Weak imposition of constraints for structural membranes in transient geometrically nonlinear isogeometric analysis on multipatch surfaces

Abstract

Membranes have been extensively used for the design of architectural and general structural models due to their low cost and high load carrying capacity. Traditionally such models were discretized using the standard low order Finite Element Method (FEM) which typically results in a compromised description of the geometry. However, the accurate geometric description of membrane structures is essential as for instance bifurcation points in geometrically nonlinear analysis may be inaccurately predicted when the geometric description of the model is not accurate enough. Moreover, the design of membrane structures typically requires several cycles of form-finding and subsequent structural analysis under various loads which can benefit from a direct connection to the Computer-Aided Design (CAD) environment using its exact geometric description. In this contribution, the form-finding analysis using the Updated Reference Strategy (URS) and the geometrically nonlinear transient analysis of membranes is extended to Isogeometric Analysis (IGA) on multipatch surfaces with Non-Uniform Rational B-Splines (NURBS). As typical in IGA for real CAD geometries, multiple patches with non-matching parametrizations are considered and therefore the continuity of the solution field along with the application of weak Dirichlet boundary conditions need to be addressed. Thus, four different constraint enforcement methods are elaborated and compared, namely, the Penalty, the Lagrange Multipliers, the augmented Lagrange Multipliers and a Nitsche-type method. For the latter method, a solution dependent stabilization approach is employed in order to render the Nitsche-type method coercive. All methods are elaborated and systematically compared in both form-finding analysis, whenever necessary, and subsequently in geometrically nonlinear transient analysis. It should be noted that the Nitsche-type method is more computationally demanding amongst these methods due to the additional nonlinear terms. However, the results suggest that the Nitsche-type method is advantageous for these kinds of problems as no parameter or discretization other than the isogeometric discretization within each patch needs to be specified prior to the analysis.