Delaunay Graph Mapping Research Articles

Hybrid Mesh Deformation for Aerodynamic-Structural Coupled Adjoint Optimization

This paper demonstrates that a discrete coupled-adjoint aeroelastic shape optimization can be made more efficient with the use of different mesh-deformation algorithms for the fluid-structure interaction (FSI) simulations and the coupled-adjoint calculations within the optimization loop. Mesh deformation using radial basis functions (RBF) with only a subset of surface points is popular due to the efficiency and mesh quality produced by the technique. However, this technique can reduce the rate of convergence of the coupled-adjoint and even prevent the coupled-adjoint equations from converging. This paper proposes a hybrid mesh-deformation strategy to improve the efficiency of coupled-adjoint optimizations: use the RBF method with a data-reduction algorithm when deforming the mesh within FSI simulations but use the Delaunay graph mapping (DGM) method in the coupled-adjoint procedure. The DGM method increases the rate of convergence of the coupled-adjoint matrix, relative to the RBF approach with a data-reduction algorithm, with the additional benefit of being a faster method. Using this hybrid approach, an optimization in which lift and pitching-moment constraints are satisfied within the FSI simulation is performed. The results of the optimization, and the effects of the hybrid approach, are presented.

Robust optimization for a wing at drag divergence Mach number based on an improved PSO algorithm

In this study, an improved PSO (particle swarm optimization) algorithm is proposed and applied to the robust optimization of a wing at drag divergence Mach number. In order to reduce the number of design variables, a six-order CST (class/shape function transformation) method is employed for airfoil parameterization. For the purpose of improving the optimization efficiency, Delaunay graph mapping method is adopted for mesh deformation in each iteration of the airfoil optimization, and NURBS (non-uniform rational B-splines)-FFD (free-form deformation) method is employed for mesh deformation in each iteration of the wing optimization. For improving the standard PSO algorithm, CVTs (centroidal Voronoi tessellations) method is introduced to generate original positions of the particles more dispersedly, a second-order oscillating scheme is used and an FDR (fitness distance ratio) item is added for updating velocities and positions of the particles. By virtue of the improved PSO algorithm, single point optimization and robust optimization are conducted for both airfoil and wing. The results indicate that, comparing with the single point optimizations, the robust optimizations not only reduce drag coefficients of the airfoil and the wing at cruise Mach numbers, but also attenuate the drag increments as the Mach number increases up to drag divergence Mach numbers.

Delaunay graph-based moving mesh method with damping functions

The fluid–structure interaction and aerodynamic shape optimization usually involve the moving or deforming boundaries, thus the dynamic mesh techniques are the key techniques to cope with such deformation. A novel dynamic mesh method was developed based on the Delaunay graph in this paper. According to the Delaunay graph, the mesh points were divided into groups. In each group, a factor ranging from 0 to 1 was calculated based on the area/volume ratio. By introducing a proper function for this factor, this method can control the mesh quality with high efficiency. Several test cases were compared with other dynamic mesh methods regarding mesh quality and CPU time, such as radial basis function method and Delaunay graph mapping method.

Nonconsistent Mesh Movement and Sensitivity Calculation on Adjoint Aerodynamic Optimization

This paper presents an investigation of the influence of a nonconsistent approach in terms of mesh movement and mesh sensitivity calculation in a discrete adjoint-based optimization. Some mesh movement methods are more robust or of higher quality, whereas others can be more efficient for calculating mesh sensitivity. It is found that a nonconsistent approach gives comparable results when compared to a consistent approach. Therefore, an appropriate combination of nonconsistent approaches can be achieved for efficient adjoint optimization. This paper investigates and compares various consistent and nonconsistent combinations by using linear elasticity, Delaunay graph mapping, and radial basis function mesh movement methods. An investigation is presented, using a lift-constrained drag minimization, to assess which step of the chain introduces a deviation, if any, and to which degree this affects the final result.

