Aerodynamic design optimization by using a continuous adjoint method

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

Design optimization of unsteady airfoils with continuous adjoint method

Inverse design optimization of the spatial distribution of emissivity for radiative transfer problems based on adjoint method

This paper presents an adjoint method for the optimal control of unsteady flows. The goal is to develop the continuous and discrete unsteady adjoint equations and their corresponding boundary conditions for the time-accurate method. First, this paper presents the complete formulation of the time-dependent optimal design problem. Second, we present the time-accurate unsteady continuous and discrete adjoint equations. Third, we present results that demonstrate the application of the theory to a two-dimensional oscillating airfoil. The results are compared with a multipoint approach to illustrate the added benefit of performing full unsteady optimization.

In this article, the gradient of aerodynamic objective functions with respect to design variables, in problems governed by the incompressible Navier–Stokes equations coupled with the k–ω SST turbulence model, is computed using the continuous adjoint method, for the first time. Shape optimization problems for minimizing drag, in external aerodynamics (flows around isolated airfoils), or viscous losses in internal aerodynamics (duct flows) are considered. Sensitivity derivatives computed with the proposed adjoint method are compared to those computed with finite differences or a continuous adjoint variant based on the frequently used assumption of frozen turbulence; the latter proves the need for differentiating the turbulence model. Geometries produced by optimization runs performed with sensitivities computed by the proposed method and the ‘frozen turbulence’ assumption are also compared to quantify the gain from formulating and solving the adjoint to the turbulence model equations.

This paper is a further study of the authors’ previous work on the continuous adjoint method based on the variation in grid node coordinates and Jacobi Matrices of the flow fluxes. This method simplifies the derivation and expression of the adjoint system, and reduces the computation cost. In this paper, the differences between the presented and the traditional methods are analyzed in details by comparing the derivation processes and the adjoint systems. In order to demonstrate the reliability and accuracy of the adjoint system deduced by the authors, the presented method is applied to different optimal problems, which include two inverse designs and two shape optimizations in both 2D and 3D cascades. The inverse designs are implemented by giving the isentropic Mach number distributions along the blade wall for 2D inviscid flow and 3D laminar flow. The shape optimizations are implemented with the objective function of the entropy generation in flow passage for 2D and 3D laminar flows. In the 3D optimal case, this method is validated by supersonic turbine design case with and without mass flow rate constraint. The numerical results testify the accuracy of this adjoint method, which includes only the boundary integrals, and furthermore, the universality and portability of this adjoint system for inverse designs and shape optimizations are demonstrated.

In this research, the continuous adjoint method is applied to optimize an airfoil in subsonic and transonic flows. An inverse design problem is solved to evaluate the ability of the optimization algorithm and then, two types of optimizations, constrained and non-constrained, are investigated in a drag minimization problem. In the non-constrained drag minimization problem, the optimization is performed in a fixed angle of attack with neither geometric nor aerodynamic constraint, but in the constrained drag minimization problem, the optimization is performed in a fixed lift coefficient. Comparison of the results of these two optimizations shows the effects of the constraint on the optimization trend and the optimized geometry. Moreover, imposing the aerodynamic constraint increased the computational costs of the adjoint method. In constrained and non-constrained drag minimization problems, the surface points are adopted as design variables to show the performance of the adjoint equations approach in problems with numerous design variables.

*† This paper deals with the use of the adjoint equation for aerodynamic shape optimization of complex configurations with overset grids methods. The use of overset grid eases the grid generation process, but the nontrivial task of ensuring communication between the overlapping grids needs careful attention. This need is effectively addressed by using a practically useful technique known as Implicit Hole Cutting (IHC) method first introduced by Lee and Baeder 1 and further demonstrated by the authors 2 in a previous study. The method depends on the simplicity of the cell selection process based on the main criterion of cell size, and all grid points including hole interior points and hole fringe points are treated indiscriminately in the flow computation. The simplicity of the IHC method is exploited for the adjoint equation solver. Similar to the flow solver, the adjoint equations are solved on the conventional point-matched grids and overlapped grids within a multi-block framework. Parallel computing with MPI is also used to improve the overall efficiency of the design optimization process. The method is successfully applied to several two and threedimensional shape optimization cases for both external and internal flows problems.

In a general sense, inverse airfoil design can be described as the means by which desired aerodynamic characteristics along with certain geometric constraints are specified and the resulting airfoil shape that satisfies the specifications is determined. In this paper, an approach is presented in which an airfoil can be designed by specifying the boundary-layer transition curve. By specifying the transition curve, the desired airfoil performance can be more readily achieved by taking advantage of the strong relationship between the boundary-layer transition curve and the drag polar. This approach builds on the inverse design method, PROFOIL. This paper contains a brief description of the PROFOIL code as well as an overview and a detailed formulation of the current approach. In the approach, a multidimensional Newton iteration scheme is used to adjust the velocity distribution over a segment of the airfoil until the transition lift coefficient is within a specified tolerance. This paper also contains examples demonstrating both the validity and usefulness of this method. The ability to specify the transition curve represents as enhancement over existing single and multipoint design methods in that the current approach allows for a single specification that spans multiple operating points.

