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

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 C0 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 C1 continuity.


Summary of computational approach
First we sketch the concept of SB-IGA, then we show how to integrate the C 1 coupling across interface patches. For this purpose we introduce special basis functions for the approximation in the scaling center. Finally, we point out opportunities for generalization including trimmed geometries.

Scaled boundary isogeometric analysis
The underlying idea of SB-IGA, see e.g. [5], [6], fits to the fact that in CAD applications the computational domain is often represented by means of its boundary. For a star-convex domain Ω one chooses a scaling center x 0 ∈ R d and the domain is then defined by a scaling of the boundary w.r.t. to x 0 . In the planar case, which is the one we focus on here, and in view of the isogeometric analysis we have some boundary NURBS curve γ(ζ) = n1 i=1 C iN r i,p (ζ) and define the SB parametrization of Ω through F : Ω := (0, 1) 2 → Ω , (ζ, ξ) → ξ γ(ζ) − x 0 + x 0 .
Above,N r i,p denote the basis functions corresponding to a NURBS space N r p of degree p and global regularity C r and they are associated to proper control points C i ∈ R 2 . γ (1) γ (2) γ (3) γ (4) F 1 Fig. 1: On the left we have a single-patch SB parametrization and a multi-patch domain on the right. The C 1 coupling of the basis functions in the latter case requires a special consideration.
Such a SB parametrization can be interpreted as an element of a suitable NURBS space (N r p ⊗ S r p ) 2 with S r p denoting an univariate B-spline space. Following the concept of isogeometric analysis one introduces discrete spaces on the domain Ω by means of a push-forward operation, i.e. we deal with spaces of the form However, in general the domain boundary is prescribed by multiple boundary curves; see Fig. 1. Then supposing SB parametrizations F k : Ω → Ω k for the patches Ω k , the uncoupled multi-patch spaces are straightforwardly given by stands for the k-th patch space and Ω = ∪ k Ω k . A basic assumption of our approach is the conforming patch interface property, which requires matching control points of meeting patches. Obviously, depending on the application one needs test and ansatz functions with a specific regularity. Whereas a continuous coupling, meaning spaces V 0 h := V M h ∩ C 0 (Ω), can be achieved easily within SB-IGA, the situation becomes more difficult if higher regularity is requested.
Compared to a classical two-patch geometry on the left we get a singularity at x0 in the SB context. In return we obtain a geometry that is closely related to so-called analysis-suitable G 1 parametrizations.

C 1 coupling
Here we look at the construction of globally C 1 smooth basis functions. In other words we focus on V 1 h := V M h ∩ C 1 (Ω) which implies r ≥ 1, p ≥ 2. Starting point of our considerations is the work by Collin et al. [2] which coined the theoretical foundation for the C 1 coupling of isogeometric spaces. It turns out that the constraints coming along with the C 1 condition are in general too restrictive and one observes C 1 locking i.e. the worsening or loss of convergence in numerical applications. Nevertheless, for the class of analysis-suitable G 1 multi-patch (ASG 1 ) parametrizations these order reductions can be avoided. In the case of a simple two-patch scenario like in Fig. 2 (a) one has a ASG 1 parametrization if we find polynomial functions [2] states that for B-spline parametrizations latter condition guarantees enough interface functions in V 1 h for p > r + 1 > 1 and C 1 locking is not an issue. Thus, it is natural to prefer these special geometry representations if smoothness is crucial. An analogous two-patch geometry within SB-IGA, see Fig. 2 (b), fulfills the ASG 1 condition except at the scaling center, where a singularity appears. However, this connection between SB-IGA and ASG 1 geometries suggests good approximation properties of V 1 h and since w.l.o.g. the NURBS weight functions are constant along SB interfaces, the SB-IGA ansatz seems appropriate for coupling also for general NURBS boundary curves. But, on the one hand, we have to clarify how one constructs C 1 basis functions in the scaling center. On the other hand, the actual implementation of the coupling conditions should be practicable. Regarding the former, we can exploit the isoparametric paradigm observing that in 2D we only need three scaling center basis functions that determine the value and the first derivatives at x 0 . Let the k-th patch be parametrized through And since i,sc , we see directly that the global mappings ϕ i,sc determine a linearly independent Hermite data set in x 0 . For the implementation of the coupling conditions one can adapt the approach from [2] which results for two-patch geometries in the following steps.
1. Remove basis functios with non-vanishing values or derivatives at x 0 .
2. Add the scaling center basis functions ϕ i,sc .
3. Do a C 0 coupling step, i.e. we obtain globally continuous basis functions ϕ 0 i . 4. Incorporate possible boundary conditions. 5. Compute C 1 coupled basis functions by means of the null space of the interface derivative jump matrix with n L as the normal to the interface Γ and [[·]] as the jump at Γ. The generalization to multi-patch geometries is straightforward and consequently various shapes Ω are feasible.

