A minimal stabilization procedure for Isogeometric methods on trimmed geometries

Abstract

Trimming is a common operation in CAD, and, in its simplest formulation, consists in removing superfluous parts from a geometric entity described via splines (a spline patch). After trimming the geometric description of the patch remains unchanged, but the underlying mesh is unfitted with the physical object. We discuss the main problems arising when solving elliptic PDEs on a trimmed domain. First we prove that, even when Dirichlet boundary conditions are weakly enforced using Nitsche's method, the resulting method suffers lack of stability. Then, we develop novel stabilization techniques based on a modification of the variational formulation, which allow us to recover well-posedness and guarantee accuracy. Optimal a priori error estimates are proven, and numerical examples confirming the theoretical results are provided.

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• Extension Operators for Trimmed Spline Spaces
• A localized reduced basis approach for unfitted domain methods on parameterized geometries
• Fast parametric analysis of trimmed multi-patch isogeometric Kirchhoff-Love shells using a local reduced basis method
• Vanquishing volumetric locking in quadratic NURBS-based discretizations of nearly-incompressible linear elasticity: CAS elements
• A theoretical study on the effect of mass lumping on the discrete frequencies in immersogeometric analysis
• A CFL condition for the finite cell method
• Generalized eigenvalue stabilization for immersed explicit dynamics