The Development and Verification of a Fully Turbulent Discrete Adjoint Solver Using Algorithmic Differentiation

2021 ◽  
Author(s):  
Hangkong Wu ◽  
Shenren Xu ◽  
Xiuquan Huang ◽  
Dingxi Wang
Keyword(s):  
Computational Cost ◽  
Adjoint Equations ◽  
Nonlinear Flow ◽  
Algorithmic Differentiation ◽  
Asymptotic Rate ◽  
Discrete Adjoint ◽  
Design Variables ◽  
Sensitivity Evaluation ◽  
Design Cycle ◽  
History Of

Abstract This paper presents the development and verification of a discrete adjoint solver using algorithmic differentiation (AD). The computational cost of sensitivity evaluation using the adjoint method is largely independent of the number of design variables, making it attractive for optimization applications where the design variables are far more than objectives and constraints. To obtain the gradients of a single objective function or constraint with respect to many design variables, the nonlinear flow and the adjoint equations need to be solved once at every design cycle. This paper makes a detailed presentation of how AD is used to develop a discrete adjoint solver. The data flow diagrams of the nonlinear flow, linear and adjoint solvers are compared. Moreover, a comparison of convergence history of sensitivity, asymptotic rate of residual convergence and computational cost between the linear and adjoint solvers is also made. Two cases — the subsonic Durham turbine and transonic NASA Rotor 67 are studied in this paper. The results show that the adjoint solver has the same asymptotic rate of residual convergence and produces consistent convergence history of sensitivity as the linear solver, but the adjoint solver consumes more time and memory.

Employing the Time-Domain Unsteady Discrete Adjoint Method for Shape Optimization of Three-Dimensional Multirow Turbomachinery Configurations

2018 ◽  
Vol 140 (8) ◽  
Author(s):  
Georgios Ntanakas ◽  
Marcus Meyer ◽  
Kyriakos C. Giannakoglou
Keyword(s):  
Time Domain ◽  
Adjoint Method ◽  
Transient Flow ◽  
Three Dimensional ◽  
Adjoint Equations ◽  
Objective Functions ◽  
Discrete Adjoint ◽  
Design Variables ◽  
Frequency Domain Methods ◽  
The Time Domain

In turbomachinery, the steady adjoint method has been successfully used for the computation of derivatives of various objective functions with respect to design variables in gradient-based optimization. However, the continuous advances in computing power and the accuracy limitations of the steady-state assumption lead toward the transition to unsteady computational fluid dynamics (CFD) computations in the industrial design process. Previous work on unsteady adjoint for turbomachinery applications almost exclusively rely upon frequency-domain methods, for both the flow and adjoint equations. In contrast, in this paper, the development the discrete adjoint to the unsteady Reynolds-averaged Navier–Stokes (URANS) solver for three-dimensional (3D) multirow applications, in the time-domain, is presented. The adjoint equations are derived along with the adjoint to the five-stage Runge–Kutta scheme. Communication between adjacent rows is achieved by the adjoint sliding interface method. An optimization workflow that uses unsteady flow and adjoint solvers is presented and tested in two cases, with objective functions accounting for the transient flow in a turbine vane and the periodic flow in a compressor three-row setup.

Efficient Gradient-Based Algorithms for the Construction of Pareto Fronts

2011 ◽  
Author(s):  
Sriram Shankaran ◽  
Brian Barr
Keyword(s):  
Pareto Front ◽  
Computational Cost ◽  
High Fidelity ◽  
Simulation Tools ◽  
Pareto Points ◽  
Uniform Spacing ◽  
Design Variables ◽  
Gradient Based ◽  
Pareto Fronts ◽  
Sampling Approach

