Adjoint Boundary Conditions Research Articles

Adjoint-based shape optimization for compressible flow based on volume penalization method

AbstractReducing the resistance of compressible flow around a blunt body is of great interest in engineering applications, while an efficient shape optimization method for compressible flows remains far from well established, especially for high Mach numbers. To this end, a volume penalization method for simulating compressible flows past a no-slip and isothermal solid is established by introducing an artificial body force and a heat sink into the governing equations. The level-set functions are introduced as design variables, and the cost functional is defined as the total drag acting on the solid. Then, a continuous adjoint-based shape optimization algorithm for drag reduction is developed by deriving the adjoint equations, the adjoint boundary conditions, and the shape update formula. Both the forward and adjoint simulations are verified by existing studies. The results show that the relative deviations of the drag coefficients obtained in the present study from those reported in the reference studies are around 5% at most, and also a comparable drag reduction rate and also optimal shapes can be reproduced by the present optimization scheme for benchmark problems at relatively low Mach numbers considered in previous studies. Finally, the present method is applied to shape optimization of an initially two-dimensional cylinder and also a three-dimensional sphere in the transonic regime of Ma∞ = 1.2. The drag reduction of over 20% is achieved for both two-dimensional and three-dimensional cases.

Mathematical derivation of a unified equations for adjoint lattice Boltzmann method in airfoil and wing aerodynamic shape optimization

Unified equations of the adjoint lattice Boltzmann method (ALBM) are derived for five applicable objective functions in 2D/3D aerodynamic shape optimization problems. The derived equations include the adjoint equation, boundary condition, terminal condition and gradient of the cost function. In this research, firstly, these relations are extracted for each objective in details and then the general form of ALBM equations are presented for all defined practical aerodynamic objective function. Five applicable cost functions which are the most important objectives in optimization of aerodynamic geometries include desired pressure and viscous shear stress (VSS) inverse design, drag and moment at fixed lift and finally lift to drag ratio at fixed angle of attack. The new extracted relations are based on the circular and spherical function scheme, and are valid for viscous/inviscid, compressible/incompressible and 2D/3D flows in all continuous flow regimes. Proof of new extracted general relations have been performed by authors.

Study on continuous adjoint-based optimization method for internal cooling structures in turbine blades with conjugate heat transfer

A continuous adjoint-based shape optimization system is presented in this paper for conjugate heat transfer (CHT) problems and is applied to the optimization for internal cooling structures of gas turbine blades. With the consideration for the variation of heat conduction equation in the solid regions, the adjoint equations and adjoint boundary conditions in different regions are derived. The adjoint energy correction equation is also derived successfully based on the SIMPLE algorithm, and the adjoint CHT solver is implemented by imposing the values of adjoint energy at the interfaces. Two cases of full 3D optimization for the C3X hollow blade and cooling channel with pin fins are conducted by using the optimization system. In both cases, their flow and heat transfer characteristics are optimized and analyzed in detail. This study demonstrates the potential advantages and basic capabilities of the continuous adjoint method in dealing with CHT optimization problems.

Airfoil inverse design based on laminar compressible adjoint lattice Boltzmann method

AbstractA new optimization techniques based on the adjoint lattice Boltzmann method is derived for airfoil inverse design in laminar compressible flows. In this study, the developed adjoint lattice Boltzmann scheme based on the circular function (CF) is extended for airfoil inverse design problems in laminar incompressible and compressible flows. New mathematical derivation based on compressible lattice Boltzmann equations (LBE) is developed which can find target shape of an airfoil with available desired pressure distribution. The adjoint lattice Boltzmann method is extended for both the incompressible and compressible flows by selecting the circular function idea for calculating the equilibrium distribution functions. So, the adjoint equation is also expanded based on CF idea for calculation of objective function gradient vector. The steepest decent technique is utilized as gradient optimizer. Also, a novel solution is presented to remove singularity problem of the adjoint boundary condition. In order to validate the developed optimization algorithm, results are presented for both incompressible and compressible inverse problem in steady and unsteady flow and accurate results are obtained.

