Continuous Adjoint Research Articles

The continuous adjoint approach to the k–ε turbulence model for shape optimization and optimal active control of turbulent flows

The continuous adjoint to the incompressible Reynolds-averaged Navier–Stokes equations coupled with the low Reynolds number Launder–Sharma k–ε turbulence model is presented. Both shape and active flow control optimization problems in fluid mechanics are considered, aiming at minimum viscous losses. In contrast to the frequently used assumption of frozen turbulence, the adjoint to the turbulence model equations together with appropriate boundary conditions are derived, discretized and solved. This is the first time that the adjoint equations to the Launder–Sharma k–ε model have been derived. Compared to the formulation that neglects turbulence variations, the impact of additional terms and equations is evaluated. Sensitivities computed using direct differentiation and/or finite differences are used for comparative purposes. To demonstrate the need for formulating and solving the adjoint to the turbulence model equations, instead of merely relying upon the ‘frozen turbulence assumption’, the gain in the optimization turnaround time offered by the proposed method is quantified.

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

An adjoint formulation for the gradient-based optimization of oil–gas compositional reservoir simulation problems is presented. The method is implemented within an automatic differentiation-based compositional flow simulator (Stanford’s Automatic Differentiation-based General Purpose Research Simulator, AD-GPRS). The development of adjoint procedures for general compositional problems is much more challenging than for oil–water problems due to the increased complexity of the code and the underlying physics. The treatment of nonlinear constraints, an example of which is a maximum gas rate specification in injection or production wells, when the control variables are well bottom-hole pressures, poses a particular challenge. Two approaches for handling these constraints are presented—a formal treatment within the optimizer and a simpler heuristic treatment in the forward model. The relationship between discrete and continuous adjoint formulations is also elucidated. Results for four example cases of increasing complexity are presented. Improvements in the objective function (cumulative oil produced) relative to reference solutions range from 4.2 to 11.6 %. The heuristic treatment of nonlinear constraints is shown to offer a cost-effective means for obtaining feasible solutions, which are, in some cases, better than those obtained using the formal constraint handling procedure.

Optimal shape design for divertors

Owing to the complex physics of the plasma edge, numerical simulation tools are indispensable for the evaluation of novel divertor concepts. Simulation-based divertor design is computationally extremely expensive, however, not in the least due to the large number of design variables. In this work, we show how shape optimisation methods can prove very valuable in partially automating the design process. We apply the continuous adjoint method to design divertor targets for maximum power load spreading. Shape sensitivities are derived using the material derivative approach. The resulting sensitivities depend on boundary data only, and can be computed very efficiently. Through numerical tests, we also prove that their accuracy is good. Next, the performance of two gradient-based optimisation algorithms and a so-called one-shot method is compared. Using the latter method, optimal solutions are obtained in a computational effort less than five times the time needed for a single analysis simulation.

Three-dimensional tool design for steady-state electrochemical machining by continuous adjoint-based shape optimization

In electrochemical machining (ECM), it is important to design the shape of an appropriate tool capable of producing a workpiece of desired shape. This work presents a numerical approach to solving the two- and three-dimensional tool design problem in steady-state ECM. The tool design problem is transformed into a shape optimization problem and then solved using the continuous adjoint method combined with elements of shape calculus, ensuring high efficiency and the maximum possible degrees of freedom. A numerical experiment on a two-dimensional Gaussian-shaped workpiece shows a good agreement of the calculated tool shape with the exact analytical solution. Tool design is carried out for a series of two- and three-dimensional workpiece shapes to investigate the influence of the curvature of the desired workpiece on the front gap width in steady-state ECM.

Shape optimization in the Navier‐Stokes flow with thermal effects

In this article, we consider the shape optimization problem of a body immersed in the incompressible fluid governed by Navier‐Stokes equations coupling with a thermal model. Based on the continuous adjoint method, we formulate and analyze the shape optimization problem. Then we derive the structure of shape gradient for the cost functional by using the differentiability of a minimax formulation involving a Lagrange functional with the function space parametrization technique. Moreover, we present a gradient‐type algorithm to the shape optimization problem. Finally, numerical examples demonstrate the feasibility and effectiveness of the proposed algorithm. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1700–1715, 2014

Sequential Quadratic Programming (SQP) for optimal control in direct numerical simulation of turbulent flow