Discrete boundary smoothing using control node parameterisation for aerodynamic shape optimisation

• A novel method of shape deformation using discrete boundary smoothing. • Geometry and domain coupled via a control nodes approach. • Volume mesh is deformed via a Delaunay Graph Mapping technique. • Modified Cuckoo Search - an evolutionary optimisation algorithm - is employed. • Successfully applied to a range of geometries and objective functions. This paper presents an automated aerodynamic optimisation algorithm using a novel method of parameterising the search domain and geometry by employing user–defined control nodes. The displacement of the control nodes is coupled to the shape boundary movement via a ‘discrete boundary smoothing’. This is initiated by a linear deformation followed by a discrete smoothing step to act on the boundary during the mesh movement based on the change in its second derivative. Implementing the discrete boundary smoothing allows both linear and non-linear shape deformation along the same boundary dependent on the preference of the user. The domain mesh movement is coupled to the shape boundary movement via a Delaunay graph mapping. An optimisation algorithm called Modified Cuckoo Search (MCS) is used acting within the prescribed design space defined by the allowed range of control node displacement. In order to obtain the aerodynamic design fitness a finite volume compressible Navier-Stokes solver is utilized. The resulting coupled algorithm is applied to a range of case studies in two dimensional space including the optimisation of a RAE2822 aerofoil and the optimisation of an intake duct under subsonic, transonic and supersonic flow conditions. The discrete mesh–based optimisation approach outlined is shown to be effective in terms of its generalised applicability, intuitiveness and design space definition.

Delaunay Graph Based Inverse Distance Weighting for Fast Dynamic Meshing

AbstractA novel mesh deformation technique is developed based on the Delaunay graph mapping method and the inverse distance weighting (IDW) interpolation. The algorithm maintains the advantages of the efficiency of Delaunay graph mapping mesh deformation while it also possesses the ability of better controlling the near surface mesh quality. The Delaunay graph is used to divide the mesh domain into a number of sub-domains. On each sub-domain, the inverse distance weighting interpolation is applied, resulting in a similar efficiency as compared to the fast Delaunay graph mapping method. The paper will show how the near-wall mesh quality is controlled and improved by the new method

Efficient Method to Eliminate Mesh Sensitivity in Adjoint-Based Optimization

A method based on the Delaunay graph mapping mesh movement is proposed to eliminate mesh sensitivity, which is required in the discrete adjoint optimization framework. The method makes use of a one-to-one explicit algebraic mapping between the volume and surface mesh nodes, for which the relative volume coefficients based on the Delaunay graph are computed only once. This procedure also results in a straightforward computation of the mesh sensitivity without the need to invert the large sparse matrix generally associated with implicit mesh movements such as the spring analogy, torsional spring, and linear elastic analogy. The advantage of the method comes from the fact that the solution of the second linear mesh adjoint system of equations is no longer required. The method has been verified for the two-dimensional RAE 5243 airfoil and for the three-dimensional ONERA M6 wing for two cases: 1) sensitivities of the objective function with respect to each surface mesh point as a design variable compared with those computed using finite differences, and 2) sensitivities against incidence as a design variable compared with those derived analytically. A comparison of the surface sensitivities with the mesh adjoint method using linear elasticity is also presented.

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.

FAST DYNAMIC MESHING METHOD BASED ON DELAUNAY GRAPH AND INVERSE DISTANCE WEIGHTING INTERPOLATION

A novel mesh deformation technique is developed based on the Delaunay graph mapping method and the inverse distance weighting (IDW) interpolation. The algorithm maintains the advantages of the efficiency of Delaunay-graph-mapping mesh deformation while possess the ability for better controlling the near surface mesh quality. The Delaunay graph is used to divide the mesh domain into a number of sub-domains. On each of the sub-domains, the inverse distance weighting interpolation is applied to build a much smaller sized translation matrix between the original mesh and the deformed mesh, resulting a similar efficiency for the mesh deformation as compared to the fast Delaunay graph mapping method. The paper will show how the near-wall mesh quality is controlled and improved by the new method while the computational time is compared with the original Delaunay graph mapping method.

