Forced Response Sensitivity Analysis Using an Adjoint Harmonic Balance Solver

2018 ◽  
Author(s):  
Anna Engels-Putzka ◽  
Jan Backhaus ◽  
Christian Frey
Keyword(s):  
Frequency Domain ◽  
Harmonic Balance ◽  
Unsteady Aerodynamics ◽  
Harmonic Balance Method ◽  
Forced Response ◽  
Algorithmic Differentiation ◽  
Design And Optimization ◽  
Initial Application ◽  
Reverse Mode ◽  
Geometric Changes

This paper describes the development and initial application of an adjoint harmonic balance solver. The harmonic balance method is a numerical method formulated in the frequency domain which is particularly suitable for the simulation of periodic unsteady flow phenomena in turbomachinery. Successful applications of this method include unsteady aerodynamics as well as aeroacoustics and aeroelasticity. Here we focus on forced response due to the interaction of neighboring blade rows. In the CFD-based design and optimization of turbomachinery components it is often helpful to be able to compute not only the objective values — e.g. performance data of a component — themselves, but also their sensitivities with respect to variations of the geometry. An efficient way to compute such sensitivities for a large number of geometric changes is the application of the adjoint method. While this is frequently used in the context of steady CFD, it becomes prohibitively expensive for unsteady simulations in the time domain. For unsteady methods in the frequency domain, the use of adjoint solvers is feasible, but still challenging. The present approach employs the reverse mode of algorithmic differentiation (AD) to construct a discrete adjoint of an existing harmonic balance solver in the framework of an industrially applied CFD code. The paper discusses implemen-tational issues as well as the performance of the adjoint solver, in particular regarding memory requirements. The presented method is applied to compute the sensitivities of aeroelastic objectives with respect to geometric changes in a turbine stage.

Forced Response Sensitivity Analysis Using an Adjoint Harmonic Balance Solver

2019 ◽  
Vol 141 (3) ◽  
Author(s):  
Anna Engels-Putzka ◽  
Jan Backhaus ◽  
Christian Frey
Keyword(s):  
Frequency Domain ◽  
Harmonic Balance ◽  
Unsteady Aerodynamics ◽  
Forced Response ◽  
Algorithmic Differentiation ◽  
Design And Optimization ◽  
Initial Application ◽  
Reverse Mode ◽  
The Time Domain ◽  
Geometric Changes

This paper describes the development and initial application of an adjoint harmonic balance (HB) solver. The HB method is a numerical method formulated in the frequency domain which is particularly suitable for the simulation of periodic unsteady flow phenomena in turbomachinery. Successful applications of this method include unsteady aerodynamics as well as aeroacoustics and aeroelasticity. Here, we focus on forced response due to the interaction of neighboring blade rows. In the simulation-based design and optimization of turbomachinery components, it is often helpful to be able to compute not only the objective values—e.g., performance data of a component—themselves but also their sensitivities with respect to variations of the geometry. An efficient way to compute such sensitivities for a large number of geometric changes is the application of the adjoint method. While this is frequently used in the context of steady computational fluid dynamics (CFD), it becomes prohibitively expensive for unsteady simulations in the time domain. For unsteady methods in the frequency domain, the use of adjoint solvers is feasible but still challenging. The present approach employs the reverse mode of algorithmic differentiation (AD) to construct a discrete adjoint of an existing HB solver in the framework of an industrially applied CFD code. The paper discusses implementational issues as well as the performance of the adjoint solver, in particular regarding memory requirements. The presented method is applied to compute the sensitivities of aeroelastic objectives with respect to geometric changes in a turbine stage.

scholarly journals Comparison of Common Methods in Dynamic Response Predictions of Rotor Systems with Malfunctions

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hongliang Yao ◽  
Qian Zhao ◽  
Qi Xu ◽  
Bangchun Wen
Keyword(s):  
Frequency Domain ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Newmark Method ◽  
Incremental Harmonic Balance Method ◽  
Rotor Systems ◽  
Double Frequency ◽  
Frequency Domain Methods ◽  
Incremental Harmonic Balance

