Adjoint harmonic balance method for forced response analysis in turbomachinery

This paper describes the derivation and implementation of the discrete adjoint equations based on frequency domain methods (linear harmonics and harmonic balance) within a turbomachinery CFD code. Applications to model problems are presented which demonstrate the potential of the method for multidisciplinary turbomachinery problems, e.g. aeroelastics or aeroacoustics

An Efficient Implementation of Second Quantization-Based Many-Body Methods for Electrons and its Application to Coupled-Cluster with Arbitrary Excitation Level

This thesis deals with selected aspects of a new implementation of many-body methods which can be formulated in the framework of second quantization, in particular the coupled-cluster (CC) method with arbitrary excitation level. Coupled-cluster is one of the most successful and widely used quantum chemical methods for accurate calculations on small to medium-sized molecules. Since it employs a nonlinear parametrization of the wave function, its implementation is rather difficult, in particular if higher (i.e. more than double) excitations are to be included. The latter is necessary to obtain highly accurate results and also as a prerequisite for the generalization to multi-reference cases. The implementation described here has a twofold focus. One is on generality and flexibility regarding the method to be implemented, the other is on efficiency. To achieve flexibility, it is useful to have a machinery which automatically derives working equations for a given method. We realize this by applying techniques of second quantization. This work treats in particular the last step of this procedure, namely the simplification of the resulting equations by the identification of equivalent terms. The algorithm used here is based on the interpretation of algebraic terms as graphs. The derived CC equations then have to be solved iteratively. The efficiency of the program is mainly determined by the evaluation of the occurring expressions, which has to be done in each iteration step. The evaluation is split up in a sequence of tensor contractions. Their generic implementation is complicated by the particular structure of the involved tensors. We reduce each contraction to a sequence of matrix multiplications, which requires a previous data rearranging. But since matrix multiplication is the most efficient operation on modern computers, this additional effort pays off. Preliminary tests show that our program is at least as fast as the most efficient general coupled-cluster implementation so far, and the relation is expected to improve for calculations with larger basis sets where the matrix multiplication becomes the time-determining step. Finally, we give an outlook to possible further developments. In particular, the preparation of the equations before the actual evaluation (factorization) offers much potential for optimizations which we do not exploit at the moment, in contrast to the program with which we compare, which employs at least a partial optimization at this point

Adjoint-based Shape Sensitivities for Turbomachinery Design Optimizations

An adjoint preprocess for an adjoint-based turbomachinery design process is described and applied. Shape sensitivities are calculated using an adjoint elliptic mesh deformation tool. Calculation of sensitivities for the CAD parameters is performed using these shape sensitivities and deformed surface grids. The method is applied to the shape optimization process for a counter-rotating fan. Integrating the adjoint preprocess into the gradient evaluation results in a significantly reduced dependency of the computation time on the number of design parameters

Adjoint mesh deformation and adjoint-based sensitivities with respect to boundary values

This paper presents recent developments related to an adjoint RANS solver, which is part of the TRACE flow solver developed at DLR in particular for applications in turbomachinery. On the one hand, the implementation of an adjoint preprocess is described, which consists of two separate parts. First, the adjoint solution is transformed into a field of sensitivities with respect to grid point coordinates by calculating the derivative of the flow equation residual with respect to these coordinates. Second, an elliptic grid deformation procedure is adjoined in order to translate these sensitivities to surface values. The aim is to reduce the computational cost for sensitivity evaluations in applications with a large number of parameters by avoiding the generation of deformed volume grids. Furthermore, the evaluation of sensitivities with respect to boundary values, i.e. average quantities (for example pressure) which are prescribed at entry and exit surfaces, is discussed

On the usage of finite differences for the development of discrete linearised and adjoint CFD solvers

In this paper we discuss the usage of finite differences for the computation of the flux Jacobian in the framework of a discrete adjoint or time-linearised flow solver, in particular the associated choice of an appropriate step size. For comparison, we apply algorithmic differentiation to obtain an exact flux Jacobian. It turns out that the results depend strongly on the choice of the slope limiter. A careful choice of this function is crucial for computations with exact flux linearisations as well as for finite difference approximations

Adjoint process chain for forced response analysis using a harmonic balance method

This paper describes the setup and validation of a process chain for aeroelastic sensitivity analysis using adjoint methods. In particular, the adjoint of an harmonic balance solver is applied to capture unsteady effects. The interaction between different blade rows is simulated by an "external" coupling of the adjoint harmonic balance solver with a steady adjoint solver. The adjoint process is applied to a turbine stage and the results are compared to finite differences of forward computations

Sensitivity Analysis for Forced Response in Turbomachinery using an adjoint Harmonic Balance Method

The analysis of aeroelastic properties is an important aspect in the design of turbomachinery components. In this study we focus on vibrations caused by the interaction of adjacent blade rows (forced response). This is an inherently unsteady phenomenon. But due to its periodic nature it can be efficiently treated by numerical methods formulated in the frequency domain, e.g. the harmonic balance method. When going from the analysis of individual designs using CFD to CFD-based optimisation it is desirable to compute also sensitivities of objective functions (targets and restrictions for the optimisation) with respect to design parameters. Since in typical applications the number of design parameters is much larger than the number of objective functions, it is advantageous to use the adjoint method for the computation of these sensitivities. An adjoint solver based on the harmonic balance method has been implemented in the framework of the flow solver TRACE. This is now extended and used to compute the sensitivities of aeroelastic objective functions to the amplitudes of a harmonic perturbation at the entry or exit of the respective blade row. These sensitivities can be validated by comparing to finite differences obtained from harmonic balance computations with different perturbations. We apply the method to model problems which are representative for turbomachinery configurations

Forced response sensitivity analysis using an adjoint harmonic balance solver

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

Development of an adjoint harmonic balance solver for turbomachinery applications using algorithmic differentiation techniques

This contribution describes the development of a discrete adjoint solver for the harmonic balance method using the technique of algorithmic differentiation by operator overloading. Harmonic balance is a nonlinear frequency domain method which is suitable to simulate time dependent periodic phenomena as they occur for example in turbomachinery flows. The adjoint solver is implemented in the framework of an industrial turbomachinery flow solver and applied in this work to compute sensitivities of aeroelastic forces with respect to geometric changes of the blades in a turbine