The current work focuses on the development and application of an efficient algorithm for optimization of three-dimensional turbulent flows, simulated using Direct Numerical Simulation (DNS) or Large-Eddy Simulations, and further characterized by large-dimensional optimization-parameter spaces. The optimization algorithm is based on Sequential Quadratic Programming (SQP) in combination with a damped formulation of the limited-memory BFGS method. The latter is suitable for solving large-scale constrained optimization problems whose Hessian matrices cannot be computed and stored at a reasonable cost. We combine the algorithm with a line-search merit function based on an L1-norm to enforce the convergence from any remote point. It is first shown that the proposed form of the damped L-BFGS algorithm is suitable for solving equality constrained Rosenbrock type functions. Then, we apply the algorithm to an optimal-control test problem that consists of finding the optimal initial perturbations to a turbulent temporal mixing layer such that mixing is improved at the end of a simulation time horizon T. The controls are further subject to a non-linear equality constraint on the total control energy. DNSs are used to resolve all turbulent scales of motion, and a continuous adjoint formulation is employed to calculate the gradient of the cost functionals. We compare the convergence speed of the SQP L-BFGS algorithm to a conventional non-linear conjugate-gradient method (i.e. the current standard in DNS-based optimal control), and find that the SQP algorithm is more than an order of magnitude faster than the conjugate-gradient method.

Adjoint-based constrained topology optimization for viscous flows, including heat transfer

In fluid mechanics, topology optimization is used for designing flow passages, connecting predefined inlets and outlets, with optimal performance based on selected criteria. In this article, the continuous adjoint approach to topology optimization in incompressible ducted flows with heat transfer is presented. A variable porosity field, to be determined during the optimization, is the means to define the optimal topology. The objective functions take into account viscous losses and the amount of heat transfer. Turbulent flows are handled using the Spalart–Allmaras model and the proposed adjoint is exact, i.e. the adjoint to the turbulence model equation is formulated and solved, too. This is an important novelty in this article which extends the porosity-based method to account for heat transfer flow problems in turbulent flows. In problems such as the design of manifolds, constraints on the outlet flow direction, rates and mean outlet temperatures are imposed.

The continuous adjoint method as a guide for the design of flow control systems based on jets

PurposeThe purpose of this paper is to propose the use of the continuous adjoint method as a tool to identify the appropriate location and “type” (suction or blowing) of steady jets used in active flow control systems.Design/methodology/approachThe method is based on continuous adjoint and covers both internal and external aerodynamics. The adjoint equations, including the adjoint to the SpalartAllmaras turbulence model and their boundary conditions are formulated. At the cost of solving the flow and adjoint equations just once, the sensitivity derivatives of the objective function with respect to hypothetical (normal) jet velocities at all wall nodes are computed. Comparisons of the computed sensitivities with finite differences and parametric studies to assess the present method are included.FindingsThough the sensitivities are computed for zero jet velocities, they adequately support decision making on: the recommended location of jet(s), at boundary nodes with high absolute valued sensitivities; and the selection between suction or blowing jets, based on the sign of the computed sensitivities. Regarding adjoint methods, two important findings of this work are: the role of the adjoint pressure which proves to be an excellent sensor in flow control problems; and the prediction accuracy of the proposed adjoint method compared to the commonly made assumption of “frozen turbulence”.Originality/valueFirst use of the continuous adjoint method using full differentiation of the turbulence model, in flow control optimization. A low‐cost design tool for recommending some of the most important jet characteristics.

Shape optimization of a general bypass duct for tone noise reduction using continuous adjoint method

A novel continuous adjoint-based acoustic propagation method is proposed for low-noise turbofan duct design. A fan bypass duct tonal noise propagation model that is verified by comparison with an analytical solution of the modal radiation from a semi-infinite duct with the shear layer is enhanced with its continuous adjoint formulation, having been applied to design the bypass duct. First, this article presents the complete formulation of the time-dependent optimal design problem. Second, a continuous adjoint-based acoustic propagation method for two-dimensional bypass duct configurations is derived and presented. This article aims at describing the potential of the adjoint technique for aeroacoustic shape optimization. The implementation of the unsteady aeroacoustic adjoint method is validated by comparing the sensitivity derivative with that obtained by finite differences. Using a continuous adjoint formulation, the necessary aerodynamic gradient information is obtained with large computational savings over traditional finite-difference methods. The examples presented demonstrate that the combination of a continuous-adjoint algorithm with a noise prediction method can be an efficient design tool in the bypass duct noise design problem.