Unsteady blade row interactions play an important role in the performance of the compressor stages. However, due to the large cost of the unsteady flow simulation, most aerodynamic optimizations of the compressor are based on the steady flow simulation. This paper adopts the time spectral method to reduce the cost of the unsteady flow simulation and a discrete adjoint solver based on the unsteady flow solver has been developed. The unsteady flow equations and the adjoint equations are solved using an in-house code. The in-house code is based on the finite volume method and solves the URANS (Unsteady Reynolds Averaged Navier-Stokes) equations on the multi-block structured mesh. For spatial discretization the 3rd order WENO (Weighted Essentially Nonoscillatory) upwind scheme is used for reconstruction and the convective flux is computed with Roe’s approximate Riemann solver. The widely used one-equation Spalart-Allmaras turbulence model is adopted for the flow simulation. For the adjoint solution, the constant-eddy viscosity assumption is adopted and only the main flow adjoint equations are solved. The adjoint equations are formed in a discrete manner, which leads to more accurate discrete objective function sensitivity than the continuous adjoint method. The present work serves as an essential part of the system for efficient unsteady aerodynamic optimization of turbomachinery.

Topology optimization with a finite strain nonlocal damage model using the continuous adjoint method

Introduction A very efficient formulation for the inverse design of two-dimensional turbomachinery blades in turbulent transonic flow conditions is presented. The proposed optimization technique employs a highly accurate space discretization scheme for the accurate computation of the flow solution and of the objective function. The sensitivity derivatives are evaluated very efficiently by using a discrete adjoint formulation. The use of an auxiliary dissipative flow solver in the adjoint equations, rather than the accurate basic flow solver, allows a robust evaluation of the sensitivity derivatives, even in presence of shocks and of numerical noise. Moreover, if the variable prescribed on the blade surface is the pressure, as usual, a simplified inviscid formulation of the adjoint problem can be used, even in the real case of turbulent flows. The entire procedure is termed progressive optimization, because both the flow computation and the iterative solution of the adjoint problem, based on the addition of an artificial time-dependent term in the adjoint equations, are brought to convergence while approaching the optimum. The idea of the simultaneous convergence of all iterative solutions suggests also the use of progressively finer grids for the progressive computation of the flow field; the cost of the adjoint problem is strongly reduced by solving the adjoint equations always at the coarsest level. The proposed approach has been tested on the inverse design of a two-dimensional turbine blade in turbulent transonic flow conditions, both with fixed stagger angle and with fixed outlet angle of the camberline. The numerical results demonstrate that the methodology is robust and highly efficient, with a converged design optimization produced in no more than the amount of computational work needed to perform from 0.7 to 1.3 converged flow analyses on the finest grid. 'Associate Professor. ^Professor and Department Head, AIAA senior member. Copyright © 2000 by L. A. Catalano and A. Dadone. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. Until few years ago, the optimization of the aerodynamic efficiency of turbomachinery bladings has been mostly approached by subsequent improvements based on the experience of the designers, each one of them followed by expensive and timeconsuming wind tunnel tests. For such a reason, the recent development of automatic tools based on the numerical solution of the flow equations and on an optimization process for the shape changes has greatly attracted the interest of turbomachinery designers. Due to the complex flow patterns about turbomachinery bladings, these numerical methods have been first developed and applied to the inverse and the direct design of airfoils; some of them have been later extended to the optimization of turbine and compressor blades. Nevertheless, many of the inverse design methods for turbomachinery bladings currently used are based on the use of a transpiration model', which cannot be extended to the direct design of the blade. For such a reason, the authors find very attractive to extend more general metodologies, already developed and tested for the optimization of airfoils or wings, to the design of turbomachinery bladings. In the application of these methods to transonic flows, typical of high-load bladings, a major difficulty commonly arises: the objective function is generally noisy or non-smooth, so that the convergence of standard optimization techniques may be compromised or even prevented, as shown in Ref. 3 for the case of a transonic nozzle. In such situations a standard finite-difference approach for design sensitivities or an alternative route based on the solution of a set of sensitivity equations' may result in an optimization process hung up at one of the many local minima artificially created by the shock shifting. The adjoint procedure proposed by Jameson et al overcomes this problem by effectively smoothing the objective function, which however may change the result of the optimization process, as outlined in Ref. 7. An alternative smoothing procedure which operates only on the design sensitivities and not on the objective function itself is proposed in Ref. 8, where it is (c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. shown to be effective and efficient in designing transonic diffusers and airfoils by using two-dimensional inviscid flow equations. A second, very important challenge is the development of an optimization technique able to converge to the optimum in a computational work corresponding to few flow analyses. Indeed, most existing methodologies are based on a sequence of global iterations, each one of them requiring a computation of the converged flow field, a solution for the sensitivity derivatives and an appropriate update of the design variables. The repeated solution of the flow equations typical of such a serial approach results in a high computational cost of the entire design procedure. The efficiency of the smoothing procedure developed in Ref. 8 has been recently improved by introducing a progressive optimization strategy''', whereby the optimization process is based on partially converged flow solutions with the aim to converge the flow solution while converging the design problems. In particular, in Ref. 11 this efficient approach is applied to the inverse design of turbine blades in inviscid transonic flow conditions. In Refs. 9 — 1 2 , the adjoint equations are solved using a direct solver, appropriate for two-dimensional flow problems, but not effective for three-dimensional flow problems. The extension to three dimensions has been considered in Ref. 13, where the progressive procedure suggested in Ref. 9 is modified by adding a time derivative in the adjoint equations, and by solving them iteratively. in a progressive manner quite similar to the one employed for the flow equations. The smoothing procedure and the use of partially converged flow solutions as well as of partially converged adjoint equation solutions in the progressive optimization implies that the optimization process is based on approximate values of the gradient of the objective function. Another example of approximate design sensitivity has been discussed by Matsuzawa and Hafez, where adjoint equations based on an inviscid flow formulation have been used successfully to determine approximate gradients of the objective function for airfoil inverse design in laminar flow conditions. The aim of the present paper is to apply all of the aforementioned strategies that contribute to improve the robustness and the efficiency of the optimization process to the inverse design of two-dimensional cascades in turbulent flow conditions. The final goal is to validate the robustness and efficiency of the outlined procedure also for this important class of design problems and, more importantly, to perform a first important step towards the development of an automatic tool for the direct optimization of turbomachinery bladings in real flow conditions. The adjoint equations will be derived using an inviscid solver involving a traditional centered scheme with added numerical viscosity. The inviscid formulation will be used in order to preserve the simplicity of the adjoint equations. Moreover, an additional time derivative will be introduced in order to solve the time dependent adjoint equations progressively in time. Finally, the flow equations will be solved on a progressively finer mesh, while the adjoint equations will be always solved on the coarsest mesh in order to reduce the cost of the computation of the gradient of the objective function to a few percents of the total cost of the optimization procedure. The following sections will be devoted to the description of the main ingredients of the proposed approach, that will be tested on the inverse design of a two-dimensional turbine blade in turbulent transonic flow conditions, both with fixed stagger angle and with fixed outlet angle of the camberline.

