A priori penalty factor determination for (trimmed) NURBS-based shells with Dirichlet and coupling constraints in isogeometric analysis

The penalty method has proven to be a well-suited approach for the application of coupling and boundary conditions on (trimmed) multi-patch NURBS shell structures within isogeometric analysis. Beside its favorable simplicity and efficiency, the main challenge is the appropriate choice of the underlying penalty factor — choosing the penalty factor too low yields a poor constraint accuracy, while choosing it too high causes numerical issues like ill-conditioned system matrices or a small infeasible time step size in explicit dynamics. Although recommendations for penalty values exist, profound methods for its determination are still an active field of research. We address this issue and provide formulas allowing an a priori determination of penalty factors for NURBS-based Reissner–Mindlin shells with penalty-based coupling and boundary conditions. The underlying approach is inspired by a methodology previously used for conventional finite elements, for which penalty factors are derived through a comparison with exact Lagrange multiplier solutions. In that way, penalty formulas consisting of a problem-dependent factor and a problem-independent intensity factor are obtained. The fact that the latter is a direct measure of the penalty-induced error is the main advantage of this approach and enables a problem-independent definition of the penalty factor as a function of the desired accuracy. We demonstrate the validity of the derived formulas and the corresponding error measure with benchmark problems in linear elasticity including trimmed non-matching NURBS shells. Furthermore we show that the mesh-adaptivity of the penalty formulas improves the convergence behavior and conditioning of penalty methods.