Anna Engels-Putzka

a code-coupling approach to the implementation of discrete adjoint solvers based on automatic differentiation

We propose a method for selectively applying automatic differentiation (AD) by operator overloading to develop the discrete adjoint of a turbomachinery flow solver. A fully differentiated version of the solver is generated by operator overloading using the tapeless tangent mode of ADOL-C. The differentiated solver is coupled to an undifferentiated version of the same code using message passing. The automatic differentiation is used to calculate derivatives of the flux calculation routines. The flux derivatives depending on inner cell states are sparse, and this sparsity is exploited using analytical differentiation of the spatial discretization scheme. Subsequently the sparse matrix is communicated to the undifferentiated code for solution. Turbomachinery boundary conditions may have dense Jacobians and are therefore only evaluated during the solution process. The solution of the adjoint system of equations is achieved through a preconditioned GMRES, implemented inside the undifferentiated code. A modern three dimensional contra-rotating fan stage with engineering parameterization serves as application example in order to demonstrate the technique and to perform numerical validations. The validation of gradient results is performed by comparing against results from finite differences, and the tangent forward mode.

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 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 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.

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.