GPU-based assembly of stiffness matrices in the parallel multilevel partition of unity method

Many real world problems can be modeled with Partial Differential Equations (PDEs). Since for many PDEs no exact solution can be found, there exists a variety of methods which give an approximate solution to those PDEs. One method which can be applied to find an approximate solution for elliptic PDEs is the Parallel Multilevel Partition of Unity Method (PMPUM). The major computational effort in this method is needed for the discretization of the differential operator. In this work we focus on the applicability of General-purpose computing on graphics processing units (GPGPU) on this task. A GPGPU implementation of the PMPUM is and a comparison to a given CPU implementation is presented. It is shown that the implementation using a GPGPU approach can be applied to many cases arising in the PMPUM to improve the performance

Partition of Unity Methode für dreidimensionale Probleme

Ziel dieser Studienarbeit ist die Implementierung und Bewertung einer Erweiterung für die institutseigene Partition of Unity Method-Software, die es ermöglicht, die Methode sowohl für zwei- als auch für dreidimensionale Geometrien direkt auf der Computer Aided Design-Geometrie anzuwenden

Stable Generalized Finite Element Method (SGFEM)

The Generalized Finite Element Method (GFEM) is a Partition of Unity Method (PUM), where the trial space of standard Finite Element Method (FEM) is augmented with non-polynomial shape functions with compact support. These shape functions, which are also known as the enrichments, mimic the local behavior of the unknown solution of the underlying variational problem. GFEM has been successfully used to solve a variety of problems with complicated features and microstructure. However, the stiffness matrix of GFEM is badly conditioned (much worse compared to the standard FEM) and there could be a severe loss of accuracy in the computed solution of the associated linear system. In this paper, we address this issue and propose a modification of the GFEM, referred to as the Stable GFEM (SGFEM). We show that the conditioning of the stiffness matrix of SGFEM is not worse than that of the standard FEM. Moreover, SGFEM is very robust with respect to the parameters of the enrichments. We show these features of SGFEM on several examples.Comment: 51 pages, 4 figure

Cache-effiziente Block-Matrix-Löser für die Partition of Unity Methode

Die Partition of Unity Methode findet Anwendung in gitterlosen Diskretisierungsverfahren zum Lösen elliptischer partieller Differentialgleichungen. Die bei der Diskretisierung entstehenden Gleichungssysteme besitzen eine Blockstruktur, die sich mittels der Multilevel Partition of Unity Methode asymptotisch optimal lösen lassen. Ein alternatives Verfahren zum Lösen dieser Gleichungssysteme stellen die vorkonditionierten Krylow- Unterraumverfahren dar. In dieser Arbeit wird ein auf der ILU-Zerlegung basierenders CG-Verfahren für Block-Matrizen implementiert, das auf Cache-effizienten Algorithmen basiert. Der Ausgangspunkt stellt die Bibliothek TifaMMy dar. Die in den letzten Jahren entwickelte Bibliothek basiert auf inhärent Cache-effiziente Algorithmen für dicht- und dünnbesetzte Matrizen. Dabei wird eine neue Datenstruktur für Blockmatrizen (BCRS) und die nötigen Algorithmen implementiert. Die Leistung des Block-Matrix-Löser wird mit der Multilevel Partition of Unity Methode verglichen

The Weak Coupling Method for Coupling Continuum Mechanics with Molecular Dynamics

