Continuous Adjoint Method Research Articles

Effect of grid displacement models on sensitivity derivatives computed by the continuous adjoint method in aerodynamic and conjugate heat transfer shape optimization

ABSTRACT For aerodynamic and conjugate heat transfer problems, the work reported in this article investigated the effect of the grid displacement model on the derivatives of an objective function with respect to the design variables. A comparison was made in the first step of the study between reference sensitivity derivatives computed by finite differences and different grid displacement models (volumetric B-splines, Laplace partial differential equations, Delaunay graphs and inverse distance weighting). Then, a continuous adjoint method including the adjoint to the Laplace grid displacement model was used to recompute sensitivity derivatives. To shed more light and to support the authors' findings, an alternative continuous adjoint method for computing the objective function gradient in terms of the field integrals of grid sensitivities was also formulated and programmed. Finally, the effect of an inconsistent differentiation of the grid displacement model on the optimized solution was investigated. The choice of the grid displacement model did not affect the sensitivity derivatives in any of the cases examined.

A complexity-driven framework for waveform tomography with discrete adjoints

Summary We develop a new approach for computing Fréchet sensitivity kernels in full waveform inversion by using the discrete adjoint approach in addition to the widely used continuous adjoint approach for seismic waveform inversion. This method is particularly well suited for the forward solver AxiSEM3D, a combination of the spectral-element method (SEM) and a Fourier pseudo-spectral method, which allows for a sparse azimuthal wavefield parametrization adaptive to wavefield complexity, leading to lower computational costs and better frequency scaling than conventional 3-D solvers. We implement the continuous adjoint method to serve as a benchmark, additionally allowing for simulating off-axis sources in axisymmetric or 3-D models. The kernels generated by both methods are compared to each other, and benchmarked against theoretical predictions based on linearized Born theory, providing an excellent fit to this independent reference solution. Our verification benchmarks show that the discrete adjoint method can produce exact kernels, largely identical to continuous kernels. While using the continuous adjoint method we lose the computational advantage and fall back on a full-3-D frequency scaling, using the discrete adjoint retains the speedup offered by AxiSEM3D. We also discuss the creation of a data-coverage based mesh to run the simulations on during the inversion process, which would allow to exploit the flexibility of the Fourier parametrization and thus the speedup offered by our method.

An Iterative Method for Shape Optimal Design of Stokes–Brinkman Equations with Heat Transfer Model

This work is concerned with the shape optimal design of an obstacle immersed in the Stokes–Brinkman fluid, which is also coupled with a thermal model in the bounded domain. The shape optimal problem is formulated and analyzed based on the framework of the continuous adjoint method, with the advantage that the computing cost of the gradients and sensitivities is independent of the number of design variables. Then, the velocity method is utilized to describe the domain deformation, and the Eulerian derivative for the cost functional is established by applying the differentiability of a minimax problem based on the function space parametrization technique. Moreover, an iterative algorithm is proposed to optimize the boundary of the obstacle in order to reduce the total dissipation energy. Finally, numerical examples are presented to illustrate the feasibility and effectiveness of our method.

Influence of shroud profiling on the compressor diffuser: Frozen-eddy approach and mode decomposition

