Discrete Adjoint Research Articles

익형 주변 박리 유동 예측 향상을 위한 데이터 기반 난류 모델 개선

A data-driven approach is investigated to improve the Spalart-Allmaras (SA) turbulence model. The field inversion and machine learning framework is chosen for the model improvement. Flow around the S809 airfoil is chosen for the test case. In this approach, a spatial-varying correction term is obtained using the discrete adjoint method in the field inversion process. Then, an artificial neural network is constructed to generalize the correction term with relevant flow features. This study shows that the corrected SA model reduces the turbulence production near the separation point, which leads to the improved prediction of the stalled airfoil at high angles of attack. Detailed investigation on modified flow fields and airfoil pressure distribution is conducted in this study to explain how the machine-learned model improves the turbulence model for separated flow.

Adjoint-based unsteady shape optimization to suppress transonic buffet

The traditional aerodynamic shape optimization mainly aims at the enhancement of lift-drag ratio in the steady state. In this study, the onset of transonic buffet is improved by the unsteady aerodynamic shape optimization. To achieve this goal, the variance of lift coefficient in a period is adopted as the optimization objective. In addition, the gradients of the objective function with respect to shape parameters are efficiently calculated by the unsteady discrete adjoint method. The proposed method is verified by two optimization examples of NACA0012 and OAT15A. Not only the onsets of transonic buffet are increased by 0.9∘ and 1.7∘ respectively, but also the aerodynamic performance is improved. According to the shape change of optimization examples, we summarize three airfoil geometric characteristics: the position of maximum thickness of the upper airfoil, the thickness near the leading edge, and the thickness near the trailing edge. The influence of the geometric characteristics on the flow stability has been studied.

Data-driven turbulence modeling in separated flows considering physical mechanism analysis

Accurate simulation of turbulent flow with separation is an important but challenging problem. In this paper, a data-driven Reynolds-averaged turbulence modeling approach, field inversion and machine learning is implemented to modify the Spalart–Allmaras model separately on three cases, namely, the S809 airfoil, a periodic hill and the GLC305 airfoil with ice shape 944. Field inversion based on a discrete adjoint method is used to quantify the model-form uncertainty with limited experimental data. An artificial neural network is trained to predict the model corrections with local flow features to extract generalized modeling knowledge. Physical knowledge of the nonequilibrium turbulence in the separating shear layer is considered when setting the prior model uncertainty. The results show that the model corrections from the field inversion demonstrate strong consistency with the underlying physical mechanism of nonequilibrium turbulence. The quantity of interest from the observation data can be reproduced with relatively high accuracy by the augmented model. In addition, the validation in similar flow conditions shows a certain extent of generalization ability.

New design of U-turn type minichannel cold plate with hybrid fins for high temperature uniformity

The thermo-hydraulic performance of cold plates is strongly influenced by the channel profiles and fin shapes. First, a shape optimization of U-turn type liquid cold plates is performed employing a discrete adjoint method in this work. The better uniformity of flow rate in minichannel cold plates is obtained by applying the optimized design of structures for entrance and baffle regions. Compared with the baseline design, the standard deviation of the flow rate for the optimal design decreases by 82.76%. The temperature uniformity of liquid cold plates can be further improved by using an asymmetric design of the double flow paths. The standard deviation of the temperature of the heating wall surface in the optimal design decreased by 20.75% compared with that of the conventional design. Furthermore, the roles of fin shape (rectangular, discontinuous, and wavy), fin thickness, and amplitude, wavelength and phase difference of sinusoidal wavy fin on the thermal performance of liquid cold plates having the asymmetric design with the curved baffle proposed are evaluated in detail. The results reveal that the larger amplitude, the smaller wavelength, the greater phase difference of wavy units may contribute to the greater heat transfer performance and the better temperature uniformity of cold plates. The heat transfer augmentation is attributed to the formation of vortices in the channel caused by the wavy fins and the increase of heat transfer surface area caused by the corrugated walls. As a result, two novel designs of minichannel cold plate were presented. New design of minichannel cold plate with wavy fins exhibits better heat transfer performance, while the other design with hybrid fins has more advantages to enhance temperature uniformity. The obtained results demonstrate that the new designs proposed have better temperature uniformity than the initial design of cold plates having straight rectangular minichannels.

