A projected super-penalty method for the $$C^1$$-coupling of multi-patch isogeometric Kirchhoff plates

Abstract

This work focuses on the development of a super-penalty strategy based on the $$L^2$$ L 2 -projection of suitable coupling terms to achieve $$C^1$$ C 1 -continuity between non-conforming multi-patch isogeometric Kirchhoff plates. In particular, the choice of penalty parameters is driven by the underlying perturbed saddle point problem from which the Lagrange multipliers are eliminated and is performed to guarantee the optimal accuracy of the method. Moreover, by construction, the method does not suffer from boundary locking, especially on very coarse meshes. We demonstrate the applicability of the proposed coupling algorithm to Kirchhoff plates by studying several benchmark examples discretized by non-conforming meshes. In all cases, we recover the optimal rates of convergence achievable by B-splines where we achieve a substantial gain in accuracy per degree-of-freedom compared to other choices of the penalty parameters.

Coupling of non-conforming trimmed isogeometric Kirchhoff-Love shells via a projected super-penalty approach

“…Remark 4. The p/p − 2 pairing has been proven to be inf-sup stable in the context of isogeometric mortar methods in [13] and it has been extended to the coupling of non-trimmed Kirchhoff plates in [19]. Although its stability for trimmed geometries has not been rigorously studied, we verify numerically its applicability to the coupling of trimmed Kirchhoff-Love shells.…”

“…In this section, we extend the method studied by the authors in [19] for coupling non-conforming Kirchhoff plates to the analysis of trimmed multi-patch Kirchhoff-Love shells. Motivated by the work presented in [13] in the context of isogeometric mortar methods, our strategy leverages the L 2 projection of the coupling terms at the interface, typically defined in terms of the degree p of the solution space related to the corresponding patch, onto a reduced space of B-splines of degree p − 2 defined on the so-called active side of the interface.…”

“…where we have introduced the parameters α disp and α rot corresponding to the displacements and normal rotations, respectively. For a rigorous derivation of the singularly-perturbed saddle point formulation in the scope of Kirchhoff plates we refer to [19]. As highlighted in [28,45], these coefficients depend in general on the problem definition, e.g.…”

“…, L. However, the well-posedness of the underlying problem is insensitive to the choice of the parameters α disp and α rot . Therefore, our method is inherently free from boundary locking, independently of the choice of penalty values, see [19] for further details in the context of isogeometric Kirchhoff plates. This allows us to select α disp and α rot to guarantee the high-order convergence rates achievable by B-splines.…”

“…Our contribution seeks to mitigate the aforementioned issues. Inspired by [28] and taking [19] as our starting point, we present an algorithm for achieving displacement and rotational continuity in trimmed shells which is inherently locking-free and where the penalty parameters are automatically defined by the problem setup, namely material properties, geometry of the structure and underlying discretization. Moreover, the aforementioned penalty factors are suitably built to retain the higherorder accuracy of B-splines.…”