The efficiency and accuracy of common time and frequency domain methods that are used to simulate the response of a rotor system with malfunctions are compared and analyzed. The Newmark method and the incremental harmonic balance method are selected as typical representatives of time and frequency domain methods, respectively. To improve the simulation efficiency, the fixed interface component mode synthesis approach is combined with the Newmark method and the receptance approach is combined with the incremental harmonic balance method. Numerical simulations are performed for rotor systems with single and double frequency excitations. The inherent characteristic that determines the efficiency of the two methods is analyzed. The results of the analysis indicated that frequency domain methods are suitable single and double frequency excitation rotor systems, whereas time domain methods are more suitable for multifrequency excitation rotor systems.

scholarly journals Dual Time Stepping Algorithms With the High Order Harmonic Balance Method for Contact Interfaces With Fretting-Wear

2011 ◽  
Author(s):  
Loi¨c Salles ◽  
Laurent Blanc ◽  
Fabrice Thouverez ◽  
Alexander M. Gouskov ◽  
Pierrick Jean
Keyword(s):  
Frequency Domain ◽  
Dry Friction ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Contact Forces ◽  
Balance Method ◽  
Fretting Wear ◽  
Rate Equations ◽  
Time Stepping ◽  
Dual Time Stepping

Contact interfaces with dry friction are frequently used in turbomachinery. Dry friction damping produced by the sliding surfaces of these interfaces reduces the amplitude of bladed-disk vibration. The relative displacements at these interfaces lead to fretting-wear which reduces the average life expectancy of the structure. Frequency response functions are calculated numerically by using the multi-Harmonic Balance Method (mHBM). The Dynamic Lagrangian Frequency-Time method is used to calculate contact forces in the frequency domain. A new strategy for solving non-linear systems based on dual time stepping is applied. This method is faster than using Newton solvers. It was used successfully for solving Nonlinear CFD equations in the frequency domain. This new approach allows identifying the steady state of worn systems by integrating wear rate equations a on dual time scale. The dual time equations are integrated by an implicit scheme. Of the different orders tested, the first order scheme provided the best results.

On the Use of the Automatic Differentiation for Developing a Linear Harmonic Solver

2018 ◽  
Author(s):  
Wei Zhang ◽  
Dingxi Wang ◽  
Xiuquan Huang ◽  
Tianxiao Yang ◽  
Hong Yan ◽  
...  
Keyword(s):  
Harmonic Balance ◽  
Automatic Differentiation ◽  
Harmonic Balance Method ◽  
Source Codes ◽  
Reverse Mode ◽  
Frequency Domain Methods ◽  
The Cost ◽  
Time Periodic ◽  
Small Disturbances

The linear and nonlinear harmonic methods are efficient frequency domain methods for analyzing time periodic unsteady flow fields. They have been widely used in both academia and industry. But the cost and complexity of developing a linear harmonic solver has been limiting its wider applications. On the other hand, the automatic differentiation (AD) has long been used in the CFD community with a focus on generating adjoint codes in a reverse mode. All those AD tools can do a much better job in generating linearized codes in a tangent mode, but so far very little, if any, attention is paid to using AD for developing linear harmonic solvers. The linear harmonic method, in comparison with the harmonic balance method, has its own advantages. For example, it can capture small disturbances very effectively, and avoids aliasing errors which can lead to solution instability since each wave component is solved for separately. This paper presents the effort of using an AD tool to generate major source codes for the development of a linear harmonic solver for analyzing time periodic unsteady flows. It includes the procedures and advice of using AD for such a purpose. A case study is also presented to validate the developed linear harmonic solver.