EFFICIENT CFD/CSD COUPLING METHODS FOR AEROELASTIC APPLICATIONS

A fast aeroelastic numerical simulation method using CFD/CSD coupling are developed. Generally, aeroelastic numerical simulation costs much time and significant hardware resources with CFD/CSD coupling. In this paper, dynamic grid method, full implicit scheme, parallel technology and improved coupling method are researched for efficiency simulation. An improved Delaunay graph mapping method is proposed for efficient dynamic grid deform. Hybrid grid finite volume method is used to solve unsteady flow fields. The dual time stepping method based on parallel implicit scheme is used in temporal discretization for efficiency simulation. An approximate system of linear equations is solved by the GMRES algorithm with a LU-SGS preconditioner. This method leads to a significant increase in performance over the explicit and LU-SGS implicit methods. A modification of LU-SGS is proposed to improve the parallel performance. Parallel computing overs a very effective way to improve our productivity in doing CFD/CFD coupling analysis. Improved loose coupling method is an efficiency way over the loose coupling method and tight coupling method. 3D wing's aeroelastic phenomenon is simulated by solving Reynolds-averaged Navier–Stokes equations using improved loose coupling method. The flutter boundary is calculated and agrees well with experimental data. The transonic hole is very clear in numerical simulation results.

A novel implementation of computational aerodynamic shape optimisation using Modified Cuckoo Search

This paper outlines a new computational aerodynamic design optimisation algorithm using a novel method of parameterising a computational mesh using ‘control nodes’. The shape boundary movement as well as the mesh movement is coupled to the movement of user–defined control nodes via a Delaunay Graph Mapping technique. A Modified Cuckoo Search algorithm is employed for optimisation within the prescribed design space defined by the allowed range of control node displacement. A finite volume compressible Navier–Stokes solver is used for aerodynamic modelling to predict aerodynamic design ‘fitness’. The resulting coupled algorithm is applied to a range of test cases in two dimensions including aerofoil lift–drag ratio optimisation intake duct optimisation under subsonic, transonic and supersonic flow conditions. The discrete (mesh–based) optimisation approach presented is demonstrated to be effective in terms of its generalised applicability and intuitiveness.

A Hybrid Method for Dynamic Mesh Generation Based on Radial Basis Functions and Delaunay Graph Mapping

AbstractAiming at complex configuration and large deformation, an efficient hybrid method for dynamic mesh generation is presented in this paper, which is based on Radial Basis Functions (RBFs) and Delaunay graph mapping. Based on the computational mesh, a set of very coarse grid named as background grid is generated firstly, and then the computational mesh can be located at the background grid by Delaunay graph mapping technique. After that, the RBFs method is applied to deform the background grid by choosing partial mesh points on the boundary as the control points. Finally, Delaunay graph mapping method is used to relocate the computational mesh by employing area or volume weight coefficients. By applying different dynamic mesh methods to a moving NACA0012 airfoil, it can be found that the RBFs-Delaunay graph mapping hybrid method is as accurate as RBFs and is as efficient as Delaunay graph mapping technique. Numerical results show that the dynamic meshes for all test cases including one two-dimensional (2D) and two three-dimensional (3D) problems with different complexities, can be generated in an accurate and efficient manner by using the present hybrid method.

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.

Parallelized Aeroelastic Investigation Based on Delaunay Graph Mapping

In this paper, an effective and well robust dynamic grid deformation method based on Delaunay graph mapping is developed to solve the deformation of the 3-D hybrid multiblocks grids which is combined by near-wall viscosity grids and unstructured grids. Further more, the static aeroelastics problems of the standard model M6 elastic wing is investigated altogether by coupled with structure dynamic equation. The comparison, analysis and investigation were done as well. The CFD grids domain is subdivided into subdomains for parallel computation. And the program is carried out by MPI parallel computation standards.

Parallelized Aeroelastic Investigation Based on Delaunay Graph Mapping

Parallelized Aeroelastic Investigation Based on Delaunay Graph Mapping

Aerodynamic design optimization of nacelle/pylon position on an aircraft

The arbitrary space-shape free form deformation (FFD) method developed in this paper is based on non-uniform rational B-splines (NURBS) basis function and used for the integral parameterization of nacelle-pylon geometry. The multi-block structured grid deformation technique is established by Delaunay graph mapping method. The optimization objects of aerodynamic characteristics are evaluated by solving Navier–Stokes equations on the basis of multi-block structured grid. The advanced particle swarm optimization (PSO) is utilized as search algorithm, which combines the Kriging model as surrogate model during optimization. The optimization system is used for optimizing the nacelle location of DLR-F6 wing-body-pylon-nacelle. The results indicate that the aerodynamic interference between the parts is significantly reduced. The optimization design system established in this paper has extensive applications and engineering value.