The article presents results of multi-criteria optimization of air nozzle topology. The optimization in the CFD has been recently developed since equations of flow in porous media were applied among others governing equations. Optimization is a seeking for extremum of an objective function with respect to the function constraints. With this definition in mind, the optimization by using a continuous adjoint for the current cases is a finding such channel topology which minimizes for example pressure or energy loss when the constraints of objective function are in the form of flow governing equations of momentum and continuity. This methodology of optimization makes a design process faster comparing to the methods related to Design of Experiments (DoE). However, for the sake of flow governing equations nonlinearity, the continuous adjoint method can be successfully applied only in relatively simply and steady-state cases. The reason is of possibility of finding the global extremum of the objective function only for that kind of cases. The results of optimization of two selected cases are presented in the article and show advantages and limitations of the method applied. The continuous adjoint simulation results indicate the nozzles design directions and can be applied in industry with limited reliability. The object of research reported in the article is the nozzle which is augmented equipment used with a pneumatic pulsator. The pulsators are devices that utilize an air stream to destruct vaults created in loose material structure. The pulsator productivity equipped with a nozzle depends on outlet pressure. Therefore, the optimization problem was stated so that pressure loss is to be as low as possible.

Most of the inverse design methods of turbomachinery experience the shortcoming where the target aerodynamic parameters need to be manually specified depending on the designers’ experience and insight, making the design result aleatory and even deviated from the real optimal solution. To tackle this problem, an experience-independent inverse design optimization method is proposed and applied to the redesign of a compressor cascade airfoil in this study. The experience-independent inverse design optimization method can automatically obtain the target pressure distribution along the cascade airfoil through the genetic algorithm, rather than through the manual specification approach. The shape of cascade airfoil is then solved by the adjoint method. The effectiveness of the experience-independent inverse design optimization method is demonstrated by two inverse design cases of the compressor cascade airfoil, i.e. the inverse design of only the suction surface and the inverse design of both the suction and pressure surfaces. The results show that the proposed inverse design method is capable of significantly improving the aerodynamic performance of the compressor cascade. At the examined flow condition, a thin airfoil profile is beneficial to flow accelerations near the leading edge and flow separation avoidance near the trailing edge. The proposed inverse design method is quite generic and can be extended to the three-dimensional inverse design of advanced compressor blades.

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