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A higher-order accurate meshing algorithm for non-smooth surfaces defined via Boolean set operations from smooth surfaces is presented. Input data are a set of level-set functions. This geometry definition allows an easy detection of non-smooth features such as corners and edges. Furthermore, edges can be treated as intersection curves. At first, a linear reconstruction of the surface is subsequently generated with octree decomposition, corner- and edge-detection and surface meshing. Based on this, first the edges and then the interior nodes of higher-order Lagrangian elements are projected onto the implicit surface. Various projection methods are studied for nodes of edge elements and surface elements. Special attention is paid to the accurate meshing of tangential intersection curves. Optimal convergence properties for approximation problems are confirmed in numerical studies.

A new higher-order accurate method is proposed that combines the advantages of the classical $p$-version of the FEM on body-fitted meshes with embedded domain methods. A background mesh composed by higher-order Lagrange elements is used. Boundaries and interfaces are described implicitly by the level set method and are within elements. In the elements cut by the boundaries or interfaces, an automatic decomposition into higher-order accurate sub-elements is realized. Therefore, the zero level sets are detected and meshed in a first step which is called reconstruction. Then, based on the topological situation in the cut element, higher-order sub-elements are mapped to the two sides of the boundary or interface. The quality of the reconstruction and the mapping largely determines the properties of the resulting, automatically generated conforming mesh. It is found that optimal convergence rates are possible although the resulting sub-elements are not always well-shaped.

In this paper the isogeometric Nystr��m method is presented. It's outstanding features are: it allows the analysis of domains described by many different geometrical mapping methods in computer aided geometric design and it requires only pointwise function evaluations just like isogeometric collocation methods. The analysis of the computational domain is carried out by means of boundary integral equations, therefor only the boundary representation is required. The method is thoroughly integrated into the isogeometric framework. For example, the regularization of the arising singular integrals performed with local correction as well as the interpolation of the pointwise existing results are carried out by means of Bezier elements. The presented isogeometric Nystr��m method is applied to practical problems solved by the Laplace and the Lame-Navier equation. Numerical tests show higher order convergence in two and three dimensions. It is concluded that the presented approach provides a simple and flexible alternative to currently used methods for solving boundary integral equations, but has some limitations., 21 Figures

A method is presented which combines features of meshfree and meshbased methods in order to enable the simulation of complex flow problems involving large deformations of the domain or moving and rotating objects. Conventional meshbased methods like the finite element method have matured as standard tools for the simulation of fluid and structure problems. They offer efficient and reliable approximations, provided that a conforming mesh with sufficient quality can be maintained throughout the simulation. This, however, may not be guaranteed for complex fluid and fluid-structure interaction problems. Meshfree methods on the other hand approximate partial differential equations based on a set of nodes without the need for an additional mesh. Therefore, these methods are frequently used for problems where suitable meshes are prohibitively expensive to construct and maintain. This advantage of meshfree methods comes at the price of being considerably more time-consuming than their meshbased counterparts. A coupled meshfree/meshbased fluid solver is developed which combines the advantages of both methodologies. A meshfree method, closely related to the element-free Galerkin method, is used in small parts of the domain where a mesh is difficult to maintain, whereas the efficient meshbased finite element method is employed in the rest of the domain. Major steps in the development of the coupled flow solver are the extension of standard stabilization methods to meshfree approximations, and the realization of the coupling on the level of the shape functions, which are involved in the approximation of the governing equations. Concerning the stabilization, it is found that the same structure of the standard stabilization schemes may be used for meshfree approximations, however, the aspect of the stabilization parameter, weighing the stabilization terms, requires special care. For the coupling, these requirements for a reliable stabilization lead to modifications of the existing coupling approaches. The stabilized and coupled flow solver is verified and used for the simulation of geometrically complex fluid-structure interaction problems. Conventional meshbased methods are not suitable for the approximation of these test cases due to the prohibitively large deformations of the geometry., Es wird ein Verfahren vorgestellt, das Eigenschaften netzfreier und netzbasierter Methoden nutzt, um die Simulation komplexer Strömungsprobleme zu ermöglichen, bei denen große Verformungen des Gebietes oder sich bewegende und rotierende Objekte in der Strömung berücksichtigt werden können. Konventionelle netzbasierte Verfahren wie die Finite Element Methode haben sich zu Standardwerkzeugen bei der numerischen Analyse von Strömungen und Festkörpern entwickelt. Sie ermöglichen eine schnelle und zuverlässige Approximation, vorausgesetzt, daß ein konformes Netz mit geeigneter Qualität während der gesamten Simulation aufrecht erhalten werden kann. Dies kann jedoch bei komplexen Strömungs- und Fluid-Struktur-Interaktionsproblemen eine entscheidende Einschränkung in der Anwendbarkeit dieser Verfahren darstellen. Netzfreie Verfahren approximieren dagegen die zugrundeliegenden Modellgleichungen nur in Abhängigkeit einer gegebenen Knotenverteilung, ohne ein Netz zu erfordern, das die Konnektivität a priori festlegt. Deshalb werden diese Verfahren häufig dort eingesetzt, wo die Generierung oder Erhaltung geeigneter Netze nicht mit vertretbarem Aufwand möglich ist. Allerdings geht der Vorteil der Netzunabhängigkeit bei netzfreien Verfahren einher mit deutlich aufwendiger zu konstruierenden Ansatzfunktionen, was eine empfindliche Erhöhung des Rechenaufwandes bei der Integration der zugrundeliegenden Gleichungen erfordert. Hierin wird ein gekoppelter netzfreier/netzbasierter Strömungslöser entwickelt, der die Vorteile beider Verfahren kombiniert. Ein netzfreies Verfahren, das eng mit der "element-free" Galerkin Methode verwandt ist, wird nur in kleinen Teilen des Gebietes verwendet, wo ein Netz zu Schwierigkeiten führt, und im gesamten Rest des Gebiets wird die Finite Element Methode als effizientes netzbasiertes Standardverfahren eingesetzt. Wichtige Schritte bei der Entwicklung der gekoppelten Methode sind die Erweiterung von Standard-Stabilisierungsansätzen auf netzfreie Verfahren und die Umsetzung der Kopplung auf der Ebene der Ansatzfunktionen, die für die Approximation eingesetzt werden. Bezüglich der Stabilisierung wird gezeigt, daß die Struktur der verschiedenen Stabilisierungsmethoden direkt auf netzfreie Verfahren anwendbar ist, allerdings erfordert die Wahl geeigneter Stabilisierungsparameter, die den Stabilisierungseinfluß wichten, besondere Beachtung. Bei der Kopplung führen die Voraussetzungen, die für eine zuverlässige Stabilisierung gegeben sein müssen, zu Modifikationen der Standardansätze. Der gekoppelte Strömungslöser wird verifiziert und für die Simulation geometrisch komplexer Fluid-Struktur-Interaktionsprobleme eingesetzt. Die Fähigkeiten des gekoppelten Verfahrens werden dabei sichtbar, denn klassische netzbasierte Standardverfahren versagen bei den gezeigten Anwendungsbeispielen., Braunschweiger Schriften zur Mechanik, vol. 59 - 2005

