# Parametric multi-block grid generation and application to adaptive flow simulations

- Parametrische, blockstrukturierte Gittergenerierung und Anwendung auf adaptive Strömungssimulationen

Lamby, Philipp; Dahmen, Wolfgang (Thesis advisor)

*Aachen : Publikationsserver der RWTH Aachen University (2007)*

Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2007

Abstract

Many difficulties encountered in the numerical solution of partial differential equations emanate from the approximate, polygonal based geometry that usually underlies the philosophy of finite element or finite volume methods. This is especially true for adaptive codes. Here, for instance, one faces the problem that once a standard polygonal grid has been constructed, subsequent grid refinement requires information from the exact geometry model which istypically unaccessible by the PDE solver. Furthermore, in the case of moving grids sophisticated and computationally expensive remeshing algorithms have to be developed. In this thesis we address these problems with a new block-structured grid generation concept. The central idea is to describe the geometry of the physical domain by parametric mappings from which consistent discretizations at arbitrary level of resolution can be generated simply by function evaluation. The mappings themselves can ideally be represented by only a small number of design parameters thus reducing the complexity of the grid generation and deformation task to the genuine complexity of the geometric problem instead to the complexity of the discretization. As standard representation for the grid mappings we use B-spline tensor products, however, other types of mappings are admissible. In order to retain sufficient flexibility the grid mappings are bundled into a multiblock structure. This allows us to combine techniques from classical structured grid generation and computer aided geometric design to build a fully functional grid generation system. These methods include, in particular, fast interpolation, approximation and fairing algorithms for tensor product B-splines, algebraic grid generation based on transfinite interpolation, and elliptic grid generation with harmonicmappings. A central task in the implementation of a grid generation system is the design of a data structure that can serve both as interface to the PDE solver and as flexible tool for the modeling of grids. We present the implementation of such a framework based on the object-oriented programming paradigm. The information about the block connectivity and the geometry are separated as far as possible. Thus our method allows for widely different ways of describing the geometry. In particular it can handle both parametric and discrete grids. In view of applications to aerodynamics, where the Navier-Stokes equations have to be solved for high Reynolds numbers, specialized methods to generate high-quality boundary layer grids have been developed. These are based on the successive generation of curvature dependent offset curves and can roughly be counted among the class of hyperbolic methods. This method is applied to a simple, but realistic wing configuration. The new grid generation system has been integrated into the adaptive flow solver QUADFLOW which aims at large scale simulations of compressible fluid flow and fluid-structure interaction. This solver employs an unstructured finite volume scheme for the time integration and a multiresolution analysis in order to construct locally refined grids. These ingredients have to be carefully coordinated with the mesh generation strategy. We analyze which requirements this solution strategy poses to the grid generator. In particular we derive the so-called geometric conservation laws. These are consistency conditions which stem from the simple supposition that a constant, homogeneous flow field should be exactly reproduced by the numerical scheme even if the grid changes due to deformations induced by moving boundaries or due to grid refinement. We present a general guideline how to fulfill these conditions in the parametric setting and present its concrete realization for B-spline mappings. Finally we summarize the overall structure of the QUADFLOW-solver and present several numerical examples, proving the practicability of the new concept and demonstrating some of its features.

Institutions

- Department of Mathematics [110000]
- Chair of Mathematics and Institute for Geometry and Applied Mathematics [111410]

### Identifier

- URN: urn:nbn:de:hbz:82-opus-19990
- RWTH PUBLICATIONS: RWTH-CONV-124025