The stability prediction and unsteady-flow characteristic explanations of centrifugal compressors mostly focus on radial vaneless diffusers with parallel walls (PWs). However, vaneless diffusers with curved shroud profiling are more often adopted in a practical compression system owing to their better performance or larger stability margin. In this work, the stability properties of the unsteady flow in the radial vaneless diffusers with parallel walls and curved shroud profiling were studied in the context of numerical simulations and global modes. First, the unsteady simulations of a full annular compressor, including vaneless diffusers with parallel walls and constant area walls, were carried out. The simulation results were validated against experimental data, and the dominant modes of the diffuser flow under stall conditions were captured using dynamic mode decomposition (DMD) analysis. Then, a frozen eddy stability analysis was performed on the vaneless diffusers, and the global mode associated with the flow perturbations was investigated. Finally, a continuous adjoint method was adopted to reveal the receptivity of the instability perturbations. By focusing on the effect of the curved shroud profiling on the stability characteristics, the global and adjoint eigenmodes corresponding to the onset of instability were used to provide a physical explanation for the stall mechanism. The simulation results indicated that the stability margin for diffuser with constant area walls was 2.15% wider than that with PWs. The DMD and global stability-analysis results showed that the diffuser with constant area walls had a stall frequency of 298 Hz, which was larger than that of diffuser with PWs. But both cases shared the same stall number of 4. The value of growth rate came to its minimum at 30% span and maximum at 95% span in both cases, indicating the dominant flow fields were near the shroud walls. For the vaneless diffuser with PWs, the reverse flow at 95% span within the radius range of r = 1.3–1.4 was the dominant reason for the occurrence of instability with large value of direct and adjoint mode. Meanwhile, flow mode indicated that flow instability of the diffuser with constant area walls was caused by the reverse flow at 90% span with the radius range of r = 1.02–1.1.

Shape Optimization of a Two-Fluid Mixing Device Using Continuous Adjoint

In this paper, the continuous adjoint method is used for the optimization of a static mixing device. The CFD model used is suitable for the flow simulation of the two miscible fluids that enter the device. The formulation of the adjoint equations, which allow the computation of the sensitivity derivatives is briefly demonstrated. A detailed analysis of the geometry parameterization is presented and a set of different parameterization scenarios are investigated. In detail, two different parameterizations are combined into a two-stage optimization algorithm which targets maximum mixture uniformity at the exit of the mixer and minimum total pressure losses. All parameterizations are in conformity with specific manufacturability constraints of the final shape. The non-dominated front of optimal solutions is obtained by using the weighted sum of the two objective functions and executing a set of optimization runs. The effectiveness of the proposed synthetic parameterization schemes is assessed and discussed in detail. Finally, a reduced length mixer is optimized to study the impact of the length of the tube on the device’s performance.

Optimization of a static mixing device using the continuous adjoint to a two-phase mixing model

The continuous adjoint method is formulated and utilized for the optimization of a static mixing device. The CFD tool used for the simulations is based on a two-phase model governing flows of two miscible fluids. The formulation of the corresponding continuous adjoint problem is presented and the computed gradients are utilized in an optimization loop. In specific, a multi-objective optimization problem is formulated and solved for maximum mixture uniformity at the outlet and minimum total pressure losses inside a static mixing device. The weighted sum of these two quantities of interest is the objective function to be minimized by solving a single-objective problem. Through the solution of a number of optimization problems, with different weights each, the Pareto front of optimal solutions is computed. Two optimization approaches are employed taking the manufacturability of the final shape into consideration, giving rise to different optimal designs to be discussed and compared. Differences in the efficiency and the optimal shapes between the two approaches are thoroughly discussed and compared.

Rate optimization of fractional flow reservoir model based on the continuous adjoint method

In this paper, continuous adjoint method is applied to research the rate optimization of fractional flow reservoir model. The continuous adjoint equations are derived by the variational theory and the stable numerical schemes of adjoint equations are obtained through numerical methods. Firstly, the continuous adjoint equations are linear equations and evolve from the end simulation time. The convection velocity direction of the continuous adjoint saturation equation is opposite to the convection direction of the saturation equation. Secondly, the total simulation time has significant impact on the adjoint variables and the adjoint gradient. After water breakthrough in production well, the adjoint variables exhibit spatial variety that gradually changes from production well to injection well. With this observation, we find that this variety is related to the fractional flow function and its derivative value. The sharp change of fractional flow function and its derivative at the end of the simulation time can lead to large fluctuation of the adjoint gradient along the time step. Thirdly, the influence of time interval on optimization results is studied. Increasing the number of time steps will increase the optimal value, but the value of the difference between optimal and initial value will not increase significantly. When the variable number of time steps is adopted, this difference decreases as the number of time steps increases, which indicates that the coarse time step optimization can provide a better initial control value for the fine time step optimization. An adaptive time step from coarse step to fine step is thus recommended for optimization.

