33,637 research outputs found
Weakly Enforced Boundary Conditions for the NURBS-Based Finite Cell Method
In this paper, we present a variationally consistent formulation for the weak enforcement
of essential boundary conditions as an extension to the finite cell method, a fictitious
domain method of higher order. The absence of boundary fitted elements in fictitious domain or
immersed boundary methods significantly restricts a strong enforcement of essential boundary
conditions to models where the boundary of the solution domain coincides with the embedding
analysis domain. Penalty methods and Lagrange multiplier methods are adequate means to
overcome this limitation but often suffer from various drawbacks with severe consequences for
a stable and accurate solution of the governing system of equations. In this contribution, we
follow the idea of NITSCHE [29] who developed a stable scheme for the solution of the Laplace
problem taking weak boundary conditions into account. An extension to problems from linear
elasticity shows an appropriate behavior with regard to numerical stability, accuracy and an
adequate convergence behavior. NURBS are chosen as a high-order approximation basis to
benefit from their smoothness and flexibility in the process of uniform model refinement
Hybrid finite difference/finite element immersed boundary method
The immersed boundary method is an approach to fluid-structure interaction that uses a Lagrangian
description of the structural deformations, stresses, and forces along with an Eulerian description of the
momentum, viscosity, and incompressibility of the fluid-structure system. The original immersed boundary
methods described immersed elastic structures using systems of flexible fibers, and even now, most
immersed boundary methods still require Lagrangian meshes that are finer than the Eulerian grid. This
work introduces a coupling scheme for the immersed boundary method to link the Lagrangian and Eulerian
variables that facilitates independent spatial discretizations for the structure and background grid. This
approach employs a finite element discretization of the structure while retaining a finite difference scheme
for the Eulerian variables. We apply this method to benchmark problems involving elastic, rigid, and actively
contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases
in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors
that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes.
The Lagrangian-Eulerian coupling approach developed in this work enables the effective use of these coarse
structural meshes with the immersed boundary method. This work also contrasts two different weak forms
of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations
facilitated by our coupling approach
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
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