Deformable Overset Grid for Multibody Unsteady Flow Simulation

A deformable overset grid method is proposed to simulate the unsteady aerodynamic problems with multiple flexible moving bodies. This method uses an unstructured overset grid coupled with local mesh deformation to achieve both robustness and efficiency. The overset grid hierarchically organizes the subgrids into clusters and layers, allowing for overlapping/embedding of different type meshes, in which the mesh quality and resolution can be independently controlled. At each time step, mesh deformation is locally applied to the subgrids associated with deforming bodies by an improved Delaunay graph mapping method that uses a very coarse Delaunay mesh as the background graph. The graph is moved and deformed by the spring analogy method according to the specified motion, and then the computational meshes are relocated by a simple one-to-one mapping. An efficient implicit hole-cutting and intergrid boundary definition procedure is implemented fully automatically for both cell-centered and cell-vertex schemes based on the wall distance and an alternative digital tree data search algorithm. This method is successfully applied to several complex multibody unsteady aerodynamic simulations, and the results demonstrate the robustness and efficiency of the proposed method for complex unsteady flow problems, particularly for those involving simultaneous large relative motion and self-deformation

A new approach to solving multiorder time-fractional advection-diffusion-reaction equations using BEM and Chebyshev matrix

In this paper, the boundary element method is combined with Chebyshev operational matrix technique to solve two-dimensional multiorder time-fractional partial differential equations: nonlinear and linear in respect to spatial and temporal variables, respectively. Fractional derivatives are estimated by Caputo sense. Boundary element method is used to convert the main problem into a system of a multiorder fractional ordinary differential equation. Then, the produced system is approximated by Chebyshev operational matrix technique, and its condition number is analyzed. Accuracy and efficiency of the proposed hybrid scheme are demonstrated by solving three different types of two-dimensional time-fractional convection-diffusion equations numerically. The convergent rates are calculated for different meshing within the boundary element technique. Numerical results are given by graphs and tables for solutions and different type of error norms

Delaunay graph and radial basis function for fast quality mesh deformation

A novel mesh deformation technique is developed based on the Delaunay graph mapping (DGM) method and the radial basis function (RBF) method. The algorithm combines the advantages of the efficiency of DGM mesh deformation and the better control of the near surface mesh quality from the RBF method. The Delaunay graph is used to divide the mesh domain into a number of sub-domains. On each of the sub-domains, the radial basis function is applied to build a much smaller sized translation matrix between the original mesh and the deformed mesh, resulting in a similar efficiency for the mesh deformation as compared to the fast Delaunay graph mapping method. Furthermore, by separating the translation and rotation motion, the mesh quality near the wall can be substantially improved for extremely large rotational deformation. The paper will show how the near-wall mesh quality is controlled and improved by the new method while the computational time is maintained to be comparable to the original Delaunay graph mapping method

Efficiency Improvements to Adjoint-Based Aeroelastic Optimisations using a Trim-Corrected and Hybrid Mesh Deformation Strategy

This purpose of this research is to increase the efficiency of the aeroelastic shape optimisation process for commercial aircraft. Aeroelastic simulations capture the interaction between aerodynamic loading and structural displacements. High-fidelity aeroelastic simulations are computationally expensive, hence an adjoint-based approach to aircraft shape optimisation is the most suitable approach when large numbers of design parameters are present. The coupled nature of the fluid-structure interaction (FSI) is reflected in the resulting adjoint equations that are used to find the gradient. Previous coupled-adjoint optimisations performed in literature have used high-fidelity solvers for both computational fluid dynamics (CFD) and computational structural mechanics (CSM) while also satisfying the trim constraints within the FSI simulation. This project builds on those studies by proposing a simple yet powerful control surface parameterisation method for satisfying the trim constraints within the FSI simulation. An additional contribution of this work is an investigation into the effects that different mesh deformation algorithms have on the rate of convergence of the coupled-adjoint. An important aspect of capturing the FSI is an effective mesh deformation strategy. The algorithm used for deforming the mesh in an FSI simulation needs to be robust to large deformations but also efficient due to the large number of times it will be required. The radial basis function (RBF) mesh deformation strategy with a data-reduction algorithm is a popular method for achieving robust and efficient deformations within FSI simulations. A key contribution of this work is the finding that the application of a data-reduction algorithm to the input field of the mesh deformation strategy has a significantly negative effect on the convergence of the coupled-adjoint whilst having only a negligible effect on the convergence of the FSI simulation. The Delaunay Graph Mapping (DGM) mesh deformation algorithm is employed to obtain faster convergence of the coupled-adjoint than the RBF approach. To increase the efficiency of optimisation process, a hybrid mesh deformation strategy is proposed by using the RBF approach within the FSI simulation and the DGM approach within the coupled-adjoint. The gradients that are obtained via the hybrid mesh deformation approach are successfully validated. The hybrid mesh deformation strategy is then applied to two optimisation scenarios in the transonic flow region. The first is a lift constrained wing optimisation. The second is a lift and trim constrained optimisation performed on a full transport aircraft configuration. The developed trim-corrected and hybrid mesh deformation optimisation strategy is shown to demonstrate a more efficient coupled-adjoint aeroelastic shape optimisation process