Hydrodynamic Shape Optimization of an Autonomous Underwater Vehicle With an Integrated Lifting Line and Viscous Continuous Adjoint Approach

The designs of the optimal shape of an autonomous underwater vehicle (AUV) and its aft propeller's optimal circulation can interact with each other. We propose an optimization approach that integrates the lifting line theory and the continuous adjoint method to address this conflict. The optimal circulation of a propeller is obtained via lifting line theory. The corresponding propeller force is introduced into the Reynolds-averaged flow equations as a source term and is considered in the development of a continuous adjoint formulation for the AUV shape optimization. Adjoint equations, with boundary conditions and sensitivity derivative expressions for the drag reduction problem, are derived in detail. The shapes of a small AUV and its aft propeller are simultaneously optimized using the proposed approach. Numerical results show that the proposed method decrease AUV drag by 17% at a rated speed. And a prototype test further confirms the effectiveness of the method.

The unsteady continuous adjoint method for minimizing flow-induced sound radiation

The unsteady continuous adjoint method for minimizing flow-induced sound radiation

The unsteady continuous adjoint method for minimizing flow-induced sound radiation

This paper develops the unsteady continuous adjoint method for aeroacoustic optimization problems governed by time-dependent turbulent flows. To predict the flow-induced sound radiation from a body in free-stream, an incompressible Improved Delayed Detached Eddy Simulation is firstly performed. The generated noise is then propagated with the Kirchhoff Integral method, which uses the pressure distribution on the body surface to compute the sound pressure at selected receivers. The continuous adjoint method for the aforementioned process is developed and presented for the first time. The differentiated Kirchhoff Integral is used to compute the boundary condition of the adjoint velocity on the body surface and, then, the unsteady adjoint equations are solved backwards in time. The time window during which the simulation is performed and the one during which the objective function is evaluated do not coincide. This is reflected on the adjoint boundary conditions along the body and the time integration of the sensitivity derivatives. To ensure the consistency of the continuous adjoint-based gradients, grid sensitivities are taken into account. By incorporating the grid displacement equations during the mathematical development of the adjoint method, the surface integral of the residuals of the flow PDEs, which is commonly omitted, is replaced with a volume integral of the grid sensitivities and gives rise to the adjoint grid displacement equations along with an accurate sensitivity derivatives expression. The proposed method is verified against finite differences in a 3D turbulent flow around a cylinder and, then, applied to a real-world test case, concerned with the flow-induced sound radiation of the side mirror of the generic SAE vehicle.

Adjoint characteristic decomposition of one-dimensional waves

Adjoint methods enable the accurate calculation of the sensitivities of a quantity of interest. The sensitivity is obtained by solving the adjoint system, which can be derived by continuous or discrete adjoint strategies. In acoustic wave propagation, continuous and discrete adjoint methods have been developed to compute the eigenvalue sensitivity to design parameters and passive devices (Aguilar et al., 2017, [1]). In this short communication, it is shown that the continuous and discrete adjoint characteristic decompositions, and Riemann invariants, are connected by a similarity transformation. The results are shown in the Laplace domain. The adjoint characteristic decomposition is applied to a one-dimensional acoustic resonator, which contains a monopole source of sound. The proposed framework provides the foundation to tackle larger acoustic networks with a discrete adjoint approach, opening up new possibilities for adjoint-based design of problems that can be solved by the method of characteristics.

PFOSM: An efficient algorithm for aerodynamic robust design based on continuous adjoint and matrix-vector products

This article proposes a novel robust design tool based on the Method of Moments (MoM), demonstrated in problems governed by laminar flows of incompressible fluids. The objective function of the robust design problem consists of the weighted sum of the mean value and standard deviation of the Quantity of Interest (QoI), computed based on the value and gradient of the latter with respect to the uncertain variables. First-order derivatives with respect to the uncertain and design variables are computed using the continuous adjoint method. Using a first-order MoM to compute the objective function and by minimizing it with a gradient-based approach leads to the need of computing up to second-order derivatives of the QoI with respect to the design and uncertain variables. This can lead to a high computational cost in problems with a great number of uncertain variables. The proposed robust design algorithm avoids computing second-order derivatives by computing matrix-vector products instead, using a combination of adjoint and direct differentiation. This makes the cost per gradient computation independent of the numbers of both the design and uncertain variables. The proposed method is demonstrated through the constrained shape optimization of an isolated 2D airfoil under uncertain farfield flow conditions.

