Discrete Adjoint Equations Research Articles

Discrete Adjoint Optimization Method for Low-Boom Aircraft Design Using Equivalent Area Distribution

This paper introduces a low-boom aircraft optimization design method guided by equivalent area distribution, which effectively improves the intuitiveness and refinement of inverse design. A gradient optimization method based on discrete adjoint equations is proposed to achieve the fast solution of the gradient information of target equivalent area distribution relative to design variables and to drive the aerodynamic shape update to the optimal solution. An optimization experiment is carried out based on a self-developed supersonic civil aircraft configuration with engines. The results show that the equivalent area distribution adjoint equation can accurately solve the gradient information. After optimization, the sonic boom level of the aircraft was reduced by 13.2 PLdB, and the drag coefficient was reduced by 60.75 counts. Moreover, the equivalent area distribution adjoint optimization method has outstanding advantages, such as high sensitivity and fast convergence speed, and can take both the low sonic boom and the low drag force of the aircraft into account, providing a powerful tool for the comprehensive optimization design of supersonic civil aircraft by considering sonic boom and aerodynamic force.

A discrete adjoint gradient approach for equality and inequality constraints in dynamics

The optimization of multibody systems requires accurate and efficient methods for sensitivity analysis. The adjoint method is probably the most efficient way to analyze sensitivities, especially for optimization problems with numerous optimization variables. This paper discusses sensitivity analysis for dynamic systems in gradient-based optimization problems. A discrete adjoint gradient approach is presented to compute sensitivities of equality and inequality constraints in dynamic simulations. The constraints are combined with the dynamic system equations, and the sensitivities are computed straightforwardly by solving discrete adjoint algebraic equations. The computation of these discrete adjoint gradients can be easily adapted to deal with different time integrators. This paper demonstrates discrete adjoint gradients for two different time-integration schemes and highlights efficiency and easy applicability. The proposed approach is particularly suitable for problems involving large-scale models or high-dimensional optimization spaces, where the computational effort of computing gradients by finite differences can be enormous. Three examples are investigated to validate the proposed discrete adjoint gradient approach. The sensitivity analysis of an academic example discusses the role of discrete adjoint variables. The energy optimal control problem of a nonlinear spring pendulum is analyzed to discuss the efficiency of the proposed approach. In addition, a flexible multibody system is investigated in a combined optimal control and design optimization problem. The combined optimization provides the best possible mechanical structure regarding an optimal control problem within one optimization.

Aerodynamic Optimization Design of Supersonic Wing Based on Discrete Adjoint

Reducing fuel consumption and improving the economy by effectively reducing cruising drag is the main objective of the aerodynamic design of supersonic civil aircraft. In this paper, the aerodynamic optimization design system based on the Reynolds-Averaged Navier–Stokes (RANS) equation and discrete adjoint theory is applied to supersonic wing design. Based on this system, a single-point optimization design study of aerodynamic drag reduction in cruise conditions was carried out for two typical supersonic wing layouts, subsonic leading edge and supersonic leading edge, and the drag reduction reached 3.78% and 4.53%, respectively. The aerodynamic design characteristics of different types of supersonic wings were explored from the perspectives of wing load, twist angle distribution, pressure distribution, airfoil shape characteristics, and flow field characteristics. The optimization results show that the drag reduction of the subsonic leading edge configuration is dominated by the induced drag, while the optimizer mainly focuses on reducing the shock wave drag for the supersonic leading edge configuration. By comparing the sensitivity analysis of lift and drag coefficients to airfoil deformation with the optimization results, the optimized dominant directions of two types of supersonic wings are qualitatively analyzed. The derivatives obtained from discrete adjoint equations are useful to elaborate the design tendency and the reason for the trade-off generation of supersonic wings under specific layouts and engineering constraints, which provides a reference for the design of supersonic wings in the future.

