A novel convergence enhancement method based on Online Dimension Reduction Optimization

Iterative steady-state solvers are widely used in computational fluid dynamics. Unfortunately, it is difficult to obtain steady-state solution for unstable problem caused by physical instability and numerical instability. Optimization is a better choice for solving unstable problem because steady-state solution is always the extreme point of optimization regardless of whether the problem is unstable or ill-conditioned, but it is difficult to solve partial differential equations (PDEs) due to too many optimization variables. In this study, we propose an Online Dimension Reduction Optimization (ODRO) method to enhance the convergence of the traditional iterative method to obtain the steady-state solution of unstable problem. This method performs proper orthogonal decomposition (POD) on the snapshots collected from a few iteration steps, optimizes PDE residual in the POD subspace to get a solution with lower residual, and then continues to iterate with the optimized solution as the initial value, repeating the above three steps until the residual converges. Several typical cases show that the proposed method can efficiently calculate the steady-state solution of unstable problem with both the high efficiency and robustness of the iterative method and the good convergence of the optimization method. In addition, this method is easy to implement in almost any iterative solver with minimal code modification

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

CAD-based CFD shape optimisation using discrete adjoint solvers

Computational fluid dynamics is reaching a level of maturity that it can be used as a predictive tool. Consequently, simulation-driven product design and optimisation is starting to be deployed for industrial applications. When performing gradient-based aerodynamic shape optimisation for industrial applications, adjoint method is preferable as it can compute the design gradient of a small number of objective functions with respect to a large number of design variables efficiently. However, for certain industrial cases, the iterative calculation of steady state nonlinear flow solver based on the Reynolds-averaged Navier{Stokes equations tends to fail to converge asymptotically. For such cases, the adjoint solver usually diverges exponentially, due to the inherited linear instability from the non-converged nonlinear flow. A method for stabilising both the nonlinear flow and the adjoint solutions via an improved timestepping method is developed and applied successfully to industrial relevant test cases. Another challenge in shape optimisation is the shape parametrisation method. A good parametrisation should represent a rich design space to be explored and at the same time be flexible to take into account the various geometric constraints. In addition, it is preferable to be able to transform from the parametrisation to a format readable by most CAD software, such as the STEP le. A novel NURBS-based parametrisation method is developed that uses the control points of the NURBS patches as design variables. In addition, a test-point approach is used to impose various geometric constraints. The parametrisation is fully compatible with most CAD software. The NURBS-based parametrisation is applied to several industrial cases

Solving an inverse coupled conjugate heat transfer problem by an adjoint approach

A framework for obtaining adjoint gradients for coupled conjugate heat transfer problems is presented. The framework is tailored to partitioned approaches in which separate solvers are used for the fluid and solid domains. The exchange of sensitiv- ities between adjoint fluid and solid solvers is necessary in order to obtain gradients and how this is achieved is described. The effectiveness of the procedure is demonstrated by solving a conjugate heat transfer problem using a gradient based approach. The presented method can be extended to sensitivity analysis of multidisciplinary problems where both solvers offer adjoint derivatives

Robust and stable discrete adjoint solver development for shape optimisation of incompressible flows with industrial applications

PhD, 156ppThis thesis investigates stabilisation of the SIMPLE-family discretisations for incompressible flow and their discrete adjoint counterparts. The SIMPLE method is presented from typical \prediction-correction" point of view, but also using a pressure Schur complement approach, which leads to a wider class of schemes. A novel semicoupled implicit solver with velocity coupling is proposed to improve stability. Skewness correction methods are applied to enhance solver accuracy on non-orthogonal grids. An algebraic multi grid linear solver from the HYPRE library is linked to flow and discrete adjoint solvers to further stabilise the computation and improve the convergence rate. With the improved implementation, both of flow and discrete adjoint solvers can be applied to a wide range of 2D and 3D test cases. Results show that the semi-coupled implicit solver is more robust compared to the standard SIMPLE solver. A shape optimisation of a S-bend air flow duct from a VW Golf vehicle is studied using a CAD-based parametrisation for two Reynolds numbers. The optimised shapes and their flows are analysed to con rm the physical nature of the improvement. A first application of the new stabilised discrete adjoint method to a reverse osmosis (RO) membrane channel flow is presented. A CFD model of the RO membrane process with a membrane boundary condition is added. Two objective functions, pressure drop and permeate flux, are evaluated for various spacer geometries such as open channel, cavity, submerged and zigzag spacer arrangements. The flow and the surface sensitivity of these two objective functions is computed and analysed for these geometries. An optimisation with a node-base parametrisation approach is carried out for the zigzag con guration channel flow in order to reduce the pressure drop. Results indicate that the pressure loss can be reduced by 24% with a slight reduction in permeate flux by 0.43%.Queen Mary-China Scholarship Council Co-funded Scholarship No. 201206280018

Iterative Methods for Problems in Computational Fluid Dynamics

We discuss iterative methods for solving the algebraic systems of equations arising from linearization and discretization of primitive variable formulations of the incompressible Navier-Stokes equations. Implicit discretization in time leads to a coupled but linear system of partial differential equations at each time step, and discretization in space then produces a series of linear algebraic systems. We give an overview of commonly used time and space discretization techniques, and we discuss a variety of algorithmic strategies for solving the resulting systems of equations. The emphasis is on preconditioning techniques, which can be combined with Krylov subspace iterative methods. In many cases the solution of subsidiary problems such as the discrete convection-diffusion equation and the discrete Stokes equations plays a crucial role. We examine iterative techniques for these problems and show how they can be integrated into effective solution algorithms for the Navier-Stokes equations

A Mixed Hybrid Finite Volumes Solver for Robust Primal and Adjoint CFD

PhDIn the context of gradient-based numerical optimisation, the adjoint method is an e cient way of computing the gradient of the cost function at a computational cost independent of the number of design parameters, which makes it a captivating option for industrial CFD applications involving costly primal solves. The method is however a ected by instabilities, some of which are inherited from the primal solver, notably if the latter does not fully converge. The present work is an attempt at curbing primal solver limitations with the goal of indirectly alleviating adjoint robustness issues. To that end, a novel discretisation scheme for the steady-state incompressible Navier- Stokes problem is proposed: Mixed Hybrid Finite Volumes (MHFV). The scheme draws inspiration from the family of Mimetic Finite Di erences and Mixed Virtual Elements strategies, rid of some limitations and numerical artefacts typical of classical Finite Volumes which may hinder convergence properties. Derivation of MHFV operators is illustrated and each scheme is validated via manufactured solutions: rst for pure anisotropic di usion problems, then convection-di usion-reaction and nally Navier-Stokes. Traditional and novel Navier-Stokes solution algorithms are also investigated, adapted to MHFV and compared in terms of performance. The attention is then turned to the discrete adjoint Navier-Stokes system, which is assembled in an automated way following the principles of Equational Di erentiation, i.e. the di erentiation of the primal discrete equations themselves rather than the algorithm used to solve them. Practical/computational aspects of the assembly are discussed, then the adjoint gradient is validated and a few solution algorithms for the MHFV adjoint Navier-Stokes are proposed and tested. Finally, two examples of full shape optimisation procedures on internal ow test cases (S-bend and U-bend) are reported.European Union's Seventh Framework Programme grant agreement number 317006