Über die Implementierung der verallgemeinerten Finite-Element-Methode

The Generalized Finite Element Method (GFEM) combines desirable features of the standard Finite Element Method and the meshless methods. The key difference of the GFEM compared to the traditional FEM is the construction of the ansatz space. Each node of the finite element mesh carries a number of ansatz functions, expressed in terms of the global coordinate system. Those ansatz functions are multiplied by a partition of unity and serve as element ansatz functions in the patch constituted by the elements incident at the node. Using this technique to create the ansatz space allows for arbitrary ansatz functions. C0-continuity is enforced by construction. The ansatz is enriched using analytical functions or numerical approximations derived from side calculations containing a-priori knowledge of the solution close to singularities. The performance of GFEM with a higher order of polynomial ansatz functions is compared to traditional h-, p- and hp-extensions of the FEM. Most of the efficient solvers, e.g. multi-grid or cg, cannot be applied to the semi-definite systems resulting from a GFEM discretization. Several solving strategies are evaluated for higher order GFEM. The work concludes with a description of the implementation of the GFEM with a flexible object-oriented framework using C++.Die verallgemeinerte Finite-Element-Methode (GFEM) kombiniert Vorteile der klassischen Finite-Element-Methode mit Vorteilen der netzfreien Methoden. Hauptunterschied beim Vergleich der GFEM mit der FEM ist die Konstruktions des Ansatzes. Jeder Knoten des FE-Netzes trägt eine Anzahl an Ansatzfunktionen, die in globalen Koordinaten ausgedrückt werden. Diese Ansatzfunktionen werden mit einer Partition of Unity multipliziert und dienen als Elementansatzfunktionen für den Patch, der aus den angrenzenden Elementen des Knotens gebildet wird. Durch diese Art des Ansatzes wird die C0-Stetikeit für beliebige Ansatzfunktionen gewährleistet. Der Ansatz wird mit analytischen Funktionen und numerischen Näherungsrechnungen angereichert und enthält somit a-priori Wissen der Lösung in der Nähe von Singularitäten. Die Performance der GFEM mit Ansätzen höhere Ordnung wird mit klassischen h-, p- und hp-Diskretisierungen der FEM verglichen. Die meisten effizienten Löser, z.B. Multi Grid Verfahren oder die CG-Methode, können nicht für das semi-definite Gleichungssystem verwendet werden, dass aus der GFEM-Diskretisierung resultiert. Verschiedene Lösungsstrategien für GFEM-Diskretisierungen höhere Ordnungen werden untersucht. Die Arbeit schließt mit einer Beschreibung der Implementierung der Methode in Form eines Objekt-orientierten Frameworks in C++ ab

A Knowledge Integration Framework for 3D Shape Reconstruction

The modern emergence of automation in many industries has given impetus to extensive research into mobile robotics. Novel perception technologies now enable cars to drive autonomously, tractors to till a field automatically and underwater robots to construct pipelines. An essential requirement to facilitate both perception and autonomous navigation is the analysis of the 3D environment using sensors like laser scanners or stereo cameras. 3D sensors generate a very large number of 3D data points in sampling object shapes within an environment, but crucially do not provide any intrinsic information about the environment in which the robots operate with. This means unstructured 3D samples must be processed by application-specific models to enable a robot, for instance, to detect and identify objects and infer the scene geometry for path-planning more efficiently than by using raw 3D data. This thesis specifically focuses on the fundamental task of 3D shape reconstruction and modelling by presenting a new knowledge integration framework for unstructured 3D samples. The novelty lies in the representation of surfaces by algebraic functions with limited support, which enables the extraction of smooth consistent shapes from noisy samples with a heterogeneous density. Moreover, many surfaces in urban environments can reasonably be assumed to be planar, and the framework exploits this knowledge to enable effective noise suppression without loss of detail. This is achieved by using a convex optimization technique which has linear computational complexity. Thus is much more efficient than existing solutions. The new framework has been validated by critical experimental analysis and evaluation and has been shown to increase the accuracy of the reconstructed shape significantly compared to state-of-the-art methods. Applying this new knowledge integration framework means that less accurate, low-cost 3D sensors can be employed without sacrificing the high demands that 3D perception must achieve. This links well into the area of robotic inspection, as for example regarding small drones that use inaccurate and lightweight image sensors

Radial Basis Function Methods in Fluid-Structure Interaction

This thesis focuses on the use of a number of radial basis function (RBF) methods in computational fluid-structure interaction (FSI). RBF provide a general interpolation framework and have been applied in a number of different ways in various areas of FSI, including bi-directional fluid-structure coupling, mesh or point cloud motion, and in the numerical solution of PDE themselves. This thesis presents novel techniques in all three of these areas. Firstly, an efficiency improvement to the state-of-the-art in RBF mesh motion. Secondly, improvement and simplification of the handing of both moving and static boundaries in RBF-Finite Difference (RBF-FD) methods for numerically solving PDE. Lastly, a partitioned FSI solver is developed using a new coupling method, which extends the current applicability of RBF-FD to a more general class of FSI problems