Spatial sensitivities of human health risk to intercontinental and high-altitude pollution

We perform the first long-term (>1 year) continuous adjoint simulations with a global atmospheric chemistry–transport model focusing on population exposure to fine particulate matter (PM2.5) and associated risk of early death. Sensitivities relevant to intercontinental and high-altitude PM pollution are calculated with particular application to aircraft emissions. Specifically, the sensitivities of premature mortality risk in different regions to NOx, SOx, CO, VOC and primary PM2.5 emissions as a function of location are computed. We apply the resultant sensitivity matrices to aircraft emissions, finding that NOx emissions are responsible for 93% of population exposure to aircraft-attributable PM2.5. Aircraft NOx accounts for all of aircraft-attributable nitrate exposure (as expected) and 53% of aircraft-attributable sulfate exposure due to the strong “oxidative coupling” between aircraft NOx emissions and non-aviation SO2 emissions in terms of sulfate formation. Of the health risk-weighted human PM2.5 exposure attributable to aviation, 73% occurs in Asia, followed by 18% in Europe. 95% of the air quality impacts of aircraft emissions in the US are incurred outside the US. We also assess the impact of uncertainty or changes in (non-aviation) ammonia emissions on aviation-attributable PM2.5 exposure by calculating second-order sensitivities. We note the potential application of the sensitivity matrices as a rapid policy analysis tool in aviation environmental policy contexts.

A Discrete Adjoint Model for OpenFOAM

We present a discrete adjoint version of the open-source fluid-dynamics package OpenFOAM obtained by operator overloading which yields, in comparison to continuous adjoint versions, a greater flexibility and robustness. We discuss our implementation and how the discrete adjoint version of OpenFOAM differs from existing continuous implementations. To reduce the inherent memory requirement of discrete adjoint code we introduce a checkpointing scheme to trade computation time for memory. Moreover we show results from a relevant reference case.

Topology optimization of steady Navier–Stokes flow with body force

This paper presents the topology optimization of steady Navier–Stokes flows with body forces that influence the optimal shape and topology of fluid flows. Based on the implicit expression of the fluid flow with the level set method, an optimization problem is formulated and analyzed using the continuous adjoint method. The shape and topological sensitivities are computed based on the adjoint and asymptotic analysis of the optimization problem. In the optimization procedure, the level set surface is evolved based on the shape sensitivity and nucleated based on the topological sensitivity simultaneously. Three kinds of body forces that are commonly used in the design of fluid devices, i.e. constant, nonuniform, and solution-dependent body forces, are considered in the two-dimensional and three-dimensional numerical examples. Numerical results demonstrate that this method can effectively achieve the topology optimization of the Navier–Stokes flows with body forces.

Derivation of an adjoint high‐Reynolds number model for thermally driven flows

AbstractThis paper presents a high‐Reynolds number (high‐Re) approach to the adjoint Navier‐Stokes‐Fourier equations based upon modified high‐Re boundary conditions for buoyancy‐driven flows. For reasons of efficiency, the computational framework adheres to the continuous adjoint method using a frozen‐turbulence assumption. Opposed to the state of the art, the present model uses a high‐Re model for the primal equations and a high‐Re model for the adjoint equations. The adjoint high‐Re model can easily be implemented and be used for flow problems where wall functions are needed. The impact of this approach is an improved consistency of primal and adjoint equations which leads to stable and realistic shape optimisation results for industrial flows. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Investigation of a continuous adjoint-based optimization procedure for aeroacoustic control of plane jets

A control optimization technique using the continuous adjoint of the compressible Navier–Stokes equations is implemented for aeroacoustic optimization of plane jet flows. The purpose of the adjoint equations is to provide sensitivity information, which is afterwards used in a gradient-based minimization of a prescribed cost functional, designed to describe the far-field sound pressure level (SPL). The objective of the present paper is to demonstrate the ability to reduce the sound in the near far-field of plane jets. Furthermore, as the continuous adjoint approach can become inaccurate, due to inconsistencies between the continuous and the discretized system, the accuracy of the continuous adjoint approach is investigated. The considered cases exhibit a nozzle exit Reynolds number of Rejet=ρujetD/μ=2000 and a Mach number of Mjet=0.9, performed using two-dimensional direct numerical simulation and three-dimensional large-eddy simulation, respectively. A comparison of the obtained gradient via adjoint and finite differences is presented and it is shown, that in order to obtain reliable gradient directions, the length of the optimization time needs to be restricted. Furthermore, a receding horizon optimization for the two-dimensional plane jet simulation is used to obtain a sound reduction over much longer time intervals. The influence of different formulations of the viscosity in the adjoint equations is finally investigated.