Heat and mass transfer in two-phase annular flows in channels with capillary-porous walls under first-type boundary conditions

In order to create and perfect modern technologies, it is necessary to have a deep understanding of the processes they are based upon. The classical approach to the study of heat and mass transfer using mathematical modeling involves solving a system of differential equations under appropriate boundary conditions. When studying gas–liquid systems, one needs to determine the driving force of the process. The differential equations for each phase are considered under adjoint boundary conditions, which complicates the mathematical model. Since the solution involves determining local parameters with a required degree of accuracy, it is almost impossible to obtain an exact solution of this mathematical model analytically.The numerical research method requires certain data on the characteristics of the process, namely the local values of heat and mass transfer coefficients. This paper presents the results of a physical study of one of the possible liquid–gas interaction types: cooling of the gas phase with isothermal liquid film using passive intensification methods. Having analyzed the obtained results, the authors determine the system parameters that affect the processes intensity the most and obtain similarity equations for determining local and average heat and mass transfer coefficients at the initial heat section and the stabilized heat transfer section in the range of Reynolds numbers Re=1140⋯8490. Using the obtained dependences allows solving the mathematical model of the process numerically and improving the existing methods of calculating contact heat and mass exchangers.

DIRECT AND INVERSE PROBLEMS FOR THERMAL GROOVING BY SURFACE DIFFUSION WITH TIME DEPENDENT MULLINS COEFFICIENT

We consider the Mullins’ equation of a single surface grooving when the surface diffusion is not considered as very slow. This problem can be formed by a surface grooving of profiles in a finite space region. The finiteness of the space region allows to apply the Fourier series analysis for one groove and also to consider the Mullins coefficient as well as slope of the groove root to be time-dependent. We also solve the inverse problem of finding time-dependent Mullins coefficient from total mass measurement. For both of these problems, the grooving side boundary conditions are identical to those of Mullins, and the opposite boundary is accompanied by a zero position and zero curvature which both together arrive at self adjoint boundary conditions.

Derivation of the Adjoint Drift Flux Equations for Multiphase Flow

The continuous adjoint approach is a technique for calculating the sensitivity of a flow to changes in input parameters, most commonly changes of geometry. Here we present for the first time the mathematical derivation of the adjoint system for multiphase flow modeled by the commonly used drift flux equations, together with the adjoint boundary conditions necessary to solve a generic multiphase flow problem. The objective function is defined for such a system, and specific examples derived for commonly used settling velocity formulations such as the Takacs and Dahl models. We also discuss the use of these equations for a complete optimisation process.

Entropy and Adjoint Methods

Aerodynamic drag can be partially approximated by the entropy flux across fluid domain boundaries with a formula due to Oswatitsch. In this paper, we build the adjoint solution that corresponds to this representation of the drag and investigate its relation to the entropy variables, which are linked to the integrated residual of the entropy transport equation. For inviscid isentropic flows, the resulting adjoint variables are identical to the entropy variables, an observation originally due to Fidkowski and Roe, while for non-isentropic flows there is a significant difference that is explicitly demonstrated with analytic solutions in the shocked quasi-1D case. Both approaches are also investigated for viscous and inviscid flows in two and three dimensions, where the adjoint equations and boundary conditions are derived. The application of both approaches to mesh adaptation is investigated, with especial emphasis on inviscid flows with shocks.

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 analysis of Buckley-Leverett and two-phase flow equations

This paper analyzes the adjoint equations and boundary conditions for porous media flow models, specifically the Buckley-Leverett equation, and the compressible two-phase flow equations in mass conservation form. An adjoint analysis of a general scalar hyperbolic conservation law whose primal solutions include a shock jump is initially presented, and the results are later specialized to the Buckley-Leverett equation. The non-convexity of the Buckley-Leverett flux function results in adjoint characteristics that are parallel to the shock front upstream of the shock and emerge from the shock front downstream of the shock. Thus, in contrast to the behavior of Burgers’ equation where the adjoint is continuous at a shock, the Buckley-Leverett adjoint, in general, contains a discontinuous jump across the shock. Discrete adjoint solutions from space-time discontinuous Galerkin finite element approximations of the Buckley-Leverett equation are shown to be consistent with the derived closed-form analytical solutions. Furthermore, a general result relating the adjoint equations for different (though equivalent) primal equations is used to relate the two-phase flow adjoints to the Buckley-Leverett adjoint. Adjoint solutions from space-time discontinuous Galerkin finite element approximations of the two-phase flow equations are observed to obey this relationship.

