A robust patch coupling method for NURBS-based isogeometric analysis of non-conforming multipatch surfaces

With the emergence of isogeometric analysis (IGA), the Galerkin rotation-free discretization of Kirchhoff-Love shells is facilitated, enabling more efficient thin shell structural analysis. High-order shape functions used in IGA also allow the collocation of partial differential equations, avoiding the time-consuming numerical integration of the Galerkin technique. The goal of the present work is to apply this method to NURBS-based isogeometric Kirchhoff-Love plates and shells, under the assumption of small deformations.Since Kirchhoff-Love plate theory yields a fourth-order formulation, two boundary conditions are required at each location on the contour, generating some conflicts at the corners where there are more equations than needed. To remedy this overdetermination, we provide priority and averaging rules that cover all the possible combinations of adjacent edge boundary conditions (i.e. the clamped, simplysupported, symmetric and free supports). Greville and alternative superconvergent points are used for NURBS basis of even and odd degrees, respectively. For square, circular, and annular flat plates, convergence orders are found to be in agreement with a-priori error estimates. The proposed isogeometric collocation method is then validated and benchmarked against a Galerkin implementation by studying a set of problems involving Kirchhoff-Love shells.

Pantographic sheets are metamaterials constituted by two interconnected layers of straight fibers. One of the great features of these structures is that they are extremely elastically compliant toward large nonlinear deformations. To model pantographic lattices, Kirchhoff rods based on Euler-Bernoulli assumptions can be used. Otherwise, if the fibers are sufficiently dense, homogenization of the micro-structure results to a two-dimensional second-order gradient continuum model.The discrete and continuum models have in common the fact that their energy terms depend on the second-order derivatives of the displacement field, such that the classical finite element method cannot be directly employed. We propose instead the use of the isogeometric analysis where the basis functions are endowed with a higher inter-element continuity, allowing the rotation-free discretization of these problems. The discrete and continuum models are compared to existing benchmarks and are found in excellent agreement, validating the proposed approach.

The finite element method is the reference technique in the simulation of metal forming and provides excellent results with both Eulerian and Lagrangian implementations. The latter approach is more natural and direct but the large deformations involved in such processes require remeshing-rezoning algorithms that increase the computational times and reduce the quality of the results. Meshfree methods can better handle large deformations and have shown encouraging results. However, viscoplastic flows are nearly incompressible, which poses a challenge to meshfree methods. In this paper we propose a simple model of viscoplasticity, where both the pressure and velocity fields are discretized with maximum entropy approximants. The inf-sup condition is circumvented with a numerically consistent stabilized formulation that involves the gradient of the pressure. The performance of the method is studied in some benchmark problems including metal forming and orthogonal cutting.