The objective of this study is to develop and assess a gradient-based algorithm that efficiently traverses the Pareto front for multi-objective problems. We use high-fidelity, computationally intensive simulation tools (for eg: Computational Fluid Dynamics (CFD) and Finite Element (FE) structural analysis) for function and gradient evaluations. The use of evolutionary algorithms with these high-fidelity simulation tools results in prohibitive computational costs. Hence, in this study we use an alternate gradient-based approach. We first outline an algorithm that can be proven to recover Pareto fronts. The performance of this algorithm is then tested on three academic problems: a convex front with uniform spacing of Pareto points, a convex front with non-uniform spacing and a concave front. The algorithm is shown to be able to retrieve the Pareto front in all three cases hence overcoming a common deficiency in gradient-based methods that use the idea of scalarization. Then the algorithm is applied to a practical problem in concurrent design for aerodynamic and structural performance of an axial turbine blade. For this problem, with 5 design variables, and for 10 points to approximate the front, the computational cost of the gradient-based method was roughly the same as that of a method that builds the front from a sampling approach. However, as the sampling approach involves building a surrogate model to identify the Pareto front, there is the possibility that validation of this predicted front with CFD and FE analysis results in a different location of the “Pareto” points. This can be avoided with the gradient-based method. Additionally, as the number of design variables increases and/or the number of required points on the Pareto front is reduced, the computational cost favors the gradient-based approach.

Parameter assessment for virtual Stackelberg game in aerodynamic shape optimization

2018 ◽  
Vol 32 (12n13) ◽  
pp. 1840044
Author(s):  
Jing Wang ◽  
Fangfang Xie ◽  
Yao Zheng ◽  
Jifa Zhang
Keyword(s):  
Shape Optimization ◽  
Stackelberg Game ◽  
Critical Parameters ◽  
Aerodynamic Shape Optimization ◽  
Aerodynamic Optimization ◽  
Aerodynamic Shape ◽  
Design Variables ◽  
Design Cycle ◽  
Parametric Studies ◽  
The Impact

In this paper, parametric studies of virtual Stackelberg game (VSG) are conducted to assess the impact of critical parameters on aerodynamic shape optimization, including design cycle, split of design variables and role assignment. Typical numerical cases, including the inverse design and drag reduction design of airfoil, have been carried out. The numerical results confirm the effectiveness and efficiency of VSG. Furthermore, the most significant parameters are identified, e.g. the increase of design cycle can improve the optimization results but it will also add computational burden. These studies will maximize the productivity of the effort in aerodynamic optimization for more complicated engineering problems, such as the multi-element airfoil and wing-body configurations.

A Hybrid Adjoint Network Model for Thermoacoustic Optimization

2021 ◽  
Author(s):  
Fellcitas Schäfer ◽  
Luca Magri ◽  
Wolfgang Polifke
Keyword(s):  
Network Model ◽  
Hybrid Approach ◽  
Adjoint System ◽  
Scattering Matrix ◽  
Adjoint Equations ◽  
Jump Conditions ◽  
Discrete Adjoint ◽  
Direct Scattering ◽  
Annular Combustor ◽  
Continuous Adjoint

Abstract A method is proposed that allows the computation of the continuous adjoint of a thermoacoustic network model based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. In this way, the need to derive the adjoint equations for every element of the network model is eliminated. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved whilst maintaining the flexibility of the discrete adjoint. It is demonstrated how the obtained adjoint system may be utilized to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.

The aerodynamic optimisation of a low-Reynolds paper plane with adjoint method

2020 ◽  
pp. 1-15
Author(s):  
Y. Zhang ◽  
X. Zhang ◽  
G. Chen
Keyword(s):  
Adjoint Method ◽  
Low Cost ◽  
Aerodynamic Performance ◽  
Discrete Adjoint ◽  
Design Variables ◽  
Limited Variation ◽  
Aerial Vehicle ◽  
Paper Plane ◽  
Drag Ratio ◽  
Lift To Drag Ratio