Probability of Initiation in Neutron Transport

We discuss the numerical solution of the nonlinear integro-differential equation for the probability of a divergent neutron chain in a stationary system (i.e., the probability of initiation (POI)). We follow the development described in Bell’s classic paper on the stochastic theory of neutron transport. As noted by Bell, the linearized form of this equation resembles the linear adjoint neutron transport equation. A matrix formalism for the discretized steady state (or forward) neutron equation in slab geometry is first developed, and is then used to derive the discrete adjoint equation. A main advantage of this discrete development is that the resulting discrete adjoint equation does not depend upon how the multigroup cross sections for the forward problem are obtained. That is, we derive the discrete adjoint directly from the discrete forward equations rather than discretizing directly the adjoint equation. This also guarantees that the discrete adjoint operator is consistent with the inner product used to define the adjoint operator. We discuss three approaches for the numerical solution of the POI equations, and present numerical results on several test problems. The three solution methods are a simple fixed point iteration, a second approach that is akin to a nonlinear Power iteration, and a third approach which uses a Newton-Krylov nonlinear solver. We also give sufficient conditions to guarantee the existence and uniqueness of nontrivial solutions to our discrete POI equations when the discrete system is supercritical, and that only the trivial solution exists when the discrete system is subcritical. Our approach is modeled after the analysis presented for the continuous POI equations by Mokhtar-Kharroubi and Jarmouni-Idrissi, and by Pazy and Rabinowitz.

Topology optimization for rarefied gas flow problems using density method and adjoint IP-DSMC

In this paper, a topology optimization method based on direct simulation Monte Carlo (DSMC) is presented for rarefied gas flows. Distribution of fluid and solid in the design domain is characterized by a pseudo density. The traditional DSMC algorithm is extended to include the pseudo density by interpreting solid as an imaginary gas with fixed temperature and zero macroscopic velocity. Treating the pseudo density as the design variable, design sensitivity is obtained through the Lagrangian multiplier method and the adjoint state method. A discrete version of the adjoint equations is used, so that the adjoint variables can be evaluated using the information stored during the forward DSMC process. The information preservation (IP) method for subsonic flow is also considered in order to reduce random noise and simplify the discrete adjoint equations. The extended DSMC algorithm is verified by numerical examples, and optimized design for bent pipe is also included.

Adjoint optimisation for wind farm flow control with a free-vortex wake model

Wind farm flow control aims to improve wind turbine performance by reducing aerodynamic wake interaction between turbines. Dynamic, physics-based models of wind farm flows have been essential for exploring control strategies such as wake redirection and dynamic induction control. Free-vortex methods can provide a computationally efficient way to model wind turbine wake dynamics for control optimisation. We present a control-oriented free-vortex wake model of a 2D and 3D actuator disc to represent wind turbine wakes. The novel derivation of the discrete adjoint equations allows efficient gradient evaluation for gradient-based optimisation in an economic model-predictive control algorithm. Initial results are presented for mean power maximisation in a two-turbine case study. An induction control signal is found using the 2D model that is roughly periodic and supports previous results on dynamic induction control to stimulate wake mixing. The 3D model formulation effectively models a curled wake under yaw misalignment. Under time-varying wind direction, the optimisation finds solutions demonstrating both wake steering and a smooth transition to greedy control. The free-vortex wake model with gradient information shows potential for efficient optimisation and provides a promising way to further explore dynamic wind farm flow control.

Analysis of finite-volume discrete adjoint fields for two-dimensional compressible Euler flows

This work deals with a number of questions related to the discrete and continuous adjoint fields associated with the compressible Euler equations and classical aerodynamic functions. The consistency of the discrete adjoint equations with the corresponding continuous adjoint partial differential equations is one of them. It has been established or at least discussed only for some numerical schemes and a contribution of this article is to give the adjoint consistency conditions for the 2-D Jameson-Schmidt-Turkel scheme in cell-centered finite-volume formulation. The consistency issue is also studied here from a new heuristic point of view by discretizing the continuous adjoint equation for the discrete flow and adjoint fields. Both points of view prove to provide useful information. Besides, it has been often noted that discrete or continuous inviscid lift and drag adjoint exhibit numerical divergence close to the wall and stagnation streamline for a wide range of subsonic and transonic flow conditions. This is analyzed here using the physical source term perturbation method introduced in reference [1]. With this point of view, the fourth physical source term of [1] appears to be the only one responsible for this behavior. It is also demonstrated that the numerical divergence of the adjoint variables corresponds to the response of the flow to the convected increment of stagnation pressure and diminution of entropy created at the source and the resulting change in lift and drag.

