21 research outputs found
A note on the penalty parameter in Nitsche's method for unfitted boundary value problems
Nitsche's method is a popular approach to implement Dirichlet-type boundary
conditions in situations where a strong imposition is either inconvenient or
simply not feasible. The method is widely applied in the context of unfitted
finite element methods. From the classical (symmetric) Nitsche's method it is
well-known that the stabilization parameter in the method has to be chosen
sufficiently large to obtain unique solvability of discrete systems. In this
short note we discuss an often used strategy to set the stabilization parameter
and describe a possible problem that can arise from this. We show that in
specific situations error bounds can deteriorate and give examples of
computations where Nitsche's method yields large and even diverging
discretization errors
Critical time-step size analysis and mass scaling by ghost-penalty for immersogeometric explicit dynamics
In this article, we study the effect of small-cut elements on the critical
time-step size in an immersogeometric context. We analyze different
formulations for second-order (membrane) and fourth-order (shell-type)
equations, and derive scaling relations between the critical time-step size and
the cut-element size for various types of cuts. In particular, we focus on
different approaches for the weak imposition of Dirichlet conditions: by
penalty enforcement and with Nitsche's method. The stability requirement for
Nitsche's method necessitates either a cut-size dependent penalty parameter, or
an additional ghost-penalty stabilization term is necessary. Our findings show
that both techniques suffer from cut-size dependent critical time-step sizes,
but the addition of a ghost-penalty term to the mass matrix serves to mitigate
this issue. We confirm that this form of `mass-scaling' does not adversely
affect error and convergence characteristics for a transient membrane example,
and has the potential to increase the critical time-step size by orders of
magnitude. Finally, for a prototypical simulation of a Kirchhoff-Love shell,
our stabilized Nitsche formulation reduces the solution error by well over an
order of magnitude compared to a penalty formulation at equal time-step size
Residual-based error estimation and adaptivity for stabilized immersed isogeometric analysis using truncated hierarchical B-splines
We propose an adaptive mesh refinement strategy for immersed isogeometric
analysis, with application to steady heat conduction and viscous flow problems.
The proposed strategy is based on residual-based error estimation, which has
been tailored to the immersed setting by the incorporation of appropriately
scaled stabilization and boundary terms. Element-wise error indicators are
elaborated for the Laplace and Stokes problems, and a THB-spline-based local
mesh refinement strategy is proposed. The error estimation .and adaptivity
procedure is applied to a series of benchmark problems, demonstrating the
suitability of the technique for a range of smooth and non-smooth problems. The
adaptivity strategy is also integrated in a scan-based analysis workflow,
capable of generating reliable, error-controlled, results from scan data,
without the need for extensive user interactions or interventions.Comment: Submitted to Journal of Mechanic
Preconditioned iterative solution techniques for immersed finite element methods: with applications in immersed isogeometric analysis for solid and fluid mechanics
Efficient solution methods for numerical simulations of complex geometrie
Preconditioned iterative solution techniques for immersed finite element methods:with applications in immersed isogeometric analysis for solid and fluid mechanics
Efficient solution methods for numerical simulations of complex geometrie
A note on the stability parameter in Nitsche's method for unfitted boundary value problems
Nitsche's method is a popular approach to implement Dirichlet-type boundary conditions in situations where a strong imposition is either inconvenient or simply not feasible. The method is widely applied in the context of unfitted finite element methods. Of the classical (symmetric) Nitsche's method it is well-known that the stabilization parameter in the method has to be chosen sufficiently large to obtain unique solvability of discrete systems. In this short note we discuss an often used strategy to set the stabilization parameter and describe a possible problem that can arise from this. We show that in specific situations error bounds can deteriorate and give examples of computations where Nitsche's method yields large and even diverging discretization errors. \u3cbr/\u3