Jörg Hörner

Finite element analysis with b-splines: weighted and isogeometric methods

Weighted and isogeometric methods use b-splines to construct bases for FEM. They combine the computational efficiency of regular grids with the geometric flexibility of CAD representations. We give a brief description of the key ideas of the two approaches, presenting them in a unified framework. In particular, we use b-spline nodes, to visualize the free parameters. Moreover, we explain how to combine features of both techniques by introducing weighted isogeometric finite elements. An error estimate for the resulting mixed method is given, and the performance of weighted approximations is illustrated by numerical examples.

Parallel Finite Element Methods with Weighted Linear B-Splines

Weighted extended B-splines (web-splines) combine the computational efficiency of B-splines and the geometric flexibility of standard finite elements on unstructured meshes. These new finite elements on uniform grids (cf. [5] and www.webspline. de) are ideally suited for vectorization, parallelization and multilevel techniques. In this project we explore the potential of the web-method for large scale applications with performance tests on the NEC SX-8 cluster of the HLRS. We implement a new minimal degree variant which uses predefined instruction sequences for matrix assembly and is almost as efficient as a difference scheme on rectangular domains.

Collocation with WEB---Splines

We describe a collocation method with weighted extended B–splines (WEB–splines) for arbitrary bounded multidimensional domains, considering Poisson’s equation as a typical model problem. By slightly modifying the B–spline classification for the WEB–basis, the centers of the supports of inner B–splines can be used as collocation points. This resolves the mismatch between the number of basis functions and interpolation conditions, already present in classical univariate schemes, in a simple fashion. Collocation with WEB–splines is particularly easy to implement when the domain boundary can be represented as zero set of a weight function; sample programs are provided on the website http://www.web-spline.de. In contrast to standard finite element methods, no mesh generation and numerical integration is required, regardless of the geometric shape of the domain. As a consequence, the system equations can be compiled very efficiently. Moreover, numerical tests confirm that increasing the B–spline degree yields highly accurate approximations already on relatively coarse grids. Compared with Ritz-Galerkin methods, the observed convergence rates are decreased by 1 or 2 when using splines of odd or even order, respectively. This drawback, however, is outweighed by a substantially smaller bandwidth of collocation matrices.

Finite Element Approximation with Hierarchical B-Splines

We review the definition of hierarchical spline spaces and their application to finite element methods. Then we discuss how hierarchical techniques can be implemented using the FEMB program package. Subdivision algorithms play a crucial role and lead to a very simple program structure. A numerical example illustrates the substantial gains in accuracy for the adaptive strategy, in particular for higher degree B-splines.

Programming finite element methods with weighted B-splines

We discuss the main components of the recently developed FEMB program package, which implements finite element methods with weighted B-splines for basic linear elliptic boundary value problems in two and three dimensions. A three-dimensional implementation without topological restrictions has not been available before. We describe in particular the mathematical background for the recursive quadrature/cubature over boundary cells and explain how to utilize the regular data structure of uniform B-splines efficiently. Considering the Lame-Navier equations of linear elasticity as a typical example, we illustrate the performance of the main FEMB routines. The numerical tests confirm that, due to the new integration routines, the weighted B-spline solvers have become considerably more efficient.

Rotationskörper, Schwerpunkt und Trägheitsmoment

Volumen von Rotationskorpern bezuglich unterschiedlicher Achsen Profil und Volumen einer Vase Volumen und Mantelflache eines Rotationskorpers Geometrischer Schwerpunkt einer Eistute Flachenschwerpunkt eines Paraboloids Schwerpunkt und Tragheitsmoment eines Kegelstumpfes Masse, Schwerpunkt und Tragheitsmoment eines Paraboloids

Vektorräume, Skalarprodukte und Basen

Unterraume des Vektorraums der Polynome Eigenschaften reeller Skalarprodukte Lineare Unabhangigkeit von Vektoren im \( {\mathbb{R}}^{4} \) Basis mit Parameter Basis eines Polynomraums Erganzung zu einer komplexen orthogonalen Basis und Koffzientenbestimmung Orthogonale Basis einer Hyperebene und Projektion

Stetigkeit, partielle Ableitungen und Jacobi-Matrix

Stetigkeit im Ursprung Partielle Ableitungen bivariater Funktionen Partielle Ableitungen trivariater Funktionen Hohere partielle Ableitungen von trivariaten Funktionen Partielle Ableitungen eines Polynoms Partielle Ableitungen erster und zweiter Ordnung einer trivariaten Funktion Partielle Ableitungen bis zur dritten Ordnung einer bivariaten Funktion Jacobi-Matrizen (2 x 1, 1 x 2, 2 x 3) Jacobi-Matrizen (2 x 4, 3 x 3)

Längen, Winkel und Skalarprodukt

Skalarprodukt, Betrag und Winkel fur Vektoren Grosen im Dreieck (WSW) Grosen im Dreieck (SSW) Geometrie eines Sechsecks Erganzung zu einer Orthonormalbasis in der Ebene und Koeffizientenbestimmung Seitenlangen, Winkel und Flacheninhalt eines Parallelogramms Rechnen mit Epsilon-Tensor und Kronnecker-Symbol Nahtlange eines Fusballs

Differentialgleichungen erster Ordnung

Richtungsfeld einer Differentialgleichung Lineare Differentialgleichung erster Ordnung Parameterabhangige lineare Differentialgleichung erster Ordnung Bernoullische Differentialgleichung Separable Differentialgleichung Ahnlichkeitsdifferentialgleichung Exakte Differentialgleichung Integrierender Faktor Substitution bei einer Differentialgleichung erster Ordnung