Optimization of unsteady incompressible Navier–Stokes flows using variational level set method

SUMMARYThis paper presents the optimization of unsteady Navier–Stokes flows using the variational level set method. The solid–liquid interface is expressed by the level set function implicitly, and the fluid velocity is constrained to be zero in the solid domain. An optimization problem, which is constrained by the Navier–Stokes equations and a fluid volume constraint, is analyzed by the Lagrangian multiplier based adjoint approach. The corresponding continuous adjoint equations and the shape sensitivity are derived. The level set function is evolved by solving the Hamilton–Jacobian equation with the upwind finite difference method. The optimization method can be used to design channels for flows with or without body forces. The numerical examples demonstrate the feasibility and robustness of this optimization method for unsteady Navier–Stokes flows.Copyright © 2012 John Wiley & Sons, Ltd.

Third‐order sensitivity analysis for robust aerodynamic design using continuous adjoint

SUMMARYRobust design problems in aerodynamics are associated with the design variables, which control the shape of an aerodynamic body, and also with the so‐called environmental variables, which account for uncertainties. In this kind of problems, the set of design variables, which leads to optimal performance, taking into account possible variations in the environmental variables, is sought. One of the possible ways to solve this problem is by means of the second‐order second‐moment approach, which requires first‐order and second‐order derivatives of the objective function with respect to the environmental variables. Should the minimization problem be solved using a gradient‐based method, algorithms for the computation of up to third‐order sensitivity derivatives (twice with respect to the environmental variables and once with respect to the shape controlling design variables) must be devised. In this paper, a combination of the continuous adjoint variable method and direct differentiation to compute the third‐order sensitivities is proposed. This is shown to be the most efficient among all alternative methods provided that the environmental variables are much less than the design ones. Apart from presenting the method formulation, this paper focuses on the assessment of the so‐computed up‐to third‐order mixed derivatives through comparison with costly finite‐difference schemes. To this end, the robust design of a two‐dimensional duct is performed. Then, using the validated method, the robust design of a two‐dimensional cascade airfoil is demonstrated. Although both cases are handled as inverse design problems, the method can be extended to other objective functions or three‐dimensional problems in a straightforward manner. Copyright © 2012 John Wiley & Sons, Ltd.

Continuous Adjoint Approach for the Spalart-Allmaras Model in Aerodynamic Optimization

a new variable, solving the Eikonal equation. The accuracy of the sensitivity derivatives obtained with the complete turbulent approach is assessed by comparison with finite difference computations and the classical continuous adjoint with frozen viscosity, showing substantial improvements in the convergence properties of the method and in the quality of the obtained gradients. The validity of the overall methodology is illustrated with several design examples, including the optimization of three-dimensional geometries in combination with advanced freeform techniques for mesh deformation.

How to verify optimal controls computed by direct shooting methods? – A tutorial

For the solution of optimal control problems, direct methods have been established in the process engineering community. If set up correctly they robustly provide more or less accurate approximations of the exact solution. In the usual engineering practice, neither the distance to the exact solution is reflected, nor the compliance with the continuous necessary conditions in form of Pontryagin's Minimum Principle is checked. At the end, some approximate solution is available but its quality is at question.This tutorial addresses the problem of the verification of optimal controls computed by direct shooting methods. We focus on this popular transcription method though the results are also relevant for other solution strategies. We review known results spread in the mathematical literature on optimal control to show how the output of the nonlinear programs (NLPs) resulting from single shooting transcriptions of optimal control problems can be interpreted in the context of Pontryagin's Minimum Principle. In particular, we show how to approximate continuous adjoint variables by means of the dual information provided by the NLP solver. Based on this adjoint approximation we use a multi-level setting to construct an estimate of the distance to a true extremal solution satisfying the continuous necessary conditions of optimality. A comprehensive case study illustrates the theoretical results.

Goal-Based Flow Optimisation for Automotive Design

The recent progress achieved in the EU-funded FlowHead project with development of shape and topology optimisation for fluid flow with application to automotive design is demonstrated. The advances with continuous and discrete adjoint solvers are described, as well as a mesh-based and a CAD-based parametrisation approach used in the project. Application examples in automotive design are presented.

Oil Reservoir Production Optimization using Single Shooting and ESDIRK Methods

Conventional recovery techniques enable recovery of 10-50% of the oil in an oil field. Advances in smart well technology and enhanced oil recovery techniques enable significant larger recovery. To realize this potential, feedback model-based optimal control technologies are needed to manipulate the injections and oil production such that flow is uniform in a given geological structure. Even in the case of conventional water flooding, feedback based optimal control technologies may enable higher oil recovery than with conventional operational strategies. The optimal control problems that must be solved are large-scale problems and require specialized numerical algorithms. In this paper, we combine a single shooting optimization algorithm based on sequential quadratic programming (SQP) with explicit singly diagonally implicit Runge-Kutta (ESDIRK) integration methods and a continuous adjoint method for sensitivity computation. We demonstrate the procedure on a water flooding example with conventional injectors and producers.