165 research outputs found
Diffuse interface models of locally inextensible vesicles in a viscous fluid
We present a new diffuse interface model for the dynamics of inextensible
vesicles in a viscous fluid. A new feature of this work is the implementation
of the local inextensibility condition in the diffuse interface context. Local
inextensibility is enforced by using a local Lagrange multiplier, which
provides the necessary tension force at the interface. To solve for the local
Lagrange multiplier, we introduce a new equation whose solution essentially
provides a harmonic extension of the local Lagrange multiplier off the
interface while maintaining the local inextensibility constraint near the
interface. To make the method more robust, we develop a local relaxation scheme
that dynamically corrects local stretching/compression errors thereby
preventing their accumulation. Asymptotic analysis is presented that shows that
our new system converges to a relaxed version of the inextensible sharp
interface model. This is also verified numerically. Although the model does not
depend on dimension, we present numerical simulations only in 2D. To solve the
2D equations numerically, we develop an efficient algorithm combining an
operator splitting approach with adaptive finite elements where the
Navier-Stokes equations are implicitly coupled to the diffuse interface
inextensibility equation. Numerical simulations of a single vesicle in a shear
flow at different Reynolds numbers demonstrate that errors in enforcing local
inextensibility may accumulate and lead to large differences in the dynamics in
the tumbling regime and differences in the inclination angle of vesicles in the
tank-treading regime. The local relaxation algorithm is shown to effectively
prevent this accumulation by driving the system back to its equilibrium state
when errors in local inextensibility arise.Comment: 25 page
A Finite Element Approach For Modeling Biomembranes In Incompressible Power-Law Flow
We present a numerical method to model the dynamics of inextensible
biomembranes in a quasi-Newtonian incompressible flow, which better describes
hemorheology in the small vasculature. We consider a level set model for the
fluid-membrane coupling, while the local inextensibility condition is relaxed
by introducing a penalty term. The penalty method is straightforward to
implement from any Navier-Stokes/level set solver and allows substantial
computational savings over a mixed formulation. A standard Galerkin finite
element framework is used with an arbitrarily high order polynomial
approximation for better accuracy in computing the bending force. The PDE
system is solved using a partitioned strongly coupled scheme based on
Crank-Nicolson time integration. Numerical experiments are provided to validate
and assess the main features of the method
An adaptive octree finite element method for PDEs posed on surfaces
The paper develops a finite element method for partial differential equations
posed on hypersurfaces in , . The method uses traces of
bulk finite element functions on a surface embedded in a volumetric domain. The
bulk finite element space is defined on an octree grid which is locally refined
or coarsened depending on error indicators and estimated values of the surface
curvatures. The cartesian structure of the bulk mesh leads to easy and
efficient adaptation process, while the trace finite element method makes
fitting the mesh to the surface unnecessary. The number of degrees of freedom
involved in computations is consistent with the two-dimension nature of surface
PDEs. No parametrization of the surface is required; it can be given implicitly
by a level set function. In practice, a variant of the marching cubes method is
used to recover the surface with the second order accuracy. We prove the
optimal order of accuracy for the trace finite element method in and
surface norms for a problem with smooth solution and quasi-uniform mesh
refinement. Experiments with less regular problems demonstrate optimal
convergence with respect to the number of degrees of freedom, if grid
adaptation is based on an appropriate error indicator. The paper shows results
of numerical experiments for a variety of geometries and problems, including
advection-diffusion equations on surfaces. Analysis and numerical results of
the paper suggest that combination of cartesian adaptive meshes and the
unfitted (trace) finite elements provide simple, efficient, and reliable tool
for numerical treatment of PDEs posed on surfaces
A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces
The closest point method (Ruuth and Merriman, J. Comput. Phys.
227(3):1943-1961, [2008]) is an embedding method developed to solve a variety
of partial differential equations (PDEs) on smooth surfaces, using a closest
point representation of the surface and standard Cartesian grid methods in the
embedding space. Recently, a closest point method with explicit time-stepping
was proposed that uses finite differences derived from radial basis functions
(RBF-FD). Here, we propose a least-squares implicit formulation of the closest
point method to impose the constant-along-normal extension of the solution on
the surface into the embedding space. Our proposed method is particularly
flexible with respect to the choice of the computational grid in the embedding
space. In particular, we may compute over a computational tube that contains
problematic nodes. This fact enables us to combine the proposed method with the
grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024,
[2009]) to obtain a numerical method for approximating PDEs on moving surfaces.
We present a number of examples to illustrate the numerical convergence
properties of our proposed method. Experiments for advection-diffusion
equations and Cahn-Hilliard equations that are strongly coupled to the velocity
of the surface are also presented
Trace Finite Element Methods for PDEs on Surfaces
In this paper we consider a class of unfitted finite element methods for
discretization of partial differential equations on surfaces. In this class of
methods known as the Trace Finite Element Method (TraceFEM), restrictions or
traces of background surface-independent finite element functions are used to
approximate the solution of a PDE on a surface. We treat equations on steady
and time-dependent (evolving) surfaces. Higher order TraceFEM is explained in
detail. We review the error analysis and algebraic properties of the method.
The paper navigates through the known variants of the TraceFEM and the
literature on the subject
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