On the Application of Frequency-Domain Methods to Multistage Turbomachinery

2015 ◽  
Author(s):  
Laura Junge ◽  
Graham Ashcroft ◽  
Peter Jeschke ◽  
Christian Frey
Keyword(s):  
Frequency Domain ◽  
Time Domain ◽  
Harmonic Balance ◽  
Axial Compressor ◽  
Harmonic Balance Method ◽  
Nonlinear Frequency ◽  
Solution Quality ◽  
Multi Stage ◽  
Frequency Domain Methods ◽  
Blade Row

Due to the relative motion between adjacent blade rows the aerodynamic flow fields within turbomachinery are normally dominated by deterministic, periodic phenomena. In the numerical simulation of such unsteady flows (nonlinear) frequency-domain methods are therefore attractive as they are capable of fully exploiting the given spatial and temporal periodicity, as well as capturing or modelling flow nonlinearity. Central to the efficiency and accuracy of such frequency-domain methods is the selection of the frequencies and the circumferential modes to be resolved in simulations. Whilst trivial in the context of the simulation of a single compressor- or turbine-stage, the choice of solution modes becomes substantially more involved in multi-stage configurations. In this work the importance of mode scattering, in the context of the unsteady aerodynamic field, is investigated and quantified. It is shown that scattered modes can substantially impact the unsteady flow field and are essential for the accurate modelling of wake propagation within multistage configurations. Furthermore, an iterative approach is outlined, based on the spectral analysis of the circumferential modes at the interfaces between blade rows, to identify the dominant solution modes that should be resolved in the adjacent blade row. To demonstrate the importance of mode scattering and validate the approach for their identification the unsteady blade row interaction within a 4.5 stage axial compressor is computed using both the harmonic balance method and, based on a full annulus midspan simulation, a time-domain method. Through the inclusion of scattered modes it is shown that the solution quality of the harmonic balance results is comparable to that of the nonlinear time-domain simulation.

Forced Response Prediction of Blades With Loosely Assembled Dovetail Attachment by HBM

2009 ◽  
Author(s):  
Shangguan Bo ◽  
Zili Xu ◽  
Qilin Wu ◽  
XianDing Zhou ◽  
ShouHong Cao
Keyword(s):  
Friction Force ◽  
Centrifugal Force ◽  
Dry Friction ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Forced Response ◽  
Balance Method ◽  
Force Model ◽  
Equivalent Stiffness ◽  
Equivalent Damping

To understand the mechanism of interfacial damping of axial loosely assembled dovetail to suppress blade vibration, a dry friction force model is presented by the Coulomb friction law and the macroslip model, and the mathematical expression of the friction force is derived. The nonlinear friction force is linearized as an equivalent stiffness and an equivalent damping through the one-term harmonic balance method. The effect of centrifugal force on the equivalent stiffness and the equivalent damping is studied. The forced response of one simplified blade with loosely assembled dovetail attachment is predicted by the harmonic balance method, in which the blade is described by the lumped mass and spring model, and the friction contact joints is simplified as a ideal friction damper. The results show that the equivalent stiffness of loosely assembled dovetail attachment increases with blade centrifugal force, gradually reaches a certain value, and there exists the maximum value for the equivalent stiffness. The equivalent damping increases at the beginning and then decreases with blade centrifugal force increasing, there exists a maximum too. The resonant frequency of blade rises with blade centrifugal force, but it no longer increases when the centrifugal force exceed a certain value. There exists a special centrifugal force on which the effect of dry friction damping is the best.

Simulations of Unsteady Blade Row Interactions Using Linear and Non-Linear Frequency Domain Methods

2015 ◽  
Author(s):  
Christian Frey ◽  
Graham Ashcroft ◽  
Hans-Peter Kersken
Keyword(s):  
Frequency Domain ◽  
Harmonic Balance ◽  
Measurement Data ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Frequency Domain Methods ◽  
Blade Row ◽  
Non Linear ◽  
Rotor Stator Interaction ◽  
Unsteady Simulation

