5,028 research outputs found
Comparison of Wide and Compact Fourth Order Formulations of the Navier-Stokes Equations
In this study the numerical performances of wide and compact fourth order
formulation of the steady 2-D incompressible Navier-Stokes equations will be
investigated and compared with each other. The benchmark driven cavity flow
problem will be solved using both wide and compact fourth order formulations
and the numerical performances of both formulations will be presented and also
the advantages and disadvantages of both formulations will be discussed
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
An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary
Common efficient schemes for the incompressible Navier-Stokes equations, such
as projection or fractional step methods, have limited temporal accuracy as a
result of matrix splitting errors, or introduce errors near the domain
boundaries (which destroy uniform convergence to the solution). In this paper
we recast the incompressible (constant density) Navier-Stokes equations (with
the velocity prescribed at the boundary) as an equivalent system, for the
primary variables velocity and pressure. We do this in the usual way away from
the boundaries, by replacing the incompressibility condition on the velocity by
a Poisson equation for the pressure. The key difference from the usual
approaches occurs at the boundaries, where we use boundary conditions that
unequivocally allow the pressure to be recovered from knowledge of the velocity
at any fixed time. This avoids the common difficulty of an, apparently,
over-determined Poisson problem. Since in this alternative formulation the
pressure can be accurately and efficiently recovered from the velocity, the
recast equations are ideal for numerical marching methods. The new system can
be discretized using a variety of methods, in principle to any desired order of
accuracy. In this work we illustrate the approach with a 2-D second order
finite difference scheme on a Cartesian grid, and devise an algorithm to solve
the equations on domains with curved (non-conforming) boundaries, including a
case with a non-trivial topology (a circular obstruction inside the domain).
This algorithm achieves second order accuracy (in L-infinity), for both the
velocity and the pressure. The scheme has a natural extension to 3-D.Comment: 50 pages, 14 figure
Simplex space-time meshes in thermally coupled two-phase flow simulations of mold filling
The quality of plastic parts produced through injection molding depends on
many factors. Especially during the filling stage, defects such as weld lines,
burrs, or insufficient filling can occur. Numerical methods need to be employed
to improve product quality by means of predicting and simulating the injection
molding process. In the current work, a highly viscous incompressible
non-isothermal two-phase flow is simulated, which takes place during the cavity
filling. The injected melt exhibits a shear-thinning behavior, which is
described by the Carreau-WLF model. Besides that, a novel discretization method
is used in the context of 4D simplex space-time grids [2]. This method allows
for local temporal refinement in the vicinity of, e.g., the evolving front of
the melt [10]. Utilizing such an adaptive refinement can lead to locally
improved numerical accuracy while maintaining the highest possible
computational efficiency in the remaining of the domain. For demonstration
purposes, a set of 2D and 3D benchmark cases, that involve the filling of
various cavities with a distributor, are presented.Comment: 14 pages, 11 Figures, 4 Table
An Eulerian projection method for quasi-static elastoplasticity
A well-established numerical approach to solve the Navier--Stokes equations
for incompressible fluids is Chorin's projection method, whereby the fluid
velocity is explicitly updated, and then an elliptic problem for the pressure
is solved, which is used to orthogonally project the velocity field to maintain
the incompressibility constraint. In this paper, we develop a mathematical
correspondence between Newtonian fluids in the incompressible limit and
hypo-elastoplastic solids in the slow, quasi-static limit. Using this
correspondence, we formulate a new fixed-grid, Eulerian numerical method for
simulating quasi-static hypo-elastoplastic solids, whereby the stress is
explicitly updated, and then an elliptic problem for the velocity is solved,
which is used to orthogonally project the stress to maintain the
quasi-staticity constraint. We develop a finite-difference implementation of
the method and apply it to an elasto-viscoplastic model of a bulk metallic
glass based on the shear transformation zone theory. We show that in a
two-dimensional plane strain simple shear simulation, the method is in
quantitative agreement with an explicit method. Like the fluid projection
method, it is efficient and numerically robust, making it practical for a wide
variety of applications. We also demonstrate that the method can be extended to
simulate objects with evolving boundaries. We highlight a number of
correspondences between incompressible fluid mechanics and quasi-static
elastoplasticity, creating possibilities for translating other numerical
methods between the two classes of physical problems.Comment: 49 pages, 20 figure
Sobolev gradients and image interpolation
We present here a new image inpainting algorithm based on the Sobolev
gradient method in conjunction with the Navier-Stokes model. The original model
of Bertalmio et al is reformulated as a variational principle based on the
minimization of a well chosen functional by a steepest descent method. This
provides an alternative of the direct solving of a high-order partial
differential equation and, consequently, allows to avoid complicated numerical
schemes (min-mod limiters or anisotropic diffusion). We theoretically analyze
our algorithm in an infinite dimensional setting using an evolution equation
and obtain global existence and uniqueness results as well as the existence of
an -limit. Using a finite difference implementation, we demonstrate
using various examples that the Sobolev gradient flow, due to its smoothing and
preconditioning properties, is an effective tool for use in the image
inpainting problem
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