ABSTRACT The aerodynamic performance of a deployable and low-cost unmanned aerial vehicle (UAV) is investigated and improved in present work. The parameters of configuration, such as airfoil and winglet, are determined via an optimising process based on a discrete adjoint method. The optimised target is locked on an increasing lift-to-drag ratio with a limited variation of pitching moments. The separation that will lead to a stall is delayed after optimisation. Up to 128 design variables are used by the optimised solver to give enough flexibility of the geometrical transformation. As much as 20% enhancement of lift-to-drag ratio is gained at the cruise angle-of-attack, that is, a significant improvement in the lift-to-drag ratio adhering to the preferred configuration is obtained with increasing lift and decreasing drag coefficients, essentially entailing an improved aerodynamic performance.

The Dynamic Response and Sensitivity Analysis of the SFD-Sliding Bearing Flexible Rotor System

2012 ◽  
Vol 472-475 ◽  
pp. 1460-1464
Author(s):  
Ji Yan Wang ◽  
Yu Cheng Zhao ◽  
Chao Wang
Keyword(s):  
Dynamic Response ◽  
Periodic Motion ◽  
Period Doubling ◽  
Rotor System ◽  
Flexible Rotor ◽  
Sliding Bearing ◽  
Runge Kutta Method ◽  
Design Variables ◽  
Design Cycle ◽  
Stable Periodic

The paper established the mechanical model of SFD-sliding bearing flexible rotor system, adopting Runge-Kutta method to solve nonlinear differential equation, thus acquiring the dynamic response and the unbalanced response curve. The study has shown: from stable periodic motion, the route of the flexible rotor system to go into chaos is: periodic motion—quasi-periodic motion—chaos—period doubling bifurcation—chaos. The paper analyzed the sensitivity of the first two critical speeds of flexible rotor system, offering design variables for optimization analysis, improving the efficiency of optimization and shortening the design cycle.

scholarly journals Reliability-Based Robust Design Optimization of Structures Considering Uncertainty in Design Variables

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Shujuan Wang ◽  
Qiuyang Li ◽  
Gordon J. Savage
Keyword(s):  
Sensitivity Analysis ◽  
Design Optimization ◽  
Robust Design ◽  
Structural Reliability ◽  
Computational Cost ◽  
Optimization Process ◽  
Robust Design Optimization ◽  
Global Approximation ◽  
Design Variables ◽  
Uncertainty In Design

This paper investigates the structural design optimization to cover both the reliability and robustness under uncertainty in design variables. The main objective is to improve the efficiency of the optimization process. To address this problem, a hybrid reliability-based robust design optimization (RRDO) method is proposed. Prior to the design optimization, the Sobol sensitivity analysis is used for selecting key design variables and providing response variance as well, resulting in significantly reduced computational complexity. The single-loop algorithm is employed to guarantee the structural reliability, allowing fast optimization process. In the case of robust design, the weighting factor balances the response performance and variance with respect to the uncertainty in design variables. The main contribution of this paper is that the proposed method applies the RRDO strategy with the usage of global approximation and the Sobol sensitivity analysis, leading to the reduced computational cost. A structural example is given to illustrate the performance of the proposed method.

Prediction and Screening of Truss Spar VIM With CFD

2013 ◽  
Author(s):  
Yiannis Constantinides ◽  
Owen H. Oakley
Keyword(s):  
Design Process ◽  
Large Scale ◽  
Computational Cost ◽  
Fatigue Loading ◽  
Response Curves ◽  
Detail Design ◽  
Design Cycle ◽  
Order Of Magnitude ◽  
Offshore Industry ◽  
High Reynolds

