Manifold-based isogeometric analysis basis functions with prescribed sharp features

Abstract

We introduce manifold-based basis functions for isogeometric analysis of surfaces with arbitrary smoothness, prescribed C 0 continuous creases and boundaries. The utility of the manifold-based surface construction techniques in isogeometric analysis was demonstrated in Majeed and Cirak (CMAME, 2017). The respective basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. The connectivity of a given unstructured quadrilateral control mesh in R 3 is used to define a set of overlapping charts. Each vertex with its attached elements is assigned a corresponding conformally parametrised planar chart domain in R 2 so that a quadrilateral element is present on four different charts. On the collection of unconnected chart domains, the partition of unity method is used for approximation. The transition functions required for navigating between the chart domains are composed out of conformal maps. The necessary smooth partition of unity, or blending, functions for the charts are assembled from tensor-product B-spline pieces and require in contrast to earlier constructions no normalisation. Creases are introduced across user tagged edges of the control mesh. Planar chart domains that include creased edges or are adjacent to the domain boundary require special local polynomial approximants in the partition of unity method. Three different types of chart domain geometries are necessary to consider boundaries and arbitrary number and arrangement of creases. The new chart domain geometries are chosen so that it becomes trivial to establish local polynomial approximants that are always C 0 continuous across the tagged edges. The derived non-rational manifold-based basis functions correspond to the vertices of the mesh and may have an arbitrary number of creases and prescribed smoothness. This makes them particularly well suited for isogeometric analysis of Kirchhoff-Love thin shells with kinks, which require C 1 continuous basis functions that are C 0 continuous across the kinks. We demonstrate the convergence and utility of the new basis functions with linear and nonlinear beam, plate and shell examples.

Mollified finite element approximants of arbitrary order and smoothness

“…It is possible to apply a different scaling factor and to choose the scaling in each cell differently. For any given point x ∈ Ω in the domain the mollified basis functions N i (x) are evaluated by computing the convolution integral (13). Evidently, when the chosen local polynomial basis p i (x) is a monomial basis the mollified basis functions are simply the moments of the mollifier.…”

“…In CAGD a range of ingenious constructions has been conceived to generate smooth high-order approximants around the extraordinary vertices, see the books [4,5] for an overview. Unfortunately, most of these constructions, including [6][7][8][9][10][11][12][13], target bivariate manifolds and do not generalise to the arbitrary variate case. Indeed, there are currently no sufficiently flexible and intuitive non-tensor-product arbitrary-variate constructions that can yield smooth polynomial high-order basis functions.…”

“…Combining the preceding two estimates we obtainv h − u W p,l (Ω) ≤ v h − u W p,l (Ω) + u − u W p,l (Ω) Bh k+1−l m |u| W p,k+1 (Ω) . (A.12)This bound is minimised by takingh m = lA (k + 1 − l)B 1/(k+1) h ≡ h * m , (A 13).…”

An optimally convergent smooth blended B-spline construction for semi-structured quadrilateral and hexahedral meshes

“…In computer-aided design numerous constructions have been proposed to deal with extraordinary vertices in a surface mesh, including geometrically G k and parametrically C k continuous constructions [24][25][26][27][28][29][30][31][32][33], subdivision surfaces [34][35][36][37][38][39][40] and manifold constructions [41][42][43][44][45][46][47]. There is, however, a very limited number of constructions for volume meshes, including [48][49][50][51][52][53]; most likely because conventional computer-aided design representations do not require a volume parametrisation.…”

“…As widely reported, most constructions from computer-aided design do not lead in isogeometric analysis to optimally convergent finite elements, especially when applied to higher-order partial differential equations, see the discussion in [29]. There are, however, constructions for unstructured quadrilateral meshes, including [30,32,38,45,46], which yield optimal or nearly optimal convergence rates. In contrast, there are no B-spline based optimally convergent smooth constructions for unstructured hexahedral meshes.…”