Dual and approximate dual basis functions for B-splines and NURBS – Comparison and application for an efficient coupling of patches with the isogeometric mortar method

Inverse mass matrix for isogeometric explicit transient analysis via the method of localized Lagrange multipliers

“…with a trivial inverse that permits to compute the inverse mass matrix directly from the momentum matrix. Different local and global dual interpolation functions of this kind, originally designed and tested for mortar methods, have been recently proposed in the literature [30][31][32] for NURBS and B-spline interpolations. Specifically for the construction of RMMs, biorthogonal basis functions have been proposed under the framework of FEM 8,9 and also IGA.…”

“…These dual basis functions entail minimal support, ie, the support of the dual basis is equal to the support of the original B-spline basis, that for the case of B-splines covers p + 1 knot spans. Analytical expressions for the construction of local continuous (LC) biorthogonal dual basis exist in the literature for 1D B-splines, [32][33][34] but they are difficult to generalize for multidimensions, restricting their application to our case. For higher dimensions, the simplest alternative is the use of a local discontinuous (LD) biorthogonal dual basis 31,35 obtained via an elementwise linear combination of the primal shape functions, providing discontinuous reciprocal basis functions that fulfill the partition of unity with the same support than their primal counterparts.…”

“…Another possibility to determine GC dual basis functions for B-spline basis functions is the inversion of the Gram matrix. 32,37 In this case, the first step in the calculation of the dual basis requires the computation of the Gramian matrix for the primal basis…”

“…We incorporate into this final category bases that are not strictly orthogonal but can simplify the computation of the mass inverse due to the exploitation of a convenient property. One example is the approximate dual basis based on wavelets proposed in the work of Dornisch et al 32 for mortar coupling methods that are able to produce semicompact approximations of an orthogonal basis.…”

Moment Preserving Local Spline Projection Operators

“…In mortar methods for instance [3], the use of local dual bases spanning M. Campos Pinto Sorbonne Université, Université Paris-Diderot SPC, CNRS, Laboratoire Jacques-Louis Lions, LJLL, F-75005 Paris Tel. : +33-1-44275405 E-mail: campos@ljll.math.upmc.fr high order polynomials allows to localize the matching conditions at the interfaces without loosing the optimality of the method, see [23,20,17]. Another framework where such a tool is useful is the extension of classical commuting diagrams [18,1,2,11] to non-conforming discretization spaces, see [13,12,14] and the presentation below.…”