inverse problem is based on a misfit or likelihood objective function. Adjoint methods have proved robust and efficient for gradient calculation of the objective function in this type of problems. However, if time-lapse electrical resistivity measurements during the well test are included in the objective function, both the forward and inverse problems become multiphysical, and straightforward application of the adjoint method is problematic. In this paper we present a novel adjoint algorithm for the inverse problems considered. It takes into account the structure of cross dependencies between flow and electrical equations and variables, as well as specifics of the equations (mixed parabolic-hyperbolic for flow and elliptic for electricity), numerical discretizations and grids, and measurements in the inverse problem. Decomposition is proposed for the adjoint problem which makes possible step-wise solution of the electric adjoint equations, like in the forward problem, after which a cross-term is computed and added to the right-hand side of the flow adjoint equations at this timestep. The overall procedure provides accurate gradient calculation for the multiphysical objective function while preserving robustness and efficiency of the adjoint methods. Example cases of the adjoint gradient calculation are presented and compared to straightforward difference-based gradient calculation in terms of accuracy and efficiency.

Since the traditional anti-statistical algorithms in the inversion of refractivity from radar clutter （RFC） need too much calculation, a new physical algorithm （variation adjoint combined with the regularization method） is proposed instead. In the variation adjoint method, the tangent model, adjoint equations and accompanying boundary conditions, the solving expression of adjoint equations, and the mathematical functional expression of the gradient were derived respectively, and the problem of how to coordinate functional gradient of the complex equations is solved. In the regularization method, to take into account the characteristics of radar electromagnetic wave propagation, we choose a suitable regularization term to solve the inversion of the ill-posed problems. Finally, the implementation of the iterative inversion algorithm was derived.

This paper concerns a numerical optimization method for designing airfoils based on adjoint method. The goal of present work is to reduce the compressibility drag or pitching moment of transonic airfoils without compromising on the lift coefficient. A new cost function based on this requirement is defined and the corresponding adjoint equations are discussed in details. At the end, by demonstrating some numerical results, we show that this technique is capable of converging to the optimum design point corresponding to the initial geometry of the airfoil.