815 research outputs found
Essentially nonoscillatory postprocessing filtering methods
High order accurate centered flux approximations used in the computation of numerical solutions to nonlinear partial differential equations produce large oscillations in regions of sharp transitions. Here, we present a new class of filtering methods denoted by Essentially Nonoscillatory Least Squares (ENOLS), which constructs an upgraded filtered solution that is close to the physically correct weak solution of the original evolution equation. Our method relies on the evaluation of a least squares polynomial approximation to oscillatory data using a set of points which is determined via the ENO network. Numerical results are given in one and two space dimensions for both scalar and systems of hyperbolic conservation laws. Computational running time, efficiency, and robustness of method are illustrated in various examples such as Riemann initial data for both Burgers' and Euler's equations of gas dynamics. In all standard cases, the filtered solution appears to converge numerically to the correct solution of the original problem. Some interesting results based on nonstandard central difference schemes, which exactly preserve entropy, and have been recently shown generally not to be weakly convergent to a solution of the conservation law, are also obtained using our filters
High order filtering methods for approximating hyberbolic systems of conservation laws
In the computation of discontinuous solutions of hyperbolic systems of conservation laws, the recently developed essentially non-oscillatory (ENO) schemes appear to be very useful. However, they are computationally costly compared to simple central difference methods. A filtering method which is developed uses simple central differencing of arbitrarily high order accuracy, except when a novel local test indicates the development of spurious oscillations. At these points, the full ENO apparatus is used, maintaining the high order of accuracy, but removing spurious oscillations. Numerical results indicate the success of the method. High order of accuracy was obtained in regions of smooth flow without spurious oscillations for a wide range of problems and a significant speed up of generally a factor of almost three over the full ENO method
Moving-boundary problems solved by adaptive radial basis functions
The objective of this paper is to present an alternative approach to the conventional level set methods for solving two-dimensional moving-boundary problems known as the passive transport. Moving boundaries are associated with time-dependent problems and the position of the boundaries need to be determined as a function of time and space. The level set method has become an attractive design tool for tracking, modeling and simulating the motion of free boundaries in fluid mechanics, combustion, computer animation and image processing. Recent research on the numerical method has focused on the idea of using a meshless methodology for the numerical solution of partial differential equations. In the present approach, the moving interface is captured by the level set method at all time with the zero contour of a smooth function known as the level set function. A new approach is used to solve a convective transport equation for advancing the level set function in time. This new approach is based on the asymmetric meshless collocation method and the adaptive greedy algorithm for trial subspaces selection. Numerical simulations are performed to verify the accuracy and stability of the new numerical scheme which is then applied to simulate a bubble that is moving, stretching and circulating in an ambient flow to demonstrate the performance of the new meshless approach. (C) 2010 Elsevier Ltd. All rights reserved
Some flows in shape optimization
Geometric flows related to shape optimization problems of Bernoulli type are
investigated. The evolution law is the sum of a curvature term and a nonlocal
term of Hele-Shaw type. We introduce generalized set solutions, the definition
of which is widely inspired by viscosity solutions. The main result is an
inclusion preservation principle for generalized solutions. As a consequence,
we obtain existence, uniqueness and stability of solutions. Asymptotic behavior
for the flow is discussed: we prove that the solutions converge to a
generalized Bernoulli exterior free boundary problem
Characterizing Width Uniformity by Wave Propagation
This work describes a novel image analysis approach to characterize the
uniformity of objects in agglomerates by using the propagation of normal
wavefronts. The problem of width uniformity is discussed and its importance for
the characterization of composite structures normally found in physics and
biology highlighted. The methodology involves identifying each cluster (i.e.
connected component) of interest, which can correspond to objects or voids, and
estimating the respective medial axes by using a recently proposed wavefront
propagation approach, which is briefly reviewed. The distance values along such
axes are identified and their mean and standard deviation values obtained. As
illustrated with respect to synthetic and real objects (in vitro cultures of
neuronal cells), the combined use of these two features provide a powerful
description of the uniformity of the separation between the objects, presenting
potential for several applications in material sciences and biology.Comment: 14 pages, 23 figures, 1 table, 1 referenc
A parametric level-set method for partially discrete tomography
This paper introduces a parametric level-set method for tomographic
reconstruction of partially discrete images. Such images consist of a
continuously varying background and an anomaly with a constant (known)
grey-value. We represent the geometry of the anomaly using a level-set
function, which we represent using radial basis functions. We pose the
reconstruction problem as a bi-level optimization problem in terms of the
background and coefficients for the level-set function. To constrain the
background reconstruction we impose smoothness through Tikhonov regularization.
The bi-level optimization problem is solved in an alternating fashion; in each
iteration we first reconstruct the background and consequently update the
level-set function. We test our method on numerical phantoms and show that we
can successfully reconstruct the geometry of the anomaly, even from limited
data. On these phantoms, our method outperforms Total Variation reconstruction,
DART and P-DART.Comment: Paper submitted to 20th International Conference on Discrete Geometry
for Computer Imager
A new ghost cell/level set method for moving boundary problems:application to tumor growth
In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics—an effect observed in real tumor growth
Following multi-dimensional Type Ia supernova explosion models to homologous expansion
The last years have witnessed a rapid development of three-dimensional models
of Type Ia supernova explosions. Consequently, the next step is to evaluate
these models under variation of the initial parameters and to compare them with
observations. To calculate synthetic lightcurves and spectra from numerical
models, it is mandatory to follow the evolution up to homologous expansion. We
report on methods to achieve this in our current implementation of
multi-dimensional Type Ia supernova explosion models. The novel scheme is
thoroughly tested in two dimensions and a simple example of a three-dimensional
simulation is presented. We discuss to what degree the assumption of homologous
expansion is justified in these models.Comment: 15 pages, 16 figures, resolution of some figures reduced to meet
astro-ph file size restriction, submitted to A&
Full-star Type Ia supernova explosion models
We present full-star simulations of Type Ia supernova explosions on the basis
of the standard Chandrasekhar-mass deflagration model. Most simulations so far
considered only one spatial octant and assumed mirror symmetry to the other
octants. Two full-star models are evolved to homologous expansion and compared
with previous single-octant simulations. Therefrom we analyze the effect of
abolishing the artificial symmetry constraint on the evolution of the flame
surface. It turns out that the development of asymmetries depends on the chosen
initial flame configuration. Such asymmetries of the explosion process could
possibly contribute to the observed polarization of some Type Ia supernova
spectra.Comment: 11 pages, 10 figures, resolution of some figures reduced to meet
astro-ph file size restriction, submitted to A&
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
- …