Investigation of Trailing-Edge Flap Gap Effects on Rotor Performance Using High-Fidelity Analysis

Effects of trailing-edge flap gaps on rotor performance are investigated using a high-fidelity coupled computational fluid dynamics computational structural dynamics analysis. Both integral flap (the flap is an integral part of the blade such that there are no physical gaps at the flap ends) and discrete flap (the flap is a separate entity with physical gaps in the span and chord directions) are examined on an UH-60A rotor at high-speed forward-flight conditions. A novel grid deformation scheme based on the Delaunay graph mapping is developed and implemented to allow the computational fluid dynamics modeling of the gaps with minimal distortion of mesh around the flap gap regions. This method offers an alternative to the traditional approach of modeling such configurations using overset meshes. The simulation results show that the effectiveness of the flap is minimally affected with span gaps; the penalty on rotor performance is of the order of 1% compared to the integral flap. On the other hand, the chord gaps significantly degrade the benefits of active flap on rotor performance due to the flow penetration between the upper and lower surfaces of the flap.

A study of gridless method with dynamic clouds of points for solving unsteady CFD problems in aerodynamics

AbstractWhen solving unsteady computational fluid dynamics problems in aerodynamics with a gridless method, a cloud of points is usually required to be regenerated due to its accommodation to moving boundaries. In order to handle this problem conveniently, a fast dynamic cloud method based on Delaunay graph mapping strategy is proposed in this paper. A dynamic cloud method makes use of algebraic mapping principles and therefore points can be accurately redistributed in the flow field without any iteration. In this way, the structure of the gridless clouds is not necessarily changed so that the clouds regeneration can be avoided successfully. The spatial derivatives of the mathematical modeling of the flow are directly determined by using weighted least‐squares method in each cloud of points, and then numerical fluxes can be obtained. A dual time‐stepping method is further implemented to advance the two‐dimensional Euler equations in arbitrary Lagarangian–Eulerian formulation in time. Finally, unsteady transonic flows over two different oscillating airfoils are simulated with the above method and results obtained are in good agreement with the experimental data. Copyright © 2009 John Wiley & Sons, Ltd.

COMBINATION OF MESH DEFORMATION AND OVERSET GRID FOR SIMULATING UNSTEADY FLOW OF INSECT FLIGHT

As numerical simulation of unsteady flows due to moving boundaries such as flexible flapping-wings is difficult by conventional approaches, an effective strategy which combines mesh deformation based on Delaunay graph mapping and unstructured overset grids is proposed in this paper. A Delaunay graph is generated for each body-fitted grid cluster which overlaps or is embedded within an off-body background grid cluster. At each time step, the graph moves according to the wing's motion and deformation, and the grids move to new positions according to a one-to-one mapping between the graph and the grid. Then, intergrid-boundary definition is implemented automatically for computation.

Fast dynamic grid deformation based on Delaunay graph mapping

A simple and efficient dynamic grid deformation technique is proposed for computing unsteady flow problems with geometrical deformation, relative body movement or shape changes due to aerodynamic optimisation and fluid–structure interaction. A Delaunay graph of the solution domain is first generated, which can be moved easily during the geometric dynamic deformation, even for very large distortion. A one to one mapping between the Delaunay graph and the computational grid is maintained during the movement. Therefore the new computational grid after the dynamic movement can be generated efficiently through the mapping while maintaining the primary qualities of the grid. While most dynamic grid deformation techniques are iterative based on the spring analogy, the present method is non-iterative and much more efficient. On the other hand, in comparison with dynamic grid techniques based on transfinite interpolation for structured grids, it offers both geometric and cell topology flexibility, which is crucial for many unsteady flow problems involving geometric deformation and relative motions. The method is demonstrated through some typical unsteady flow test cases, including a pitching aerofoil in a fixed domain boundary, relative movement between multi-element aerofoils due to flap deployment for high lift, a deformable sphere in a fixed cube, and a three-dimensional flexible wing with large deformation.