In part I of this work, meshfree Galerkin methods have been used for the approximation of the incompressible Navier-Stokes equations in Eulerian or arbitrary Lagrangian-Eulerian formulation. The problem of stabilization of meshfree methods is addressed there. Analogously, in the meshbased context, the finite element method is frequently used in similar stabilized formulations for the simulation of flow problems. In order to combine the advantages of both methods, different coupling techniques are examined in this part of the work. Standard coupling approaches are modified in order to fulfill the requirements for a reliable stabilization found in part I of this work. The resulting stabilized and coupled meshfree/meshbased flow solver employs the comparatively costly meshfree Galerkin method only where it is needed---i.e. in areas of the domain, where a mesh is difficult to maintain---, and the efficient meshbased finite element method in the rest of the domain. This enables the solution of complex flow problems, as thus involving large deformations of the physical domain and/or moving and rotating obstacles., Informatik-Berichte der Technischen Universität Braunschweig, vol. 2005-03

This paper gives an overview over Meshfree Methods. Starting point is an extended and modified classification of Meshfree Methods due to three aspects: The construction of a partition of unity, the choice of an approximation either with or without using an extrinsic basis and the choice of test functions, resulting into a collocation, Bubnov-Galerkin or Petrov-Galerkin Meshfree Method. Most of the relevant Meshfree Methods are described taking into account their different origins and viewpoints as well as their advantages and disadvantages. Typical problems arising in meshfree methods like integration, treatment of essential boundary conditions, coupling with mesh-based methods etc.~are discussed. Some valuing comments about the most important aspects can be found at the end of each section. This text was revised in 2004, Informatik-Berichte der Technischen Universität Braunschweig, vol. 2003-03

This paper gives a detailed review of popular stabilization approaches that have developed to be standard tools in the numerical world. The need for stabilization is outlined and stabilization ideas based on the Petrov-Galerkin concept are discussed. Stabilization methods are explained on the one hand in an illustrative approach with help of the artificial diffusion idea and on the other hand in a theoretical approach by outlining the mathematical background of stabilization. A generalization to meshfree methods is investigated. We find that the structure of stabilizing perturbation terms can be applied in the same manner to meshfree methods. However, the weighting of the stabilizing terms, defined with the stabilization parameter, requires special attention. Using standard formulas for the stabilization parameter, raised in the meshbased finite element context, is only suitable for meshfree shape functions with small dilatation parameters. This is confirmed with numerical experiments for the advection-diffusion equation and incompressible Navier-Stokes equations., Informatik-Berichte der Technischen Universität Braunschweig, vol. 2004-01