1,405 research outputs found
Adaptive time-stepping for incompressible flow. Part II: Navier-Stokes equations
We outline a new class of robust and efficient methods for solving the Navier- Stokes equations. We describe a general solution strategy that has two basic building blocks: an implicit time integrator using a stabilized trapezoid rule with an explicit Adams-Bashforth method for error control, and a robust Krylov subspace solver for the spatially discretized system. We present numerical experiments illustrating the potential of our approach. © 2010 Society for Industrial and Applied Mathematics
An adaptive preconditioner for steady incompressible flows
This paper describes an adaptive preconditioner for numerical continuation of
incompressible Navier--Stokes flows. The preconditioner maps the identity (no
preconditioner) to the Stokes preconditioner (preconditioning by Laplacian)
through a continuous parameter and is built on a first order Euler
time-discretization scheme. The preconditioner is tested onto two fluid
configurations: three-dimensional doubly diffusive convection and a reduced
model of shear flows. In the former case, Stokes preconditioning works but a
mixed preconditioner is preferred. In the latter case, the system of equation
is split and solved simultaneously using two different preconditioners, one of
which is parameter dependent. Due to the nature of these applications, this
preconditioner is expected to help a wide range of studies
Spectral methods for CFD
One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched
A fluctuating boundary integral method for Brownian suspensions
We present a fluctuating boundary integral method (FBIM) for overdamped
Brownian Dynamics (BD) of two-dimensional periodic suspensions of rigid
particles of complex shape immersed in a Stokes fluid. We develop a novel
approach for generating Brownian displacements that arise in response to the
thermal fluctuations in the fluid. Our approach relies on a first-kind boundary
integral formulation of a mobility problem in which a random surface velocity
is prescribed on the particle surface, with zero mean and covariance
proportional to the Green's function for Stokes flow (Stokeslet). This approach
yields an algorithm that scales linearly in the number of particles for both
deterministic and stochastic dynamics, handles particles of complex shape,
achieves high order of accuracy, and can be generalized to three dimensions and
other boundary conditions. We show that Brownian displacements generated by our
method obey the discrete fluctuation-dissipation balance relation (DFDB). Based
on a recently-developed Positively Split Ewald method [A. M. Fiore, F. Balboa
Usabiaga, A. Donev and J. W. Swan, J. Chem. Phys., 146, 124116, 2017],
near-field contributions to the Brownian displacements are efficiently
approximated by iterative methods in real space, while far-field contributions
are rapidly generated by fast Fourier-space methods based on fluctuating
hydrodynamics. FBIM provides the key ingredient for time integration of the
overdamped Langevin equations for Brownian suspensions of rigid particles. We
demonstrate that FBIM obeys DFDB by performing equilibrium BD simulations of
suspensions of starfish-shaped bodies using a random finite difference temporal
integrator.Comment: Submitted to J. Comp. Phy
An inexact Newton-Krylov algorithm for constrained diffeomorphic image registration
We propose numerical algorithms for solving large deformation diffeomorphic
image registration problems. We formulate the nonrigid image registration
problem as a problem of optimal control. This leads to an infinite-dimensional
partial differential equation (PDE) constrained optimization problem.
The PDE constraint consists, in its simplest form, of a hyperbolic transport
equation for the evolution of the image intensity. The control variable is the
velocity field. Tikhonov regularization on the control ensures well-posedness.
We consider standard smoothness regularization based on - or
-seminorms. We augment this regularization scheme with a constraint on the
divergence of the velocity field rendering the deformation incompressible and
thus ensuring that the determinant of the deformation gradient is equal to one,
up to the numerical error.
We use a Fourier pseudospectral discretization in space and a Chebyshev
pseudospectral discretization in time. We use a preconditioned, globalized,
matrix-free, inexact Newton-Krylov method for numerical optimization. A
parameter continuation is designed to estimate an optimal regularization
parameter. Regularity is ensured by controlling the geometric properties of the
deformation field. Overall, we arrive at a black-box solver. We study spectral
properties of the Hessian, grid convergence, numerical accuracy, computational
efficiency, and deformation regularity of our scheme. We compare the designed
Newton-Krylov methods with a globalized preconditioned gradient descent. We
study the influence of a varying number of unknowns in time.
The reported results demonstrate excellent numerical accuracy, guaranteed
local deformation regularity, and computational efficiency with an optional
control on local mass conservation. The Newton-Krylov methods clearly
outperform the Picard method if high accuracy of the inversion is required.Comment: 32 pages; 10 figures; 9 table
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