Brief remarks on generalizations
Up to now our approach is quite restrictive since we assumed star-convex domains. But this restriction can be overcome.
If Ω is not star-shaped we need a decomposition into star-convex blocks, which is possible e.g. utilizing a quadtree partition. Provided that the different star-convex blocks are separated by straight interfaces, like in Fig. 3 (a), we preserve the (quasi) ASG 1 structure, because the parametrizations of two blocks meet as two bilinear B-spline patches, which are always ASG 1 ; see Fig. 3  3. In case of star-convexity of Ω T one calculates a new scaling center, otherwise a preliminary decomposition step has to be applied; see Fig. 4(c).
After the mentioned steps, we can couple the basis functions analogous to the untrimmed setting. www.gamm-proceedings.com with the bending stiffness D = Et 3 12(1−ν 2 ) . The classical weak form of the problem for a suitable test function space V h that depends on the boundary conditions reads: The discretization of the solution field, meaning the definition of V h , is performed by the previously defined SB-IGA test functions V 1 h including the coupling approach at the patch boundaries, which ensure C 1 -continuity within the whole domain that is required for the plate formulation. In the following, the theory is applied to a rotational symmetric plate, to combine the power of the boundary representation and the exact approximation of the geometry and solution field for a circular plate, both. Afterwards, the application of trimming in scaled boundary is evaluated by trimming the center of the structure to a ring.

Rotational symmetric plate
At first, the plate formulation is checked on its general performance. A rotational symmetric plate of Ω with radius R = 2 is subjected to a smoothly distributed source function g that is chosen such that the exact solution is u = cos(π ρ/4) 2 , with ρ denoting the radial coordinate. A homogeneous and isotropic plate of elastic material with Young's modulus E = 10 6 and Poisson's ratio ν = 0 is considered and the thickness is chosen as t ≈ 0.0229 such that D = 1. The Dirichlet boundary conditions are clamped on the whole boundary of the domain. The body is discretized with four SB-IGA patches. The scaling center is placed in the center of the plate. Fig. 5 exemplary shows the mesh for h = 1/4 and the corresponding deformation plot.
The example is evaluated in terms of the H 2 seminorm and the L 2 norm. The results are shown in Fig. 6. The model shows good results for all orders and converges smoothly towards the reference solution. Furthermore, the convergence rate coincides for both error estimations with the optimal convergence rate C · h p−1 for the H 2 seminorm and C · h p+1 for the L 2 norm. Note that C differs for each order and error estimation.

Trimmed plate -ring
In the second example, the rotational symmetric plate from 2.1 is trimmed by a circular trimming curve defined entirely inside the domain. The resulting ring is subjected to a uniform load g = 1. All other properties stay the same. Since the structure is not inherently star shaped, the ring is divided manually into four star shaped domains, having four scaling centers. Consequently, the C 1 condition needs to be fulfilled also across block interfaces. The reference solution of the deformation in   Fig. 7: On the left, the original disk mesh for h = 1/4 is pictured including the trimming curve (red) and cut lines (magenta) which are used to obtain the SB mesh for the ring on the right. radial direction using a Kirchhoff approach for rotational symmetric plates yields a smooth solution w.r.t. the radial variable ρ. The results show that the trimming is applicable and the trimmed geometry is described properly. Moreover, in Fig. 8 the corresponding deformation plot for the mesh from Fig. 7 (b) is displayed, also along the radial direction. The results show good agreement with the analytical solution. www.gamm-proceedings.com

Conclusion and outlook
We showed that C 1 coupling in the context of SB-IGA is possible, despite the singularity in the scaling center. Since the underlying parametrizations fit to the class of ASG 1 geometries from [2], we should not suffer from C 1 locking which is also confirmed by different numerical experiments. Still there are different aspects which are objects of ongoing research, which should be considered in the future, respectively. First of all, we think that the C 1 coupling is appropriate to study also (trimmed) Kirchhoff-Love shells [3], [4]. For this purpose, one applies the coupling in the planar parametric domain of the shell and uses the C 1 basis functions to approximate the geometry as well as the shell displacement. On the downside of our approach we have the restriction of conforming patch interfaces. Another point is the need of additional stabilization in case of fine SB meshes as the mesh elements near the scaling center degenerate.