The understanding and prediction of spar vortex induced motion (VIM) is an important area for the offshore industry as it affects the integrity of riser and mooring systems related to strength and fatigue loading. With exploration moving to deeper water, there is a desire not only to asses spar VIM early in the design cycle, but also to optimize the design. The methodology to model spar VIM with computational fluid dynamics (CFD) was pioneered by the authors and is utilized in this study to demonstrate how CFD can aid the design process, as well as conducting additional benchmark cases to improve confidence in CFD. Comparisons against experiments of a large scale hull at high Reynolds number are performed to validate CFD and establish an efficient methodology that is subsequently used for screening. This new method utilizes a boundary condition that allows simulation under arbitrary headings. As a result the overall computational cost and setup overhead is reduce by an order of magnitude enabling screening for VIM well within design cycle requirements. Simulations of spar geometries with increasing detail are performed starting from a hull with strakes, adding pipes and mooring components and finally incorporating the truss. The effects of each component are discussed with their impact on VIM response. Overall the computations show that high quality predictions are possible and it is practical to incorporate CFD into the design process. Screening and the generation of response curves can be done during initial design and more targeted assessments during detail design.

Aerofoil profile and sweep optimisation for a blended wing-body aircraft using a discrete adjoint method

2006 ◽  
Vol 110 (1111) ◽  
pp. 589-604 ◽  
Author(s):  
A. Le Moigne ◽  
N. Qin
Keyword(s):  
Sequential Quadratic Programming ◽  
Adjoint Method ◽  
Navier Stokes ◽  
High Fidelity ◽  
Sweep Angle ◽  
Sqp Method ◽  
Discrete Adjoint ◽  
Design Variables ◽  
Variable Fidelity ◽  
Blended Wing Body

Abstract Aerodynamic optimisations of a blended wing-body (BWB) aircraft are presented. A discrete adjoint solver is used to calculate efficiently the gradients, which makes it possible to optimise for a large number of design variables. The optimisations employ either a variable-fidelity method that combines low- and high-fidelity models or a direct sequential quadratic programming (SQP) method. Four Euler optimisations of a BWB aircraft are then presented. The optimisation is allowed to change a series of master sections defining the aircraft geometry as well as the sweep angle on the outer wing for two of the optimisations. Substantial improvements are obtained, not only in the Euler mode but also when the optimised geometries are evaluated using Reynolds-averaged Navier-Stokes solutions. Some interesting features of the optimised wing profiles are discussed.

scholarly journals Brass Instruments Design Using Physics-Based Sound Simulation Models and Surrogate-Assisted Derivative-Free Optimization

2016 ◽  
Author(s):  
Robin Tournemenne ◽  
Jean-François Petiot ◽  
Bastien Talgorn ◽  
Michael Kokkolaras
Keyword(s):  
Design Optimization ◽  
Search Algorithm ◽  
Rank Order ◽  
Computational Cost ◽  
Simulation Models ◽  
Function Evaluation ◽  
Brass Instruments ◽  
Design Variables ◽  
Direct Search Algorithm ◽  
Sound Simulation

This paper presents a method for design optimization of brass wind instruments. The shape of a trumpet’s bore is optimized to improve intonation using a physics-based sound simulation model. This physics-based model consists of an acoustic model of the resonator (input impedance), a mechanical model of the excitator (the lips of a virtual musician) and a model of the coupling between the excitator and the resonator. The harmonic balance technique allows the computation of sounds in a permanent regime, representative of the shape of the resonator according to control parameters of the virtual musician. An optimization problem is formulated, in which the objective function to be minimized is the overall quality of the intonation of the different notes played by the instrument (deviation from the equal-tempered scale). The design variables are the physical dimensions of the resonator. Given the computationally expensive function evaluation and the unavailability of gradients, a surrogate-assisted optimization framework is implemented using the mesh adaptive direct search algorithm (MADS). Surrogate models are used both to obtain promising candidates in the search step of MADS and to rank-order additional candidates generated by the poll step of MADS. The physics-based model is then used to determine the next design iterate. Two examples (with two and five design optimization variables, respectively) are presented to demonstrate the approach. Results show that significant improvement of intonation can be achieved at reasonable computational cost. The implementation of this method for computer-aided instrument design is discussed, considering different objective functions or constraints based on intonation but also on the timbre of the instrument.

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