734 research outputs found
Stability of the B-spline basis via knot insertion
Abstract We derive the stability inequality C γ i c i b i for the B-splines b i from the formula for knot insertion. The key observation is that knot removal increases the norm of the B-spline coefficients C = {c i } i∈Z at most by a constant factor, which is independent of the knot sequence. As a consequence, stability for splines follows from the stability of the Bernstein basis
Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: explicit time-stepping and efficient mass matrix inversion
We present a class of spline finite element methods for time-domain wave
propagation which are particularly amenable to explicit time-stepping. The
proposed methods utilize a discontinuous Galerkin discretization to enforce
continuity of the solution field across geometric patches in a multi-patch
setting, which yields a mass matrix with convenient block diagonal structure.
Over each patch, we show how to accurately and efficiently invert mass matrices
in the presence of curved geometries by using a weight-adjusted approximation
of the mass matrix inverse. This approximation restores a tensor product
structure while retaining provable high order accuracy and semi-discrete energy
stability. We also estimate the maximum stable timestep for spline-based finite
elements and show that the use of spline spaces result in less stringent CFL
restrictions than equivalent piecewise continuous or discontinuous finite
element spaces. Finally, we explore the use of optimal knot vectors based on L2
n-widths. We show how the use of optimal knot vectors can improve both
approximation properties and the maximum stable timestep, and present a simple
heuristic method for approximating optimal knot positions. Numerical
experiments confirm the accuracy and stability of the proposed methods
Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes
The focus on locally refined spline spaces has grown rapidly in recent years
due to the need in Isogeoemtric analysis (IgA) of spline spaces with local
adaptivity: a property not offered by the strict regular structure of tensor
product B-spline spaces. However, this flexibility sometimes results in
collections of B-splines spanning the space that are not linearly independent.
In this paper we address the minimal number of B-splines that can form a linear
dependence relation for Minimal Support B-splines (MS B-splines) and for
Locally Refinable B-splines (LR B-splines) on LR-meshes. We show that the
minimal number is six for MS B-splines, and eight for LR B-splines. The risk of
linear dependency is consequently significantly higher for MS B-splines than
for LR B-splines. Further results are established to help detecting collections
of B-splines that are linearly independent
TiGL - An Open Source Computational Geometry Library for Parametric Aircraft Design
This paper introduces the software TiGL: TiGL is an open source high-fidelity
geometry modeler that is used in the conceptual and preliminary aircraft and
helicopter design phase. It creates full three-dimensional models of aircraft
from their parametric CPACS description. Due to its parametric nature, it is
typically used for aircraft design analysis and optimization. First, we present
the use-case and architecture of TiGL. Then, we discuss it's geometry module,
which is used to generate the B-spline based surfaces of the aircraft. The
backbone of TiGL is its surface generator for curve network interpolation,
based on Gordon surfaces. One major part of this paper explains the
mathematical foundation of Gordon surfaces on B-splines and how we achieve the
required curve network compatibility. Finally, TiGL's aircraft component module
is introduced, which is used to create the external and internal parts of
aircraft, such as wings, flaps, fuselages, engines or structural elements
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