Three dimensional vibroacoustic topology optimization of hearing instruments using cut elements

The focus of this article is on level set based topology optimization of vibroacoustic hearing instruments. The goal is to demonstrate the applicability of the proposed framework for optimization of 3D industrial scale hearing instruments. The framework employs an immersed boundary cut element method to handle the modeling of complex design geometries on fixed unstructured meshes. Utilization of unstructured meshes allows for optimizing small parts of a complex hearing instrument without disturbing the overall geometry. The remaining parts of the model which are not included in the optimization process are modeled with segregated approach. The design parameterization is based on an explicit level set approach, for which the nodal level set values are linked to the mathematical design variables. The optimization problem is solved using mathematical programming and the sensitivities are obtained with a discrete adjoint approach. A validation study is carried out comparing the proposed cut element model to a body fitted mesh for a large range of frequencies. The optimization framework is then applied on the tube and the suspension structures of a hearing instrument system considering two sets of material properties for the design parts. The optimization considers the minimization of sound pressure on the microphone surface for discrete frequencies aiming to reduce the feedback paths and to increase the amplification that the device can deliver to the user. Both optimization cases improve the performance of the hearing instrument system by effectively reducing the sound pressure on the microphone surface for the considered frequency range.

Data-adaptive global full-waveform inversion

SUMMARYWe present a novel approach to global-scale full-waveform inversion (FWI) that can reduce computational cost by over an order of magnitude, compared to previously published methods, without sacrificing physical and mathematical rigour. This is based on data-adaptation, and thereby application-oriented specialization, on two complementary levels. On the simulation level, we exploit the approximate azimuthal symmetry of seismic wavefields by implementing wavefield-adapted meshes and discrete adjoints, thereby lowering numerical simulation cost. On the measurement level, we use a quasi-stochastic approach where variable mini-batches of data are used during an iterative misfit minimization in order to promote a parsimonious exploitation of data. In addition to the methodological developments, we present an inversion of long-period (100–200 s) seismic waveforms from 1179 earthquakes for 3-D whole-mantle structure. The computational cost of the 72 iterations in the inversion approximately equals one third of a single iteration using an FWI approach with widely used cubed-sphere-based meshes and non-stochastic gradient optimization. The resulting LOng-Wavelength earth model (LOWE) constitutes the first global FWI constructed entirely from a spherically symmetric initial mantle structure. While mostly serving as a showcase for the method, LOWE contains a wealth of regional-scale structures that compare well to earlier tomographic images. Being conservatively smooth and based on minimal assumptions, it may therefore serve as starting model for future inversions at shorter period or smaller scales.

Aerodynamic shape optimization using a time spectral coupled adjoint for nonlinear aeroelastic problems

In this work, prediction of flutter and Limit Cycle Oscillations (LCO) for the aircraft wing has been carried out using a coupled, time spectral computational fluid dynamics (CFD) and computational structural dynamics (CSD) method. Flutter related shape design has also been carried out using a coupled, adjoint based sensitivity analysis framework. A discrete adjoint solver was developed in time-spectral form to calculate coupled sensitivity of the dynamic aeroelastic motion. The total structural energy was constrained at different levels to determine the flutter onset conditions or different LCO solutions. Compared to the time-marching approach, the time-spectral method not only enables faster convergence for unsteady problems, but also leads to significant improvement in implementation of adjoint based sensitivity analysis. Unlike the time-marching approach, there is no high memory requirement for time-spectral approach. The proposed method was applied to a pitch-plunge airfoil section at transonic flight conditions. The flutter and LCO results showed great agreements with time-marching results. The adjoint based sensitivity values were validated by finite difference based results, showing great agreement with errors less than 2%. Finally, aerodynamic shape optimization was carried out with the design objective of maximizing the flutter velocity, with and without constraint on the drag coefficient, respectively. Significant increase in flutter boundary velocity was observed in both cases.

Inversion learning of turbulent thermal diffusion for film cooling

Film cooling is a typical three-dimensional fluid phenomenon, where the coolant with lower temperature is ejected from discrete holes to protect metal walls from being burnt by the hot mainstream. It is a great challenge for Reynolds-averaged Navier–Stokes (RANS) methods to accurately predict the coolant coverage on the wall because the turbulent thermal diffusion tends to be under-predicted due to inherent assumptions behind RANS models. In this paper, a framework of integrated field inversion and machine learning is built to enhance RANS prediction of turbulent thermal diffusion. A neural network (NN) is trained in this framework to predict the spatially varying turbulent Prandtl number (Prt) and to improve the prediction of RANS models. The temperature distribution obtained from the large eddy simulation is used as the learning target, and the discrete adjoint method is used as the inverse model that helps calculate derivatives of mean square error of the temperature distribution to NN parameters. The training process of NN shows good convergence properties. The results show that the obtained NN effectively increases the insufficient turbulent thermal diffusion by predicting much lower Prt than the commonly used value of 0.9. The NN-enhanced RANS provides significant improvements on predicting experimental temperature distributions compared with general RANS models not only on the training data but also on the unseen testing data. In addition, the obtained NN can be implemented into general-purpose software with minimal effort and no numerical stability problem.