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.

On the numerical implementation of continuous adjoint sensitivity for transient heat conduction problems using an isogeometric approach

A crucial problem of continuous adjoint shape sensitivity analysis is the numerical implementation of its lengthy formulations. In this paper, the numerical implementation of continuous adjoint shape sensitivity analysis is presented for transient heat conduction problems using isogeometric analysis, which can serve as a tutorial guide for beginners. Using the adjoint boundary and loading conditions derived from the design objective and the primary state variable fields, the numerical analysis procedure of the adjoint problem, which is solved backward in time, is demonstrated. Following that, the numerical integration algorithm of the shape sensitivity using a boundary approach is provided. Adjoint shape sensitivity is studied with detailed explanations for two transient heat conduction problems to illustrate the numerical implementation aspects of the continuous adjoint method. These two problems can be used as benchmark problems for future studies.

Solving Third-Order Singularly Perturbed Problems: Exponentially and Polynomially Fitted Trial Functions

For the third-order linearly singularly perturbed problems under four different types boundary conditions, we develop a weak-form integral equation method (WFIEM) to find the singular solution. In the WFIEM the exponentially and polynomially fitted trial functions are designed to satisfy the boundary conditions automatically, while the test functions satisfy the adjoint boundary conditions exactly. The WFIEM provides accurate and stable solutions to the highly singular third-order problems.

A comparison of the continuous and discrete adjoint approach extended based on the standard lattice Boltzmann method in flow field inverse optimization problems

The objective of this research is the numerical implementation and comparison between the performance of the continuous and discrete adjoint Lattice Boltzmann (LB) methods in optimization problems of unsteady flow fields. For this purpose, a periodic two-dimensional incompressible channel flow affected by the constant and uniform body forces is considered as the base flow field. The standard LB method and D2Q9 model are employed to solve the flow field. Moreover, the inverse optimization of the selected flow field is defined by considering the body forces as the design variables and the sum of squared errors of flow field variables on the whole field as the cost function. In this regard, the continuous and discrete adjoint approaches extended based on the LB method are used to achieve the gradients of the cost function with respect to the design variables. Finally, the numerical results obtained from the continuous adjoint LB method are compared with the discrete one, and the accuracy and efficiency of them are discussed. In addition, the validity of the obtained cost function gradients is investigated by comparing with the results of the standard forward finite difference and complex step methods. The numerical results show that regardless of the implementation cost of the two approaches, the computational cost to evaluate the gradients in each optimization cycle for the discrete adjoint LB approach is slightly more than the other one but has a little higher convergence rate and needs a smaller number of cycles to converge. Besides, the gradients obtained from the discrete version have a better agreement with those of the complex step method. Eventually, based on the structural similarities of the continuous LB equation and its corresponding adjoint one and using the simple periodic and complete bounce-back boundary conditions for the LB equation, the improved boundary conditions for the continuous adjoint LB equation are presented. The numerical results show that the use of these boundary conditions instead of the original adjoint boundary conditions significantly improves the relative accuracy and also the convergence rate of the continuous adjoint LB method.

Optimal control of a transitional jet using a continuous adjoint method

The use of active flow control with unsteady means gains increasing interest in engineering designs. The main bottleneck of the methodology is the strong dependence on trial and error to find the right set of control parameters. In this context, adjoint-based control using high-fidelity simulations is a promising method to explore optimal values in large parameter spaces. However, the applicability of the methodology to complex engineering geometries remains extremely limited. In this work, we employ adjoint-based optimal control using unsteady high-fidelity simulations in a generic unstructured grid framework. To this end, an optimal flow control study is conducted in OpenFOAM® using the continuous adjoint method and DNS simulations. To demonstrate the methodology, we study control of an incompressible axisymmetric jet at ReD=2000, with focus on improving its mixing properties. The gradient of the cost functional is calculated with a newly developed unsteady-adjoint solver based on a classical incremental projection scheme. Particular attention is paid into the presentation of mathematical and algorithmic details. Moreover, we address three main issues that remained relatively undiscussed in common practise: the choice of adjoint boundary conditions on computational boundaries, the failure of the adjoint methodology for long optimization horizons in turbulent flows and the treatment of the additional transposed convective term in the adjoint equations. Practical solutions are employed for these issues. Two optimization cases with different initial conditions are designed. To this end, we considered maximization of enstrophy in the near field, for which increments of 10.5% and 5.6% are obtained.

