Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given

Three Dimensional Numerical Analysis of Screw Compressor Performance

Modern manufacturing methods enable screw compressors to be constructed to very close tolerances where full 3-D numerical calculation of the heat and fluid flow through them is then required to obtain the maximum possible improvements in their design. An independent stand-alone interface program has been developed by the authors in order to generate a numerical grid for this purpose. The interface employs a procedure to produce rotor profiles and an analytical transfinite interpolation method with adaptive meshing to obtain a fully structured 3-D numerical mesh, which is directly transferable to a CFD code. This was required to overcome problems associated with moving, stretching and sliding rotor domains and with robust calculations in domains with significantly different geometry ranges. Some changes had to be made within the solver functions both to enable calculations and to make them faster. These include a means to maintain constant pressures at the inlet and outlet ports and consideration of two-phase flow resulting from oil injection in the working chamber. Modifications implemented to the CFD procedure improved solutions in complex domains with strong pressure gradients. The pre-processor code and calculating method have been tested on a commercial CFD solver to obtain flow simulations and integral parameter calculations. The results of calculations on an oil injected screw compressor are presented in this paper and compared with experimental results

Efficient Near-Field to Mid-Field Sonic Boom Propagation Using a High-Order Space Marching Method

An efficient strategy for propagating sonic boom signatures from a near-field Computational Fluid Dynamics (CFD) solution to the mid-field is presented. The method is based on a high-order accurate finite-difference discretization of the 3D Euler equations on a specially designed curvilinear grid and a single sweep space marching solution algorithm. The new approach leads to more than a factor of two reduction in overall computational resources compared to the current method used to propagate near-field sonic booms to the ground. Accuracy and efficiency of the near-field to mid-field process is demonstrated using a selection of test cases from the AIAA Sonic Boom Prediction Workshops. Azimuthal dependence of nonlinear wave propagation from the near-field to mid-field is analyzed along with its effects on the ground level noise

Development of an unstructured solution adaptive method for the quasi-three-dimensional Euler and Navier-Stokes equations

A general solution adaptive scheme based on a remeshing technique is developed for solving the two-dimensional and quasi-three-dimensional Euler and Favre-averaged Navier-Stokes equations. The numerical scheme is formulated on an unstructured triangular mesh utilizing an edge-based pointer system which defines the edge connectivity of the mesh structure. Jameson's four-stage hybrid Runge-Kutta scheme is used to march the solution in time. The convergence rate is enhanced through the use of local time stepping and implicit residual averaging. As the solution evolves, the mesh is regenerated adaptively using flow field information. Mesh adaptation parameters are evaluated such that an estimated local numerical error is equally distributed over the whole domain. For inviscid flows, the present approach generates a complete unstructured triangular mesh using the advancing front method. For turbulent flows, the approach combines a local highly stretched structured triangular mesh in the boundary layer region with an unstructured mesh in the remaining regions to efficiently resolve the important flow features. One-equation and two-equation turbulence models are incorporated into the present unstructured approach. Results are presented for a wide range of flow problems including two-dimensional multi-element airfoils, two-dimensional cascades, and quasi-three-dimensional cascades. This approach is shown to gain flow resolution in the refined regions while achieving a great reduction in the computational effort and storage requirements since solution points are not wasted in regions where they are not required

Implicitly extrapolated geometric multigrid on disk-like domains for the gyrokinetic Poisson equation from fusion plasma applications

The gyrokinetic Poisson equation arises as a subproblem of Tokamak fusion reactor simulations. It is often posed on disk-like cross sections of the Tokamak that are represented in generalized polar coordinates. On the resulting curvilinear anisotropic meshes, we discretize the differential equation by finite differences or low order finite elements. Using an implicit extrapolation technique similar to multigrid tau-extrapolation, the approximation order can be increased. This technique can be naturally integrated in a matrix-free geometric multigrid algorithm. Special smoothers are developed to deal with the mesh anisotropy arising from the curvilinear coordinate system and mesh grading

