Imposing Dirichlet boundary conditions with Nitsche's method and spline‐based finite elements

Abstract

SUMMARYA key challenge while employing non-interpolatory basis functions in finite-element methods is the robust imposition of Dirichlet boundary conditions. The current work studies the weak enforcement of such conditions for B-spline basis functions, with application to both second-and fourth-order problems. This is achieved using concepts borrowed from Nitsche's method, which is a stabilized method for imposing constraints on surfaces. Conditions for the stability of the system of equations are derived for each class of problem. Stability parameters in the Nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the Dirichlet boundary. The approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not. Through several numerical examples, the approach is shown to yield optimal rates of convergence.

A parameter-free variational coupling approach for trimmed isogeometric thin shells

“…On the embedded domain side, the importance of geometrically faithful quadrature of trimmed elements and corresponding techniques have been discussed in a series of recent papers [28,27,[29][30][31][32][33][34][35]. For the weak enforcement of boundary and interface conditions at trimming curves and surfaces, variational methods such as Lagrange multiplier [36][37][38] or Nitsche techniques [39][40][41][42][43][44] have been successfully developed.…”

“…Symmetric variants of Nitsche methods are accurate and robust, but their performance crucially depends on appropriate estimates of the stabilization parameters involved [40,44,45]. If estimates are too large, the method degrades to a penalty method, which adversely influences consistency, accuracy and robustness.…”

Essential boundary and interface conditions in the meshless analysis of shells.

“…where, as it was done for the initial deformation, one can de ne 16) as the cross-sectional strain vectors. Once again, their back-rotated counterparts are more suitable for an objective theory.…”

“…the equilibrium on the natural boundary between the tractions and the imposed external forces, 16) which impose the equilibrium between the internal generalized stresses and the generalized reactions and the compatibility on the kinematic boundary, which are the independently approximated Lagrange Multipliers for the Dirichlet boundary.…”

“…It has been applied with eXtended Finite Elements (XFEM) for EBCs and ICs in single-eld problems [15,25,28], to enforce weak discontinuities between elastic materials [51,52,2,1]; and applied to non conforming embedded FE meshes for elastostatics [53,54]. B-splines and NURBs-base FEs share similar features with meshless approximations and can also take advantage of Nitsche's Method, as in [16] for 2nd and 4th order problems, [46,33,49,50] for elasticity, [27,26] for thin plates and [22,23] for thin shells. The theory has also been extended for contact with FEs [59] and XFEM [4,3].…”