Karl-Heinz Brakhage

Algebraic–hyperbolic grid generation with precise control of intersection of angles

A new approach to construct high quality meshes for flow calculations is investigated in the context of algebraic grid generation, where the angle of intersection between two alternate co-ordinate lines can be specified by the user. From an initial grid, new co-ordinate lines in one parameter direction are determined by solving a set of ordinary differential equations (ODE). These trajectories intersect the alternate co-ordinate lines of the initial grid in a prescribed angle. The resulting grid function is proven to be folding-free. The advantages with respect to grid quality achieved by prescribing the angles are gained at the expense that the boundary distribution can only be prescribed on three of four boundaries. In view of a sparse representation of the grid function, the points of intersection are interpolated by a B-spline surface. Thus, the grid can be accessed evaluating the generated parametric representation of the computational domain

Application of B-spline techniques to the modeling of airplane wings and numerical grid generation

In the present paper we give account of an effort that aimed at the unification of the whole geometric preprocessing that preceded the wind tunnel readings with a realistic airplane wing model in a recent research project. This preprocessing includes the automated generation of the CAD models which were used for the manufacturing of the multi-parted wing-fuselage configuration and the generation of the numerical grids for the corresponding numerical simulations. Due to the constraints of the project it was decided to employ only exact, watertight, untrimmed B-Spline representations.

A Unified Approach to the Modeling of Airplane Wings and Numerical Grid Generation Using B-Spline Representations

In this article we summarize the development of a unified platform for treating the entire range of geometric preprocessing tasks that preceded the wind tunnel readings and the numerical simulations performed in the collaborative research center SFB 401. In particular, this includes the automated generation of the CAD models which were used for manufacturing multi-parted wing-fuselage configurations as well as the generation of the numerical grids for the corresponding, adaptive numerical simulations.

Aero-structural Dynamics Experiments at High Reynolds Numbers

The elastic wing model, its excitation and comprehensive high frequency measuring equipment for the High Reynolds Number Aero-Structural Dynamics (HIRENASD) tests in the European Transonic Windtunnel (ETW) are shortly described. Some of the stationary polars are presented in terms of wing deformation, as well as aerodynamic coefficients and pressure distributions. Then unsteady processes observed in the measurements of static aerodynamic coefficients, are regarded with focus on small amplitude pressure waves travelling upstream from the trailing edge and triggering periodically break-down and redeployment of the local supersonic domains with transonic shock waves to run upstream and to disappear. Another focus is on stochastic vibrations excitation while moving forward during nominally static experiments. Emphasis is put on measured variations of pressure distribution on the wing surface caused by defined vibration excitation applying internal force couples at the wing root, whereby the exciter frequencies were chosen close to natural frequencies of the wing model. Phase and magnitude of measured local lift fluctuations as well as real and imaginary parts of pressure distributions are presented.

Mathematical modeling of ceramic bond bridges in grinding wheels

Ceramic-bonded grinding wheels with cubic boron nitride (CBN) as grain material belong to the most efficient grinding tools available. They feature a high hardness combined with a high thermal stability and the applicability to grinding ferrous materials. However, the appropriate volumetric composition of grain and bonding material is an expensive and time-consuming process based on experience. Our objective is the mathematical modeling of grinding wheel structures for the prediction of compositions which fulfill given grinding requirements such that using trial and error methods can be avoided. For this purpose, we focus on a three-dimensional element of a grinding wheel which we call volumetric structure element. In this paper, we briefly describe our overall modeling approach and present in detail how we model the ceramic bond. For the latter, we combine analytical and discrete calculations, embedded into an iterative algorithm which ensures to meet bond volume fractions prescribed by grinding wheel specifications.

Reparametrization and Volume Mesh Generation for Computational Fluid Dynamics Using Modified Catmull-Clark Methods

A new technique is presented for using the Catmull-Clark subdivision method, modified for modeling sharp creases, to generate volume meshes used in computational fluid dynamics. Given a target surface of arbitrary genus, e.g., defined by a collection of trimmed B-spline patches, which represents an object in a flow, a simple polyhedron is constructed roughly approximating this target surface. After one Catmull-Clark subdivision, the polyhedron exclusively consists of quadrilaterals and its Catmull-Clark limit surface can be pre-computed. Points of the limit surface are projected onto the target surface and the control points of the polyhedron are adjusted by approximating the projected points. An iterative process of alternating subdivisions, projections and approximations leads to a watertight mesh consisting of untrimmed surface patches matching the given target surface. By attaching an offset mesh and a far-field mesh, a block-structured volume mesh is obtained, being well-suited for adaptive flow solvers.

Numerical treatment of surface-surface intersection and contouring

Abstract Three Surface–Surface–Intersection (SSI) algorithms will be introduced. Surfaces may be given implicitly ( f ( x ) = 0) or parametically ( x = x( u, v )). Thus we have to find the solution of F ( y ) = 0 with a function F : ℝ n +1 → ℝ n and n ∈ {1, 2, 3}. Contouring is a problem of the same type ( n = 1). The first algorithm is for the special case F : ℝ 2 → ℝ and the others are marching methods with different high order approximants for taking steps of variable length. They proceed in two stages. The first is to find an initial guess of the curve or a point on the curve, and the second is to refine the curve or to march along the curve respectively.

Analytical investigations for the design of fast approximation methods for fitting curves and surfaces to scattered data

Abstract We present an analytical framework for linear and nonlinear least squares methods and adopt it to the construction of fast iterative methods for fitting curves and surfaces to scattered data. The results are directly applicable to curves and surfaces that have a representation as a linear combination of smooth basis functions associated with the control points. Standard Bezier and B-spline curves / surfaces as well as subdivision schemes have this property. In the global approximation step for the control points our approach couples the standard linear approximation part with the reparameterization to heavily reduce the number of overall steps in the iteration process. This can be formulated in such a way that we have a standard least squares problem in each step. For the local nonlinear parameter corrections our results allow for an optimal choice of the methods used in different stages of the process. Furthermore, regularization terms that express the fairness of the intermediate and / or final result can be added. Adaptivity is easily integrated in our concept. Moreover our approach is well suited for reparameterization occurring in grid generation.