A discrete adjoint based level set topology optimization method for stress constraints

This paper proposes a new methodology for computing boundary sensitivities in level set topology optimization using the discrete adjoint method. The adjoint equations are constructed using the discretized governing field equations. The objective function is differentiated with respect to the boundary point movement for computing boundary sensitivities using the discrete adjoint equations. For this purpose, we present a novel approach where we perturb the boundary implicitly by locally modifying the level set function around a given boundary point. These local perturbations are combined with the derivatives of the objective function with respect to the volume fractions of individual elements to compute boundary sensitivities. This enables the circumvention of smoothing or interpolation methods typically used in level set topology optimization to compute sensitivities; and improves the accuracy of the sensitivities and the convergence characteristics. We demonstrate the effectiveness of our method in the context of stress minimization and stress constrained topology optimization problems for orthogonal bracket design under multiple load cases.

On Entropy-Stable Discretizations and the Entropy Adjoint

For conservation laws that admit a convex entropy, Fidkowski and Roe (SIAM J Sci Comput 32(3):1261–1287, 2010) showed that the entropy variables satisfy an adjoint equation corresponding to an entropy-balance functional. In general, this relationship between the entropy variables and the entropy adjoint becomes an approximation when the conservation law is discretized. However, this work shows that the relationship is mimicked discretely for entropy-stable discretizations based on summation-by-parts (SBP) operators and two-point entropy-conservative flux functions; for these discretizations, the discrete entropy variables exactly satisfy the discrete adjoint equation for a particular discretization of the entropy-balance functional. A detailed proof of this result is presented for first-order conservation laws that are semi-discretized using SBP operators with diagonal norm and boundary operators. Subsequently, generalizations to second-order conservation laws, temporal discretizations, and non-diagonal SBP operators are described. The result is verified for steady inviscid and viscous flows.

Adjoint based state estimation of compressible flow in porous media

Abstract In this paper, we study the state estimation of compressible single phase flow in compressible porous media. The initial pressure distribution is estimated according to discrete adjoint approach based on the collected well pressure data. The first-order Tykhonov regularization method is used to obtain reasonable estimation. By analyzing the optimality condition of estimation problem, the discrete adjoint state equation and discrete adjoint gradient are derived based on the numerical scheme of the continuous equations. A quasi-Newton numerical optimization method related to adjoint gradient is proposed to solve the estimation problem. The estimation results with different regularization coefficients are compared and analyzed by numerical experiments. The deviation between the estimated pressure obtained without regularization and the real pressure is large. Estimation result with smaller deviation and higher smoothness can be obtained through appropriate regularization coefficient. When the observation error is large, the observed values generated by the estimated pressure fit well with the real pressure.

Three-Dimensional Aerodynamic/Stealth Optimization Based on Adjoint Sensitivity Analysis for Scattering Problem

A radar cross section (RCS) gradient calculation approach based on the discrete adjoint equation of Maxwell integral equation is proposed. The adjoint method can obtain the gradients of all design variables by a single solution of the scattering problem if electric field integral equation is adopted, and by one solution of the scattering problem and one adjoint problem if magnetic field integral equation or combined field integral equation is adopted. Multilevel fast multipole method (MLFMA) is employed in the solution of the scattering and adjoint problem. The accuracy and reliability of the adjoint approach are verified with a double-ogive model. Optimizations considering aerodynamic, stealth, and aerodynamic/stealth performance of a blended-wing–body aircraft with a gradient-based optimizer are conducted. Stealth optimization results indicate that a small leading-edge radius and curvature on the forepart are beneficial to RCS reduction. Aerodynamic performance of aerodynamic optimized model and aero/stealth optimization result are comparable, but stealth performance of aero/stealth optimized result is inferior to stealth optimization. Conflictions between small leading edge and drag reduction are not very prominent based on the current study, whereas curvature is harmful to aerodynamic performance.

Efficient gHMC Reconstruction of Contaminant Release History

We present a generalized hybrid Monte Carlo (gHMC) method for fast, statistically optimal reconstruction of release histories of reactive contaminants. The approach is applicable to large-scale, strongly nonlinear systems with parametric uncertainties and data corrupted by measurement errors. The use of discrete adjoint equations facilitates numerical implementation of gHMC, without putting any restrictions on the degree of nonlinearity of advection-dispersion-reaction equations that are used to describe contaminant transport in the subsurface. To demonstrate the salient features of the proposed algorithm, we identify the spatial extent of a distributed source of contamination from concentration measurements of a reactive solute.

