Algorithmic differentiation (AD) is a widely-used approach to compute derivatives of numerical models. Many numerical models include an iterative process to solve non-linear systems of equations. To improve efficiency and numerical stability, AD is typically not applied to the linear solvers. Instead, the differentiated linear solver call is replaced with hand-produced derivative code that exploits the linearity of the original call. In practice, the iterative linear solvers are often stopped prematurely to recompute the linearisation of the non-linear outer loop. We show that in the reverse-mode of AD, the derivatives obtained with partial convergence become inconsistent with the original and the tangent-linear models, resulting in inaccurate adjoints. We present a correction term that restores consistency between adjoint and tangent-linear gradients if linear systems are only partially converged. We prove the consistency of this correction term and show in numerical experiments that the accuracy of adjoint gradients of an incompressible flow solver applied to an industrial test case is restored when the correction term is used.

CAD geometries are most often exchanged between analysis tools using NURBS patches to represent the boundary (BRep). We present a method where the control points of the BRep are used to automatically derive parametrisations suitable for shape optimisation with gradient-based methods. Particular focus is on ensuring geometric continuity between the NURBS patches, which is achieved through formulation of discrete constraints. Design variables then arise from formulating an orthogonal basis to the remaining design space using a singular value decomposition (SVD). The manuscript presents the extension of earlier work on B-spline surfaces to full NURBS surfaces and investigates the effect of the cutoff threshold of the SVD on the optimisation results. To enable routine automatic use, an estimation of the effective rank is proposed which allows to automatically determine the suitable cutoff threshold. The effectiveness of the algorithm is demonstrated for the minimisation of total pressure loss over a section of an automotive climate duct, and a U-bend cooling channel.

The algebraic fixed-point form of the equation of the discrete adjoint of a segregated SIMPLE-like incompressible solver is derived and an implementation using building blocks produced with automatic differentiation (AD) software tools is presented which reuses a substantial amount of code for matrix building and matrix-vector products from the original linearised flow code (primal). The approach reduces the memory requirement compared to code from 'brute-force' application of AD. Most importantly, the adjoint solution can be 'hot-started' from an earlier design iteration which is important for use in 'one-shot' design optimisation loops.

Abstract. A continuous adjoint solver is developed for calibration of the inlet velocity profile boundary condition (BC) for computational fluid dynamics (CFD) simulations of the neutral atmospheric boundary layer (ABL). The adjoint solver uses interior domain wind speed observations to compute the gradient of a calibration function with respect to inlet velocity speed and wind direction. The solver has been implemented in the open-source CFD package OpenFOAM coupled with the local gradient-based “CONMIN-frcg” solver of the DAKOTA optimization package. The feasibility of the optimizer output is continuously monitored during the calibration process. The inlet flow profile is considered acceptable only if it can be fitted to a logarithmic or power law function with a tolerance of 3 %. Otherwise, the optimization takes the last fitted profile and asks for a new gradient evaluation. The newly developed framework has been applied in two cases, namely the Ishihara case and Kassel domain. By using the measurements over the hill in the Ishihara case, the method was able to predict the velocity profiles upstream and downstream of the hill accurately. For the Kassel domain, despite the complexity of the site, the method managed to achieve the targeted profile within a reasonable number of the solver calls.

The pressure-correction equation and its coupling to the momentum equation as used in the popular incompressible approaches such as SIMPLE algorithm often exhibit poor stability and only converge to limit-cycle oscillations. Steady-state discrete adjoint solvers inherit the spectral properties of the flow solver from which they are derived and typically diverge if the flow discretisation is not stable. The paper discusses derivation of the discrete adjoint solver 1 using Automatic Differentiation (AD) and demonstrates the lack of stability of the SIMPLE scheme. An alternative strategy for the coupling of the flow equations is proposed that offers improved stability of the flow solver, as well as its adjoint. Numerical experiments comparing the standard SIMPLE scheme to the new stabilisation technique demonstrate the effectiveness and improved convergence of primal and adjoint system.