Analysis of Adjoint Error Correction for Superconvergent Functional Estimates

Earlier work introduced the notion of adjoint error correction for obtaining superconvergent estimates of functional outputs from approximate PDE solutions. This idea is based on a posteriori error analysis suggesting that the leading order error term in the functional estimate can be removed by using an adjoint PDE solution to reveal the sensitivity of the functional to the residual error in the original PDE solution. The present work provides a priori error analysis that correctly predicts the behaviour of the remaining leading order error term. Furthermore, the discussion is extended from the case of homogeneous boundary conditions and bulk functionals, to encompass the possibilities of inhomogeneous boundary conditions and boundary functionals. Numerical illustrations are provided for both linear and nonlinear problems.\ud \ud This research was supported by EPSRC under grant GR/K91149, and by NASA/Ames Cooperative Agreement No. NCC 2-5431

An introduction to the adjoint approach to design

Optimal design methods involving the solution of an adjoint system of equations are an active area of research in computational fluid dynamics, particularly for aeronautical applications. This paper presents an introduction to the subject, emphasising the simplicity of the ideas when viewed in the context of linear algebra. Detailed discussions also include the extension to p.d.e.'s, the construction of the adjoint p.d.e. and its boundary conditions, and the physical significance of the adjoint solution. The paper concludes with examples of the use of adjoint methods for optimising the design of business jets.\ud \ud This research was supported by funding from Rolls-Royce plc, BAe Systems plc and EPSRC grants GR/K91149 and GR/L95700

Analytic Adjoint Solutions for the Quasi-1D Euler Equations

The analytic properties of adjoint solutions are examined for the quasi-1D Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is required at the shock. A Green's function approach is used to derive the analytic adjoint solutions corresponding to supersonic, subsonic, isentropic and shocked transonic flows in a converging-diverging duct of arbitrary shape. This analysis reveals a logarithmic singularity at the sonic throat and confirms the expected properties at the shock.\ud \ud This research was supported by EPSRC under grant GR/K9114

Adjoint recovery of superconvergent functionals from PDE approximations

Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasi-one-dimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multi-dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations

Inversion of spinning sound fields

A method is presented for the reconstruction of rotating monopole source distributions using acoustic pressures measured on a sideline parallel to the source axis. The method requires no \textit{a priori} assumptions about the source other than that its strength at the frequency of interest vary sinusoidally in azimuth on the source disc so that the radiated acoustic field is composed of a single circumferential mode. When multiple azimuthal modes are present, the acoustic field can be decomposed into azimuthal modes and the method applied to each mode in sequence. The method proceeds in two stages, first finding an intermediate line source derived from the source distribution and then inverting this line source to find the radial variation of source strength. A far-field form of the radiation integrals is derived, showing that the far field pressure is a band-limited Fourier transform of the line source, establishing a limit on the quality of source reconstruction which can be achieved using far-field measurements. The method is applied to simulated data representing wind-tunnel testing of a ducted rotor system (tip Mach number~0.74) and to control of noise from an automotive cooling fan (tip Mach number~0.14), studies which have appeared in the literature of source identification.Comment: Revised version of paper submitted to JASA; five more figures; expanded content with more discussion of error behaviour and relation to Nearfield Acoustical Holograph

A formal soundness proof of region-based memory management for object-oriented paradigm.

Region-based memory management has been proposed as a viable alternative to garbage collection for real-time applications and embedded software. In our previous work we have developed a region type inference algorithm that provides an automatic compile-time region-based memory management for object-oriented paradigm. In this work we present a formal soundness proof of the region type system that is the target of our region inference. More precisely, we prove that the object-oriented programs accepted by our region type system achieve region-based memory management in a safe way. That means, the regions follow a stack-of-regions discipline and regions deallocation never create dangling references in the store and on the program stack. Our contribution is to provide a simple syntactic proof that is based on induction and follows the standard steps of a type safety proof. In contrast the previous safety proofs provided for other region type systems employ quite elaborate techniques

Algorithm Developments for Discrete Adjoint Methods

This paper presents a number of algorithm developments for adjoint methods using the 'discrete' approach in which the discretisation of the non-linear equations is linearised and the resulting matrix is then transposed. With a new iterative procedure for solving the adjoint equations, exact numerical equivalence is maintained between the linear and adjoint discretisations. The incorporation of strong boundary conditions within the discrete approach is discussed, as well as a new application of adjoint methods to linear unsteady flow in turbomachinery

Simulation model of erosion and deposition on a barchan dune

Erosion and deposition over a barchan dune near the Salton Sea, California, are modeled by bookkeeping the quantity of sand in saltation following streamlines of transport. Field observations of near surface wind velocity and direction plus supplemental measurements of the velocity distribution over a scale model of the dune are combined as input to Bagnold type sand transport formulas corrected for slope effects. A unidirectional wind is assumed. The resulting patterns of erosion and deposition compare closely with those observed in the field and those predicted by the assumption of equilibrium (downwind translation of the dune without change in size or geometry). Discrepancies between the simulated results and the observed or predicted erosional patterns appear to be largely due to natural fluctuations in the wind direction. The shape of barchan dunes is a function of grain size, velocity, degree of saturation of the oncoming flow, and the variability in the direction of the oncoming wind. The size of the barchans may be controlled by natural atmospheric scales, by the age of the dunes, or by the upwind roughness. The upwind roughness can be controlled by fixed elements or by sand in the saltation. In the latter case, dune scale is determined by grain size and wind velocity

High-fidelity simulation of an ultrasonic standing-wave thermoacoustic engine with bulk viscosity effects

We have carried out boundary-layer-resolved, unstructured fully-compressible Navier--Stokes simulations of an ultrasonic standing-wave thermoacoustic engine (TAE) model. The model is constructed as a quarter-wavelength engine, approximately 4 mm by 4 mm in size and operating at 25 kHz, and comprises a thermoacoustic stack and a coin-shaped cavity, a design inspired by Flitcroft and Symko (2013). Thermal and viscous boundary layers (order of 10 $\mathrm{\mu}$m) are resolved. Vibrational and rotational molecular relaxation are modeled with an effective bulk viscosity coefficient modifying the viscous stress tensor. The effective bulk viscosity coefficient is estimated from the difference between theoretical and semi-empirical attenuation curves. Contributions to the effective bulk viscosity coefficient can be identified as from vibrational and rotational molecular relaxation. The inclusion of the coefficient captures acoustic absorption from infrasonic ($\sim$10 Hz) to ultrasonic ($\sim$100 kHz) frequencies. The value of bulk viscosity depends on pressure, temperature, and frequency, as well as the relative humidity of the working fluid. Simulations of the TAE are carried out to the limit cycle, with growth rates and limit-cycle amplitudes varying non-monotonically with the magnitude of bulk viscosity, reaching a maximum for a relative humidity level of 5%. A corresponding linear model with minor losses was developed; the linear model overpredicts transient growth rate but gives an accurate estimate of limit cycle behavior. An improved understanding of thermoacoustic energy conversion in the ultrasonic regime based on a high-fidelity computational framework will help to further improve the power density advantages of small-scale thermoacoustic engines.Comment: 55th AIAA Aerospace Sciences Meeting, AIAA SciTech, 201