This paper compares various approaches to simulate unsteady blade row interactions in turbomachinery. Unsteady simulations of turbomachinery flows have gained importance over the last years since increasing computing power allows the user to consider 3D unsteady flows for industrially relevant configurations. Furthermore, for turbomachinery flows, the last two decades have seen considerable efforts in developing adequate CFD methods which exploit the rotational symmetries of blade rows and are therefore up to several orders of magnitude more efficient than the standard unsteady approach for full wheel configurations. This paper focusses on the harmonic balance method which has been developed recently by the authors. The system of equations as well as the iterative solver are formulated in the frequency domain. The aim of this paper is to compare the harmonic balance method with the time-linearized as well as the non-linear unsteady approach. For the latter the unsteady flow fields in a fan stage are compared to reference results obtained with a highly resolved unsteady simulation. Moreover the amplitudes of the acoustic modes which are due to the rotor stator interaction are compared to measurement data available for this fan stage. The harmonic balance results for different sets of harmonics in the blade rows are used to explain the minor discrepancies between the time-linearized and unsteady results published by the authors in previous publications. The results show that the differences are primarily due to the neglection of the two-way coupling in the time-linearized simulations.

Vibration Control for Integrally Bladed Disks Using Friction Ring Dampers

2007 ◽  
Author(s):  
Denis Laxalde ◽  
Fabrice Thouverez ◽  
Jean-Pierre Lombard
Keyword(s):  
Frequency Domain ◽  
Forced Response ◽  
Reduced Order Model ◽  
Order Model ◽  
Bladed Disks ◽  
Reduced Order ◽  
Design And Optimization ◽  
Beam Elements ◽  
Parametric Studies ◽  
Frequency Domain Approach

A damping strategy for integrally bladed disks (blisks) is discussed in this paper; this involves the use of friction rings located underside the wheel of bladed disks. The forced response of the blisk with friction rings is derived in the frequency domain using a frequency domain approach known as Dynamic Lagrangian Frequency-Time method. The blisk is modeled using a reduced-order model and the rings are modeled using beam elements. The results of some numerical simulations and parametric studies are presented. The range of application of this damping device is discussed. Parametric studies are presented and allow to understand the dissipation phenomena. Finally some design and optimization guidelines are given.

On the Development of Harmonic Balance Methods for Multiple Fundamental Frequencies

2018 ◽  
Author(s):  
Laura Junge ◽  
Graham Ashcroft ◽  
Hans-Peter Kersken ◽  
Christian Frey
Keyword(s):  
Frequency Domain ◽  
Fundamental Frequency ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Cross Coupling ◽  
Nonlinear Frequency ◽  
Fundamental Frequencies ◽  
Frequency Domain Methods ◽  
Sampling Points ◽  
The Common

Due to the relative motion between adjacent blade rows the aerodynamic flow fields within turbomachinery are usually dominated by deterministic, periodic phenomena. In the numerical simulation of such unsteady flows, (nonlinear) frequency-domain methods are therefore attractive as they are capable of fully exploiting the given spatial and temporal periodicity, as well as modelling flow nonlinearities. A nontrivial issue in the application of frequency-domain methods to turbomachinery flows is to simultaneously capture disturbances with multiple fundamental frequencies in one relative system. In case of harmonically related frequencies, the interval spanned by the sampling points typically resolves the common fundamental frequency. To avoid signal aliasing the highest harmonic of the common frequency should be sampled with an appropriate number of sampling points. However, when the common fundamental frequency is very low in relation to the frequencies of primary interest, equidistant time sampling leads to a high number of sampling points, hence frequency-domain methods can become computationally inefficient. Furthermore, when a problem can no longer be described by harmonic perturbations that are integer multiples of one fundamental frequency, as it may occur in two-shaft configurations, the standard discrete Fourier transform is no longer suitable and the basic harmonic balance method requires extension. In this article two nonlinear frequency-domain approaches for dealing with the accounted issues are demonstrated and compared. The first approach is a generalized harmonic balance method based on almost periodic Fourier transforms with non-equidistant time sampling. Then the so-called harmonic set approach, developed by the authors, is evaluated. Based on the neglection of the nonlinear, quadratic cross-coupling terms between higher harmonics of different fundamental frequencies, the harmonic set approach allows the superposition of periodic disturbances with different fundamental frequencies as well as the separated, equidistant sampling of the highest harmonic of each base frequency. The aim of this paper is to compare the computational efficiency and accuracy of the two methods and assess the impact of neglecting the quadratic cross-coupling terms.

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