For the global behavior of solids in structural mechanics of nonlinear processes, local effects on the atomistic level play an important role. Often a direct numerical simulation of the macroscopic behavior by a complete resolution of the microscale is for computational reason not possible. Thus, employing a multiscale strategy for an efficient and accurate modelling seems favorable since by separating the problem into two different frameworks, the accuracy of a fine scale model can be combined with the advantages of a computationally efficient model. More precisely a comparably small region of atoms e.g. surrounding the tip of a crack is modelled by molecular dynamics. Outside of this region, we take advantage of the fact that the displacement is almost homogeneous and can thus be modelled efficiently by a linear elastic continuum dynamical simulation. Clearly, both scales offer fundamentally different descriptions of the matter and they use different simulation methods. Whereas on the continuum scale the finite element method and a function space setting is used, the molecular dynamics is based on the movement of particles in the Euclidean space. Additionally, dynamical simulations with a transition zone (handshake region) between atomistic systems and the coarser finite element mesh suffer from unwanted (spurious) reflections, since the finite element method can not represent short wave length vibrational modes. Here a completely new approach is presented, which takes advantage of an infinite dimensional function space for the information transfer between the scales. Starting from a handshake region, the key idea is to construct a transfer operator between the different scales. This transfer operator is based on local averaging taken values. In order to construct the local weight functions, a partition of unity is assigned to the molecular degree of freedom. This allows us to decompose the micro scale displacement in the handshake region into a small and large wave number part by means of a weighted $L^2$ projection. In the first instance, this function space oriented interpretation of the atomistic displacement is applied in the context of a completely overlapping decomposition. More precisely, we consider the case, when the domain of the handshake region is conform with the domain of the molecular dynamics. In order to identify the displacements pertaining to the atomistic or continuum level respectively, we employ a multiscale decomposition. In particular, we decompose the micro scale displacement into a "low frequency'' and a "high frequency'' part in a weak sense. This new approach is also used in the context of a partly overlapping decomposition. Therein, the coarse and the fine scale simulation are matched by constraining the two displacements in the handshake region. The key issue in this context is, that our function space oriented approach allows us to interpret the constraints in a weak sense. Thus the "low frequent'' part can be captured by the coarse scale, whereas the "high frequent'' part of the displacement which has no meaning on the coarse scale is damped in the handshake region. Moreover numerical examples in 1d,2d and 3d show that this approach allows molecular displacements for entering into the continuum domain and the other way round flawlessly

Robuste Multilevel-Lösung elliptischer partieller Differentialgleichungen mit springenden Koeffizienten

In dieser Arbeit wird eine robuste Multilevel-Lösung für elliptische partielle Differentialgleichungen mit springenden Koeffizientenfunktionen im Kontext der Partition of Unity Methode realisiert und analysiert. Bei dieser gitterfreien Methode müssen die Koeffizientensprünge nicht auf dem gröbsten Level geometrisch aufgelöst sein, vielmehr kann durch geeignete Anreicherungsfunktionen mittels algebraischer Verfeinerung die Approximationsqualität verbessert werden. Die Implementierung eines stabilen und robusten Multilevel-Lösers sowie die Realisierung verschiedener Anreicherungsfunktionen sind Kernbereich dieser Arbeit. Insbesondere werden verschiedene Anreicherungsfunktionen entwickelt und deren Auswirkung im Hinblick auf die Robustheit des Lösers untersucht. Mögliche Ursachen für nicht robustes Verhalten des Lösers werden in diesem Zusammenhang detailliert diskutiert und Ansätze für Verbesserungen gegeben. Der realisierte Multilevel-Löser zeigt im Vergleich zu früheren Arbeiten effizienteres und für viele Fälle durchaus robustes Verhalten. Einfache Distanzfunktionen führen dabei i. A. zu den besten Ergebnissen jedoch lässt sich durch alleinige Verbesserung der Approximationsqualität der lokalen Anreicherungsräume die Effizienz und Robustheit des Lösers aufgrund schlechterer Glättungseigenschaften nicht beliebig steigern

GPU-based numerical integration in the partition of unity method

In this thesis, we present a CUDA-implementation of two sub-steps of the Parallel Multilevel Partition of Unity Method (PMPUM). The PMPUM is a method for the approximation of Partial Differential Equations (PDEs) whose main computational effort is caused by the integration of the weak formulation. Therefore, an efficient CUDA-implementation of the required steps could speed up a given PMPUM-implementation. The core of this thesis is the analysis of the applicability of CUDA in the PMPUM. To this end the required steps, the decomposition of the domain and the integration, were implemented using CUDA. The analysis showed, that the usage of CUDA can speed up the implementation and identified the limitations of the implementation. We give recommendations how to improve these limitations and expect the performance to increase further with these recommendations applied