Adjoint-Based Geometrically Constrained Aerodynamic Optimization of a Transonic Compressor Stage

Efficient method to handle the geometric constraints in the optimization of turbomachinery blade profile is required. Without constraints on the blade thickness, optimal designs typically yield thinner blade to reduce the friction loss, however, at the risk of degraded strength and stiffness. This issue is seldom discussed and existing literature always treat the blade thickness constraint in an indirect manner. In this work, two different geometric constraints on the blade thickness are proposed and applied in the adjoint optimization: one is on the maximum blade thickness and the other is on the blade area. Methods to compute sensitivities of both constraints are proposed and they are integrated into an optimization system based on a finite volume code and a solver for the discrete adjoint equation. Adjoint optimization is conducted to minimize the entropy production in a transonic compressor stage. Results for the adjoint optimization without geometry constraint and two comparative cases are detailed. It is indicated that three cases yield similar performance improvement; however, if geometry constraints are properly handled, the optimal designs have almost the same maximum thickness as the original design, compared to a thinner blade profile with 14% reduction of maximum thickness in the case without geometry constraint. The cases considering geometry constraints also consume slightly reduced Central Processing Unit (CPU) cost. Result of this work verifies the effectiveness of the proposed method to treat geometric constraints in adjoint optimization.

Discrete adjoint of fractional-step incompressible Navier-Stokes solver in curvilinear coordinates and application to data assimilation

The discrete adjoint of an incompressible Navier-Stokes algorithm in generalized coordinates is derived and applied to estimate the states of saturated and turbulent circular Couette flows. The forward Navier-Stokes model is based on the fractional-step algorithm in curvilinear coordinates on a structured grid [1], which has been widely adopted in direct numerical simulations of transitional and turbulent flows. The discrete adjoint equations adopt the same stencil and temporal scheme as the forward discretization, and expressions are derived that relate the discrete adjoint variables to their continuous counterpart. The key ingredients of the forward algorithm can be retained in the adjoint, including the computation of the cell geometry, the approximate factorization method, and the parallelization strategy. The accuracy, efficiency, and stability of the adjoint solver are also commensurate with the forward model. In addition, a novel symmetric projector is proposed to guarantee that the outcome of the adjoint algorithm is divergence free. The implementation of the algorithm in double precision satisfies the forward-adjoint relation up to eight significant figures, and further validation is performed using circular Couette flow. The forward and adjoint growth rates of instability modes from linear theory are accurately reproduced. In addition, an adjoint-variational data-assimilation algorithm (4DVar) is adopted to reconstruct the initial condition of circular Couette flows from limited measurements, obtained from an independent simulation. When the flow is comprised of saturated wavy vortices, wall measurements are sufficient to reconstruct an initial condition that latches onto the target state after a short time. For the more challenging turbulent case, coarse-grained velocity data are used to estimate the initial condition.

A gradient-based aero-stealth optimization design method for flying wing aircraft

Flying wing layout has both favorable aerodynamic performance and outstanding radar scattering characteristics, which is regarded as an ideal stealth aircraft layout. In this paper, a gradient-based aerodynamic and stealth optimization design method is established by coupling the Free-Form Deformation approach (FFD), Radial Basis Function algorithm (RBF), Computational Fluid Dynamics (CFD), Physical Optics (PO) and the Sequential Quadratic Programming algorithm (SQP). The gradient of the aerodynamic characteristic parameters could be obtained by solving the discrete adjoint equations. With PO code differentiated by the automatic differentiation tool, the Radar Cross Section (RCS) and its gradient would be collected. Based on the aero-stealth optimization design method, three kinds of optimization strategies are applied to design a certain flying wing layout aircraft: 1) The drag coefficient is chosen as the objective function while the RCS is set as a constraint condition; 2) The RCS is selected as the objective function while the drag coefficient is set as a constraint condition; 3) The objective function is composed of the drag coefficient and the RCS through the Weighted Sum method. The optimization results show that when choosing the first two strategies, the optimization algorithm would push single discipline performance to the edge. Alternatively, it would obtain several trade-off configurations when selecting the last strategy. The results demonstrate that the gradient-based aero-stealth optimization design method can deal with the multidisciplinary optimization problems which require a large number of design variables. Without a large number of function evaluations, this method could rapidly and efficiently obtain the shape that meets the requirements of aerodynamic and stealth performances.