Unstructured Meshes for Large Rigid Body Motions Using Mapping Operators

RÉSUMÉ Cette thèse propose une approche originale pour le contrôle des maillages autours d’objets rigides en mouvement. L’approche proposée permet de maintenir fixe la topologie du maillage, et ainsi d’éliminer le recours à un processus d’interpolation des solutions entre les pas de temps lors de simulations en régime transitoire. Afin de simplifier le traitement du mouvement des objets, leur évolution est décrite dans un espace de calcul, qui est par la suite transformé vers l’espace physique grâce à des opérateurs différentiels. Deux types d’opérateurs différentiels ont été étudiés, les premiers inspirés de fonctionnelles de forme des éléments (Longueur, Aire et Orthogonalité), et les seconds des équations de Winslow. L’une des contributions principales de cette étude est l’extension des équations d’Euler-Lagrange aux fonctionnelles de forme, ainsi qu’à leurs combinaisons, et l’application de ces fonctionnelles au traitement de maillages non-structurés. Deux techniques distinctes de discrétisation de ces équations aux dérivées partielles ont été étudiées. La première technique est basée sur un schéma de différences finies à neuf points, et la seconde sur un schéma de volumes finis utilisant une linéarisation des opérateurs. Une seconde contribution a consisté à introduire la notion de glissement des noeuds du maillage sur les frontières des objets en mouvement. En intégrant les techniques de glissement des noeuds, gérées dans l’espace de calcul, et les techniques de transformation de l’espace de calcul vers l’espace physique, une approche robuste de contrôle des maillages pour de très grands déplacements des objets a pu être mise au point. Cette approche, combinée à une seconde méthode de gestion du mouvement dans l’espace physique utilisant les fonctions à bases radiales, a permis de traiter des configurations d’objets en mouvement le long de trajectoires complexes. La méthodologie globale a ainsi pu être utilisée pour traiter des configurations représentatives d’applications en ingénierie.----------ABSTRACT The main objective of this thesis was to generate unstructured meshes with fixed connectivity for large rigid body motion. The proposed approach consists in generating a mesh in computational space for a generic configuration of the moving body. The management of body and mesh motion is then carried out in computational space using a sliding mesh paradigm. Afterwards, the mesh in physical space is obtained through PDE mapping operators. Two different mapping operators based on functionals and Winslow equations have been investigated to recover the physical space by the computational mesh. One of the main contributions of this study is extending the Euler-Lagrange equations of Length, Area, Orthogonality functionals and their combinations, to unstructured grid technologies. Two new discritization techniques are implemented, validated and compared for performing different mapping operators on unstructured grids. The first approach used a 9-point cartesian stencil inside each patch of the computational mesh and discritizes the mapping operators on that using conventional finite difference schemes. The second approach used finite volume discritization technique by linearizing the system of mapping equations. Finally, the Radiad basis Functions interpolation technique can be used as a secondary mesh motion technique and after the mesh sliding procedure. Combination of these two techniques allow us to handle more complex trajectories of the boundaries in physical space. The overall methodology is applied to complex geometric configurations representative of engineering applications

Adaptive Grid Solution Procedure for Elliptic Flows.

This thesis deals with the formulation of a computationally efficient adaptive grid system for two-dimensional elliptic flow and heat transfer problems. The formulation is in a curvilinear coordinate system so that flow in irregular geometries can be easily handled. An equal order pressure-velocity scheme is formulated in this thesis to solve the flow equations. An adaptive grid solution procedure is developed in which the grid is automatically refined in regions of high errors and consecutive calculations are performed between the coarse grid and adapted grid regions in the same spirit as that of a Multi-Grid method. In orthogonal coordinate systems, checkerboard pressure and velocity fields are avoided by using staggered grids. In curvilinear coordinates however, the geometric complications associated with staggered grids are overwhelming and therefore a non-staggered grid arrangement is desirable. To this end, an equal order pressure-velocity interpolation scheme is developed in this thesis. This scheme is termed as the SIMPLEM algorithm and is shown to have good convergence characteristics, and to suppress checkerboard pressure and velocity fields. The adaptive grid technique developed flags the important regions in the calculation domain from an initial coarse grid calculation. Then, adaptation is performed by generating a nonuniform mesh in the flagged region using Poisson\u27s equations in which the nonhomogeneous terms are chosen so that a denser clustering of grid points is obtained where needed most in the flagged region. Coarse grid calculations in the whole domain, and fine grid calculations in the flagged region are consecutively performed until convergence, with correction terms from the fine grid solution added to the coarse grid equations in the flagged region in every cycle of calculation. Thus, the solution in the non-refined regions improves due to the influence of the correction terms added to the coarse grid equations. The effectiveness of the method is demonstrated by solving a variety of test problems and comparing the results with those obtained on a uniform or fixed grid. The adaptive grid solutions are shown to be more accurate than the fixed uniform grid solutions for the same level of computational effort