Meshfree Methods for PDEs on Surfaces

This dissertation focuses on meshfree methods for solving surface partial differential equations (PDEs). These PDEs arise in many areas of science and engineering where they are used to model phenomena ranging from atmospheric dynamics on earth to chemical signaling on cell membranes. Meshfree methods have been shown to be effective for solving surface PDEs and are attractive alternatives to mesh-based methods such as finite differences/elements since they do not require a mesh and can be used for surfaces represented only by a point cloud. The dissertation is subdivided into two papers and software. In the first paper, we examine the performance and accuracy of two popular meshfree methods for surface PDEs:generalized moving least squares (GMLS) and radial basis function-finite differences (RBF-FD). While these methods are computationally efficient and can give high orders of accuracy for smooth problems, there are no published works that have systematically compared their benefits and shortcomings. We perform such a comparison by examining their convergence rates for approximating the surface gradient, divergence, and Laplacian on the sphere and a torus as the resolution of the discretization increases. We investigate these convergence rates also as the various parameters of the methods are changed. We also compare the overall efficiencies of the methods in terms of accuracy per computation cost. The second paper is focused on developing a novel meshfree geometric multilevel (MGM) method for solving linear systems associated with meshfree discretizations of elliptic PDEs on surfaces represented by point clouds. Multilevel (or multigrid) methods are efficient iterative methods for solving linear systems that arise in numerical PDEs. The key components for multilevel methods: \grid coarsening, restriction/ interpolation operators coarsening, and smoothing. The first three components present challenges for meshfree methods since there are no grids or mesh structures, only point clouds. To overcome these challenges, we develop a geometric point cloud coarsening method based on Poisson disk sampling, interpolation/ restriction operators based on RBF-FD, and apply Galerkin projections to coarsen the operator. We test MGM as a standalone solver and preconditioner for Krylov subspace methods on various test problems using RBF-FD and GMLS discretizations, and numerically analyze convergence rates, scaling, and efficiency with increasing point cloud resolution. We finish with several application problems. We conclude the dissertation with a description of two new software packages. The first one is our MGM framework for solving elliptic surface PDEs. This package is built in Python and utilizes NumPy and SciPy for the data structures (arrays and sparse matrices), solvers (Krylov subspace methods, Sparse LU), and C++ for the smoothers and point cloud coarsening. The other package is the RBFToolkit which has a Python version and a C++ version. The latter uses the performance library Kokkos, which allows for the abstraction of parallelism and data management for shared memory computing architectures. The code utilizes OpenMP for CPU parallelism and can be extended to GPU architectures

2nd International Conference on Numerical and Symbolic Computation

The Organizing Committee of SYMCOMP2015 – 2nd International Conference on Numerical and Symbolic Computation: Developments and Applications welcomes all the participants and acknowledge the contribution of the authors to the success of this event. This Second International Conference on Numerical and Symbolic Computation, is promoted by APMTAC - Associação Portuguesa de Mecânica Teórica, Aplicada e Computacional and it was organized in the context of IDMEC/IST - Instituto de Engenharia Mecânica. With this ECCOMAS Thematic Conference it is intended to bring together academic and scientific communities that are involved with Numerical and Symbolic Computation in the most various scientific area

A Model Integrated Meshless Solver (mims) For Fluid Flow And Heat Transfer

Numerical methods for solving partial differential equations are commonplace in the engineering community and their popularity can be attributed to the rapid performance improvement of modern workstations and desktop computers. The ubiquity of computer technology has allowed all areas of engineering to have access to detailed thermal, stress, and fluid flow analysis packages capable of performing complex studies of current and future designs. The rapid pace of computer development, however, has begun to outstrip efforts to reduce analysis overhead. As such, most commercially available software packages are now limited by the human effort required to prepare, develop, and initialize the necessary computational models. Primarily due to the mesh-based analysis methods utilized in these software packages, the dependence on model preparation greatly limits the accessibility of these analysis tools. In response, the so-called meshless or mesh-free methods have seen considerable interest as they promise to greatly reduce the necessary human interaction during model setup. However, despite the success of these methods in areas demanding high degrees of model adaptability (such as crack growth, multi-phase flow, and solid friction), meshless methods have yet to gain notoriety as a viable alternative to more traditional solution approaches in general solution domains. Although this may be due (at least in part) to the relative youth of the techniques, another potential cause is the lack of focus on developing robust methodologies. The failure to approach development from a practical perspective has prevented researchers from obtaining commercially relevant meshless methodologies which reach the full potential of the approach. The primary goal of this research is to present a novel meshless approach called MIMS (Model Integrated Meshless Solver) which establishes the method as a generalized solution technique capable of competing with more traditional PDE methodologies (such as the finite element and finite volume methods). This was accomplished by developing a robust meshless technique as well as a comprehensive model generation procedure. By closely integrating the model generation process into the overall solution methodology, the presented techniques are able to fully exploit the strengths of the meshless approach to achieve levels of automation, stability, and accuracy currently unseen in the area of engineering analysis. Specifically, MIMS implements a blended meshless solution approach which utilizes a variety of shape functions to obtain a stable and accurate iteration process. This solution approach is then integrated with a newly developed, highly adaptive model generation process which employs a quaternary triangular surface discretization for the boundary, a binary-subdivision discretization for the interior, and a unique shadow layer discretization for near-boundary regions. Together, these discretization techniques are able to achieve directionally independent, automatic refinement of the underlying model, allowing the method to generate accurate solutions without need for intermediate human involvement. In addition, by coupling the model generation with the solution process, the presented method is able to address the issue of ill-constructed geometric input (small features, poorly formed faces, etc.) to provide an intuitive, yet powerful approach to solving modern engineering analysis problems