Michael Helmut Gfrerer

arXiv2022

The Numerical Assembly Technique for arbitrary planar systems based on an alternative homogeneous solution

The Numerical Assembly Technique is extended to investigate arbitrary planar frame structures with the focus on the computation of natural frequencies. This allows us to obtain highly accurate results without resorting to spatial discretization. To this end, we systematically introduce a frame structure as a set of nodes, beams, bearings, springs, and external loads and formulate the corresponding boundary and interface conditions. As the underlying homogeneous solution of the governing equations, we use a novel approach recently presented in the literature. This greatly improves the numerical stability and allows the stable computation of very high natural frequencies accurately. We show this numerically at two frame structures by investigation of the condition number of the system matrix and also by the use of variable precision arithmetic.

math.NA↗

arXiv2018

Isogeometric Parametrization Inspired by Large Elastic Deformation

The construction of volumetric parametrizations for computational domains is a key step in the pipeline of isogeometric analysis. Here, we investigate a solution to this problem based on the mesh deformation approach. The desired domain is modeled as a deformed configuration of an initial simple geometry. Assuming that the parametrization of the initial domain is bijective and that it is possible to find a locally invertible displacement field, the method yields a bijective parametrization of the target domain. We compute the displacement field by solving the equations of nonlinear elasticity with the neo-Hookean material law, and we show an efficient variation of the incremental loading algorithm tuned specifically to this application. In order to construct the initial domain, we simplify the target domain's boundary by means of an L2-projection onto a coarse basis and then apply the Coons patch approach. The proposed methodology is not restricted to a single patch scenario but can be utilized to construct multi-patch parametrizations with naturally looking boundaries between neighboring patches. We illustrate its performance and compare the result to other established parametrization approaches on a range of two-dimensional and three-dimensional examples.

math.NA↗