Mesh Sensitivity Investigation in the Discrete Adjoint Framework

Aerodynamic optimisation using gradient-based methods has found a wide range of academic applications in the last 30 years. This framework is also becoming more and more popular in the industrial world where, most of the time, unstructured grids are largely used. In this framework, apart from the need to solve the flow field, there is the need to quickly map the aerodynamic surface in terms of some aerodynamic figure of merits such as the drag coefficient, without being limited by the computational expense related to the grid size. This is a concrete industrial need which requires the efficient computation of the grid sensitivity. A novel method based on the DGM (Delaunay Graph Mapping) mesh movement is proposed to efficiently compute the grid sensitivity required in the discrete adjoint optimisation framework. The method makes use of a one-to-one explicit algebraic mapping between the volume mesh and the solid boundary nodes. This procedure results in a straightforward computation of the gradient without the need to invert a large, sparse and stiff matrix generally associated with implicit mesh movements such as the spring or LE (Linear Elastic) analogy. The method is verified using FDs (Finite Difference) and a thorough comparison in terms of CPU time, formulation against the LE-based mesh movement and adjoint gradient is presented. The DGM-based gradient chain allows to comfortably obtain the gradient with respect to each surface mesh point. Unfortunately, these gradients cannot be used directly because of their inherent poor smoothness feature. In order to address this issue one has to use a parameterisation technique which inevitably sacrifices the design space explorablity. To bridge the gap between the free-nodes and the parameterisation approaches, a novel formulation of the CST (Class Shape Transformation) was developed and termed l-CST (local-CST). The method is based on a simple trigonometric function which works as a cut-off filter on the BPs (Bernstein Polynomials) which are used to enforce a strong on-demand local control. The method is tested on an inverse geometric fitting and its effect on the resulting aerodynamic coefficients and the pressure distribution is also analysed. The DGM-based chain allows the efficient mapping of the entire surface while the l-CST allows the combination of excellent explorablity and surface smoothness. The former is tested within the non-consistent mesh movement and sensitivity framework because there are situations where one method may be preferred over the other based on the grounds that mesh movement is a very different task than mesh sensitivity although strongly related to each other. The latter is instead tested against the free-nodes approach which offers a similar advantage in terms of discrete control although without maintaining a C2 curve unless properly smoothed

Mesh Sensitivity Investigation in the Discrete Adjoint Framework

Aerodynamic optimisation using gradient-based methods has found a wide range of academic applications in the last 30 years. This framework is also becoming more and more popular in the industrial world where, most of the time, unstructured grids are largely used. In this framework, apart from the need to solve the flow field, there is the need to quickly map the aerodynamic surface in terms of some aerodynamic figure of merits such as the drag coefficient, without being limited by the computational expense related to the grid size. This is a concrete industrial need which requires the efficient computation of the grid sensitivity. A novel method based on the DGM (Delaunay Graph Mapping) mesh movement is proposed to efficiently compute the grid sensitivity required in the discrete adjoint optimisation framework. The method makes use of a one-to-one explicit algebraic mapping between the volume mesh and the solid boundary nodes. This procedure results in a straightforward computation of the gradient without the need to invert a large, sparse and stiff matrix generally associated with implicit mesh movements such as the spring or LE (Linear Elastic) analogy. The method is verified using FDs (Finite Difference) and a thorough comparison in terms of CPU time, formulation against the LE-based mesh movement and adjoint gradient is presented. The DGM-based gradient chain allows to comfortably obtain the gradient with respect to each surface mesh point. Unfortunately, these gradients cannot be used directly because of their inherent poor smoothness feature. In order to address this issue one has to use a parameterisation technique which inevitably sacrifices the design space explorablity. To bridge the gap between the free-nodes and the parameterisation approaches, a novel formulation of the CST (Class Shape Transformation) was developed and termed l-CST (local-CST). The method is based on a simple trigonometric function which works as a cut-off filter on the BPs (Bernstein Polynomials) which are used to enforce a strong on-demand local control. The method is tested on an inverse geometric fitting and its effect on the resulting aerodynamic coefficients and the pressure distribution is also analysed. The DGM-based chain allows the efficient mapping of the entire surface while the l-CST allows the combination of excellent explorablity and surface smoothness. The former is tested within the non-consistent mesh movement and sensitivity framework because there are situations where one method may be preferred over the other based on the grounds that mesh movement is a very different task than mesh sensitivity although strongly related to each other. The latter is instead tested against the free-nodes approach which offers a similar advantage in terms of discrete control although without maintaining a C2 curve unless properly smoothed