1,388 research outputs found
Extended Smoothed Boundary Method for Solving Partial Differential Equations with General Boundary Conditions on Complex Boundaries
In this article, we describe an approach for solving partial differential
equations with general boundary conditions imposed on arbitrarily shaped
boundaries. A continuous function, the domain parameter, is used to modify the
original differential equations such that the equations are solved in the
region where a domain parameter takes a specified value while boundary
conditions are imposed on the region where the value of the domain parameter
varies smoothly across a short distance. The mathematical derivations are
straightforward and generically applicable to a wide variety of partial
differential equations. To demonstrate the general applicability of the
approach, we provide four examples herein: (1) the diffusion equation with both
Neumann and Dirichlet boundary conditions; (2) the diffusion equation with both
surface diffusion and reaction; (3) the mechanical equilibrium equation; and
(4) the equation for phase transformation with the presence of additional
boundaries. The solutions for several of these cases are validated against
corresponding analytical and semi-analytical solutions. The potential of the
approach is demonstrated with five applications: surface-reaction-diffusion
kinetics with a complex geometry, Kirkendall-effect-induced deformation,
thermal stress in a complex geometry, phase transformations affected by
substrate surfaces, and a self-propelled droplet.Comment: This document is the revised version of arXiv:0912.1288v
Smoothed Boundary Method for Solving Partial Differential Equations with General Boundary Conditions on Complex Boundaries
In this article, we describe an approach for solving partial differential
equations with general boundary conditions imposed on arbitrarily shaped
boundaries. A function that has a prescribed value on the domain in which a
differential equation is valid and smoothly but rapidly varying values on the
boundary where boundary conditions are imposed is used to modify the original
differential equations. The mathematical derivations are straight forward, and
generically applicable to a wide variety of partial differential equations. To
demonstrate the general applicability of the approach, we provide four
examples: (1) the diffusion equation with both Neumann and Dirichlet boundary
conditions, (2) the diffusion equation with surface diffusion, (3) the
mechanical equilibrium equation, and (4) the equation for phase transformation
with additional boundaries. The solutions for a few of these cases are
validated against corresponding analytical and semi-analytical solutions. The
potential of the approach is demonstrated with five applications:
surface-reaction diffusion kinetics with a complex geometry,
Kirkendall-effect-induced deformation, thermal stress in a complex geometry,
phase transformations affected by substrate surfaces, and a self-propelling
droplet.Comment: A better smooth algorithm has been developed and tested, will soon
replace Eq. 58 in page 16. We have also developed a level-set moving boundary
SBM method, and it will replace the Navier-Stokes-Cahn-Hilliard type domain
parameter tracking method in Section 5.
A Toy Model for Testing Finite Element Methods to Simulate Extreme-Mass-Ratio Binary Systems
Extreme mass ratio binary systems, binaries involving stellar mass objects
orbiting massive black holes, are considered to be a primary source of
gravitational radiation to be detected by the space-based interferometer LISA.
The numerical modelling of these binary systems is extremely challenging
because the scales involved expand over several orders of magnitude. One needs
to handle large wavelength scales comparable to the size of the massive black
hole and, at the same time, to resolve the scales in the vicinity of the small
companion where radiation reaction effects play a crucial role. Adaptive finite
element methods, in which quantitative control of errors is achieved
automatically by finite element mesh adaptivity based on posteriori error
estimation, are a natural choice that has great potential for achieving the
high level of adaptivity required in these simulations. To demonstrate this, we
present the results of simulations of a toy model, consisting of a point-like
source orbiting a black hole under the action of a scalar gravitational field.Comment: 29 pages, 37 figures. RevTeX 4.0. Minor changes to match the
published versio
Exact regularized point particle method for multi-phase flows in the two-way coupling regime
Particulate flows have been largely studied under the simplifying assumptions
of one-way coupling regime where the disperse phase do not react-back on the
carrier fluid. In the context of turbulent flows, many non trivial phenomena
such as small scales particles clustering or preferential spatial accumulation
have been explained and understood. A more complete view of multiphase flows
can be gained calling into play two-way coupling effects, i.e. by accounting
for the inter-phase momentum exchange between the carrier and the suspended
phase, certainly relevant at increasing mass loading. In such regime, partially
investigated in the past by the so-called Particle In Cell (PIC) method, much
is still to be learned about the dynamics of the disperse phase and the ensuing
alteration of the carrier flow.
In this paper we present a new methodology rigorously designed to capture the
inter-phase momentum exchange for particles smaller than the smallest
hydrodynamical scale, e.g. the Kolmogorov scale in a turbulent flow. In fact,
the momentum coupling mechanism exploits the unsteady Stokes flow around a
small rigid sphere where the transient disturbance produced by each particle is
evaluated in a closed form. The particles are described as lumped, point masses
which would lead to the appearance of singularities. A rigorous regularization
procedure is conceived to extract the physically relevant interactions between
particles and fluid which avoids any "ah hoc" assumption. The approach is
suited for high efficiency implementation on massively parallel machines since
the transient disturbance produced by the particles is strongly localized in
space around the actual particle position. As will be shown, hundred thousands
particles can therefore be handled at an affordable computational cost as
demonstrated by a preliminary application to a particle laden turbulent shear
flow.Comment: Submitted to Journal of Fluid Mechanics, 56 pages, 15 figure
The effects of discreteness of galactic cosmic rays sources
Most studies of GeV Galactic Cosmic Rays (GCR) nuclei assume a steady
state/continuous distribution for the sources of cosmic rays, but this
distribution is actually discrete in time and in space. The current progress in
our understanding of cosmic ray physics (acceleration, propagation), the
required consistency in explaining several GCRs manifestation (nuclei,
,...) as well as the precision of present and future space missions
(e.g. INTEGRAL, AMS, AGILE, GLAST) point towards the necessity to go beyond
this approximation. A steady state semi-analytical model that describes well
many nuclei data has been developed in the past years based on this
approximation, as well as others. We wish to extend it to a time dependent
version, including discrete sources. As a first step, the validity of several
approximations of the model we use are checked to validate the approach: i) the
effect of the radial variation of the interstellar gas density is inspected and
ii) the effect of a specific modeling for the galactic wind (linear vs
constant) is discussed. In a second step, the approximation of using continuous
sources in space is considered. This is completed by a study of time
discreteness through the time-dependent version of the propagation equation. A
new analytical solution of this equation for instantaneous point-like sources,
including the effect of escape, galactic wind and spallation, is presented.
Application of time and space discretness to definite propagation conditions
and realistic distributions of sources will be presented in a future paper.Comment: final version, 8 figures, accepted in ApJ. A misprint in fig 8 labels
has been correcte
Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver
Partial differential equations are often used to model various physical
phenomena, such as heat diffusion, wave propagation, fluid dynamics,
elasticity, electrodynamics and image processing, and many analytic approaches
or traditional numerical methods have been developed and widely used for their
solutions. Inspired by rapidly growing impact of deep learning on scientific
and engineering research, in this paper we propose a novel neural network,
GF-Net, for learning the Green's functions of linear reaction-diffusion
equations in an unsupervised fashion. The proposed method overcomes the
challenges for finding the Green's functions of the equations on arbitrary
domains by utilizing physics-informed approach and the symmetry of the Green's
function. As a consequence, it particularly leads to an efficient way for
solving the target equations under different boundary conditions and sources.
We also demonstrate the effectiveness of the proposed approach by experiments
in square, annular and L-shape domains
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