Scaled boundary isogeometric analysis with C¹ coupling for Kirchhoff‐Love theory

Abstract

Scaled boundary isogeometric analysis (SB-IGA) describes the computational domain by proper boundary NURBS together with a well-defined scaling center; see [5]. More precisely, we consider star convex domains whose domain boundaries correspond to a sequence of NURBS curves and the interior is determined by a scaling of the boundary segments with respect to a chosen scaling center. However, providing a decomposition into star shaped blocks one can utilize SB-IGA also for more general shapes. Even though several geometries can be described by a single patch, in applications frequently there appear multipatch structures. Whereas a C 0 continuous patch coupling can be achieved relatively easily, the situation becomes more complicated if higher regularity is required. Consequently, a suitable coupling method is inevitably needed for analyses that require global C 1 continuity.In this contribution we apply the concept of analysis-suitable G 1 parametrizations [2] to the framework of SB-IGA for the C 1 coupling of planar domains with a special consideration of the scaling center. We obtain globally C 1 regular basis functions and this enables us to handle problems such as the Kirchhoff-Love plate and shell, where smooth coupling is an issue. Furthermore, the boundary representation within SB-IGA makes the method suitable for the concept of trimming. In particular, we see the possibility to extend the coupling procedure to study trimmed plates and shells.The approach was implemented using the GeoPDEs package [1] and its performance was tested on several numerical examples. Finally, we discuss the advantages and disadvantages of the proposed method and outline future perspectives.

Cs-smooth isogeometric spline spaces over planar bilinear multi-patch parameterizations

“…Let us first recall some relations between the univariate spline spaces S p 1 ,r 1 h ([0, 1]) and S p 2 ,r 2 h ([0, 1]). Since by (1) we have p 1 ≤ p 2 and r 1 ≥ r 2 , it follows that…”

“…We first introduce the mixed degree and regularity spline space for the univariate case by explaining the construction of its basis. Recall (1). We start with all B-splines N p 1 ,r 1 i from the space S p 1 ,r 1 h ([0, 1]), remove all of them that have a non-vanishing derivative of order θ ≤ s at 0 or 1, and replace them with the B-splines N p 2 ,r 2 i from the space S p 2 ,r 2 h ([0, 1]) that have a non-vanishing derivative of order θ ≤ s at 0 or 1.…”

“…) is uniquely determined by selecting the smoothness s and degree p 1 via s + 1 ≤ p 1 ≤ 2s + 1 and by assumptions (1). Therefore, we get the two most practical and interesting cases by selecting the degree p 1 to be either the smallest or the greatest possible one: Case A: p 1 = s + 1.…”

“…Therefore, we get the two most practical and interesting cases by selecting the degree p 1 to be either the smallest or the greatest possible one: Case A: p 1 = s + 1. Since by (1) we have r 1 = s, and hence the regularities r 1 and r 2 are both equal to s, we can call the obtained underlying spline space also shortly just the mixed degree spline space, cf. Fig.…”

“…[46,55,58] and the related subdivision based approaches [44,49,62], or with singularities at so-called scaling centers, see e.g. [1,48], second on a C 1 -smooth multi-patch parameterization with a G 1 -smooth cap around extraordinary vertices, see e.g. [36-38, 45, 61], or third on a G 1 -smooth multi-patch parameterization, which means for a planar multi-patch domain, just on a C 0 -smooth multi-patch parameterization, see e.g.…”