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Papers by Robert Schneiders

Handbook of Grid Generation, 1998

This chapter explains techniques for the generation of quadrilateral and hexahedral element meshe... more This chapter explains techniques for the generation of quadrilateral and hexahedral element meshes. Since structured meshes are discussed in detail in other parts of this volume, we focus on the generation of unstructured meshes, with special attention paid to the 3D case. Quadrilateral or hexahedral element meshes are the meshes of choice for many applications, a fact that can be explained empirically more easily than mathematically. An example of a numerical experiment is presented by Benzley [1995], who uses tetrahedral and hexahedral element meshes for bending and torsional analysis of a simple bar, fixed at one end. If elastic material is assumed, second-order tetrahedral elements and first-order hexahedral elements both give good results (first-order tetrahedral elements perform worse). In the case of elastic–plastic material, a hexahedral element mesh is significantly better. A mathematical argument in favor of the hexahedral element is that the volume defined by one element must be represented by at least five tetrahedra. The construction of the system matrix is thus computationally more expensive, in particular if higher order elements are used. Unstructured hex meshes are often used in computational fluid dynamics, where one tries to fill most of the computational domain with a structured grid, allowing irregular nodes but in regions of complicated shape, and for the simulation of processes with plastic deformation, e.g., metal forming processes. In contrast to the favorable numerical quality of quadrilateral and hexahedral element meshes, mesh generation is a very difficult task. A hexahedral element mesh is a very “stiff” structure from a geometrical point of view, a fact that is illustrated by the following observation: Consider a structured grid and a new node that must be inserted by using local modifications only (Figure 21.1). While this can be done in 2D, in the three-dimensional case it is no longer possible! Thus, it is not possible to generate a hexahedral element mesh by point insertion methods, a technique that has proven very powerful for the generation of tetrahedral element meshes (Delaunay–type algorithms, Chapter 16). Many algorithms for the generation of tetrahedral element meshes are advancing front methods (Chapter 17), where a volume is meshed starting from a discretization of its surface and building the volume mesh layer by layer. It is very difficult to use this idea for hex meshing, even for very simple Robert Schneiders

Engineering with Computers

Remeshing is an important problem encountered often in the FEM-simulation of metal forming proces... more Remeshing is an important problem encountered often in the FEM-simulation of metal forming processes. This paper describes the remeshing scheme developed at LAMI and IBF for the FINEL FEM-program. A novel automatic, two-dimensional, quadrilateral mesh generator is introduced. Mesh generation is performed by means of image processing and computational geometry. Other features of the remeshing scheme are boundary simplification, smoothing, and rezoning. Examples of applications are given. I R. Schneiders and M. Becker were supported by the Deutsche Forschungsgemeinschaft under grant no. KO579/31-1.

The IMA Volumes in Mathematics and its Applications, 1995

We consider the problem of refining unstructured quadrilateral and brick element meshes. We prese... more We consider the problem of refining unstructured quadrilateral and brick element meshes. We present an algorithm which is a generalization of an algorithm developed by Cheng et. al. for structured quadrilateral element meshes. The problem is solved for the two-dimensional case. Concerning three dimensions we present a solution for some special cases and a general solution that introduces tetrahedral and pyramidal transition elements.

Computational Materials Science, 1994

Large strains in finite element simulations of metal forming require several remeshing procedures... more Large strains in finite element simulations of metal forming require several remeshing procedures during thc computation to take place. These remeshings should be performed automatically both for saving computational time and convenience. Such a remeshing algorithm has been well established for 2-dimensional applications and was recently presented for the 3-dimensional case. A special feature added to the remeshing algorithm in both 2D and 3D is the refinement which offers the possibility to use a finer grid in certain areas of the finite element mesh. The remeshing/refincment algorithm can play an important role in the simulation of microstructure.

This lecture reviews the state of the art in quadrilateral and hexahedral mesh generation. Three ... more This lecture reviews the state of the art in quadrilateral and hexahedral mesh generation. Three lines of development – block decomposition, superposition and the dual method – are described. The refinement problem is discussed, and methods for octree-based meshing are presented. 1

We present a new algorithm for the generation of hexahedral element meshes. The algorithm starts ... more We present a new algorithm for the generation of hexahedral element meshes. The algorithm starts with an octree discretization of the interior of the input object which is converted to a conforming hexahedral element mesh. Then the isomorphism technique 9] is used to adapt the mesh to the object boundary. keywords. hexahedra, mesh generation, octree 1 Introduction The last decades have seen immense progress in the development of numerical algorithms for the simulation of technical and physical processes. Finite element, nite di erence and nite volume methods are now routinely used in engineering. Therefore interest has grown in reducing simulation turnaround time, and the development of powerful, easy-to-use mesh generation programs has become an important issue. Much work has been done on algorithms for the generation of triangular, quadrilateral and tetrahedral element meshes. The state of the art is reviewed in 1], online information can be found in 2] and 3]. Mesh generators of this type have been integrated in many commercial programs. Unfortunately, the situation is worse in the eld of hex meshing. Most existing programs use mapped-meshing and multiblock techniques which require much user interaction and are therefore very time-consuming. Algorithms for the automatic generation of hexahedral element meshes have come up only recently, in essence the following techniques are used:

Engineering With Computers - EWC, 1996

An algorithm for the generation of hexahedraIelement meshes is presented. The algorithm works in ... more An algorithm for the generation of hexahedraIelement meshes is presented. The algorithm works in two steps. first the interior of the volume is filled with a regular grid; then the boundary region is meshed by using basically twodimensional operations. The algorithm was designed for use in the fern-simulation of metal forming processes where a remeshing mus~ be done very often. In principle, it can be used for meshing any geomewy with hexahedral elements and examples of meshes for geometries arising from various applications are given. The algorithm is checked against the criteria proposed by Sabin [1] (Advances in Engineering Software, 13. 220-225).

International Journal of Computational Geometry & Applications, 2000

An octree-based algorithm for the generation of hexahedral element meshes is presented. The algor... more An octree-based algorithm for the generation of hexahedral element meshes is presented. The algorithm works in three steps: (i) The geometry to be meshed is approximated by an octree structure. (ii) An unstructured hexahedral element mesh is derived from the octree. (iii) The mesh is adapted to the boundary of the geometry. We focus on step (ii) and describe an algorithm that constructs a hex mesh for a given octree structure.

We carry out a numerical study of fluctuations in the spectrum of regular graphs. Our experiments... more We carry out a numerical study of fluctuations in the spectrum of regular graphs. Our experiments indicate that the level spacing distribution of a generic k-regular graph approaches that of the Gaussian Orthogonal Ensemble of random matrix theory as we increase the number of vertices. A review of the basic facts on graphs and their spectra is included.

13th Computational Fluid Dynamics Conference, 1997

The present paper describes an unstructured hexahedral mesh generator for viscous flow simulation... more The present paper describes an unstructured hexahedral mesh generator for viscous flow simulations around complex 3D configurations. The first step of this method is the geometric adaptation of an initial non-body-fitted mesh by grid embedding. The resulting octree mesh is then fitted to the actual boundaries of the domain and its main features, such as sharp edges and corners, are captured. Degenerated cells resulting from body-fitting are removed using a splitting strategy and by insertion of buffer layers. Finally, in vicinity of solid walls, layers of highly stretched cells are marched directly from the quadrilateral surface mesh that is a by-product of the body-fitting process. Interfacing between the layer and octree meshes only requires the deformation of the octree to insert the layers. The resulting method is highly automated and significantly reduces turn-around times. To illustrate its capabilities, both internal and external applications are presented.

Computer Aided Geometric Design, 1995