20,623 research outputs found
Numerical Computation of Weil-Peterson Geodesics in the Universal Teichm\"uller Space
We propose an optimization algorithm for computing geodesics on the universal
Teichm\"uller space T(1) in the Weil-Petersson () metric. Another
realization for T(1) is the space of planar shapes, modulo translation and
scale, and thus our algorithm addresses a fundamental problem in computer
vision: compute the distance between two given shapes. The identification of
smooth shapes with elements on T(1) allows us to represent a shape as a
diffeomorphism on . Then given two diffeomorphisms on (i.e., two
shapes we want connect with a flow), we formulate a discretized energy
and the resulting problem is a boundary-value minimization problem. We
numerically solve this problem, providing several examples of geodesic flow on
the space of shapes, and verifying mathematical properties of T(1). Our
algorithm is more general than the application here in the sense that it can be
used to compute geodesics on any other Riemannian manifold.Comment: 21 pages, 11 figure
A numerical stabilization framework for viscoelastic fluid flow using the finite volume method on general unstructured meshes
A robust finite volume method for viscoelastic flow analysis on general
unstructured meshes is developed. It is built upon a general-purpose
stabilization framework for high Weissenberg number flows. The numerical
framework provides full combinatorial flexibility between different kinds of
rheological models on the one hand, and effective stabilization methods on the
other hand. A special emphasis is put on the velocity-stress-coupling on
co-located computational grids. Using special face interpolation techniques, a
semi-implicit stress interpolation correction is proposed to correct the
cell-face interpolation of the stress in the divergence operator of the
momentum balance. Investigating the entry-flow problem of the 4:1 contraction
benchmark, we demonstrate that the numerical methods are robust over a wide
range of Weissenberg numbers and significantly alleviate the high Weissenberg
number problem. The accuracy of the results is evaluated in a detailed mesh
convergence study
The finite-volume method in computational rheology
The finite volume method (FVM) is widely used in traditional computational fluid dynamics (CFD), and many commercial CFD codes are based on this technique which is typically less demanding in computational resources than finite element methods (FEM). However, for historical reasons, a large number of Computational Rheology codes are based on FEM. There is no clear reason why the FVM should not be as successful as finite element based techniques in Computational Rheology and its applications, such as polymer processing or, more recently, microfluidic systems using complex fluids. This chapter describes the major advances on this topic since its inception in the early 1990’s, and is organized as follows. In the next section, a review of the major contributions to computational rheology using finite volume techniques is carried out, followed by a detailed explanation of the methodology developed by the authors. This section includes recent developments and methodologies related to the description of the viscoelastic constitutive equations used to alleviate the high-Weissenberg number problem, such as the log-conformation formulation and the recent kernel-conformation technique. At the end, results of numerical calculations are presented for the well-known benchmark flow in a 4:1 planar contraction to ascertain the quality of the predictions by this method
High-order conservative finite difference GLM-MHD schemes for cell-centered MHD
We present and compare third- as well as fifth-order accurate finite
difference schemes for the numerical solution of the compressible ideal MHD
equations in multiple spatial dimensions. The selected methods lean on four
different reconstruction techniques based on recently improved versions of the
weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving
(MP) schemes as well as slope-limited polynomial reconstruction. The proposed
numerical methods are highly accurate in smooth regions of the flow, avoid loss
of accuracy in proximity of smooth extrema and provide sharp non-oscillatory
transitions at discontinuities. We suggest a numerical formulation based on a
cell-centered approach where all of the primary flow variables are discretized
at the zone center. The divergence-free condition is enforced by augmenting the
MHD equations with a generalized Lagrange multiplier yielding a mixed
hyperbolic/parabolic correction, as in Dedner et al. (J. Comput. Phys. 175
(2002) 645-673). The resulting family of schemes is robust, cost-effective and
straightforward to implement. Compared to previous existing approaches, it
completely avoids the CPU intensive workload associated with an elliptic
divergence cleaning step and the additional complexities required by staggered
mesh algorithms. Extensive numerical testing demonstrate the robustness and
reliability of the proposed framework for computations involving both smooth
and discontinuous features.Comment: 32 pages, 14 figure, submitted to Journal of Computational Physics
(Aug 7 2009
Quantitative Phase Field Model of Alloy Solidification
We present a detailed derivation and thin interface analysis of a phase-field
model that can accurately simulate microstructural pattern formation for
low-speed directional solidification of a dilute binary alloy. This advance
with respect to previous phase-field models is achieved by the addition of a
phenomenological "antitrapping" solute current in the mass conservation
relation [A. Karma, Phys. Rev. Lett 87, 115701 (2001)]. This antitrapping
current counterbalances the physical, albeit artificially large, solute
trapping effect generated when a mesoscopic interface thickness is used to
simulate the interface evolution on experimental length and time scales.
Furthermore, it provides additional freedom in the model to suppress other
spurious effects that scale with this thickness when the diffusivity is unequal
in solid and liquid [R. F. Almgren, SIAM J. Appl. Math 59, 2086 (1999)], which
include surface diffusion and a curvature correction to the Stefan condition.
This freedom can also be exploited to make the kinetic undercooling of the
interface arbitrarily small even for mesoscopic values of both the interface
thickness and the phase-field relaxation time, as for the solidification of
pure melts [A. Karma and W.-J. Rappel, Phys. Rev. E 53, R3017 (1996)]. The
performance of the model is demonstrated by calculating accurately for the
first time within a phase-field approach the Mullins-Sekerka stability spectrum
of a planar interface and nonlinear cellular shapes for realistic alloy
parameters and growth conditions.Comment: 51 pages RevTeX, 5 figures; expanded introduction and discussion; one
table and one reference added; various small correction
A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations
A new projection method based on radial basis functions (RBFs) is presented
for discretizing the incompressible unsteady Stokes equations in irregular
geometries. The novelty of the method comes from the application of a new
technique for computing the Leray-Helmholtz projection of a vector field using
generalized interpolation with divergence-free and curl-free RBFs. Unlike
traditional projection methods, this new method enables matching both
tangential and normal components of divergence-free vector fields on the domain
boundary. This allows incompressibility of the velocity field to be enforced
without any time-splitting or pressure boundary conditions. Spatial derivatives
are approximated using collocation with global RBFs so that the method only
requires samples of the field at (possibly scattered) nodes over the domain.
Numerical results are presented demonstrating high-order convergence in both
space (between 5th and 6th order) and time (up to 4th order) for some model
problems in two dimensional irregular geometries.Comment: 34 pages, 8 figure
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