Shape inverse problem for Stokes–Brinkmann equations

In this paper, we propose an imaging technique for the detection of porous inclusions in a stationary flow governed by Stokes–Brinkmann equations. We introduce the velocity method to perform the shape deformation, and derive the structure of shape gradient for the cost functional based on the continuous adjoint method and the function space parametrization technique. Moreover, we present a gradient-type algorithm to the shape inverse problem. The numerical results demonstrate the proposed algorithm is feasible and effective for the quite high Reynolds numbers problems.

Sensitivity assessment of optimal solution in aerodynamic design optimisation using SU2

Computational fluid dynamics has become the method of choice for aerodynamic shape optimisation of complex engineering problems. However, the sensitivity of the final aerodynamic shape to numerical parameters has been largely underestimated to date. The purpose of this work is to investigate the influence that numerical parameters have on the optimisation results for two aerofoil problems (NACA 0012 and RAE 2822) in transonic flow, and to provide compact guidelines for best practice. Numerical parameters include: a) two parameterisation methods, Hicks–Henne bump functions and free-form deformation; b) numerical settings related to the tuning of each parameterisation method; and c) closure coefficients of Spalart–Allmaras (SA) turbulence model. All optimisations were performed using the open-source software tool SU2, and gradients were computed using the continuous adjoint method. It was found that: a) the optimisation result of NACA 0012 aerofoil exhibits strong dependence on all numerical parameters investigated, whereas the optimal design of RAE 2822 aerofoil is insensitive to those numerical settings; b) the degree of sensitivity reflects the difference in the design space, particularly of the local curvature on the optimised shape; c) the closure coefficients of SA model affect the final optimisation performance, raising the need for a good calibration of the turbulence model.

The continuous adjoint method for shape optimization in Conjugate Heat Transfer problems with turbulent incompressible flows

In this paper, the development of the continuous adjoint method for shape optimization in Conjugate Heat Transfer (CHT) problems with incompressible flows is presented. The main focus of this study is to develop a continuous adjoint method which computes accurate sensitivity derivatives by (a) extending the adjoint to the Spalart-Allmaras turbulence model, by taking into consideration additional contributions arising from the differentiation of the heat transfer equation valid in the flow domain and (b) extending the Enhanced-Surface Integral adjoint formulation, recently introduced by the authors’ group for single-disciplinary problems, to CHT ones. The so-computed sensitivities are compared to the outcome of finite differences and alternative adjoint formulations in 2D turbulent flows. The proposed method is firstly validated in two simple cases and, then, used to perform the CHT optimization of a 2D internally cooled turbine blade and a 3D car piston-engine cylinder head.

Application of continuous adjoint method to steady-state two-phase flow simulations

Verification, validation and uncertainty quantification (VVUQ) have become a common practice in thermal-hydraulics analysis. An important step in the uncertainty analysis is the sensitivity analysis of various uncertainty input parameters. The common approach for computing the sensitivities, e.g. variance-based and regression-based methods, requires solving the governing equation multiple times, which is expensive in terms of computational effort. An alternative approach for computing the sensitivities is the adjoint method. The cost of solving an adjoint equation is comparable to the cost of solving the governing equation. Once the adjoint solution is obtained, the sensitivities of various parameters can be obtained with little effort. However, successful adjoint sensitivity analysis of the two-phase flow is rare. In this work, an adjoint sensitivity analysis framework is developed for the two-phase two-fluid model based on a new upwind numerical solver. The adjoint sensitivity analysis framework is tested with a steady-state boiling pipe problem. Results show that the adjoint sensitivity analysis framework is working as expected. The sensitivities obtained with the adjoint method are verified by the sensitivities obtained with a forward method.