Numerical Solution for Classical Optimal Control Problem Governing by Hyperbolic Partial Differential Equation via Galerkin Finite Element-Implicit method with Gradient Projection Method

This paper deals with the numerical solution of the discrete classical optimal control problem (DCOCP) governing by linear hyperbolic boundary value problem (LHBVP). The method which is used here consists of: the GFEIM " the Galerkin finite element method in space variable with the implicit finite difference method in time variable" to find the solution of the discrete state equation (DSE) and the solution of its corresponding discrete adjoint equation, where a discrete classical control (DCC) is given. The gradient projection method with either the Armijo method (GPARM) or with the optimal method (GPOSM) is used to solve the minimization problem which is obtained from the necessary condition for optimality of the DCOCP to find the DCC.An algorithm is given and a computer program is coded using the above methods to find the numerical solution of the DCOCP with step length of space variable , and step length of time variable . Illustration examples are given to explain the efficiency of these methods. The results show the methods which are used here are better than those obtained when we used the Gradient method (GM) or Frank Wolfe method (FWM) with Armijo step search method to solve the minimization problem.

Parallel Finite Element Framework for Rotorcraft Multibody Dynamics and Discrete Adjoint Sensitivities

A parallel finite element framework for high-fidelity structural dynamic analysis and gradient evaluation using the discrete adjoint method is presented. The framework is intended to be used for gradient-based design optimization of flexible multibody dynamic systems such as rotorcraft. The formulation of governing equations, the treatment of kinematic constraints, and the evaluation of functionals of interest and their derivatives are addressed. A minimal set of routines needed to implement the discrete adjoint method is proposed. The governing equations are integrated in time using a diagonally implicit Runge–Kutta method for second-order systems of equations. The formulation of the corresponding time-dependent discrete adjoint equations are presented and are numerically verified using the complex-step method. A verification of the dynamics, an assessment of parallel scalability of the analysis and derivative evaluation techniques, and a demonstration of the design capability are presented.

Solution of Optimal Harvesting Problem by Finite Difference Approximations of Size-Structured Population Model

We solve numerically a forest management optimization problem governed by a nonlinear partial differential equation (PDE), which is a size-structured population model. The formulated problem is supplemented with a natural constraint for a solution to be non-negative. PDE is approximated by an explicit or implicit in time finite difference scheme, whereas the cost function is taken from the very beginning in the finite-dimensional form used in practice. We prove the stability of the constructed nonlinear finite difference schemes on the set of non-negative vectors and the solvability of the formulated discrete optimal control problems. The gradient information is derived by constructing discrete adjoint state equations. The projected gradient method is used for finding the extremal points. The results of numerical testing for several real problems show good agreement with the known results and confirm the theoretical statements.

Optimization of wind plant layouts using an adjoint approach

Abstract. Using adjoint optimization and three-dimensional steady-state Reynolds-averaged Navier–Stokes (RANS) simulations, we present a new gradient-based approach for optimally siting wind turbines within utility-scale wind plants. By solving the adjoint equations of the flow model, the gradients needed for optimization are found at a cost that is independent of the number of control variables, thereby permitting optimization of large wind plants with many turbine locations. Moreover, compared to the common approach of superimposing prescribed wake deficits onto linearized flow models, the computational efficiency of the adjoint approach allows the use of higher-fidelity RANS flow models which can capture nonlinear turbulent flow physics within a wind plant. The steady-state RANS flow model is implemented in the Python finite-element package FEniCS and the derivation and solution of the discrete adjoint equations are automated within the dolfin-adjoint framework. Gradient-based optimization of wind turbine locations is demonstrated for idealized test cases that reveal new optimization heuristics such as rotational symmetry, local speedups, and nonlinear wake curvature effects. Layout optimization is also demonstrated on more complex wind rose shapes, including a full annual energy production (AEP) layout optimization over 36 inflow directions and 5 wind speed bins.

Design sensitivity analysis for transient response of non-viscously damped dynamic systems

A design sensitivity analysis for the transient response of the non-viscously damped dynamic systems is presented. The non-viscously (viscoelastically) damped system is widely used in structural vibration control. The damping forces in the system depend on the past history of motion via convolution integrals. The non-viscos damping is modeled by the generalized Maxwell model. The transient response is calculated with the implicit Newmark time integration scheme. The design sensitivity analysis method of the history dependent system is developed using the adjoint variable method. The discretize-then-differentiate approach is adopted for deriving discrete adjoint equations. The accuracy and the consistency of the proposed method are demonstrated through a single dof system. The proposed method is also applied to a multi-dof system. The validity and accuracy of the sensitivities from the proposed method are confirmed by finite difference results.