Multirow Adjoint-Based Optimization of NICFD Turbomachinery Using a Computer-Aided Design-Based Parametrization

Abstract Currently, most of the adjoint-based design systems documented in the open literature assume that the fluid behaves as an ideal gas, are restricted to the optimization of a single row of blades, or are not suited to impose geometric constraints. In response to these limitations, this paper presents a gradient-based shape optimization framework for the aerodynamic design of turbomachinery blades operating under nonideal thermodynamic conditions. The proposed design system supports the optimization of multiple blade rows, and it integrates a computer-aided design (CAD)-based parametrization with a Reynolds-averaged Navier–Stokes (RANS) flow solver and its discrete adjoint counterpart. The capabilities of the method were demonstrated by performing the design optimization of a single-stage axial turbine that employs isobutane (R600a) as working fluid. Notably, the aerodynamic optimization respected the minimum thickness constraint at the trailing edge of the stator and rotor blades and reduced the entropy generation within the turbine by 36%, relative to the baseline, which corresponds to a total-to-total isentropic efficiency increase of about 4 percentage points. The analysis of the flow field revealed that the performance improvement was achieved due to the reduction of the wake intensity downstream of the blades and the elimination of a shock-induced separation bubble at the suction side of the stator cascade.

Aerodynamic Shape Optimisation of a Camber Morphing Airfoil and Noise Estimation

In order to decrease the emitted airframe noise by a two-dimensional high-lift configuration during take-off and landing performance, a morphing airfoil has been designed through a shape design optimisation procedure starting from a baseline airfoil (NLR 7301), with the aim of emulating a high-lift configuration in terms of aerodynamic performance. A methodology has been implemented to accomplish such aerodynamic improvements by means of the compressible steady RANS equations at a certain angle of attack, with the objective of maximising its lift coefficient up to equivalent values regarding the high-lift configuration, whilst respecting the imposed structural constraints to guarantee a realistic optimised design. For such purposes, a gradient-based optimisation through the discrete adjoint method has been undertaken. Once the optimised airfoil is achieved, unsteady simulations have been carried out to obtain surface pressure distributions along a certain time-span to later serve as the input data for the aeroacoustic prediction framework, based on the Farassat 1A formulation, where the subsequent results for both configurations are post-processed to allow for a comparative analysis. Conclusively, the morphing airfoil has proven to be advantageous in terms of aeroacoustics, in which the noise has been reduced with respect to the conventional high-lift configuration for a comparable lift coefficient, despite being hampered by a significant drag coefficient increase due to stall on the morphing airfoil’s trailing edge.

Fast and reliable transient simulation and continuous optimization of large-scale gas networks

We are concerned with the simulation and optimization of large-scale gas pipeline systems in an error-controlled environment. The gas flow dynamics is locally approximated by sufficiently accurate physical models taken from a hierarchy of decreasing complexity and varying over time. Feasible work regions of compressor stations consisting of several turbo compressors are included by semiconvex approximations of aggregated characteristic fields. A discrete adjoint approach within a first-discretize-then-optimize strategy is proposed and a sequential quadratic programming with an active set strategy is applied to solve the nonlinear constrained optimization problems resulting from a validation of nominations. The method proposed here accelerates the computation of near-term forecasts of sudden changes in the gas management and allows for an economic control of intra-day gas flow schedules in large networks. Case studies for real gas pipeline systems show the remarkable performance of the new method.

A Hybrid Adjoint Network Model for Thermoacoustic Optimization

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 while 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.

Aerodynamic Shape Optimization Based on Discrete Adjoint and Rbf

Aerodynamic Shape Optimization Based on Discrete Adjoint and Rbf

Shape optimization of composite porous vapor chamber with radial grooves: A study on the minimization of maximum pressure drop

Composite porous vapor chamber (CPVC) with uniform radial grooves is a recently developed vapor chamber, exhibiting good temperature uniformity and excellent stability under high heat flux of 280 W/cm2 with the thermal resistance less than 0.15 °C/W. However, the maximum pressure drop in CPVC due to liquid flow is prone to exceed the limits of capillary pressure, and the shape design of the wicks lacks scientific and objective enough criteria. This paper proposes a shape optimization method to optimize the shape of the evaporator wick in CPVC, based on the study on the minimization of the maximum pressure drop. In this method, the discrete adjoint method is utilized to efficiently analyze the sensitivities and find the most rapid descendant directions for the minima problem. The shape of the evaporator wick is updated by the polynomial-based approach based on the sensitivity information. Results indicate that the bottom contours of the wick blocks’ surfaces surrounding the circular cavity contribute the highest sensitivities. Consequently, the mesh deformations near the contours change the shape of the lateral surfaces of wick blocks from rectangular to concave trapezoidal in both Cylindrical and Cartesian coordinate systems. This reduces the maximum velocities and flow resistance in CPVC, due to the smoother structural transition from the condenser wick to the evaporator wick, thus reducing the maximum pressure drop by about 18.5% and 23.3% in Cartesian and Cylindrical coordinate systems, respectively. The proposed method may pave a more objective and scientific alternative to design the shape of the wicks in CPVC.