Statistical evaluation of performance impact of manufacturing variability by an adjoint method

Statistical evaluations of the aerodynamic performance changes due to the manufacturing variability for a turbine blade using the first order (FO) and second order (SO) sensitivities are presented in the paper. On the premise that the geometric variation at each scanning point on the blade meets a standard normal distribution, a modeling method taking into account the spatial correlations is used to produce the contour of geometric variations, the basis modes of which are extracted by principal component analysis. Then the geometric variations of any manufactured blades can be described by a series of random variables regarding as the weights of the basis modes. Calculations of the FO and SO sensitivities by using a continuous adjoint method are firstly introduced in detail. The sensitivities of mass flow rate and adiabatic efficiency of the turbine blade to a finite number of primary basis modes are then calculated and sensitivity validations are presented, revealing the significant improvements on performance evaluations by the SO sensitivities. Finally, the statistics of performance changes due to the manufacturing variability are evaluated by Monte Carlo simulations with different probability density functions given for the weights of basis modes. The results demonstrate that by the FO sensitivities, the performance exhibits almost linear changes versus the manufacturing tolerance, while by the SO sensitivities the nonlinear dependence of performance on the geometric variations can be accurately evaluated.

On the use of the continuous adjoint method to compute nongeometric sensitivities

SummaryThis work explores an alternative approach to computing sensitivity derivatives of functionals, with respect to a broader range of control parameters. It builds upon the complementary character of Riemann problems that describe the Euler flow and adjoint solutions. In a previous work, we have discussed a treatment of the adjoint boundary problem, which made use of such complementarity as a means to ensure well‐posedness. Here, we show that the very same adjoint solution that satisfies those boundary conditions also conveys information on other types of sensitivities. In essence, then, that formulation of the boundary problem can extend the range of applications of the adjoint method to a host of new possibilities.

A continuous adjoint-based aeroacoustic shape optimization for multi-mode duct acoustics

A multi-mode adjoint-based optimization method is proposed for the noise reduction optimization in multi-mode duct acoustics problems. The objective is to minimize the amplitude of sound from an inlet duct on the wall and integral line while maintaining the aerodynamic performance. The complete detailed derivation of the adjoint equations and their corresponding adjoint boundary conditions are presented firstly based on the multi-mode linear Euler equations. With the solved adjoint variables, the final expression of the cost function gradient with respect to the design variables is formulated. The sensitivity derivative computed by the continuous adjoint method is validated by comparing with that obtained using finite difference method. Up to 50 design variables are involved in the adjoint optimization to ensurely provide an adequate design space. And a quasi-Newton Broyden–Fletcher–Goldfarb–Shanno algorithm is utilized to determine an improved intake duct geometry based on the objective function gradient provided by the adjoint solution. Finally, two multi-mode optimization of a typical inlet duct confirms the flexibility of the multi-mode adjoint-based framework and the efficiency of the multi-mode adjoint-based technique.

Topology Optimization of Capillary, Two-Phase Flow Problems

AbstractThis paper presents topology optimization of capillary, the typical two-phase flow with immiscible fluids, where the level set method and diffuse-interface model are combined to implement the proposed method. The two-phase flow is described by the diffuse-interface model with essential no slip condition imposed on the wall, where the singularity at the contact line is regularized by the molecular diffusion at the interface between two immiscible fluids. The level set method is utilized to express the fluid and solid phases in the flows and the wall energy at the implicit fluid-solid interface. Based on the variational procedure for the total free energy of two-phase flow, the Cahn-Hilliard equations for the diffuse-interface model are modified for the two-phase flow with implicit boundary expressed by the level set method. Then the topology optimization problem for the two-phase flow is constructed for the cost functional with general formulation. The sensitivity analysis is implemented by using the continuous adjoint method. The level set function is evolved by solving the Hamilton-Jacobian equation, and numerical test is carried out for capillary to demonstrate the robustness of the proposed topology optimization method. It is straightforward to extend this proposed method into the other two-phase flows with two immiscible fluids.