Optimum Shape Design for Multirow Turbomachinery Configurations Using a Discrete Adjoint Approach and an Efficient Radial Basis Function Deformation Scheme for Complex Multiblock Grids

This paper proposes a framework for fully automatic gradient-based constrained aerodynamic shape optimization in a multirow turbomachinery environment. The concept of adjoint-based gradient calculation is discussed and the development of the discrete adjoint equations for a turbomachinery Reynolds-averaged Navier–Stokes (RANS) solver, particularly the derivation of flow-consistent adjoint boundary conditions as well as the implementation of a discrete adjoint mixing-plane formulation, are described in detail. A parallelized, automatic grid perturbation scheme utilizing radial basis functions (RBFs), which is accurate and robust as well as able to handle highly resolved complex multiblock turbomachinery grid configurations, is developed and employed to calculate the gradient from the adjoint solution. The adjoint solver is validated by comparing its sensitivities with finite-difference gradients obtained from the flow solver. A sequential quadratic programming (SQP) algorithm is then utilized to determine an improved blade shape based on the gradient information from the objective functional and the constraints. The developed optimization method is used to redesign a single-stage transonic flow compressor in both inviscid and viscous flow. The design objective is to maximize the isentropic efficiency while constraining the mass flow rate and the total pressure ratio.

Adjoint-Based Optimization for Rigid Body Motion in Multiphase Navier--Stokes Flow

We consider the numerical simulation of rigid body motion in multiphase incompressible Navier--Stokes flow. The motion is formulated as an optimization problem and determined by minimizing an objective function in terms of forces and moments acting on the body, constrained by the multiphase Navier--Stokes equations. The corresponding adjoint system and boundary conditions are derived, and derivatives of the objective function are determined. Numerical experiments include the flow around a NACA hydrofoil, a box, and the standard benchmark KRISO Container Ship (KCS). We obtain a considerable speed-up compared to current state-of-the-art methods.

Adjoint Boundary Conditions for Incomplete Systems of Initial Relations Between Coefficients of the Boundary Layer

Adjoint boundary conditions for a boundary-value problem in the case of an incomplete system of initial relations between coefficients of the boundary layer are found. From those results, a theorem on asymptotic decomposition of solutions is obtained.

Turbofan duct geometry optimization for low noise using remote continuous adjoint method

In this paper, a remote continuous adjoint-based acoustic propagation (RABAP) method is proposed for low noise turbofan duct design. The goal is to develop a set of adjoint equations and their corresponding boundary conditions in order to quantify the influence of geometry modifications on the amplitude of sound pressure at a near-field location. The governing equations for the 2.5D acoustic perturbation equation solver (APE) formulation for duct acoustic propagation is first introduced. This is followed by the formulation and discretization of the remote continuous adjoint equations based on 2.5D APE. The special treatment of the adjoint boundary condition to obtain sensitivities derivatives is also discussed. The theory is applied to acoustic design of an axisymmetric fan bypass duct for two different tone noise radiations. The 2.5D APE is further validated using comparisons to an experiment data of the TURNEX nozzle geometry. The implementation of the remote continuous adjoint method is validated by comparing the sensitivity derivative with that obtained using finite difference method. The result obtained confirms the effectiveness and efficiency of the proposed RABAP framework.

A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms

We introduce a DPG method for convection-dominated diffusion problems. The choice of a test norm is shown to be crucial to achieving robust behavior with respect to the diffusion parameter (Demkowicz and Heuer 2011) [18]. We propose a new inflow boundary condition which regularizes the adjoint problem, allowing the use of a stronger test norm. The robustness of the method is proven, and numerical experiments demonstrate the method’s robust behavior.