A discrete adjoint approach based on finite differences applied to topology optimization of flow problems

Topology optimization methods have been vastly applied to fluid problems with different methods. In this work, the discrete adjoint (DA) approach is used in combination with a finite differences scheme to calculate the sensitivity of incompressible and compressible flow problems. The proposed methodology is especially useful for complex physical problems, where the derivation of the adjoint system can be a challenging task. The algorithm considers the finite volume model to solve the state equations, and uses an adaptive mesh refinement (AMR) scheme during the optimization to enhance the definition of the solid–fluid interface while trying to maintain a competitive computational cost. The implementation is done by using the OpenFOAM platform, which can be modified to address algorithm needs (DA and AMR). The results show an evaluation of the proposed methodology by considering traditional 2D cases of incompressible flow and exploring compressible flow cases with meshes using different cell types and 3D models.

Adjoint shape optimization of a duct for a wall jet film cooling setup

The paper presents the results of optimization of the geometric parameters of the simplified wall jet cooling system using a modified Adjoint Shape optimization method for algebraic systems of equations (Discrete Adjoint Optimization). The modification consists in using a linearized discrete system of equations with the replacement of derivatives by their finite-volume approximations. The jet flowed through a duct and out from a nozzle. The duct was inclined at an angle of 35 degrees to the cooled wall. The mean velocity ratio between the jet and the main flow was set to 2. The total heat flux on the cooled wall was taken as a cost function. The problem was considered in a two-dimensional stationary turbulent formulation (RANS). As a result of optimization, the shape of the duct changed significantly, affecting the flow inside it. The optimization led to the disappearance of the recirculation zone and reattaching of the jet to the cooled wall. As a result of the optimization performed, the heat flux at the wall increased by 20%.

An efficient algorithm for stochastic optimal control problems by means of a least-squares Monte-Carlo method

In this work, we provide discrete optimality conditions of the optimal control problems of stochastic differential equations. Euler and Runge–Kutta methods are used for discretization. A Lagrange multiplier method for a discrete-time stochastic optimal control problem is formulated. The discrete adjoint process is obtained in terms of conditional expectations and for both methods. To estimate these nested conditional expectations at each time step via simulation, we use a very powerful new approach, least-squares Monte-Carlo method, developed by Longstaff–Schwartz. This is the first time to solve a stochastic optimal control problem by calculating the nested conditional expectations numerically with the help of a least-squares Monte-Carlo method. Some examples are studied to test and demonstrate the efficiency of the Lagrange multiplier combined with the least-squares Monte-Carlo method.

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.

Shape optimization of the time-harmonic response of vibroacoustic devices using cut elements

This article presents a method for generalized shape optimization of time-harmonic vibroacoustic problems. The modeling approach utilizes an immersed boundary cut element method in conjunction with a level set representation of the geometry. The cut element method utilizes a fixed background mesh, a dimensionless contrast parameter and an integration scheme to realize complex geometries and obtain accurate physical solutions to the governing problem. The design parameterization is obtained using a nodal level set description, directly linked to the mathematical design variables, and the gradients of the objective and the constraints are obtained with the discrete adjoint approach. The framework is applied to the optimization of three 2D examples. A study on the effect of initial guess for the proposed optimization procedure is presented on a benchmark example of the design of an acoustic partitioner. Further optimization examples include design of a wave splitter to realize prescribed frequency dependent directivity for emitted acoustic waves and a suspension structure design to improve the performance of a simplified 2D model of a hearing instrument. The results demonstrate that, even though the final topology is strongly dictated by the initial design, modifying the shape allows for a significant improvement of the system behavior. • A generalized shape optimization methodology for 2D time-harmonic vibroacoustic problems using cut elements. • A simple, yet accurate, crisp interface cut element method without the need for stabilization. • A numerically stable level set based shape optimization procedure based on density methods. • Three numerical examples that demonstrate the strengths and weaknesses of the proposed method.

A Cure for Numerical Instability of Discrete Adjoint Methods to Quasi-1D Flow Equations Near Boundaries

A Cure for Numerical Instability of Discrete Adjoint Methods to Quasi-1D Flow Equations Near Boundaries